Inverse Spectra (North-Holland Mathematical Library)

Preface The notion of an inverse sequence and its limit first appeared in the well-known memoir by Alexandrov [3], whe...

Author: A. Chigogidze

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Preface

The notion of an inverse sequence and its limit first appeared in the well-known memoir by Alexandrov [3], where a special case of inverse spectra - the so called projective spectra - was considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz in [209] and later, in full generality, in [210]. In the meantime Freudenthal [152] introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works. The book of Eilenberg and Steenrod [143] contains a detailed discussion of (algebraic) inverse spectra and must be mentioned as well. Subsequently, inverse spectra began to be widely studied and applied not only in the various major branches of Topology, but also in Functional Analysis and Algebra. This is not surprising because of the categorical nature of inverse spectra and of the extraordinary power of the related techniques. In Topology inverse spectra have been used both for the construction of objects with special properties (synthetic applications) and for investigation of complicated objects by means of their spectral approximation by simpler ones (analytic applications). For instance, using projective spectra Alexandrov approximated compacta by complexes and this method allowed him to extend the basic results of the combinatorial topology of polyhedra to arbitrary compacta. The next significant success of analytic applications of inverse spectra was the investigation by Pontrjagin and A.Weil of the structure of compact groups using approximation by Lie groups. Pontrjagin also considered continuous transfinite s p e c t r a - the so called Lie series. Since their creation, inverse spectra have been extensively used for constructing various exotic examples of spaces and maps. We mention here the example of the open map of a one-dimensional compactum onto a two-dimensional one constructed by Kolmogorov [196]. Several counterexamples in dimension theory, as well as in the theory of cardinal invariants, have been produced using this approach (see, for example, [147]). It should be emphasized that all the special dimension-raising maps presented in Chapter 4 are also constructed using the techniques of inverse spectra. Marde~id made the next step in analytic applications of inverse spectra in

vi

PREFACE

[213]. His principal result states that every (non-metrizable) compactum admits a representation as the limit space of an inverse spectrum, consisting of metrizable compacta of the same dimension. The significance of this statement becomes clear after remarking that there exist [213] one-dimensional non-metrizable compacta which do not have such representations, consisting of polyhedra. Further work in this direction made possible the development of the dimension theory (including both Lebesgue and cohomological) of non-metrizable compact spaces. The work of Haydon [164] was a landmark in the development of the method of inverse spectra. Haydon proved that the class of compact absolute extensors in dimension 0 coincides with the class of Dugundji compacta. But undoubtedly the most significant ingredient of his work was the discovery of what are now called Haydon spectra. It should be emphasized especially that in the work of Alexandrov on the homological theory of compacta and in the works of Pontrjagin and A.Weil on compact groups all of the results are natural extensions of the results in the metric case. Marde~id's theorem, mentioned above, also has, to some extent, a "countable" counterpart - the theorem of Freudenthal [152] which states that every metrizable compactum is the limit of an inverse sequence, consisting of polyhedra of the same dimension. In this sense Haydon's method has an "uncountable" nature. It is based on a special spectral construction whose roots are in a simple set-theoretical principle known as "the effect of uncountability." This principle, extracted in its present form by S~epin (see [278]), serves as the foundation for the Spectral Theorem. This theorem is perhaps the most powerful tool of the whole theory. It states the following: if the limit spaces of two uncountable inverse spectra (with some additional non-restrictive properties) are homeomorphic, then these spectra contain isomorphic cofinal subspectra. The Spectral Theorem has no analog for inverse sequences: for example, two inverse sequences, consisting of even- and odd-dimensional cubes, produce the same limit space - the Hilbert cube - but, of course, they contain no isomorphic subspectra whatsoever. Theorems stating that "limit spaces of spectra, consisting of "good" spaces are "good"" were typical in spectral applications. The Spectral Theorem allows us to obtain theorems stating that "limit spaces of spectra, consisting of "bad" spaces or "bad" projections are "bad"." Detailed discussion of various versions (including compact and non-compact cases) of the Spectral Theorem is presented in Chapter 1. The remainder of the book illustrates the extraordinary strength of the spectral method. It shows exactly how this method works on a range of examples and what kinds of difficulties it helps to overcome. We present various problems from different areas of Topology where the Spectral Theorem is effective. The list of applications presented in Chapter 8 includes: topological characterizations of uncountable powers of the Cantor cube and of the separable Baire space; spectral representations of topological groups (including a simple proof of the existence of Lie series); topological characterization of locally convex linear topological spaces that are homeomorphic to the powers of the real line (generalization of

PREFACE

vii

the classical result of Anderson-Kadec); the Spectral Theorem in Shape Category; actions of non-metrizable topological groups and the structure of fixed point sets in non-metrizable manifolds; the Spectral Theorem for Baire maps (isomorphisms); connections with direct spectra and E-products. The reader can find several other particular applications of the spectral technique in other parts of the book as well. Manifold theory, or, more generally, the theory of absolute extensors, is one of the major areas of modern topology where the spectral approach has already shown its full strength. We discuss this theory in detail. We start with metrizable infinite-dimensional manifolds. This subject is almost completely covered in three excellent books written by C. Bessaga and A. Pelczyr~ski [32], by T . n . C h a p m a n [69] and by J. van Mill [228]. For this reason, in Chapter 2 we present an updated survey of this theory. The reader can find here proofs of several statements of Q-manifold and /2-manifold theories. We begin with the introduction of the concept of (strong) Z-sets and discuss all major results of these theories including Torur~czyk's characterization theorems of the model spaces: countable infinite powers of the closed unit interval [298] and the real line [299]. Chapter 3 contains a solution of the well-known and long standing problem concerning coincidence of Lebesgue and integral cohomological dimensions. Chapter 4 can be considered as an independent introduction to Menger manifold theory. We present a detailed discussion of this theory including the basic geometric constructions. We also present examples of n-soft maps of (n + 1)dimensional Menger compacta onto the Hilbert cube. These maps, constructions of which, as was mentioned above, are performed using the spectral techniques, are used both in Menger manifold theory and later in the general theory of absolute extensors. Unlike Menger compacta very little is known about NSbeling universal spaces. In Chapter 5 we illustrate the close ties between Menger and NSbeling spaces by showing how NSbeling spaces can be identified with the so-called pseudo-interiors of Menger compacta. Using the spectral approach we construct an n-soft map of n-dimensional NSbeling space onto the separable Hilbert space. In Chapters 6 and 7 we develop the general theory of absolute extensors in dimension n, n E w. The Spectral Theorem plays a crucial role in almost every statement here. As the culmination of this theory we present its two major elements - the topological characterizations of uncountable powers of the closed unit interval and of the real line (originally done by S~epin [277] and the author [84] respectively). Coupled with the corresponding results of Torur~czyk we therefore obtain topological characterizations of all infinite products of the closed unit interval and the real line. Each section is ended with Historical and Bibliographical notes. Throughout the text the reader can find discussions of several unsolved problems. This book significantly extends and updates the previously published (in Rus-

viii

PREFACE

sian) book written by the author [102] (with the help of V. V. Fedorchuk). Responding to a suggestion to translate [102] I instead decided to rewrite book completely in order to include several new topics and recent developments. The result of this work is before the reader. I wish to express my thanks to Dr. J. van Mill for his support. I also thank Dr. R. B. Sher who read the entire manuscript and made many valuable suggestions. Alex Chigogidze Saskatoon, CANADA

The author was supported in part by NSERC Grant # OGP0155552.

CHAPTER

1

Inverse Spectra

In most cases all necessary notions are defined in the text. Nevertheless we assume that the reader is familiar with basics of general topology as it is presented in [145] or in any other standard textbook. All spaces under consideration are assumed to be completely regular (i.e. Tychonov) and maps are continuous. D, N, I and R stand for the two-point (discrete) space, natural numbers, the closed unit segment and the real line respectively, w and wl denote the first infinite and the first uncountable cardinals respectively.

1.1. P r e l i m i n a r y

information

1.1.1. C o m p l e t e n e s s a n d c o m p a c t n e s s . We begin with the key notion of completeness of a metrizable space. A sequence {xn" n E w} of points in a metric space (X, d) is called a Cauchy sequence if for every e > 0, there exists a natural number k = k(e), such that d(xn, Xm) < e whenever n, m _ k. As can be easily seen every convergent sequence of points in any metric space is a Cauchy sequence, but not vice versa. DEFINITION 1.1.1. A metric space ( X , d ) is complete if every Cauchy sequence in (X, d) is convergent to a point of X . Not every metric, generating the topology of a given complete metrizable space, is complete. However, the very property of existence of at least one complete metric generating the given topology of a metrizable space is a topological invariant. In what follows, we call separable completely metrizable spaces the

Polish spaces. It is well-known that the product of countably many Polish spaces is a Polish space.

2

1. INVERSE SPECTRA PROPOSITION 1.1.2. The following conditions are equivalent f o r any subspace

Y of any Polish space X :

(i) Y is a Polish space. (ii) Y is a G6-subspace of X . Since the real line R is a Polish space we conclude by P r o p o s i t i o n 1.1.2 t h a t the space of rational n u m b e r s (considered as a subspace of R) is not Polish. At the same time irrational n u m b e r s form a Polish space. T h e validity of the last fact can also be seen as follows. Since the space of irrationals is h o m e o m o r p h i c to the c o u n t a b l e infinite power N W of the space N of n a t u r a l numbers, it suffices to note t h a t N itself, as a closed subset of R, is a Polish space and to observe once again t h a t the o p e r a t i o n of taking c o u n t a b l e p r o d u c t s m a i n t a i n s the p r o p e r t y of being a Polish space. Similarly, we see t h a t the countable infinite power R W of the real line R is also a Polish space. T h e spaces N W and R W will play key roles below. T h e following s t a t e m e n t is one of the most convenient definitions of a Polish space. PROPOSITION 1.1.3. The following conditions are equivalent f o r any space X : (i) X is a Polish space. (ii) X is h o m e o m o r p h i c to a closed subspace of the space R W . A space X has the Baire property if an intersection of any countable collection of dense open subsets of X is dense. This p r o p e r t y of the real line R was shown by Baire (which is why the p r o p e r t y bears his name). A much more general s t a t e m e n t is true. PROPOSITION 1.1.4. E v e r y Polish space has the Baire property. Let us discuss the properties of the space N W in more detail (by the way, this space a p p e a r s in the l i t e r a t u r e under the n a m e of the separable Baire space as well). First of all, we present a classical result of A l e x a n d r o v - U r y s o h n , which gives a topological c h a r a c t e r i z a t i o n of this space. THEOREM 1.1.5. The following conditions are equivalent f o r any space X : (i) X is h o m e o m o r p h i c to N W . (ii) X is a z e r o - d i m e n s i o n a l Polish space w i t h o u t compact open subspaces. It is very interesting t h a t the space N ~ plays the same role with respect to the class of zero-dimensional Polish spaces, as the space R W does with respect to the class of all Polish spaces. T h e following s t a t e m e n t can be viewed as an i l l u s t r a t i o n of this fact. PROPOSITION 1.1.6. The following conditions are equivalent f o r any space X : (i) X is a z e r o - d i m e n s i o n a l Polish space. (ii) X is h o m e o m o r p h i c to a closed subspace of the space N W .

1.1. PRELIMINARY INFORMATION

3

It is worth noting t h a t all Polish spaces can be described by means of the space N ~ . Indeed, let us observe, first of all, t h a t the real line R can be presented as an open image of N ~ (the reader can easily check this fact by analyzing the proof of L e m m a 6.4.1). Further, taking the countable infinite power of the such an open m a p p i n g we see t h a t there exists an open m a p p i n g f : N "~ ---, R W. Consider now an a r b i t r a r y Polish space X, which, by P r o p o s i t i o n 1.1.3, can be identified with a closed subspace of R W . Clearly, the inverse image f - l ( X ) is a zero-dimensional Polish space and consequently, the p r o d u c t f - l ( X ) x N ~ is h o m e o m o r p h i c to N ~ (apply T h e o r e m 1.1.5). Clearly, the composition of the projection f - l ( x ) x Y ~ --, f - l ( X ) and of the open m a p p i n g f - l ( X ) --, Z is open. Therefore X is an open image of N ~ . T h e converse s t a t e m e n t is also well-known. T h u s we have proved the following theorem. THEOREM 1.1.7. The following conditions are equivalent f o r any space X : (i) X is a Polish space. (ii) X is an open image o f the space N ~ . An i m p o r t a n t subclass of the class of Polish spaces is formed by the c o m p a c t (metrizable) spaces. The theory of these spaces is one of the most developed areas of general topology, and for this reason we restrict ourselves to a very brief discussion. The C a n t o r cube D ~ and the Hilbert cube I ~ are basic objects for the entire class of (zero-dimensional) metrizable compacta. Their roles are emphasized by the following s t a t e m e n t s (compare with Propositions 1.1.3, 1.1.6). PROPOSITION 1.1.8. The following conditions are equivalent f o r any space X : (i) X is a metrizable c o m p a c t u m . (ii) X is h o m e o m o r p h i c to a closed subspace o f the Hilbert cube I ~ . PROPOSITION 1.1.9. The following conditions are equivalent f o r any space X : (i) X is a z e r o - d i m e n s i o n a l metrizable c o m p a c t u m . (ii) X is h o m e o m o r p h i c to a closed subspace of the C a n t o r cube D ~ . Here is a topological characterization of D ~ . THEOREM 1.1.10. The following conditions are equivalent f o r any space X : (i) X is h o m e o m o r p h i c to D ~ . (ii) X is a z e r o - d i m e n s i o n a l metrizable c o m p a c t u m w i t h o u t isolated (i.e. open) points.

We finish our brief discussion with the following surjective characterization of metrizable compacta. THEOREM 1.1.11. The following conditions are equivalent f o r any space X : (i) X is a metrizable c o m p a c t u m . (ii) X is a surjective image of D ~ .

4 1.1.2. R e a l c o m p a c t nition.

1. INVERSE SPECTRA s p a c e s . We begin with the following important defi-

DEFINITION 1.1.12. A space X is called realcompact if it is homeomorphic to a closed subspace of some power of the real line. Realcompact spaces have been introduced by Hewitt [170] (under a somewhat different name). The same class of spaces has been independently defined in terms of uniformities by Nachbin [239]. It is necessary to note that realcompact spaces appear in the literature under different names, such as: realcomplete spaces, Hewitt spaces, Q-spaces, R-spaces. This class of spaces plays a very i m p o r t a n t role in the sequel. Therefore, for the reader's convenience, we discuss some of their properties here. The following s t a t e m e n t is the straightforward consequence of the above definition. PROPOSITION 1.1.13. Every closed subspace of a realcompact space is realcompact, as is the product of an arbitrary family of realcompact spaces. COROLLARY 1.1.14. Let {Xt : t E T} be a family of realcompact subspaces of a space X . Then the intersection M{Xt :t E T} is realcompact. PROOF. Obviously, the intersection M{Xt : t E T} is homeomorphic to the intersection of the subspaces Y I { x t : t E T} and A x (the diagonal of the product x T ) . Consequently, the indicated intersection is closed in the product I-I{Xt : t E T}. It only remains to apply Proposition 1.1.13 ['-! COROLLARY 1.1.15. A functionally open subspace of a realcompact space is realcompact. PROOF. Let G be a functionally open subspaces of a realcompact space X. Consider the embedding i - A { ~ : ~ E C ( X ) } : X ~ R c ( z ) and identify the space X with its homeomorphic image i ( X ) in R e(X). It follows from the definition of the embedding i that X is C-embedded in R C(x). Consequently, there exists a functionally open subset V of R C(X) such that G -- V MX. Further, by Corollary 1.1.14, it only remains to show that any functionally open subspace V of an arbitrary power R A of the real line is realcompact. If I A I_< w, then V, as an open subspace of a Polish space, is realcompact. Suppose now that ]A ]> w. Since any continuous real-valued function defined on R A depends on countably many coordinates (see, for example, [145]), we can easily conclude that V = VB • R A-B, where B is a countable subset of A and VB is an open subspace of the product R B. It only remains to apply Proposition 1.1.13. [:] PROPOSITION 1.1.16. A realcompact and C-embedded subspace of a space is closed.

1.1. PRELIMINARY INFORMATION

5

PROOF. Let X be a r e a l c o m p a c t a n d C - e m b e d d e d subspace of a space Y. By Definition 1.1.12, there exist a set A and a h o m e o m o r p h i s m f : X ~ X ~ onto t h e closed subspace X ~ of the p r o d u c t R A. D e n o t e by ~ra : R A ---, Tla -- R, a E A, the projection onto the a - t h coordinate. Since X is C - e m b e d d e d in Y, for each a E A, t h e r e is a function ~ a E C ( Y ) such t h a t ~ I X = ~ a " f . Let = A { ~ a :c~ E A}. Clearly ~p m a p s t h e space Y into R A and ~ I X = f . It follows from t h e c o n t i n u i t y of ~ t h a t

~ ( c I y X ) C_ c I R , ~ ( X ) = c l n , f ( X )

= clR, X' = X'.

In o t h e r words, t h e restriction g - ~ I c l y X m a p s t h e closure c I y X onto X ~. In this s i t u a t i o n t h e c o m p o s i t i o n r = . f - l . g . c l y X ----, X is a r e t r a c t i o n . Hence, c l y X = X . [7 DEFINITION 1.1.17. A subspace X of a space Y is said to be z - e m b e d d e d in Y if every functionally open (closed) subset G of X can be written as G = X M G, where G is a f u n c t i o n a l l y open (closed) subset of Y . C - e m b e d d e d subspaces (closed subspaces of n o r m a l spaces, for e x a m p l e ) are z - e m b e d d e d . In a perfectly n o r m a l space every s u b s p a c e is z - e m b e d d e d . LEMMA 1.1.18. A functionally open subspace of any space is z-embedded. PROOF. Let X be a functionally o p e n subspace of a space Y. F i x a nonnegative function ~ E C ( Y ) such t h a t Y - X = {y E Y ' ~ ( y ) = 0} a n d consider a functionally closed subspace Z in X . O u r goal is to find a f u n c t i o n a l l y closed s u b s p a c e Z of Y such t h a t Z = X M ~5. Fix a n o n - n e g a t i v e function r E C ( X ) such t h a t Z = {x E X " r

= 0}. Define the function h " Y ~ R as follows:

ifyEY--X,

0, h(y)=

min{~(y)

,r

ifyEX.

Let Z = {y E Y 9 h(y) - 0}. Obviously, Z -- X M Z. Therefore, it only r e m a i n s to show c o n t i n u i t y of t h e function h. T h e c o n t i n u i t y of h at points of X follows from our definitions. If y E Y - X a n d e > 0, t h e n

(y 9 y . h(y) < ~} = (y e y . v(y) < ~} u (~ e X . r

< ~}.

T h e c o n t i n u i t y of ~ implies t h e o p e n n e s s of the first t e r m on t h e right side of this equation. T h e second t e r m is also o p e n in Y. This i m m e d i a t e l y follows from t h e c o n t i n u i t y of r a n d t h e o p e n n e s s of X in Y a n d c o m p l e t e s t h e verification of t h e c o n t i n u i t y of h. K1 LEMMA 1.1.19. A LindelSf subspace of an arbitrary space is z-embedded.

6

1. INVERSE SPECTRA

PROOF. Let X be a LindelSf subspace of a space Y and G be a functionally open subspace of X. For each point x E G take a functionally open subspace V~ such t h a t x E V x M X c_ G. From the open cover { V x M X 9 x E G} of the LindelSf space G select a countable subcover {Vn M X 9 n E w}. Clearly, G = X M (~, where (~ = M{Vn 9 n E w}. It only remains to note t h a t the set (~, as a countable union of functionally open subspaces, is itself functionally open in Y. [::] It should be noted t h a t z-embeddability of certain subspaces completely determines the class of so-called perfectly a-normal spaces. First, we give the corresponding definition. DEFINITION 1.1.20. A space is called perfectly a-normal if the closure of each open subset is f u n c t i o n a l l y closed. Obviously, perfectly normal spaces are perfectly a-normal. It can be shown [274] t h a t the p r o d u c t of an a r b i t r a r y family of metrizable spaces is also perfectly a-normal. PROPOSITION 1.1.21. The following conditions are equivalent for any space X:

(i) X is perfectly a - n o r m a l . (ii) All dense subsets of X are z-embedded. (ii) All open subsets of X are z-embedded. PROOF. (i) =~ (ii). Let Y be a dense subset of X. It is enough to show t h a t for every pair (Z1,Z2) of disjoint functionally closed subsets of Y there are functionally closed subsets F1 and F2 of X such t h a t F1 M F2 M Y -- 0 and Zi C Fi, i = 1,2. Consider open (in Y) neighborhoods G1 and G2, of Z1 and Z2 respectively, with disjoint closures. Since Y is dense in X we conclude t h a t the sets Fi -- c l x G i , i = 1,2 are canonically closed in X. Therefore, by (i), these sets are functionally closed in X. Clearly, F1 M F2 M Y -- 0 and Zi C Fi, i -- 1,2. (ii) ~ (ii). Let Y be an open subset of X. Clearly, the subset A -- Y U ( Y - c l x Y ) is dense in X. It can be easily seen t h a t each functionally closed subset Z of Y is functionally closed in A. By (ii), there is a functionally closed subset F in X such t h a t F M A -- Z. But then F M Y -- Z. (ii) =~ (i). Let Y be an open subset of X. As above, consider the set A -- Y U ( Y - c l x Y ) . A is open in X and consequently, by(//), is z-embedded in X. T h e set Y is functionally closed in A. Therefore, there is a functionally closed subset Z of X such t h a t Y - Z M A. Clearly, Z - c l x Y . F3 PROPOSITION 1.1.22. Let Y

be a z-embedded subset of a space X .

Then,

f o r each f u n c t i o n ~o e C ( Y ) there exist a countable f a m i l y { G n " n E w} of f u n c t i o n a l l y open subsets in Z and a f u n c t i o n r E C ( M { G n " n E w}) such that Y C_ n { G n " n 6 w} and O l Y - ~o.

1.1. PRELIMINARY INFORMATION

7

PROOF. Since every space can be identified with a C - e m b e d d e d subspace of R r for some T > w (see the proof of Corollary 1.1.15) we can assume w i t h o u t loss of generality t h a t X - R r. First we consider the case of a b o u n d e d function ~, i.e. ] ~ ]_ a, where a is a non-negative real number. We follow the proof of the well-known t h e o r e m of Brouwer-Tietze-Urysohn (see, for example, [7]). Let ~0 -- ~ and consider functionally closed subsets A0 -- {y E Y" ~0(Y) ~_ - ~ } and B0 -- {y E Y" ~0(y) >__ } of Y. Since, by our assumption, Y is z - e m b e d d e d in X, there exist functionally closed subsets A~ and B~ of X such t h a t A0 - A~ M Y and B0 -- B~ M Y. Let Go = X - (A~ M B~). Obviously, Go is functionally open in X and Y _C Go. Note also t h a t the sets fi-o = A~ M Go a n d / ~ 0 = B~ M Go are disjoint a n d functionally closed in Go. Consequently, there exists a function go E C ( G o ) such a a Ao = g 0 1 ( - ~ ) and /~0 = go 1 (~). Let ~1 = ~ o - g o / Y . t h a t I go I-< 3, It can be easily seen t h a t I ~1 [_< 23~. Continuing this process we o b t a i n a sequence of functions ~n E C ( Y ) satisfying the inequalities ] ~ n ]_< (5)2ha, n E w. We also have a sequence gn E C ( G n ) , n E w, (where Gn is a functionally open subset of X containing Y and Gn C Gn+l for each n E w) of functions satisfying the following conditions: I gn i<_ (2)na_g and ~n+l = ~n - g n / Y . Let G -- M { G n ' n E w}. Straightforward verification shows t h a t the ( b o u n d e d on Y) functions Cn - go + " " + gn, n E w uniformly converge to some continuous (on Y) function r It only remains to note t h a t r extends the function ~ and is b o u n d e d by the same n u m b e r a. Suppose now t h a t the function ~ in u n b o u n d e d . T h e n the composition a r c t a n . E C ( Y ) is bounded. By the case, considered above, there is a function .f extending a r c t a n . ~ , b o u n d e d by the same n u m b e r as a r c t a n . ~ and defined on some set G = M { G n ' n E w}, where each Gn is a functionally open subset of X containing Y. Let Z = {x E G" I f ( x ) ] > _ ~}. Obviously, Z is functionally closed in G and Y C G - Z. Therefore the function t a n . . f is defined on G - Z and coincides with ~ on Y. It only remains to note t h a t the set G - Z can also be represented as the intersection of countably m a n y functionally open subsets of X.

[:]

Using a similar a r g u m e n t one can prove the following s t a t e m e n t . PROPOSITION 1.1.23. I f a f u n c t i o n ~ E C ( Z ) , defined on a f u n c t i o n a l l y closed subset Z of a space X , can be extended to a f u n c t i o n a l l y open neighborhood of Z in X , then ~ has an extension defined on X . PROPOSITION 1.1.24. Let X be a z-embedded subset of a space Y . Then the following conditions are equivalent: (i) X is realcompact. (ii) X coincides with the intersection of all functionally open subsets of Y containing X .

8

1. INVERSE SPECTRA

PROOF. (i) =# (ii). Let G d e n o t e the intersection of all functionally open subsets of Y containing X. Let us show t h a t X is dense in G. Suppose t h a t this a s s u m p t i o n is not true. T h e n t h e r e exists an open subset V of Y such t h a t V M G ~ O and V M X = 0. Take a point y0 E V M G . Choose also a functionally closed subset Z of Y such t h a t y0 E Z c V. Obviously, X c Y - Z and, consequently, by definition of G, G C_ Y - Z. On the other hand, tile point y0, as can be easily seen, does not belong to Y - Z. This shows t h a t X is dense in G. By P r o p o s i t i o n 1.1.22, X is C - e m b e d d e d in G. In this situation the r e a l c o m p a c t n e s s of X guarantees, by P r o p o s i t i o n 1.1.16, t h a t X = G. (ii) ~ (i). Apply Corollaries 1.1.14 and 1.1.15. V1 L e m m a 1.1.19 and P r o p o s i t i o n 1.1.24 imply the following s t a t e m e n t . COROLLARY 1.1.25. Every LindelSf space is realcompact. PROPOSITION 1.1.26. Suppose that a space X can be represented as the union of countable many z-embedded realcompact subspaces. Then X is realcompact. PROOF. Let X = U { X n : n E w}, where X n is a z - e m b e d d e d and realcompact subspace of X, n E w. Consider the Stone-(~ech compactification fiX of the space X and take a point p E BX - X. By P r o p o s i t i o n 1.1.24, it suffices to c o n s t r u c t a functionally closed subset Z of/~X such t h a t p E Z and Z M X = 0. If p q~ c l ~ x ( X n ) , t h e n there is & functionally closed subset Zn in ~ X such t h a t p E Zn and Zn M c l ~ x ( X n ) = @. If p E c l ~ x ( X n ) , then, applying P r o p o s i t i o n 1.1.24 (replacing in it X by Xn and Y by c l z x ( X n ) ) , we conclude t h a t there exists a functionally closed subset Fn of c l ~ x ( X n ) such t h a t p E Fn and Fn M X n = 0. Finally, let Zn denote a functionally closed subset of ~ X such t h a t Zn N cl[3x(Xn) -- Fn. It only remains to note t h a t Z = M{Zn : n E w} is the desired functionally closed subset of ~ X . V1 To each space X we can assign a c o m p l e t e l y d e t e r m i n e d realcompact space u X , the so called Hewitt realcompactification of X. Let us describe the construction of u X . Consider the canonical e m b e d d i n g i x = A { ~ : qo E C ( X ) } : X R C(x) and identify X with its image i x ( X ) . T h e closure of X in R C(x) is a realc o m p a c t space which contains X as a dense C - e m b e d d e d subspace. This closure is the Hewitt realcompactification of X . It can be shown t h a t v X is completely characterized (up to a h o m e o m o r p h i s m ) by these properties. It follows from P r o p o s i t i o n 1.1.16 t h a t r e a l c o m p a c t spaces can be defined as spaces satisfying the e q u a t i o n X = u X . Another, s o m e t i m e s more convenient description of u X can be o b t a i n e d as follows. E m b e d X as & z - e m b e d d e d subspace into a realcompact space Y (for example, the Stone-(~ech compactification ~ X ) and consider the intersection of all functionally open subsets of Y containing X. As indicated in the proof of P r o p o s i t i o n 1.1.24, X is C - e m b e d d e d in this intersection, which of course is r e a l c o m p a c t (apply Corollary 1.1.14). Using this description, we may

1.1. PRELIMINARY INFORMATION

9

assume t h a t X C v X C fiX and, consequently, dim X = dim v X . F u r t h e r , one can show that: (a) v X is the smallest r e a l c o m p a c t subspace o f / 3 X c o n t a i n i n g X . (b) v X is the largest subspace of ~ X containing X as a C - e m b e d d e d subspace. T h e correspondence X ~-~ v X is even functorial. For each m a p f " X ~ Y we can assign a m a p v f" v X --. vY which coincides with f on X. One can get this conclusion, by analyzing the proof of P r o p o s i t i o n 1.1.16. 1.1.3. D i r e c t e d s e t s . Let A be a p a r t i a l l y ordered directed set (i.e. for every two elements a, bEA t h e r e exists an element cEA such t h a t c>a and c>_b). We say t h a t a subset A1 C A of A majorates a n o t h e r subset A2 C_ A of A if for each element a2 E A2 t h e r e exists an element a l C A1 such t h a t al _ a2. A subset which m a j o r a t e s A is called cofinal in A. A subset of A is said to be a chain if every two elements of it are c o m p a r a b l e . T h e s y m b o l s u p B , where B CA , denotes the lower u p p e r b o u n d of B (if such an element exists in A ). Let T>_w be a cardinal number. A subset B of A is said to be T-closed in A if for each chain CC_B, with ]C]<_T, we have s u p C E B , whenever the element s u p C exists in A . Finally, a directed set A is said to be r-complete if for each chain B of elements of A with ]C] r . T h e s t a n d a r d e x a m p l e of a T-complete set can be o b t a i n e d as follows. For an a r b i t r a r y set A let expA denote, as usual, the collection of all subsets of A. T h e r e is a n a t u r a l p a r t i a l order on expA" A1 >_ A2 if and only if A1 D A2. W i t h this partial order expA becomes a directed set. If we consider only those subsets of the set A which have cardinality _ r, t h e n the c o r r e s p o n d i n g subcollection of expA, d e n o t e d by exprA, serves as a basic e x a m p l e of a T-complete set. Let us consider a second way of c o n s t r u c t i n g T-complete sets. Let A be an a r b i t r a r y directed set. We call two c o u n t a b l e chains in A equivalent if each of t h e m m a j o r a t e s the other. T h e collection of all equivalence classes of c o u n t a b l e chains in A is d e n o t e d by [A]~ and is called the w-completion of A. T h e order relation on [A]~ is defined by the relation of "majorantness"" one class majorates another if one of the chains in the first class majorates a chain belonging to the second class. It is easy to see that with respect to the described order the set [A]w is w-complete. Assigning to each element a E A the countable stationary chain (consisting of one element (a)) we obtain a natural embedding of the set A into its w-completion. If, for all A < T, the A-completions [A]), of A have already been defined, then the T-completion [A]r can be defined as the set of equivalence classes of chains of cardinality _< T from the union 0([A]~" A < T).An order on [A]r, as above, is generated by the relation of majorantness. Straightforward verification shows that the T-completion of an arbitrary directed set is T-complete for each infinite cardinal T.

10

1. INVERSE SPECTRA

PROPOSITION 1.1.27. Let { A t " t E T } be a collection of T-closed and cofinal subsets of a T-complete set A . T h e n the intersection M{At " t E T} is also cofinal and T-closed in A . PROOF. W i t h o u t loss of generality we can assume t h a t the set T is wellordered. Consider the p r o d u c t w x T endowed with the lexicographic order. Let a be an a r b i t r a r y element in A. In an obvious way, by transfinite induction, we define a chain B -- {a(n,t)" (n, t) E w x T} in A so t h a t the following conditions are satisfied: (i) a(n,t) >_ a for each (n, t) E w x T. (ii) a(n,t) <_ a(m,t,) whenever (n, t) < (m, t'). (iii) a(n,t) E A for each (n, t) e w x T. T h e T-completeness of A g u a r a n t e e s the existence of an element b -- sup{a(n,t)" n E w}. By closedness of At, b E At for each t E T. Therefore, b E M{At 9 t E T}. By the construction, b >_ a. This shows the cofinality of this intersection in A. Its T-closedness is obvious.

Vl

COROLLARY 1.1.28. For each subset B , with [ B I< T, of a T-complete set A there exists an e l e m e n t c E A such that c >_ b f o r each b E B .

PROOF. Apply P r o p o s i t i o n 1.1.27 to the collection {Ab" b E B}, where Ab -~ [-1

{aEA'a>_b},bEA.

PROPOSITION 1.1.29. Let A be a T-complete set, L C A 2, and suppose the following three conditions are satisfied: E x i s t e n c e : For each a E A there exists b E A such that (a,b) E L . Majorantness: I f (a, b) e L and c >_ b, then (a, c) E n . T - c l o s e d n e s s : Let {at " t e T } be a chain in A with a -- sup{at " t E T}. I f (at, c) E n f o r s o m e c e A and each t E T , then (a,c) E L . T h e n the set of all L-reflexive elements of A (an e l e m e n t a E A is said to be L-reflexive if (a, a) E L ) is cofinal and T-closed in A .

PROOF. First of all let us show t h a t the set of all L-reflexive elements is cofinal in A. Let a E A. We are going to c o n s t r u c t a chain B -- {ba" a < r}, indexed by all ordinals strictly less t h a n T and satisfying the following conditions: (a) b0 -- a. (b) (b~,ba) E L, whenever ~ < a < T. (C) b~ _< ba, whenever fl < a < T. Suppose t h a t the elements bz have already been c o n s t r u c t e d for all/~ smaller t h a n some a, where a < T. Let us c o n s t r u c t the element ba. If a is a limit ordinal t h e n let ba = sup{bz" ~ < a}. Clearly, ba > bt~ for e a c h ~ < a. Note t h a t ba >_ b~+l _> b~ and (b~,b~+l) E L. Therefore, by the m a j o r a n t n e s s condition, (b~, ba) E L for each ~ < a. If a is non-limit ordinal, t h e n a - ~ + 1. By the existence condition, there exists an element c E A such t h a t (b~, c) E L. Since A is directed we can find

1.2. INVERSE SPECTRA

11

an element b~ which majorates both bE and c, i.e. ba > bE and ba > c. The majorantness condition implies that (bE, b~) E L. This finishes the construction of the chain B. Let b = sup{ba: c~ < T). Let us show that b is a L-reflexive element. Indeed, by the majorantness condition, (b~,b) E L for each c~ < T. Therefore, by the closeness condition, (b, b) E L. This shows cofinality of the set of all L-reflexive elements in A. If C = {at: t E T ) is a chain, consisting of L-reflexive elements, I T I< T and a -- s u p C , then, by the majorantness condition, we have (at, a) E L whenever t E T. Finally, by the closeness condition, (a, a) E L. I--1

Historical and bibliographical notes 1.1. The results of subsection 1.1.1 are widely known. For example, Proposition 1.1.2 has been proven by Alexandrov in [2] (see also [208]). The characterization theorems 1.1.5 and 1.1.10 have been obtained by Alexandrov and Urysohn [8] and Brouwer [56] respectively. Theorems 1.1.7 and 1.1.11 belong respectively to Hausdorff and Alexandrov. Propositions 1.1.13, 1.1.16, as well as Corollary 1.1.14, have been proved by Shirota [281] (see also [185]). z-embedded subspaces were systematically studied in [40~ 41~ 42~ 10]. In particular, these references contain proofs of Lemmas 1.1.18 and 1.1.19. The property included in the definition of perfectly R-normal spaces was apparently known to Bokshtein (see also [144, 261]). The concept of perfect R-normality itself was independently introduced (under different names) by Blair [40] and Terada [292]. Shortly after the same notion was also defined by SEepin [274]. Propositions 1.1.21 and 1.1.22 appear in [40] and [42] respectively. Proposition 1.1.24 was in fact proved in [236]. Corollary 1.1.25 belongs to Hewitt [170]. Originally Proposition 1.1.26 was obtained in [40] and later rediscovered in [74]. Earlier, in the presence of normality of the given space coupled with closeness of summands, Proposition 1.1.26 was proved in [236]. It should be noted that the assumption of z-embeddedness of the summands in Proposition 1.1.26 is essential [237, 238]. The results of Subsection 1.1.3 were originally obtained in [278]; see also [107].

1.2. D e f i n i t i o n s a n d e l e m e n t a r y p r o p e r t i e s of i n v e r s e s p e c t r a If I ] { x a 9 c~ E A} is a Cartesian product of topological spaces Xa, c~ E A, and B _C A then ~ r B ' I - ] { X ~ ' a E A } ~ I ] { X ~ ' ~ E B } denotes the standard projection onto the corresponding subproduct. The same meaning is given to the projection 7rg 9 l l { X a 9 c~ E B} ~ l']{Xa 9 c~ E C}, C C B C A. For simplicity, sometimes we also use the following notation: X B = I ' ] { X ~ " ~ E B } , where B C_ A. We will maintain these notations throughout the whole text.

12

1. INVERSE SPECTRA

1 . 2 . 1 . D e f i n i t i o n o f i n v e r s e s p e c t r a . Let A be a directed set and suppose t h a t to each c~ E A there corresponds a topological space X a in such a way t h a t whenever/~ ~- c~, a continuous map P~a " X~ ---+ X a is also given. Suppose further that: 9 p ~ - p ~ - p~ for each triple (c~, ~, ~) of indexes with c~ ___ ~ ~ ~,.

9 pg - idx,~ for each c~ E A. In this s i t u a t i o n we say t h a t the inverse spectrum (inverse system) or simply spectrum S x = { X a , p ~ , A } is given. We call the spaces X a elements of the s p e c t r u m . T h e m a p p i n g s p~ 9 X~ ---+ X a are called projections (or bonding mappings) of the spectrum. An inverse sequence is a s p e c t r u m whose indexing set coincides wii;h w (the first infinite cardinal directed by its n a t u r a l order). A point {x~ 9 a E A} of the C a r t e s i a n p r o d u c t YI{x~ 9 c~ E A} of spaces X a is called a thread of the s p e c t r u m S x = { X a , p ~ , A } if p~(x~) = xa for any a , ~ E A such t h a t a ~ ~. T h e subspace of the p r o d u c t r I { x a 9 a E A} consisting of all t h r e a d s is called the inverse limit (or, simply, limit) of the s p e c t r u m S x = { X a , p ~ , A } and is d e n o t e d by l i m S z or, more explicitly, by lira { X a , p ~ , A } . By pc" l i m S x ~ X a , c~ E A, we denote the restriction of the projection r{~}" l-[{Xa 9 a E A} ---. X~, and we call this the a - t h limit projection of the s p e c t r u m S x . By definition of the p r o d u c t topology, the sets of the form lim S x M {~r~l(G~)" E B}, where B is a finite subset of the indexing set A and G~ is an open set in X~, c o n s t i t u t e a base of the limit space l i m S x . Let us show t h a t in fact we can be more specific. PROPOSITION 1.2.1. The collection of all sets p ~ l ( G a ) , where Ga is open in X a and ~ E A, forms a base for the topology of the limit space of the spectrum lim S x . PROOF. Let G = l i m S x Cl {~r~l(G~)" ~ E B}, where B is a finite subset of the indexing set A and G~ is open in X~. Take an index c~ E A such t h a t a ~ ~ for each ~ E B. T h e n the set G~ = M { ( p ~ ) - I ( G ~ ) 9 /3 E B} is open in Z a , and it easy to see t h a t G = p-~l(Ga).

K1

PROPOSITION 1.2.2. The limit space of a spectrum 8 x

-- {Xa,p~a,A} is a

closed subspace of the product l-I{x~ " a E A}. PROOF. Let x = {xa} E Y I { x a : a E A} - l i m S x . T h e n there exist indexes c~,~ E A such t h a t ~ > c~ and x~' - p~(xr ~ xa. Consider disjoint open subsets V and V ~ in X a such t h a t x~ E V ~ and xa E V. Clearly, the set U - ( p ~ ) - l ( Y ' ) is an open n e i g h b o r h o o d of the point x~ in X~. Therefore, the set W = 7r~-l(Y) M 7r~l(U) is an open n e i g h b o r h o o d of x in 1-I{xa 9 c~ E A}. It only remains to note t h a t W M l i m S x - 0.

K]

1.2. INVERSE SPECTRA

13

COROLLARY 1.2.3. The limit space of an inverse spectrum consisting of completely regular spaces is completely regular. Consider now a s p e c t r u m S x = { X a , p ~ , A} and a subspace Y of its limit space lira S x . T h e subspace Y defines the inverse s p e c t r u m S y = {Ya, q~, A } , where Ya = pa(Y), a E A, and q~ denotes the restriction p ~ / X ~ , ~ ~ a. In this s i t u a t i o n we say t h a t the s p e c t r u m S y = {Ya, q~, A} is induced by Y. Obviously, the p r o d u c t 1-I{Y~" a E A} is a subspace of the p r o d u c t Y I { x ~ . a E A} a n d the t h r e a d s of the s p e c t r u m S y are t h r e a d s of the s p e c t r u m S x . Therefore the limit space Y = lim S y is n a t u r a l l y e m b e d d e d in the limit space X -- lim S x . We claim t h a t = M{p~l(Va) 9 a E A}. Indeed, if x = { x a ' a E A} E M{p~I(Ya)" a E A}, t h e n x a E Ya for each a E A. This shows t h a t the t h r e a d x of the s p e c t r u m S x is in fact a t h r e a d of the s p e c t r u m By. Therefore, M{p~l(Ya) 9 a E A} C 9 . Conversely, if x E 9 , t h e n pa(x) E Ya for each a E A. Therefore, 9 C M{p~l(Ya) 9 a E A} and we have proved the following s t a t e m e n t . PROPOSITION 1.2.4. Let Y be a subspace of the limit space of a spectrum S x = { X a , p ~ , A } . Then the limit space lim S y of the induced (by Y ) spectrum SY = {Ya, q~, A} contains Y. The equality Y = lim S y holds if and only if Y = M{p~ 1 (Ya)" a E A}.

COROLLARY 1.2.5. Let Y be a subspace of the limit space X of a spectrum S x = {Xa,p~a,A} 9 Then the limit space of the spectrum {clx~(pa(Y), p~a/cIx~(pz(Y),A} coincides w~th c I x Y . PROOF. A p p l y P r o p o s i t i o n s 1.2.1 and 1.2.4

0

COROLLARY 1.2.6. Let Y be a closed subspace of the limit space of a spectrum S x = {Xa, p~, A } . Then Y = lim { p a ( Y ) , p ~ / p ~ ( Y ) , A }

and Y = lim {clx~ (pa(Y)),p~a/clxa(p~(Y)),A}. Suppose now t h a t a s p e c t r u m S z = {X~,p~a, A} is given and A' is a cofinal subset of the indexing set A. In this case we say t h a t the s p e c t r u m S~c = {Xa, p~, A'} is a cofinal subspectrum of S x . PROPOSITION 1.2.7. Let S~x = {Xa,p~a,A '} be a cofinal subspectrum of the spectrum S x = {Xa, p~, A} . Then the map .f " lim S x --~ lim S~: consisting of restricting all threads from X = lim S z to those from X ~ = lira S~: is a homeomorphism of X onto X ~.

14

1. INVERSE SPECTRA

PROOF. We c o n s t r u c t a m a p f~" X ~ --, X inverse to f. Let x ~ = {xa" For each a E A let x,~ = p ~ ( x / 3 ) , wheref~ ~ c~ a n d f ~ E A'. We claim t h a t the point x a does not d e p e n d on the choice of index f~ E A ~. Indeed, consider a t h i r d index -y E A ~ such t h a t ~/ ~ a. Since A is directed and A ~ is cofinal in A, t h e r e is a n o t h e r index 5 E A ~ such t h a t 5 __ f~ and 5 ~_ ~. T h e fact t h a t x ~ is a t h r e a d of the s p e c t r u m S ~ implies t h a t

E A'} E X'.

=

=

=

=

T h i s shows t h a t x = { x a ' a E A} is a well-defined point of the p r o d u c t 11 {Xa" a E A}. In fact, x is a t h r e a d of the s p e c t r u m 8 x . Indeed,

p/3p~ Thus, to each point x ~ E X ~ we have assigned a point x = f~(x ~) E X . It follows i m m e d i a t e l y from t h e above definition t h a t f f ~ ( x ~) -- x ~ for each x ~ E X ~ a n d f ~ f ( x ) = x for each x E X . Therefore, f is a bijection and f~ is the inverse off. Next we show t h a t f is an open m a p (the continuity of f is obvious). Let G a be an open subset of Xa, a E A, and f~ _ c~, where f~ E A ~. T h e n =

Therefore, the sets of the form p~l(Gf~), where f~ E A ~ and Gf~ is open in Xf~, c o n s t i t u t e a base for the t o p o l o g y of X (compare with P r o p o s i t i o n 1.2.1). Denote by qf~ the ~ - t h limit projection of the s p e c t r u m 8 ~ . T h e n

p ~ ( G / ~ ) - {x = { x , " a E A}" x/~ E G/~} and

q~l(a/3) - { x ' -

{xa" ~ E A'}" x/3 E a ~ } .

Consequently,

f p ~ l ( G / 3 ) -- q~l(Gf~). This shows t h a t f is a h o m e o m o r p h i s m .

[::]

COROLLARY 1.2.8. Let S x = { X a , p ~ , A }

be an inverse spectrum whose indexing set A has a maximal element ~. Then the inverse limit lira S x of S x is homeomorphic to the space X a . EXAMPLE 1.2.9. Suppose we are given a family {Xt" t E T } of spaces with I T I> - w. Obviously the set exp<,~T of all finite subsets of T is directed by inclusion, i.e. by letting T1 ~_ T2 if and only if T1 C_ T2. For each S E exp<~T, let X s = r I { X t " t E S}. Also, in cases when S -< R, S, R E exp<~T, denote by lr~" X l:t ---* X s the natural projection onto the corresponding subproduct. Then we obtain an inverse spectrum S = { X s , ~r~, exp<wT}. One can readily verify that the limit space of this spectrum is homeomorphic to the product 1-I{xt. t E

T}.

1.2. INVERSE SPECTRA

15

I f [ T [> w, then in the same way we can define the spectrum consisting of countable subproducts of the product Y l { x t . t E T } , of natural projections between them and with expwT as the indexing set. The limit space of the latter spectrum will still be homeomorphic to the whole product 1-I{Xt" t E T } . EXAMPLE 1.2.10. Let { X t ' t E T } be a family of subspaces of a space X , indexed by a directed set T. A s s u m e also that Xt, C X t , whenever t -~ t ~. "it Denote by z t ' X t, r Xt~ t ~ t ! the inclusion map. One can easily verify .t t that the limit space of the spectrum S -- { X t , zt , T } is homeomorphic to the intersection M{Xt" t E T } of the given subspaces. Using the general observation made in Example 1.2.10 as a guide, the reader can construct examples of inverse spectra consisting of non-empty spaces and projections with dense images (even surjections), the limit spaces of which are empty. On the other hand we have the following statement, the proof of which follows from the compactness of arbitrary products of compact spaces (Tychonov's theorem) coupled with the fact t h a t an intersection of any centered collection (i.e. a collection, all finite subcollections of which have non-empty intersection) of compact subspaces of any compact space is non-empty. PROPOSITION 1.2.11. The limit space of any inverse spectrum consisting of non-empty compacta is a non-empty compactum. The following technical statement is useful in various situations. PROPOSITION 1.2.12. Let Y1 and ]I2 be disjoint closed subsets of the limit space l i m S x of a spectrum S x - { X , , p ~ , A } . If Y1 is compact, then there exist an index a E A and an open subset G~ C X ~ such that Y1 C p - l ( G a ) and ]I2 M p-~l(G,) - 0. In particular, p , ( Y 1 ) N P,(Y2) - 0. PROOF. For each point y E Y1 consider an index c~y E A and an open subset G ~ C X ~ such t h a t y E p - I ( G a ~ ) C X - Y 2 , whereX--lim Sx SinceYliS compact, there is a finite subcollection {p--1 ,~ (G,~)" i -- 1, ..., k} of the collection {p-1 ,~ (G,~)" y E Y1} which still covers Y1. Take an index c~ E A such t h a t c~ ~_ c~ for each i - 1, ..., k. One can easily verify t h a t the set G , - U { ( p ~ ) - I ( G , ~ ) 9 i -1,..., k} has the desired property. F-1 --

O~y

--

~

1.2.2. M o r p h i s m s o f i n v e r s e s p e c t r a . Suppose we are given two inverse spectra S x -- {Xa, p~, A } and S y - {Y~, q'r, B } 9 A morphism of the spectrum ,.qx into the spectrum S y is a family {~, {f'r" 7 E B}} consisting of a nondecreasing function qo" B --. A such t h a t the set ~ ( B ) is cofinal in A, and of continuous maps f~r" X~(~) ~ Y~ defined for all ~/E B such that q~f6 = J'rP~(-r), ~ ~(~)

16

1. INVERSE SPECTRA

whenever 3', 5 E B and 3" __ 5. In other words, we require (in the above situation) the c o m m u t a t i v i t y of the following d i a g r a m

Xv(6)

~ Y6

p•(5) ~(~)

5 q.y

f~

Xv(7)

~ Y~

Any morphism

{~, {f~. ~ e B}}. S x --, S y induces a (continuous) map, called the limit map of the m o r p h i s m lira {W, {f.y" 3" E B}}" lira S x --~ lim S y . To see this, assign to each thread x = { x a ' a E A} of the s p e c t r u m S x the point Y = {Y'r" 3' E B} of the product l'I{Y.y" 3' E B} by letting y~ = f ~ ( z ~ ( ~ ) ) , - ~ ~ B .

It is easily seen t h a t the point y = {y~" 3' E B} is in fact a thread of the s p e c t r u m S y . Therefore, assigning to x = {xa" c~ E A} E lira S x the point Y = {Y'r" 3" E B} E lira S v , we define a map lira {~, {f.y" 3' E B}}" lira S x --~ lira S y . Straightforward verification shows that this map is continuous. T h e morphisms of inverse spectra which arise most frequently in practice are those defined on the same indexing set. In this case, the map ~" A -+ A of the definition of morphism is taken to be the identity. Below we shall mostly deal with such situations and use the following notation: {fa" X a ~ Y a ; a E A}" S x ~ S y

or sometimes the even shorter form

{f.}"

s x -~ s y .

PROPOSITION 1.2.13. Let S y = {Y(x, q~, A } be an inverse spectrum and X be a space. Suppose that f o r each c~ E A a map fa" X --~ Ya is given in such a way that f . = q~f~ whenever a, ~ E A and c~ -< ~. Then there exists a natural map f " X ~ lim S y (diagonal product A{.fc," c~ E A } ) satisfying, f o r each c~ E A, the condition fc~ = q a f .

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

17

PROOF. Indeed, we only have to note t h a t a space X, t o g e t h e r with its identity m a p i d x , forms the inverse s p e c t r u m S. So the collection { f a : a E A} is in fact a m o r p h i s m $ ~ S y . T h e rest follows from the definitions given above. [-1 T h e proof of the following s t a t e m e n t is an easy exercise. PROPOSITION 1.2.14. Let { f a } : S x ~ S y be a morphism consisting of homeomorphisms. Then the limit map f = lira {.fa}: lira S z ~ lim S y is a homeomorphism.

Historical and bibliographical notes 1.2. T h e basic definitions and introduct o r y s t a t e m e n t s presented in this Section are widely known and can be found in m a n y textbooks.

1.3. F a c t o r i z i n g s p e c t r a

and the spectral

theorem

1.3.1. F a c t o r i z i n g s p e c t r a . We say t h a t a s p e c t r u m S x = { X a , p ~ , A} is factorizing is for each real-valued continuous function ~: lira S x ~ R t h e r e exist an index a E A and a function ~ a : X a --* R such t h a t ~ = ~ a P a . PROPOSITION 1.3.1. Let 8 x = { X a , p ~ , A } be an inverse spectrum whose indexing set A is w-complete. If all limit projections pa, (~ E A, of S x are surjectire, then the following conditions are equivalent: (i) The spectrum S x is factorizing. (ii) Each functionally open subset G of lira 8 x is c~-cylindricaI for some a E A, i.e. G = p-~l(Ga), where ~ E A and Ga is a functionally open subset of X ~ . PROOF. If G is a functionally open subset of X = lira S x , t h e n t h e r e is a function ~ E C ( X ) such t h a t G = ~ - I ( R {0}). By (i), t h e r e exist an index E A and a function ~ a E C ( X a ) such t h a t ~ = ~ a P a . Obviously, the set G~ = ~ - I ( R {0}) is functionally open in X~. It only remains to note t h a t G = p;l(G.). Conversely, let ~ E C ( X ) . Fix a c o u n t a b l e open base 5/ = {Un: n E w} of the real line R. Obviously, the sets Gn = ~ - Z ( u n ) , n E w, are functionally open subsets of X. Consequently, by (ii), t h e r e exist indices a n E A, n E w and functionally open subsets VaN C_ Xa~ such t h a t Gn = pa,-l(va~) for each n E w. By Corollary 1.1.28, there exists an index a E A which m a j o r a t e s each of the indices a n , n E w. Let Vn = (Paa~)--l(va~), n E w. Define a function ~ E C ( X ~ ) by letting ~ a ( x a ) = ~(p-~l(x~)) for each point x~ E X~. First let us show t h a t the function ~ is well-defined. Surjectivity of the limit p r o j e c t i o n pa g u a r a n t e e s t h a t the set ~(p~Z(xa)) is non-empty. Suppose t h a t ~ ( p ~ Z ( x a ) ) contains two different points. Choose an element Un of the base 5 / t h a t contains

18

1. INVERSE SPECTRA

exactly one of these points. T h e n it is easy to see t h a t Gn M p'~l(xa) ~ 0 and p ~ l ( x a ) ~: Gn. T h e s e two relations, coupled with the choice of the index a, lead to the following two c o n t r a d i c t o t y relations: xa E Vn and xa ~ Vn. Therefore, the map ~a" X a --~ R is well-defined and satisfies the equality ~ = ~ a P a . By the construction, the inverse images r n E w, of elements of the b a s e / g are open in X a (because t h e y coincide with Vn). This shows the continuity of ~a.

[-1

COROLLARY 1.3.2. Let S x = {Xa, p~, A} be an inverse spectrum with surjectire limit projections and w-complete indexing set. If the limit space lim S z is Lindelhf, then the spectrum 8 x is factorizing. PROOF. Proposition for some a limit space

Let X - - lim S x and G be a functionally open subset of X. By 1.3.1, it suffices to show t h a t G is cylindrical, i.e. c~-cylindrical E A. Since functionally open cylindrical sets from a base of the of any inverse s p e c t r u m (see P r o p o s i t i o n 1.2.1), we conclude t h a t G = U{p-~l(Ga~)'t E T}, where Gat is a functionally open subset of Xa~ for each t E T. Obviously, a functionally open subspace of a Lindelhf space is Lindelhf itself. Therefore, G is Lindelhf and there exists a countable subcollection { p -1 a t ~ ( G a t . ) ' n E w} of the collection { p ~ ( G a t ' t E T}. Corollary 1.1.28 provides an index a E A m a j o r a t i n g each of the indices O~tn , n E W. Let Vn = ( p g t n ) - l ( G a t . ) , n E w. Obviously, the set V = U{Vn" n E w} is functionally open in X a and G = p-~l(v). P r o p o s i t i o n 1.3.1 finishes the proof. W! PROPOSITION 1.3.3. Let ~" >_ w and S x = { X a , p ~ , A } be an inverse spectrum whose limit projections pa are open, whose spaces X a have Suslin number <_ T and whose indexing set A is T-complete. Then the spectrum S x is factorizing. PROOF. First let us show t h a t for each open subset G C_ X = lira S x there exists an index c~ E A such t h a t c l z G - - p - ~ l ( c l x . p a ( G ) ) . To see this, represent the set G as the union {p-~(Va t ' t E T}, where Vat is open in Xat for each t E T. It is known [278] t h a t the Suslin n u m b e r of the limit space X does not exceed T. Therefore there is a subset T ~ C T of cardinality < T such t h a t

c l x G = clx(U{p'~l" t E T'}). By Corollary 1.1.28, there exists an index a E A such t h a t c~ _,L- a t for each t E T'. Let V = U { V t ' t E T'}, where Vt = ( p a a t ) - l ( v a t ) , t E T'. Obviously, by the construction, c l x G = cIxp-~l(V). Consequently, by openness of the limit projection pc, we have c l x G = p-~l(cIx~V) and, hence, cIxG = p ~ l ( c l x . p a ( G ) ) . T h e rest of the proof only differs slightly from the proof of Proposition 1.3.1. Consider a function ~ E C ( X ) and fix a c o u n t a b l e open b a s e / g = {Un" n E w} of the real line R. By the above established fact, coupled with Corollary 1.1.28, we may assume t h a t

clxv-l(u~) = p ; l ( d x o p . ( v - l ( u . ) ) )

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM for each n E w. We claim t h a t the function ~ a : X a ~

19

R, defined by letting

~a(xa) = ~ ( p ~ l ( x ~ ) , xa E Xa, is a well-defined single-valued function. Assume the contrary. Then, for some point xa E Xa, there are points Xl and x2 in the fiber p~l(xa), such t h a t ~(Xl) ~ ~(x2). Take elements U,~1 and Un2 of the base L/ such t h a t cIRUnl M clRUn2 = ~ and ~(xi) E Un~, i = 1,2. Since is continuous, we have c l x ~ - l ( u n l ) M clx~-l(Un=) = 0. By the above construction and the choice of the index a, b o t h c l x ~ - l ( u n l ) and clx~-l(Un2) are c~-cylindrical. Consequently, we have pa(~-l(Vnl))Mpa(~--l(Un~)) = 0. On the other hand, x~ E pa(~-l(Un~))N pa(~--l(Una) ). This contradiction shows t h a t ~a E C(Xa). Obviously, ~ = ~ p a . T h e continuity of ~ follows i m m e d i a t e l y from the openness of the limit projection pc.

I"-1

1.3.2. T h e S p e c t r a l T h e o r e m . In P r o p o s i t i o n 1.2.14 we have already seen t h a t any isomorphism (i.e. morphism, consisting of h o m e o m o r p h i s m s ) of inverse spectra induces a h o m e o m o r p h i s m of their limit spaces. Of course, the converse s t a t e m e n t is not true. Indeed, consider two inverse sequences Sodd {12k+l ~.k+l,,k W} and Seven _~ {i2k+2, Akk+l w} where ~.k+l 9 I2k+3 _..+ 12k+l and )~k+l. i2k+4 __+ /2k+2 denote the n a t u r a l projections onto the c o r r e s p o n d i n g subproducts. Obviously, b o t h lira Sodd and lim Seven a r e h o m e o m o r p h i c to the Hilbert cube Q. At the same time, as one can easily see, there is no isomorphism between the given inverse sequences whose limit m a p is a h o m e o m o r p h i s m . Our next goal is to show t h a t for certain types of inverse spectra, all m a p s between their limits are induced by m o r p h i s m s of these spectra (or their cofinal subspectra). First we need some definitions. The s p e c t r u m S x = {Xa, P~a, A} is said to be T-continuous if: 9 For each chain B in A with I B I< T and s u p B = ~, the diagonal p r o d u c t /~{p~" a E B} maps the space Xt~ h o m e o m o r p h i c a l l y into the space lim {Xa, P~a, B } . The s p e c t r u m S x = { X a , p ~ , A} is said to be a r-spectrum if: 9 w(Xa) <_ T for each a E A. 9 T h e s p e c t r u m 8 x is T-continuous. 9 T h e indexing set A of S x is T-complete. We call a s p e c t r u m transfinite if its indexing set is an u n c o u n t a b l e cardinal r and the s p e c t r u m is x-continuous (in the above sense) for each ~ with w _< ~ < T. THEOREM 1.3.4. If a ~'-spectrum S x = { X a , p ~ , A } with surjective limit projections is factorizing, then each map of its limit space into the limit space of another T-spectrum S y = {Ya, q~,A} is induced by a morphism of cofinal and T-closed subspectra. If two factorizing T-spectra with surjective limit projections and the same indexing set have homeomorphic limit spaces, then they contain isomorphic cofinal and T-closed subspectra.

20

1. INVERSE SPECTRA

PROOF. Let f : lim S x ~ lim S y be a given map. We perform the spectral search by means of the following relation L = {(c~,3) E A 2" there exists a map fa3" X 3 ~ Yc, such that f~aP3 = qaf}. Let us verify the conditions of Proposition 1.1.29. E x i s t e n c e C o n d i t i o n . Since the weight of the space Ya does not exceed T, there exists a subset {~0t: t E T} C_ C(Ya) of cardinality i T I< T such that the diagonal product/k{~ot : t E T} is a topological embedding of Ya into R T. Since the spectrum 8 x is factorizing, we conclude that for each t E T there exist an index 3t E A and a function r E C(X3t ) such that r t = ~otqa.f. Corollary 1.1.28, coupled with the inequality I T I_< T, guarantees the existence of an index 3 E A which majorates each of the indices /3t, t E T. Consider the diagonal product f~ = /X{C~tp~t't E T}. Obviously, f~ maps the space Z 3 into the product R T and f~P3 = qaf. The last equality, coupled with the surjectivity of the limit projection P3 of the spectrum 8 x , shows that f ~ ( X 3 ) C_ Ya. M a j o r a n t n e s s C o n d i t i o n . The verification of this condition is trivial. Indeed, it is enough to consider the composition f~ = f~p~. w-closeness C o n d i t i o n . Suppose that for some chain C = { a t : t E T} in A with I T I<_ w and c~ = sup C, the maps .f~t" X3 --~ Yat are already defined is such a way that fgtp 3 = qatf for each t E T (in other words, (cet,~) E L for each t E T). These equalities and surjectivity of the limit projection P3 of the spectrum S x imply the equalities f~t = q~ '~' f ~ , whenever at ~_ at,. Since the spectrum Sy is r-continuous, Ya is canonically homeomorphic to the limit space of the subspectrum S y / C . Therefore the diagonal product f~ = / X { f ~ ' t E T} maps the space X 3 into the space Yc, and satisfies the equalities f~P3 = qaf. Consequently, (a, 3) E L. Now denote by A ~ the set of all L-reflexive elements in A. By Proposition 1.1.29, A' is a cofinal and r-closed subset of A. One can easily see that the L-reflexivity of an element a E A is equivalent to the existence of a map fa = f~" X~ --. Y~ satisfying the equality .f~p,~ = qaf. Consequently, the collection Ot {fa: a E A ~} is a morphism of the cofinal and r-closed subspectrum S x / A ~ of the spectrum S x into the cofinal and r-closed subspectrum S y / A t of the spectrum Sy. It only remains to remark that the original map f is induced by the constructed morphism. This finishes the proof of the first part of our Theorem. The second part of the Theorem can be obtained from the first as follows. Let f : lim S x ~ lim S y be a homeomorphism. Denote by f - l : lim Sy ~ lim S x the inverse of f. By the first part proved above, there exist a cofinal and T-closed subset A1 of A and a morphism {fc~: Xa --+ Yc~: o~ E A1}: 8 x / A 1 ---* By~A1 such that .f = lim{f~: c~ E A1}. Similarly, there exist a cofinal and w-closed

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

21

subset A2 of A and a morphism {ga: Ya ~ X a : a E A2}: S y / A 2

---->S x / A 2

such that f - 1 = lim(ga : a E A2}. By Proposition 1.1.27, the set B -- A1 M A2 is still cofinal and T-closed in A. Therefore, in order to complete the proof, it suffices to show that for each a E B the map f a : X a ---+ Y~ is a homeomorphism. Indeed, take a point xc, E X a and, using surjectivity of the limit projection p~ of the spectrum ,-qx, fix a point x E lim ,-qx such that x a = p,~(x). Then g a f a ( x a ) - g a f a p a ( x ) -- g a q a f (x) - p a f - l

f (x) = p a ( x ) -- x a .

Similarly, we see that fago, - idy,,, for each a E B. It follows easily t h a t in this situation each .fa, oL E B , is a homeomorphism. [-1 The above Theorem provides a topological sense of the so-called "uncountability effect," the set-theoretical justification of which is contained in Proposition 1.1.27. As was remarked above, the Spectral Theorem h a s no counterpart in the case of inverse sequences. This means that two inverse sequences with homeomorphic limit spaces do not have to contain isomorphic cofinal subsequences. The Spectral Theorem allows us to give a practical sense to the m e t h o d of "spectral search," the roots of which, in fact, are still set-theoretical (see Proposition 1.1.29). More formally, if, for example, we need to show t h a t a given (non-metrizable) c o m p a c t u m can be represented as the limit space of some T-spectrum, consisting of open limit projections, and if we already have a T-spectrum representing the given compactum, then (if the problem is solvable) there is a cofinal and T-closed s u b s p e c t r u m of the given one whose limit projections are open and, therefore, the entire problem reduces to the problem of finding this subspectrum. Let us remark also that the requirement of surjectivity of limit projections of spectra, hypothesized in Theorem 1.3.4, is indeed essential. First observe t h a t any separable metrizable space X can be represented as the limit space of some factorizing w-spectrum with surjective projections (it is enough to consider the trivial spectrum all elements of which coincide with X and all projections of which are identity maps i d x ) . Moreover, one can easily see that every separable metrizable space, X in particular, can be represented as the limit space of factorizing w-spectrum consisting of Polish spaces with, generally, non-surjective limit projections (to see this embed X into, say, the Hilbert cube Q and consider the collection of all G6-subsets of Q containing x , together with inclusion maps; compare with Example 1.2.10). Let us now assume that X is a non-complete separable metrizable space and represent it as the limit space of the s p e c t r u m S x -- { X a , p ~ , A } consisting of Polish spaces. Assuming that T h e o r e m 1.3.4 is valid in this situation, we easily conclude that there exist an index a E A and

22

1. INVERSE SPECTRA

a map f a : Xa --, X such that fap,~ = i d x . The latter immediately implies the completeness of X, contradicting our choice of X. This example shows that spectra arising in different situations do not necessarily have surjective limit projections and, therefore, that we are somewhat restricted in using such a powerful tool of spectral analysis as the Spectral Theorem. If we wish to understand what sort of generalizations of Theorem 1.3.4 we might expect in order to cover wider classes of inverse spectra, let us first characterize the class of spaces which can be represented as the limit spaces of factorizing w-spectra (a more general problem, of characterizing the limit spaces of factorizing T-spectra, has been considered in [75]). This is done in the next statement. PROPOSITION 1.3.5. The following conditions are equivalent for any space X : (i) X is realcompact. (ii) X is homeomorphic to the limit space of a factorizing w-spectrum, consisting of Polish spaces. (iii) X is homeomorphic to the limit space of a factorizing w-spectrum. PROOF. (i) ~ (ii). Embed X as a C-embedded subspace into R A for sufficiently large A (one can take A - C ( X ) ) . By (i) and Proposition 1.1.16, X is closed in R A. Consider the standard w-spectrum ,~ = {R B, 7r~,expwA}, consisting of countable subproducts of R A and natural projections between them. Obviously, lira S - R A (see Example 1.2.9). Note also that since every continuous real-valued function, defined on R A, depends on countably many coordinates, the spectrum S is factorizing (the same conclusion follows, for example, from Proposition 1.3.3). Closeness of X in R A and Corollary 1.2.6 show that the limit space of the spectrum S x -- { X B , p ~ , exp.,A} coincides with X. Here X B -- clR, lrB(X), B 9 expwA, and p~ - 7r~/XB whenever C , B 9 exp~A and C C B. Obviously, the spectrum S x is w-continuous and consists of Polish spaces. Since X in C-embedded in R A we see that 8 x is a factorizing spectrum. (ii) =~ (iii). Trivial. (iii) ~ (i). Apply Propositions 1.2.2, 1.1.13 and Corollary 1.1.25. Kl Thus, we now have a result characterizing the class of realcompact spaces as the maximal class of spaces which can be studied using the concept of factorizing w-spectra. But here it is necessary to make an important remark. Namely, although the factorizing w-spectra, constructed in Proposition 1.3.5, have a very promising additional property - all spaces in this spectra are Polish- nevertheless, these spectra have an essential deficiency since their limit projections are not surjective. And as we have seen above, under the latter circumstances we cannot use the fundamental result of the whole spectral analysis- the Spectral Theorem. Of course, we can take spaces 1rB(X) as elements X B of the spectra S x (instead of their closures in RB; see the proof of Proposition 1.3.5). Doing so we lose

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

23

completeness in favor of surjectivity of the limit projections. But, unfortunately, the inverse spectra obtained in this way would not be continuous! A simple observation saves the situation. First of all, let us note t h a t according to Corollary 1.2.5 we may assume, without loss of generality, t h a t

all limit p r o j e c t i o n s of inverse spectra, c o n s i d e r e d b e l o w , have d e n s e images. Now we prove another version of the Spectral Theorem. THEOREM 1.3.6. Each map f : lira S x ~ lim S y between the limit spaces of two factorizing w-spectra S x -- {Za, p~, A} and S y - - {Ya, q~, A } , consisting of Polish spaces, is induced by a morphism of cofinal and w-closed subspectra. If f is a homeomorphism, we may assume that the above morphism is an isomorphism. PROOF. The proof of this s t a t e m e n t differs just slightly from the proof of Theorem 1.3.4. As in that proof we perform the spectral search with respect to the following relation: L = {(a,/3) E A 2" there exists a map f a~" Xf~ -+ Ya such t h a t f~p~ - q~f}. The majorantness and w-closedness conditions of Proposition 1.1.29 can be verified just as in the proof of Theorem 1.3.4. Let us verify the conditions of Proposition 1.1.29. Let a E A. Since, by our assumption, the space Ya is Polish, we can, by Proposition 1.1.3, identify Ya with a closed subspace of R w . Therefore there exists a countable subset {~n: n E w} C_ C(Ya) such that the diagonal product A { ~ n : n E w} is a closed embedding of Ya into R w . As in the proof of Theorem 1.3.4, we can find an index ~ E A and a map f~" X~ --* R w such that qaf = f~p~. The latter implies that fa~(p~(lim S x ) ) = q a ( f ( l i m S x ) ) C_ qa(lim S y ) C_ Ya. Now recall that the image of the limit projection Pt~ is dense in X t~. Therefore, by continuity of f ~ and closeness of Ya in R w , we have

.f~(X~) = f~(clx~p~(lim S x ) ) C claw .f~(pf~(lim ,~x)) c ClRw Va -- Ya. This finishes the verification of the existence condition and the proof of the Theorem. [:] Let us now consider some curious examples indicating that T h e o r e m 1.3.6, unlike Theorem 1.3.4, works even for separable metrizable spaces. Indeed, let X be a dense subspace of a Polish space X ~ and consider the collection of all Gs-subspaces of X ~ containing X. Obviously (compare with E x a m p l e 1.2.10), this collection, together with the corresponding inclusion maps, forms an w-spectrum S x , x , consisting of Polish spaces. L e m m a 1.1.19 and Proposition 1.1.22 show t h a t this spectrum is factorizing. Obviously, lim 8 x , x , = X . Suppose now that Y is a dense subspace of another Polish space Y~ and consider a

24

1. INVERSE SPECTRA

factorizing w-spectrum Sy, y, constructed in a similar way. Finally assume t h a t there is a h o m e o m o r p h i s m f" X ~ Y. T h e o r e m 1.3.6, applied to spectra , ~ x , x , and S y , y , , guarantees t h a t there exist G~-subspaces X c X ' and ]7 C Y' of X ' and Y' respectively and a map ] : )( ~ Y such t h a t X c_ )(, Y c_ ]7 and f - ]/X. In other words we have obtained the classical result of Lavrentieff In the next subsection we present a n o t h e r application of T h e o r e m 1.3.6 dealing with a well-known result of T u m a r k i n [301].

1.3.3. I m m e d i a t e c o n s e q u e n c e s . We begin by establishing some dimensional properties of realcompact spaces. We start with the following simple lemma, the straightforward proof of which is left to the reader. LEMMA 1.3.7. L e t S x -- { X a , p ~ , A } be a f a c t o r i z i n g s p e c t r u m and X lim S x . I f dim p a ( X ) <_ n f o r each ~ E A , then dim X < n.

--

PROPOSITION 1.3.8. L e t f : X ---, Y be a m a p o f at m o s t n - d i m e n s i o n a l compactum X

into a m e t r i z a b l e c o m p a c t u m Y .

T h e n there exist at m o s t n - d i m e n -

s i o n a l m e t r i z a b l e c o m p a c t u m Z a n d two m a p s g: X ---, Z and h: Z ---+ Y such t h a t f -- h . g.

PROOF. By Proposition 1.3.5, we can represent X as the limit space of some factorizing w-spectrum S x = {Xa, p~, A } . We shall show t h a t almost all spaces X a in the s p e c t r u m S x are at most n-dimensional, and t h a t as the c o m p a c t u m Z one can take an element of our spectrum. We perform the spectral search with respect to the following relation: L ---- {(~,/3) E A 2" c~ _
cover ]2 of X~ of order < n such t h a t ( p ~ ) - l ( U ) refines Y}. Let us the check conditions of Proposition 1.1.29. E x i s t e n c e c o n d i t i o n . Let {/gk" k E w} be a f u n d a m e n t a l sequence of finite open covers of X a . This means t h a t for each open cover U of X a there is an index k E w such t h a t / ~ k refines bt. For each k E w take a finite functionally open cover }4~k of X of order _< n such t h a t ~Vk refines p~l(/,{k). Since our s p e c t r u m is factorizing, we conclude (see Proposition 1.3.1) t h a t there exist an index/~k E A such that/~k _> a, and an open cover ~k~k of the c o m p a c t u m X~k -1 (Vk)" /~k such t h a t Wk -- P~k Observe t h a t the order of ~k~k is at most n. By Corollary 1.1.28, there is an index/~ E A which m a j o r a t e s each of the indices/~k, k E w. Let Vk -- (P~k)--l(Yk~k). Clearly Vk is a finite open cover of X~ of order at most n, k E w. It only remains to note t h a t (c~,/3) E L. M a j o r a n t n e s s c o n d i t i o n . Let (c~,/~) E L and -y ~-/~. I f / g is an open cover of X a , then there is an open cover V' of X~ of order at most n which refines

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

25

( p ~ ) - l ( / g ) . T h e n it is easy to see t h a t the open cover V -- (p~)-l(~),) verifies the fact t h a t (a,-y) E L. w - c l o s e n e s s c o n d i t i o n . Let {hi" i E w} be a c o u n t a b l e chain of elements in A such t h a t (c~i, fl) E L for each i E w and c~ - sup{hi" i E w}. L e t b / be an open cover of the c o m p a c t u m X a . Since our s p e c t r u m is w-continuous, the space X a is n a t u r a l l y h o m e o m o r p h i c to the limit space of the inverse sequence {Xa~, p~a~+l , w}. C o n s e q u e n t l y there exist an index i E w a n d open cover b/i of Xi such t h a t (p~a, ) - 1 (b/i) is refinement of b/. F u r t h e r , by a s s u m p t i o n , (ai, f~) E L. Therefore, t h e r e is an open cover 12 of X f~ of order _< n such t h a t 12 refines (p~if~)-l(b/i). Clearly, in this situation, 12 refines ( p ~ ) - l ( b / ) as well. This finishes the verification of all t h r e e conditions. Therefore, by P r o p o s i t i o n 1.1.29, the set A ~ of all L-reflexive indices is cofinal a n d w-closed in A. Observe t h a t if an index ~ is L-reflexive (i.e. ( ~ , ~ ) E L), t h e n dim X a < n. Since Y is m e t r i z a b l e and the s p e c t r u m S x / A ~ is still factorizing (see C o r o l l a r y 1.3.2) there exist an index ~ E A ~ a n d a m a p h" X~ --~ Y such t h a t f -- h . p a . This finishes the proof. 71 LEMMA 1.3.9. For a n y countable collection {Pk" X --~ X k ; k E w } o f m a p s o f an at m o s t n - d i m e n s i o n a l space X i n t o P o l i s h spaces X k , there exist an at m o s t n - d i m e n s i o n a l P o l i s h space P and m a p s g" X ~

P , gk" P --~ Pk, k E w, such

t h a t Pk -- gk " g f o r each k E w.

PROOF. Let us first consider the s i t u a t i o n in which the given collection con_ thins only one element, a map p0" X --~ P0. Let P0 denotes a m e t r i z a b l e corn_ pactification of Po. T h e n there is an extension p0" f~X ~ P0 of the m a p p0, where f~X denotes the S t o n e - C e c h compactification of X. Since dim X - dim f~X, we conclude, using P r o p o s i t i o n 1.3.8, t h a t t h e r e exist an at most n - d i m e n s i o n a l m e t r i z a b l e c o m p a c t u m K and two m a p s ~" f~X ~ K and g0" K --~ P0 such t h a t po = g0"g. Obviously the Polish space P = ( ~ 0 ) - l ( p 0 ) and m a p s g = ~ / X " X --. P and go - g o / P " P --~ Po satisfy t h e desired conditions. Next we consider the general case. T h e diagonal p r o d u c t A { p k" k E w } m a p s the space X into the Polish space P~ -- l l { P k " k E w}. By t h e above considered case, t h e r e exist an at m o s t n - d i m e n s i o n a l Polish space P and two m a p s g" X ~ P a n d h" P ~ P~ such t h a t p h.g. T h e n the space P and m a p s g and gk -- 7rk " h, k E w, where pi k" P~ --~ Pk d e n o t e s the p r o j e c t i o n onto t h e k-th coordinate, satisfy the desired conditions. D THEOREM 1.3.10. T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t f o r a n y r e a l c o m p a c t space X "

(i) d i m X _ < n . (ii) X can be r e p r e s e n t e d as the l i m i t space o f a f a c t o r i z i n g w - s p e c t r u m , c o n s i s t i n g o f at m o s t n - d i m e n s i o n a l P o l i s h spaces.

26

1. INVERSE SPECTRA

PROOF. One part of our statement follows from Lemma 1.3.7. Now we consider an at most n-dimensional realcompact space X, and we try to represent it as the limit space of a factorizing w-spectrum consisting of at most n-dimensional Polish spaces. If X itself is a Polish space, then the above conclusion is trivially true. Therefore, assuming t h a t X is C-embedded (and, by Proposition 1.1.16, closed) in R A , we only have to consider the case in which A is uncountable. Denote by ~rB" R A ~ R B, B C A, the natural projection onto the corresponding subproduct and let X B -- c I R , 3 r B ( X ) . Next, denote by exp
1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

27

Let X be a dense subspace of a Polish space X ' . Assume, additionally, t h a t dim X -- n. Let S x , x , be a factorizing w-spectrum consisting of all G6-subspaces of X ' , containing X, together with their inclusion maps (see the end of Subsection 1.3.2). By T h e o r e m 1.3.10 and Corollary 1.1.25, the space X is h o m e o m o r p h i c to the limit space of a factorizing w-spectrum ,.q consisting of n-dimensional Polish spaces. Now, applying T h e o r e m 1.3.6, we see t h a t the spectra S x , x , and S contain isomorphic cofinal and w-closed subspectra. In particular, there is an n-dimensional G~-subspace of X ' containing X. T h e following s t a t e m e n t will be needed later in C h a p t e r 5. PROPOSITION 1.3.11. Let f" P ~ Y be a map of a Polish space P into a metrizable compactum Y . Then f admits an extension F" P ---+Y to a metrizable compactification P of P such that dim / 5 _ dim P . PROOF. Let /3P denote the Stone-(Tech compactification of P.

Obviously,

dim fiR - dim P - n. By T h e o r e m 1.3.10, tiP = lim 8, where 8 - { Z a , P~a, A} is a factorizing w-spectrum consisting of n-dimensional metrizable compacta. Since P is Polish, we see t h a t P is the intersection of countably m a n y functionally open subsets o f / 3 P . Therefore there exists an index a0 E A such t h a t P Pao-l(Pao(P))" Consider now the s p e c t r u m S p = {Pa, q~,Ao}, where A0 = {a E A" a _> a0}, P~ = p~(P) and qa f~ = p~/P~ for each a , f l E Ao with a _
28

1. INVERSE SPECTRA

Z is closed in X . Suppose also that .f" Z -+ Y is a map such that C ( f ) ( C ( Y ) ) C_ C ( X ) / Z . Then f is induced by a morphism of cofinal and w-closed subspectra of the spectra S z and S y , where S z -- { c l x ~ p a ( Z ) , p ~ , A } . PROOF. We perform the spectral search with respect to the following relation" L = {(c~,fl) E A 2" c~ ~ fland there exists a map f~" cIx~pz(Z) ~ Ya

such t h a t qa" f - - f ~ ' P ~ / Z } . Let us first verify the Existence Condition of Proposition 1.1.29, i.e. let us show t h a t for each c~ E A there is /3 E A such t h a t (c~,fl) E L. Since Yc~ is a Polish space, we can assume, by Proposition 1.1.3, t h a t Ya is closed in R ~~ . Denote by 7rn the restriction to Ya of the n a t u r a l projection of R • onto the nth coordinate. Clearly, the diagonal product A{Trn : n E w} coincides with the inclusion map of Ya into R ~~. The function lrn .qa, n E w is an element of C(Ya) and, consequently, by our assumption, the function 7rn.qa.f has an extension ~on to the space X. T h e s p e c t r u m S x is factorizing. Therefore, for each n E w, there exist an index ~n E A and a function ~O~nn E C(X~n) such t h a t ~ o n - ~O~n~" p ~ . Since the indexing set A is w-complete, there is, by Corollary 1.1.28, an index fl E A which m a j o r a t e s each of the indices ~n, n E w. Let ~o~ = ~o~~ . p ~ , n E w. T h e diagonal p r o d u c t g~ = /~{~n" n E w} maps the space X~ into R'. It follows from the construction t h a t q a . . f = g ~ . p z / Z . Consequently, g~(pz(Z)) c_ Ya. By continuity of g~ and closedness of Ya in R ~ , we have g~(clz~pz(Z)) C_ Ya. It only remains to note t h a t the map f ~ = g ~ / c l x r satisfies the desired properties. This finishes the verification of the Existence Condition. T h e verification of the Majorantness and w-closedness Conditions of Proposition 1.1.29 is trivial. Therefore, by t h a t Proposition, the set of L-reflexive indices is cofinal and w-closed in A. [-I Later we will find it helpful to have a cardinal invariant distinguishing the Polish spaces a m o n g all separable metrizable spaces. DEFINITION 1.3.14. The R-weight of a space X is the minimal infinite cardinal T such that X admits a C-embedding into R r (notation: R - w ( Z ) = T). Obviously, the usual topological weight of any space X does not exceed its R-weight. T h e converse is true, for example, for any infinite compactum. T h e reader can easily verify that: 9 Rw ( X ) <1 c ( x ) I for any space X. 9 R - w ( X ) = R - w ( u X ) for any space X. Countability of the R - - w e i g h t characterizes Polish spaces (see Propositions 1.1.3 and 1.1.16). Monotonicity of the R-weight with respect to C - e m b e d d e d subspaces also follows immediately from the definition.

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

29

LEMMA 1.3.15. Let S x = { X a , p ~ , A } be a factorizing spectrum consisting of Polish spaces. Then R - w(lim S x ) _< max{w, ]A ]}. PROOF. If the indexing set A of the spectrum S x is countable, then the limit space lim S x is Polish and, consequently, in this case the conclusion of the Lemma is true. Suppose that A is uncountable. Since S x is a factorizing spectrum, we can easily see that the limit space lim S x is C-embedded in the product ~ { Z ~ " ~ 6 A} (compare with Proposition 1.2.2). But as noted before, the R-weight in monotone with respect to C-embedded subspaces. Therefore, it suffices to show that the R-weight of the product ~ { X a " a 6 A} of uncountable family of Polish spaces does not exceed I A I. W i t h o u t loss of generality we may assume that X~ is closed in R~ ---- R ~ (Proposition 1.1.3). Consequently,

x =

A} c II{R

.

e A} = R

and we only have to show that X is C-embedded in R A. Consider a function E C ( X ) . By Proposition 1.3.3, there exist a countable subset B of A and a function r 6 C ( X B ) , where X B = Y I { x a . ~ 6 B}, such that ~o = r (PB denotes the natural projection of X onto X B). The countability of B implies t h a t R B is Polish, and since X B is closed in R B there is an extension r 6 C ( R B) of r Then the function ~ = r where 7rs" R A ---+ R B denotes the projection, is the desired extension of the original function ~o to the whole R A. [3 Finally we present one more version of the Spectral Theorem. THEOREM 1.3.16. Let f " Z ~ F be a map between closed subspaces Z and F of an uncountable product Y I { x a . ~ E A } of Polish spaces. Suppose that B C_ A and there exists a map f B " Z B ~ FB such that 7rB. f = f B " 7rB/Z. T h e n the collection 1CB = {C E e x p ~ ( A -

B ) " there is a map f B u c " ZBuO --* FBuC

such that lrBuC" f = f B u C " 7 r B u c / Z } is cofinal and w-closed in e x p w ( A - B ) .

PROOF. Consider the following relation L B on the set ( e x p w ( A - B)) 2" L B = {(C, D) 6 ( e x p w ( g - B)) 2" C C D and there is a map

IO,D: ZBUD -'+ FBUC such t h a t ~rBu c 9 f -- IC, D" 7rBuD/Z}.

Now we are going to perform the spectral search with respect to LB. E x i s t e n c e C o n d i t i o n . We have to show that for each C E e x p w ( A - B ) there exists D E e x p ~ ( A - B ) such t h a t ( C , D ) E L B . Using the s t a n d a r d factorization argument we can find a countable set R 6 e x p w A and a map f R : Z R ~ F c such t h a t C C R and f R" 1rR/Z -- 7rc . f . Let

30

1. INVERSE SPECTRA

B. Clearly C C D. Define the m a p fC,D to be the diagonal product fC, D = f s " 7rBUDAfR " ~rBUD (note t h a t B U D = B U R). It only remains to note t h a t ~SuC" f = fC,D " ~BuD/Z. Therefore ( C , D ) 6 LB. M a j o r a n t n e s s C o n d i t i o n . Let (C, D) 6 LB, E E e x p ~ ( A - B) and D C E. D -- R -

We need to show t h a t (C, E ) E LB. Consider the m a p fC,E - - "~ _ SB Uu DE " f C , D " w - c l o s e d n e s s C o n d i t i o n . Let (Ci, D) 6 LB, i 6 w, where {Ci" i 6 w} is a c o u n t a b l e chain in e x p , ~ ( A - B ) . T h e n we need to show t h a t ( C , D ) 6 LB, where c = u{c~" i e ~}. Clearly F c coincides with the limit space of an inverse sequence / F B o c ~ + I ,i E w } 9 { F B u c ~ , 7rBBUCi+I uCi BOCi+ i

A straightforward verification shows that fc~,m = r B U C ~ "fC~+,,m for each i 6 w. It only remains to note t h a t the m a p fc,m -- /k{fc~,m" i e w} confirms the fact t h a t (C, D) 6 LB. By P r o p o s i t i o n 1.1.29, the set ~B is cofinal and w-closed in expw(A - B). [3 Observe t h a t the case B = 0 yields, in fact, the above discussed version ( T h e o r e m 1.3.4) of the Spectral T h e o r e m for w-spectra consisting of c o m p a c t a . COROLLARY 1.3.17. If in Theorem 1.3.16 the maps f and f B a r e homeomorphisms, then the collection i~, B contains a cofinal and w-closed subcoUection ~ such that f Buc is a homeomorphism for each C 6 IC~B. We conclude this C h a p t e r with the following s t a t e m e n t . PROPOSITION 1.3.18. Dimension dim is monotone with respect to z-embedded

subspaces. PROOF. Let Y be a z - e m b e d d e d subspace of a space X with dim X = n. Since dim X = dim v X and since X is z - e m b e d d e d (even C - e m b e d d e d ) in v X we may assume, w i t h o u t loss of generality, t h a t X is realcompact. Consider a finite functionally open cover L / - { U 1 , . . . , Uk} of the subspace Y. Since Y is ze m b e d d e d in X t h e r e is a functionally open subset Vi of X such t h a t Ui - Vi MY, k 1Vi. By T h e o r e m 1.3.10 X can be represented as the i - 1 , . . . , k . Let V - Ui= limit space of a factorizing w - s p e c t r u m ,gx = { X a , p ~ , A } consisting of at most n - d i m e n s i o n a l Polish spaces. Since Vi is functionally open in X we conclude, by P r o p o s i t i o n 1.3.1, t h a t t h e r e exist an index ~ 6 A and an open subset Wi, i - 1 , . . . ,k, of X~ such t h a t Vi - p - ~ l ( w i ) for each i - 1 , . . . , k. Now consider k 1W~ of XZ and its open cover ]4; - { W 1 , . . . , Wk}. the open subspace W - Ui= Since dim is m o n o t o n e with respect to any subspaces within the class of Polish spaces, we conclude t h a t there exists a finite open cover ]4;' of W of order < n which refines W. It only remains to r e m a r k t h a t p ~ l ( l / Y ' ) / Y is a functionally open cover of Y of order < n and which refines/4. Therefore dim Y < n. [3

1.3. FACTORIZING SPECTRA AND THE SPECTRAL THEOREM

31

Historical and bibliographical notes 1.3. The notion of factorizing T-spectrum first appeared in [278]. The Spectral Theorem for such spectra with surjective limit projections (Theorem 1.3.4) was established by S~epin [278]. Its version (Theorem 1.3.6) for spectra consisting of Polish spaces was proved in [84]. Theorem 1.3.10 is taken from [81]. In the case of compact spaces this result contains Marde~i5's theorem mentioned in the Preface. Proposition 1.3.11 was originally obtained in [153]. Theorem 1.3.16 is taken from [107]. Proposition 1.3.18 first appeared in [72, 77, 80] and was later rediscovered (in a more concrete situation) in [250] and [240]. All other statements from this Section are due to the author.

CHAPTER

Infinite-Dimensional

2.1. A b s o l u t e

extensors

2

Manifolds

and absolute

retracts

2 . 1 . 1 . S p a c e s o f m a p s . T h e collection of all o p e n (countable) covers of a Polish space X is d e n o t e d by coy(X). If a n o t h e r Polish space Y is given t h e n C (Y, X ) denotes the set of all c o n t i n u o u s m a p s from Y into X . Let f, g" Y --~ X be two m a p s and /d E coy(X). We call these m a p s ~d-close if for each point y E Y there exists an element U E L/ such t h a t f(y),g(y) E U. D e n o t e by B (f,/d) the collection of all those m a p s g E C (Y, X ) which a r e / d - c l o s e to f . We define the t o p o l o g y on the set C(Y, X ) as follows. A set T E C(Y, X) is o p e n if for each f E T t h e r e is an open cover 5/ E coy(X) such t h a t B(f, bl) C__T. This topology is precisely t h e limitation topology on C(Y, X). Everywhere below,

unless otherwise stated, we consider only the above defined topology 1. Suppose t h a t a m a p f" Y --+ X is given and 5/ E coy(Y). We say t h a t .f is a ~d-map if t h e r e exists an o p e n cover 12 E coy(X) such t h a t t h e cover . f - l ( v ) "- { f - l ( Y ) "

Y E V} refines/1/.

LEMMA 2.1.1. Let

/d E coy(Y).

Then the set of all ~d-maps is open in

c(y,z), PROOF. W i t h o u t loss of generality we m a y a s s u m e t h a t t h e set of a l l / d - m a p s is not empty. Let f : Y --~ X be an U - m a p . O u r 142 E cov(Z) such t h a t t h e set B(f, 142) consists is a / d - m a p , t h e r e is an open cover 12 E coy(X) Take an open cover l/Y E coy(X) star of which

goal is to find an o p e n cover entirely of U - m a p s . Since f such t h a t f - l ( l ~ ) refines /d. refines 12. This m e a n s t h a t

t h e star St(W, 142) -- U{W' E l/V: W ' M W -~ 0} of each e l e m e n t W E I/V is c o n t a i n e d in an e l e m e n t of the cover 12 ( p a r a c o m p a c t n e s s of t h e space X implies t h a t such refinements always exist). Let us show t h a t t h e set B(f, l/V) consists

lit should be emphasized that the sets B(f,b/) are not necessarily open (although their interiors are non-empty); see [53] for additional details. 33

34

2. INFINITE-DIMENSIONAL MANIFOLDS

of/d-maps.

First observe t h a t , by the choice of 14], it suffices to prove t h a t for each g E B ( f , 14)) and each W E 14). Indeed, Since t h e m a p s f and g are W-close, there is an e l e m e n t Wy E 14) such t h a t g(y), f ( y ) E Wy. Since g(y) E W , we conclude t h a t W M Wy ~ ~. Therefore f ( y ) E S t ( W , 1/V). Consequently, g - l ( W ) c_ f - 1 (St(W, l/V)) as desired. [:]

g-l(w)

c f-I(St(W,W)) suppose t h a t y E g - l ( w ) .

LEMMA 2.1.2. Let d be a complete metric on a Polish space Y and suppose 1 for each a sequence { b i n ' n e N } C coy(Y) is chosen so that d i a m d V < -~ U E bin and each n E N . Then a map f : Y ~ X is a closed embedding (i.e. an embedding with closed image) if and only if f is a bin-map for each n E N . In particular, the set of all closed embeddings of Y into X is a G~-subset of C(Y,X). PROOF. Let f " Y ---. X be a closed e m b e d d i n g and 34 E coy(Y). T h e n f is a bi-map. Indeed, consider an open cover f(/d) - { f ( U ) " V E /d} of the subspace f ( Y )

C X . For each U E bi select an open subset U I of X so t h a t f ( V ) = U' M f ( Y ) . Obviously, 1) = { V " U E bi} U {X - f ( Y ) } is an open cover

of X . It only r e m a i n s to note t h a t f - 1 ( 1 ) ) refines/d. Now suppose t h a t a m a p f " Y ~ X has the properties indicated in our L e m m a . Let us show t h a t f is a closed e m b e d d i n g . First we show t h a t f is a one-to-one map. Let yl, y2 E Y and Yl ~ y2. Take an integer n so large t h a t ~1 < d(yl, Y2). Since f is a bin-map, t h e r e is an open cover ~n E coy(X) such t h a t f - l ( ~ n ) refines/dn. Suppose t h a t f ( y l ) = x = f(y2) and consider an e l e m e n t V E Vn, c o n t a i n i n g x. T h e n there exists an e l e m e n t U E lAn such t h a t f - l ( V ) C_ U. In particular, yl, y2 E U. This c o n t r a d i c t s the choice of n. Thus, f is a one-to-one map. Consider now an a r b i t r a r y sequence {Xk" k E N } C f ( Y ) which converges to a point x E Z . Let Yk = f - l ( x k ) , k e N , e > 0, ~1 < e and l)n E coy(X) be an open cover of X such t h a t f - l ( ] 2 n ) refines bin. Take an element V E Vn c o n t a i n i n g t h e point x. Since lim Xk -- x, t h e r e is an integer m such t h a t Xk E V for each k > m. Let U E bin such t h a t f - l ( V ) C U. Obviously, Yk E U for each k _> m. B u t dicing(U) < e. Therefore d(yk, Yk,) < e whenever k , k ' >_ m. This shows t h a t t h e sequence {yk" k E N } is a C a u c h y sequence. Let y = lim Yk. By the c o n t i n u i t y of f , x = f ( y ) E f ( Y ) . Therefore t h e set f ( Y ) is closed in X . T h e r e m a i n i n g p a r t of our s t a t e m e n t follows from L e m m a 2.1.1. Z] T h e proof of t h e following s t a t e m e n t is trivial. LEMMA 2.1.3. Let A be a closed subset of a Polish space Y , bt E cov(Y) and f E C ( Y , X ) . If the restriction f / A : A ---. X is a b i / A - m a p ( h e r e / d / A = {U M A: U E bi} E coy(A)), then there exists an open neighborhood G of A in Y such that the restriction f / G : G ~ X is a U / G - m a p .

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

35

In order to obtain other, not so trivial, properties of the space C(Y, X ) we describe the topology introduced above in a somewhat different way. It is easy to see that every bounded metric d on the space X generates the sup-metric d on C(Y, X). The definition is as follows:

d(f,g) = sup{d(f(y),g(y)): y E Y } , f , g e C(Y,X). The topology generated by the metric d on the set C (Y, X) is called the topology

of d-uniform convergence. By M etr(X) we denote the collection of all bounded metrics on a space X generating the topology of X. The collection {d: d E M e t r ( X ) } generates a topology on C(Y, X) which is precisely the topology of uniform convergence with respect to all metrics. Obviously, this topology is stronger than the topology of d-uniform convergence for each particular metric d. LEMMA 2.1.4. The limitation topology coincides with the topology of uniform convergence with respect to all metrics. PROOF. First, suppose that G is a neighborhood of f E C(Y, X) with respect to the topology of uniform convergence. This means that there exist a number e > 0 and a bounded metric d E M etr(X) such that

S ~ {g e c(Y, x ) : ~(f, g) < ~} c_ a. It is easy to check that B(f, bl) C_ {g E C ( Y , X ) : d ( f , g ) < e}, where U = {Bd(x, ~" x E X } (here Bd(x, r) denotes the open d-ball of radius r with center at x). For the converse, consider a neighborhood B, f, L/) of the map f in the limitation topology. Take a metric d E M e t r ( X ) such t h a t {Sd(x, 1): x E X } refines L/ (see, for example, [32, Theorem 4.1]). Obviously, {g E C(Y, X ) : d(f, g) <

1} c_ B(f,U).

0

Remark 2.1.5. If the space X in the second half of the proof of L e m m a 2.1.4 is Polish, then the metric d, corresponding to a cover L/, may be assumed to be complete. The following statement gives us another description of the limitation topology. LEMMA 2.1.6. Let d E Metr(X) and f E C(Y, X). The collection [Bd(f, c~): c~ E C(X, (0, c~))}, where Bd(f,a) - {g E C ( Y , X ) : d(f(y),g(y)) ~_ (~(f(y)) for each y E Y }, forms a local basis at f in C (Y, X ) . PROOF. For a given open cover b / E coy(X), define a map c~u: X ~ as follows: 1 ~u(~) = 5 ~up{di~td(~, X - V). U e U } , x e X.

(0, c~)

36

2. INFINITE-DIMENSIONAL MANIFOLDS

Observe that Bd(f, c~u) C B ( f , lg). Conversely, if a 9 C ( X , (0, c~)), then we define an open cover L/a 9 coy(X) as b/a = {Uz" x 9 X}, where

U~=Bd(x, Note that B(f,/ga) C_ B d ( f , a ) .

~_~ ) n a-~((-T-, "(~) cr [3

Note that if the spaces X and Y are compact (and metrizable), then the limitation topology coincides with the compact-open topology. The following statement expresses a very important and frequently used property of the limitation topology. PROPOSITION 2.1.7. Let X be a Polish space, F be a subspace of the space C(Y, X ) and the set G n bee open in C(Y, X ) for each n 9 N. If the intersection G n M F is dense in F for each n 9 N , then F C clc(y,x)((nGn) nFd), where Fd denotes the closure of F in the topology of d-uniform convergence and d is any bounded metric on X . In particular, C(Y, X ) has the Baire property. We use Proposition 2.1.7 to characterize maps between Polish spaces which are approximable by homeomorphisms. Such maps are called near-homeomorphisms. More formally, a map f : Y ---, X is a near-homeomorphism if, given/g 9 coy(X), there is a homeomorphism hu: Y ~ X which is/~-close to f. THEOREM 2.1.8. A map f : Y ---, X between Polish spaces is a near-homeomorphism if and only if f ( Y ) is dense in X and the following condition is satisfied: (.) For each lg 9 coy(Y) and V 9 coy(X) there exist an open cover )fl) 9 coy(X) and a homeomorphism h: Y --. Y such that f h 9 B ( f , V ) and hf-l(w)

-4 b/.

PROOF. Let f : Y ~ X be a near-homeomorphism. Obviously, f ( Y ) is dense in X. Let us show that the condition (*) is satisfied. Let b/ E coy(Y) and V E coy(X). Choose a star-refinement 1)0 E coy(X) of V and let p: Y ~ X be a homeomorphism which is Vo-close to f. Now consider a star-refinement W E c o y ( X ) o f t h e open coverp(U)MV0 = {p(U)MVo: U 9 34, Vo 9 V0} 9 coy(X). Take a homeomorphism q: Y ~ X which is W-close to f. Let us show that the composition h -- p - l q : y ~ y satisfies the condition (*). First we show that f h and f are V-close. Indeed, let y 9 Y. Since f and q are V0-close, there is an element V0y 9 Vo such that f ( y ) , q ( y ) 9 V~. Consider the point p - l q ( y ) 9 y . Since f and p are V0-close, there is an element V~ 9 1)o such that q(y), f p - l q ( y ) 9 V~. Consequently, V0~ M V~ ~ 0. This implies that both points f p - l q ( y ) and f ( y ) belong to the set St(V~, 1;o). Since, by our choice, 1)0 is a star-refinement of V, we see that there is an element Vy 9 1) such that f p - l q ( y ) , f ( y ) 9 Vy. Therefore, the maps f h and f are V-close.

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

37

Next we show that h f - l ( W ) refines L/. Let W E 14) and y E h f - l ( W ) . Then f q - l p ( y ) E W. Since the maps S and q are 1/V-close, there is an element Wy E )/Y such that fq-lp(y),p(y) E Wy. Consequently, W N Wy ~ q} and p(y) E St(W,)42). By the construction, there is an element U E b/ such t h a t St(W, bt) c_ p(U). Hence p(y) E p(U) and y E U. This proves the inclusion h f - l ( W ) C_U as desired. Now we assume that a map f : Y -~ X, satisfying our conditions, is given. Let us show that S is a near-homeomorphism. Fix a complete metric dy on Y and denote by b/n, n E N, an open cover of Y, the diameters (with respect to 1 Let Gn denote the set of all the metric dy) of whose elements are less than n" b/n-maps in C (Y, X). Consider the following set:

F -- {fh: h is an autohomeomorphism of the space Y}. We claim that Gn M F is dense in F. To see this consider an arbitrary element in F. Such an element has the form fh, where h is an autohomeomorphism of Y. Consider also an element V E coy(X). Our goal is to show that B(Sh,)2) M Gn M F ~ 0. By the condition (,), applied to the open covers h(btn) E coy(Y) and )2 E coy(X), there exist an open cover )4; E coy(X) and a homeomorphism g: Y - . Y such that the composition f g is ])-close to f and g f - l ( y v ) refines h(lln). Then h - l g f - l ( ~ Y ) refines b/n. Therefore, S g - l h is ab/n-map. Obviously, f g - l h E F. Thus, it only remains to show t h a t f g - l h and f h are ])-close maps. In turn, it suffices to prove that fg-1 and f are )2-close. Indeed, consider a point y E Y. Since the maps .fg and f are ])-close, there is an element V E 1) such t h a t f (y) = f gg-l(y), f g-l(y) E Y. Consequently, f g-1 E B(f,)2). This implies that .fg-lh(y), fh(y) E Y and f g - l h E B ( f h , V). Therefore B(.fh, )2)MGnMF 0 as desired. This shows that the set Gn N F is dense in F for each n E N. By Lemma 2.1.1, Gn is open in C(Y, X). Therefore, by Proposition 2.1.7, the set F is contained in the closure (in the space C(Y, X)) of the intersection MGn MFdx, where Fdx denotes the closure of F in the topology of dx-uniform convergence and dx stands for an arbitrary bounded complete metric on X. Since f E F , we conclude that f E cl(MGnNFdx). Consequently, for each open cover )2 E coy(X) the intersection B(f,)2)N (NGn N Fax) is non-empty. Let p be an element of this intersection. We claim that p is a homeomorphism. Obviously this is all we need to show. Since p E NGn, we conclude, by L e m m a 2.1.2, t h a t p is a closed embedding. On the other hand, since p E Fdx, P is the d x - u n i f o r m limit of a sequence of maps, belonging to F. But maps in F have dense images in X. This is enough to conclude that p(Y) is also dense in X. Therefore, p is a homeomorphism. [-1 COROLLARY 2.1.9. A closed surjection f : Y --~ X between Polish spaces is a

near-homeomorphism iS and only iS the following condition is satisfied: (**) For each Lt E coy(Y) and each )2 E coy(X) there exists a homeomorphism h: Y -+ Y such that fh ~ B(I, V) and the collection ( h f - l ( x ) :

38

2. INFINITE-DIMENSIONAL MANIFOLDS x E X } refines lA.

PROOF. Clearly, (*) =~ (**). Let us prove the reverse implication. The closedness of f implies that each of the sets Z - f ( Y - h - l ( u ) ) , U E U, is open in Z . The collection VV - { X - f ( Y - h - l ( U ) ) 9 U e / 4 } covers X. Indeed, let x E X. Then, by (**), we can find an element U E/,4 such that h f - l ( x ) C V . Therefore f - l ( x ) N ( Y - h - l ( u ) ) = 0 and hence x e X - f ( Y - h - l ( u ) ) . It only remains to note that hf-l(YY) refines/~. [-1

2.1.2. M a i n d e f i n i t i o n s . Each map f : A ~ [0, 1], defined on a closed subspace A of a normal space B, can be extended to B. This classical result, known as the Brouwer-Tietze-Urysohn extension theorem, serves as a prototype of the entire theory, which studies, roughly speaking, the possibility of extending maps into given spaces. The important property of the unit segment (or the real line) expressed in this theorem can be formalized as follows. DEFINITION 2.1.10. A Polish space X is called an absolute (neighborhood) extensor, or shortly, an A ( N ) E - s p a c e , if any m a p f : A ---+ X , defined on a closed subspace A of an arbitrary Polish space B , can be extended to a m a p of the space B (respectively, of a neighborhood of A in B ) into X .

Obviously, the notion of absolute neighborhood extensor can be introduced for different classes of topological spaces. In each particular situation, the choice of the appropriate class depends, as a rule, on a variety of different reasons. In Chapter 6 the general definition of this notion, suitable for the class of completely regular spaces, will be given. But for now we restrict ourselves to the class of Polish spaces. The following notion is also very important. DEFINITION 2.1.11. Let n E w. A Polish space X is called an absolute (neighborhood) extensor in dimension n, or shortly, an A ( g ) E ( n ) - s p a c e , if any m a p f " A ~ X , defined on a closed subspace of a Polish space B with d i m B _< n, can be extended to a m a p of the space B (respectively, of a neighborhood of A in B ) into X .

It is not difficult to verify that a metrizable compactum X is an A ( N ) E ( n ) space (respectively, A ( N ) E - s p a c e ) if the condition, formulated in Definition 2.1.11 (respectively, in Definition 2.1.10), is satisfied only for compact metrizable spaces B. At the same time, it should be especially emphasized that there exists a Polish space, which is not an absolute extensor, such that the condition of Definition 2.1.10 is satisfied for each metrizable compactum B. This follows, as has been noted by van Mill (see [22]), from Corollary 3.2.16. The following characterizations of the classes of spaces introduced above are well-known. Their proofs can be found in several topological textbooks (see, for example,

[203, 176]).

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

39

THEOREM 2.1.12. Let n E w. The following conditions are equivalent f o r any Polish space X ; (i) X is an A g E ( n ) - s p a c e . (ii) X is an f_.gn-l-space.

In other words, f o r each point x E X

and any

neighborhood U of x in X there exists a smaller neighborhood V of x such that any map f " S k ~ V of the k - d i m e n s i o n a l sphere S k, where k <_ n - 1, can be extended to a map g" B k+l ~ U of the (k + 1)d i m e n s i o n a l disk B k+l into U.

(iii) A n y m a p f " A ~ X , defined on a closed subspace A of a Polish space B with dim(B - A ) ~ n, can be extended to a map of a neighborhood of A (in B ) into X .

(iv) For any Polish space B , containing X as a closed subspace so that d i m ( B - X ) ~_ n, there exists a retraction r" G --~ X , defined on a neighborhood G of X in B .

(v) For each point x E X and f o r any neighborhood U of x in X there exists a smaller neighborhood V such that any map f " A --~ V , defined on a closed subspace A of an at m o s t n - d i m e n s i o n a l Polish space B , can be extended to a map g" B ~ U of B into X . (vi) For each point x E X and f o r any neighborhood U of x in X there exists a smaller neighborhood V such that any map f " B ~ V of an at m o s t ( n - 1)-dimensional Polish space B into V is homotopic in U to a constant map of B into U.

(vii) For each open cover lg e c o y ( X ) there is an open cover l) 9 c o y ( X ) , refining Lt, such that any two ]2-close maps f , g " B ~ X , defined on an at m o s t ( n - 1 ) - d i m e n s i o n a l Polish space B , are Lt-homotopic (let us recall that two maps f and g are U-homotopic if there is a h o m o t o p y H" B x [0, 1] ~ X , connecting f and g such that the collection { H ({b} x [0, 1])" b 9 B } refines bl).

(viii) For each open cover lg 9 c o y ( X ) there exist a countable locally finite simplicial complex K and a map p" I K I---~ X such that f o r any map f " B ~ X , defined on an at m o s t ( n - 1 ) - d i m e n s i o n a l Polish space B , there is a map g" B ---~l K I such that the maps f and pg are Lthomotopic.

Note that an A N E ( n ) - s p a c e is an A E (n )-space if and only if it is ( n - 1)connected, i.e. belongs to the class g n - 1 . T h e latter, as usual, means t h a t any map of the k-dimensional sphere, k _< n - 1, into the given space is null-homotopic or, equivalently, admits an extension onto the (k + 1)-dimensional disk. PROPOSITION 2.1.13. Each Polish space is an AE(O)-space. One can prove this s t a t e m e n t in several ways. For example, it suffices to note that any closed subspace of any zero-dimensional Polish space is a retract of it. Alternatively, we can use one of the results dealing with the existence of

40

2. INFINITE-DIMENSIONAL MANIFOLDS

single-valued c o n t i n u o u s selections of many-valued maps. First, we need some definitions. By a m a n y - v a l u e d m a p (between Polish spaces) F : X ~ Y we u n d e r s t a n d a m a p which for each point x E X assigns a closed subset F ( x ) of a space Y. We call such a m a p a lower semi-continuous if the set

F-~(U) =

{~ e X: F ( ~ ) n U # 0}

is open in X w h e n e v e r U is open in Y. In the case w h e n t h e sets of the form

F#(U) =

{~ e X : F(~) C U}

are o p e n in X (whenever U is an open subset of Y) we say t h a t F is upper semicontinuous. A single-valued m a p f : Z ~ Y is a selection of F if f ( x ) e F ( x ) for each point x E X . Lower semi-continuous maps occur quite often in the t h e o r y of A b s o l u t e E x t e n s o r s . Consider, for example, the following situation. Suppose t h a t f : X ~ Y is an open surjection b e t w e e n Polish spaces. Suppose, in addition, t h a t A is a closed subspace of a Polish space B and t h a t two m a p s g: A ~ X and h: B ~ Y, satisfying t h e condition f g = h / A , are given. Let us define a m a n y - v a l u e d m a p F : B --~ X g e n e r a t e d by t h e d i a g r a m

X

*Y

A t

,~B

by t h e following rule:

F(b) =

g(b), f-l(h(b)),

ifbEA ifbeB-A

Obviously t h e o p e n n e s s of the m a p f implies the lower semi-continuity of F . T h e following s t a t e m e n t was proved in [225]. Note t h a t P r o p o s i t i o n 2.1.13 is an easy consequence of it. THEOREM 2.1.14. Let F: X ---, Y be a lower semi-continuous many-valued map between Polish spaces. If dim X = 0, then F has a selection. In order to f o r m u l a t e a selection t h e o r e m for positive dimensional cases we need a d d i t i o n a l definitions. Let n E w a n d .T" - { X t : t E T } be a collection of closed subsets of a Polish space X . We say t h a t .T" is uniformly locally connected in dimensions < n

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

41

(shortly, 9~ is an equi - s if for each open c o v e r / g E c o y ( X ) t h e r e is an open cover 12 E c o y ( X ) , refining/4, such t h a t the following condition is satisfied" 9 For each index t E T and for each element U E U there is an element V E 1) such t h a t V C_ U and any m a p f " S k --~ X t N V, with k _< n - 1, has an extension g" B k+l -~ X t M U. We also say t h a t ~ is c o n n e c t e d in dimensions < n (shortly, ~ is a C n - I collection) if X t E C n-1 for each t E T. T h e following i m p o r t a n t t h e o r e m holds. THEOREM 2.1.15. Let n E w and f " X --~ Y be a lower s e m i - c o n t i n u o u s m a n y - v a l u e d map between Polish spaces. I f d i m X ~_ n and F is an I : c n - I M c n - I collection, then F has a selection.

S u p p o s e now t h a t a lower semi-continuous m a n y - v a l u e d m a p F " X --~ E of a Polish space X into a locally convex Polish space E (i.e. E is a s e p a r a b l e Frgchet space) is given. We say t h a t the m a p F is convex if the image F ( x ) of any point x E X is a convex (and closed) subset of E. In this s i t u a t i o n we have t h e following s t a t e m e n t . THEOREM 2.1.16. E v e r y convex lower s e m i - c o n t i n u o u s m a n y - v a l u e d map of any Polish space into any separable Frgchet space has a selection. This t h e o r e m has several i m p o r t a n t corollaries. Here are two of t h e m . THEOREM 2.1.17. Every closed and convex subspace of a separable Fr~chet space is an absolute extensor.

It was shown by R . C a u t y [62] t h a t t h e r e q u i r e m e n t of local convexity in t h e above s t a t e m e n t is essential. THEOREM 2.1.18. A continuous linear surjection f " E --~ L between separable Frgchet spaces is topologically a trivial bundle with fiber f - l ( O ) . PROOF. By the classical t h e o r e m of Banach, f is an open map. T h e r e f o r e the m a n y - v a l u e d m a p f - 1 . L --~ E is lower semi-continuous. L i n e a r i t y of f implies t h a t the sets f - l ( / ) , l E L are convex (and obviously closed). T h e o r e m 2.1.17 g u a r a n t e e s the existence of a selection g" L --~ E of f - 1 . T h e n the desired trivialization of f (i.e. a h o m e o m o r p h i s m h" E --, L • f - l ( 0 ) such t h a t f ---~rLh) can be given according to the following formula: h(e) -- (e - g f ( e ) , f ( e ) ) , e E E . [:J T h e o r e m 2.1.17 provides an i m p o r t a n t class of Polish a b s o l u t e extensors. In particular, all E u c l i d e a n spaces R n, as well as the c o u n t a b l e infinite power R w of the real line R (which is h o m e o m o r p h i c to the separable Hilbert space 12) are a b s o l u t e extensors. Note also t h a t the class of Polish absolute ( n e i g h b o r h o o d ) extensors coincides with the class of r e t r a c t s of R ~ (respectively, r e t r a c t s of o p e n subspaces of R w). T h e l a t t e r class can easily be seen to coincide with the class of

42

2. INFINITE-DIMENSIONAL MANIFOLDS

Polish absolute (neighborhood) retracts (shortly, A ( N ) R - s p a c e s ) . every locally compact polyhedron is an A N R - s p a c e . The following criterion helps us to recognize A N R - s p a e e s .

In particular,

THEOREM 2.1.19. A Polish space X is an A N R-space if and only if for any open cover bl E c o y ( X ) there exist a countable locally finite simplicial complex K (finite-dimensional in case X is finite-dimensional) and maps p: X ~ Igl and q: IKI ---+ X such that the composition qp is lX-homotopic to the identity map i d x . In addition, we m a y assume that the cover p-l(L/0) refines hi, where blo denotes the cover of l K I consisting of open stars (with respect to the triangulation of K ) of vertices of K . Each Polish A N R - s p a c e has the homotopy type of a locally finite polyhedron [232]. Since an open subspace of a Polish A N R-space is itself a Polish A N Rspace (see [161]), we see that any open subspace of a Polish A N R - s p a c e has the homotopy type of a locally finite polyhedron. The converse statement (the validity of which was a long standing unsolved problem) has been recently proved by R. Cauty [63]. Thus, we have the following characterization of Polish A N Rspaces. THEOREM 2.1.20. The following conditions are equivalent for any Polish space X:

(i) X is an A N R-space. (ii) each open subspace of X has the homotopy type of a locally finite polyhedron.

The following two statements are also well-known. PROPOSITION 2.1.21. Let X be a Polish A N R - s p a c e . Then for each open cover Lt E c o y ( X ) there is an open cover 1) E c o y ( X ) , refining bl, such that any two ])-close maps f , g : B ---+ X , defined an a Polish space B , are lg-homotopic in X . PROPOSITION 2.1.22. Let X be a Polish A N R-space. Then for each open cover bl E c o y ( X ) there is an open cover )2 E c o y ( X ) , refining hi, such that the following condition is satisfied: 9 I f one of two P-close maps f , g : A --~ X , defined on a closed subspace of a Polish space B (say f ) has an extension F : B ---. X to B , then the other (the map g) also has an extension G: B ---+ X . Moreover, we may assume that the extensions F and G are Ll-homotopic.

COROLLARY 2.1.23. Let A be a closed subspace of a Polish space X . the "restriction" map C ( B , X ) ~ C ( A , X ) is open.

Then

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

43

2.1.3. M a p s o f s p e c i a l t y p e . Fine h o m o t o p y equivalences are very import a n t in the theory of A N R-spaces. In this Subsection we present some of their properties. DEFINITION 2.1.24. A map f : X ~ Y is a fine h o m o t o p y equivalence if for each open cover lg E coy(Y) there is a map gu : Y ~ X such that the composition f g is Ll-homotopic to i d y and the composition g f is f - l ( U ) - h o m o t o p i c to i d x . PROPOSITION 2.1.25. Let f : X ~ Y be a fine homotopy equivalence and X be an A N R - s p a c e . Then Y is also an A N R - s p a c e . PROOF. L e t / 4 E cov(Y). Take an open cover l) E coy(Y) which star refines U. Take a map g" Y ~ X such t h a t f g ~ i d y (and g f I - l ( v ) i d x ) . By T h e o r e m 2.1.19, there exist a locally compact polyhedron K and maps p: X ~ K and q: K --~ X such t h a t the composition qp is f - l ( ] ) ) - h o m o t o p i c to i d x . Consider the maps a - pg: Y ~ K and ~ = f q : K --~ Y . By the same T h e o r e m 2.1.19, it suffices to show t h a t the composition f~a is b/-homotopic to i d y . f-~v) y-l(v) v Since qp _ i d x , we see t h a t qpg ~ g. T h e n f qpg "-' f g. By construction, f g ~ idy. Therefore, flc~ = f qpg ~ i d y .

[:]

Maps introduced in the following Definition are closely related to fine h o m o t o p y equivalences. DEFINITION 2.1.26. A map f : X --. Y between Polish spaces is a p p r o x i m a t e l y soft if for each open coverLt E c o y ( Y ) , each Polish space B , each closed subspace A of B , and any two maps g: A --~ X and h: B ~ Y satisfying the condition f g -- h / A , there exists a map k: B ~ X such that g - k / A and the composition f k is l~-close to h. PROPOSITION 2.1.27. Let f : X --~ Y be an approximately soft map and Y be a Polish A N R-space. Then f is a fine homotopy equivalence. PROOF. Take an open cover/4 E c o y ( Y ) and let 5/1 E c o v ( Y ) be its star-refinement. Since Y is an A N R - s p a c e , there exists, by Proposition 2.1.21, an open refinement ]) E coy(Y) of b/1 such t h a t any two )2-close maps into Y (defined on an arbitrary space) are L41-homotopic. Since f is approximately soft, there exists a map f : Y --, X such t h a t the composition f g is ])-close to i d y . Obviously, by the choice of ]), we have t h a t f g ~ i d y and, consequently,

f g ~ idy.

Let us now show t h a t the composition g f is f - l ( L / ) - h o m o t o p i c to i d x . Since the maps f g a n d i d y are L/1-homotopic, we see t h a t the maps f g f and f are

44

2. INFINITE-DIMENSIONAL MANIFOLDS

also /Al-homotopic. Consider the corresponding h o m o t o p y h: X x [0, 1] ~ Y connecting the maps f g f and f. Define a map H ' : X x {0, 1} ~ X as follows:

H'(x,t) = { gf(z), x,

ift = 0 i f t = 1.

Obviously, f H ' = h / ( X x {0, 1}). Since f is approximately soft, there exists a m a p H : Z x [0, 1] ---, X such t h a t H / ( X x {0, 1}) = H ' and the composition f H is/Al-Close to h. It is not hard to see t h a t H is a h o m o t o p y connecting the maps fg and idx. Next we show t h a t H is a f - l ( / A ) - h o m o t o p y . Consider a point x 9 Z . It suffices to verify t h a t the set H ( { x } x [0, 1]) is contained in an element of the cover f - l ( / A ) . Since the h o m o t o p y h is limited by the cover/4], there exists an element U1 E/A1 such t h a t h({x} x [0, 1]) C_ Vl. Let t 9 [0, 1]. Since the maps f H and h are/Al-close, we can find an element U~ 9 /A1 such t h a t f H ( x , t ) , h ( x , t ) 9 U~. Consequently, U1M U~ -7(= q} for each t 9 [0,1]. But then

f H ( { x } x [0, 1]) = O { f H ( x , t ) " t 9 [0, 1]} C O{Uf" t 9 [0, 11} C 8t(Vl,/A1). Recall t h a t L(1 star refines/A. Therefore, there is an element U 9 such t h a t f H ( { x } x [0, 1]) C_ U. This obviously implies t h a t H ( { x } x [0, 1]) C_ f - l ( v ) . [Z] In should be noted t h a t even polyhedrally soft maps between Polish A N Rspaces are fine h o m o t o p y equivalences (we obtain the definition of such maps by considering only polyhedral pairs (A,B) in Definition 2.1.26). This fact has been established in [163]. PROPOSITION 2.1.28. Let f : X -* Y be a near-homeomorphism and Y be a Polish A N R-space. Then f is approximately soft. PROOF. Let A be a closed subset of a Polish space B and let maps ~: A ~ X and r B --~ Y such t h a t f ~ = r be given. Consider an arbitrary open cover U 9 coy(Y) and suppose t h a t )4) 9 coy(Y) is a star-refibement of/4. By Proposition 2.1.22, there is an open refinement 1; 9 coy(Y) of )/V such t h a t the following condition is satisfied: 9 If one of the two arbitrarily given );-close maps into Y, defined on A, has an extension to B, then the second also has an extension to B. Moreover, we may assume t h a t these extensions are W-close. Take a h o m e o m o r p h i s m g: X ~ Y which is ])-close to f and consider the composition g~: A ~ Y. Obviously, g~ is ])-close to f ~ . But the map f ~ has the extension r onto the space B. Consequently, by the choice of the cover ]), there is a W-close to r extension h: B ~ Y of the map g~. Define the map k: B --~ X as the composition k -- g-lh. Obviously, k/A -- ~. It now suffices to show t h a t the composition f k is/,/-close to r Let b 9 B. Since the maps h and r are W-close, there exists an element W 9 ~/V such t h a t h(b), r 9 W. Consider the point g-lh(b) 9 X. )/V-closeness of the maps g and f guarantees the existence of an element W b 9 14) such t h a t h(b) -- gg-lh(b), f g-lh(b) 9 W b.

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

45

Therefore, W M W b ~ 0. Then f g - l h ( b ) , r E s t ( W , 1/Y). Since s t ( W ) refines b/, there is an element U E b/such t h a t f k ( b ) = f g - l h ( b ) , r E V. This verifies that the maps f k and r are b/close. I-1 Now we introduce some notions from Shape Theory (the reader can find a comprehensive introduction to Shape Theory in several textbooks; for instance, in [47], [135] or [215]). We say that a c o m p a c t u m X has trivial shape (and we write S h ( Z ) -- .) if for some (any) embedding of X into I "~ (or into any other A R - c o m p a c t u m ) the following condition is satisfied: 9 for each open neighborhood U of X in I ~ there exists an open neighborhood V of X in I ~ such t h a t V c U and V contracts to a point in U. Several other equivalent definitions of compacta with trivial shape are known. They appear under the name of cell-like compacta as well as under the name of U V ~ 1 7 6 Proper maps all fibers of which are cell-like are known as cell-like maps. The proof of the following statement involves only elementary arguments. PROPOSITION 2.1.29. Let f " X ~ Y be a proper map between Polish A N R spaces. Then the following conditions are equivalent: (i) f is a fine h o m o t o p y equivalence. (ii) .f is cell-like. DEFINITION 2.1.30. Let n E w. We say that a c o m p a c t u m X is an U V nc o m p a c t u m if f o r some (any) embedding of X into I ~ the following condition is satisfied: 9 f o r each open neighborhood U of X in I ~ there exists an open neighborhood V of X in I ~ such that V C_ U and any map f " S k ---, V , k ~_ n, has an extension g" B k+l --, U. It is perhaps of some interest to remark t h a t the U V n - c o m p a c t a are exactly those compacta with trivial n-shape (see Subsection 4.4.3). U V n - m a p s are proper maps all fibers of which are U V n - c o m p a c t a . The following s t a t e m e n t is well known. PROPOSITION 2.1.31. Let n E N and f " X --~ Y be a proper map between locally compact L C n - l - s p a c e s . Then the following conditions are equivalent: (i) f is a u v n - l - m a p . (ii) For each at m o s t n - d i m e n s i o n a l locally compact space B , each closed subset A o r B , any two proper m a p s g" A ~ X and h" B ~ Y with f g = h / A , and any open cover 34 E c o v ( Y ) , there exists a proper m a p k" B --. X such that k / A -- g and the composition f k is U-close to h (in other words, f is approximately n-soft). (iii) The same as condition (ii), but f o r polyhedral pairs ( A , B ) (i.e. f is polyhedrally approximately n-soft).

46

2. INFINITE-DIMENSIONAL MANIFOLDS T h e following proposition holds. PROPOSITION 2.1.32. Let n E w and let f : X ---. Y be a u v n - m a p

between

compacta. T h e n the following conditions are satisfied:

(i) X E U V n if and only if Y E V Y n. (ii) I f X E L C n, t h e n Y E L C n. T h e following concept is related to Definition 2.1.26. DEFINITION 2.1.33. Let n = O, 1 , . . . , oa. We say that a map f : X ~ Y between Polish spaces is n-soft if f o r each at m o s t n - d i m e n s i o n a l Polish space B , each closed subset A of B , and any two m a p s g: A ---+ X and h: B --+ Y with f g = h / A , there exists a m a p k: B ---+ X such that g = k / A and f k = h. The oa-soft m a p s are also called soft.

Obviously, every n-soft map between c o m p a c t a is a U V n - l - m a p and every soft m a p is a fine h o m o t o p y equivalence. PROPOSITION 2.1.34. Let f : X ---. Y be a m a p between Polish spaces. T h e n the following conditions are equivalent:

(i) f is an open map. (ii) f is a O-soft map. PROOF. ( i ) ~ ( i i ) . Let A be a closed subset of a zero-dimensional Polish space B. Suppose also t h a t two maps g: A ~ X and h: B --, Y are given so t h a t f g = h / A . Consider the many-valued map F : B ~ X generated by the diagram

X

~Y

A t

.B

by the following rule: F(b) -- { g(b), f-l(h(b)),

ifbEA ifbEB-A

Obviously, by(i), F is lower semi-continuous. Since dim B - 0 we conclude, by T h e o r e m 2.1.14, t h a t F has a (continuous) selection k: B ~ X. It only remains to note t h a t g - - k / A and f k -- h.

2.2. Z-SETS IN ANR-SPACES

47

( i i ) ~ ( i ) . First observe that every 0-soft map is surjective. Let U be an open subset of X. Our goal is to show t h a t f ( U ) is open in Y. Consider an arbitrary converging sequence { y n : n E N} of points of the complement Y - f ( U ) . Suppose that y is the limit point of this sequence. We will be done if we show that y E Y - f ( U ) . Assume the contrary and take a point x E U such that f ( x ) - y. Consider the homeomorphism r

a N ---+ U{yn: n E N} U {y},

where r = y and r - Yn for each n E N (here c~N denotes the one-point compactification of N). Let ~ ( a ) = x. By the zero-dimensionality of a N and the 0-softness of f, there is a map k: c~N --+ X such that k(c~) = x and f k = r The last equality implies that each of the points k ( n ) , n E N , is contained in the complement X - U . The continuity of k implies that the sequence {k(n) : n E N} converges in X to the point x. This contradicts the openess of U. Consequently,

yeY-/(u).

D

Historical and bibliographical notes 2.1. The majority of the results of Subsection 2.1.1 are taken from [299]. Corollary 2.1.9 was proven by Bind [38] and is known as Bing's shrinking criterion. The same name is applied to Theorem 2.1.8 (see [299, Theorem 1.5]). Selection Theorems 2.1.14, 2.1.15 and 2.1.16 were proved by Michael [224], [225], [226]. Theorem 2.1.17 was in fact proved in [45] and [133]. Theorem 2.1.18 appears in [27]. Theorem 2.1.19 is due to Hanner [161] (see also [299]). Propositions 2.1.21 and 2.1.22 are widely known (see, for example, [176]). Lacher's survey [207] is an excellent introduction to the theory of UVn-compacta and UVn-maps. Proposition 2.1.31 is taken from this work (see also [33]).

2.2. Z - s e t s in A N R - s p a c e s

2.2.1. G e n e r a l p r o p e r t i e s o f Z - s e t s . If a subset A is closed and nowhere dense in a Polish space X, then it easy to see t h a t the set ( f E C ( I ~ f(I~ A -- 0} is dense in the space C ( I ~ (I ~ denotes the zero-dimensional cube, i.e. the one-point space). Of course, the converse is also true. This simple observation allows us to define higher degrees of nowhere density of subsets. This can be formalized as follows. DEFINITION 2.2.1. Let n E w. We say that a closed subset A of a Polish space X is a Z n - s e t in X if the set { f E C ( I n, Z ) : f ( I n) M A = 0} is dense in the space C (I n, X ) .

48

2. INFINITE-DIMENSIONAL MANIFOLDS

DEFINITION 2.2.2. We say that a closed subset A of a Polish space X is a Z-set in X if the set { f E C ( I ~ , X ) : f ( I r176M A : q}} is dense in the space

c(z~,x). T h u s , having in m i n d the r e m a r k m a d e above, we see t h a t the Z0-sets are precisely t h e closed a n d n o w h e r e dense sets. Z - s e t s in A N R-spaces a d m i t some useful c h a r a c t e r i z a t i o n s . PROPOSITION 2.2.3. Let A be a closed subset of a Polish ANR-space X . Then the following conditions are equivalent: (i) A is a Z-set in X . (ii) For each locally compact polyhedron P, the set { f E C ( P , X ) : f ( P ) A A : 0} is dense in the space C ( P , X ) . (iii) The set { f E C ( X , X ) : f ( X ) M A = 0} is dense in the space C ( X , X ) . (iv) For each Polish space Y, the set { f E C ( Y , X ) : f ( Y ) MA = 0} is dense in the space C (Y, X ). PROOF. (i) =:~ (ii). F i r s t let us n o t e t h a t if T is an A R - c o m p a c t u m , t h e n the set { f C C ( T , X ) : f ( T ) M A = 0} is dense in the space C ( T , X ) . Indeed, e m b e d T into I ~ a n d fix a r e t r a c t i o n r : I ~ ~ T. C o n s i d e r also an a r b i t r a r y m a p f : T ~ X a n d an o p e n cover H e coy(X). By (i), there exists a m a p g~: I • ~ X such t h a t g~(I ~) M A = 0 and the c o m p o s i t i o n f r is N-close to g~. O b v i o u s l y g = g'/t is N-close to f a n d g(T) M A = 0. S u p p o s e now t h a t K is a c o u n t a b l e locally finite simplicial complex such t h a t

I T I-- P . Let us show t h a t for each simplex a E K t h e set B a -- {f E C ( P , X ) : f(I a ] ) M A = 0} is dense in the space C ( P , X ) . C o n s i d e r any m a p f : P ~ X a n d an open c o v e r / 4 E coy(X). C o n s i d e r an o p e n cover 1) E coy(X) satisfying the c o n d i t i o n of P r o p o s i t i o n 2.1.22 relative to U. By t h e case considered above, t h e r e is a m a p g': lal ~ z such t h a t g' is )2-close to f / l a l and g'(lal) M A = 0. Consequently, by t h e choice of )2, the m a p g~ has an e x t e n s i o n g: P ~ X which is H-close to f . Clearly, g(I a [) M A = 0. N e x t we show t h a t the set B a is o p e n in the space C ( P , X ) for each a E K . Take an e l e m e n t f E Ba. Since f(I a l) a n d A are disjoint closed sets of t h e space X , we can find an o p e n n e i g h b o r h o o d U of f(I a I) which does not intersect A. It only r e m a i n s to note t h a t the n e i g h b o r h o o d B ( f , { U , X - f(I a ])}) of f in C(P, X ) is c o m p l e t e l y c o n t a i n e d in t h e set Ba. Observe now t h a t {f EC(P,X): f(P) MA----O}--M{Ba:aEK} a n d n o t e t h a t , by P r o p o s i t i o n 2.1.7, t h e set w r i t t e n in the right side of the above e q u a l i t y is dense in the space C(P, X ) . (ii) =r (ii). C o n s i d e r a m a p f : Z -~ X and an o p e n cover H E coy(X). Let 1) E coy(X) be a s t a r - r e f i n e m e n t of H. For an o p e n cover f - 1 ( ) 2 ) E coy(X), by T h e o r e m 2.1.19 t h e r e exist a locally c o m p a c t p o l y h e d r o n P a n d m a p s p: X -~ P

2.2. Z-SETS IN ANR-SPACES

49

and q: P -~ X such t h a t the composition qp if f - l ( ] ) ) - c l o s e to i d x . By (ii), there is a map g~: P ~ X which is ])-close to the composition f q and such t h a t g~(P) N A - - 0. Let g - - g~p. It is easy to check t h a t g and f are/g-close and g ( X ) N A =O. Implications (iii)=~ (iv) and (iv) ~ (i) are obvious. Fl If X happens to be locally compact, t h e n we have a stronger conclusion.

COROLLARY 2.2.4. Let A be a Z-set in a locally compact A N R - s p a c e X . Then for each space Y the set { f E C(Y, X ) : c l x f ( Y ) n A = 0} is dense in the space

c(Y,X). PROOF. Obviously, it suffices to show t h a t each neighborhood of the point i d z in the space C ( X , X ) contains a m a p f : Z ---+ X such t h a t c l . f ( X ) n A = 0. Consider an open c o v e r / 4 E coy(X). Local compactness of X guarantees the existence of an open cover 12 E c o y ( X ) such t h a t every map f : X ~ X which is ])-close to i d x has a closed image in X. By Proposition 2.2.3, the n e i g h b o r h o o d B ( i d x , l g A ] ) ) contains a map f : Z -~ X such t h a t f ( Z ) N A = q}. It only remains to note that, by the choice of 12, f ( X ) is closed in X. [:] We reserve the term strong Z-set for those closed sets A of a space X satisfying the property of Corollary 2.2.4 DEFINITION 2.2.5. We say that a closed subset A of a Polish space X is a strong Z-set in X iS the set { f E C ( X , X ) : e l f ( X ) n A = 0} is dense in the space C (X, X ) . T h u s Corollary 2.2.4 can be reformulated as follows: each Z-set in a locally c o m p a c t A N R - s p a c e is a strong Z-set. T h e following simple example shows t h a t the converse need not be true. Consider the subset

x = ([o, 1] x {o}) u ( U { { Z} n x [0, 1]}) of the plane:

(o,o)

(~, o)

(89, o)

(~, o)

50

2. INFINITE-DIMENSIONAL MANIFOLDS

It is not hard to see that X is a Polish AR-space. Moreover, X is even locally compact at each point other than the origin (0, 0). For each t E [0, 1) consider a map ht: X --+ X determined by the following properties: 9 h (1, o) = o). 9 ht(1~,1) -- (K, i 1). 9 ht linearly shrinks the segment [0, 1] x {0} onto the segment [t, 1] x {0}. 9 ht linearly expands the segment { 1} x [t, 1] onto the segment {~} x [0,1], 1 x [0, t] onto the segment [1 , t q- -1st 9 ht linearly maps the segment {~} - ]x 1 0 ) a n d ( ~' 1 0 ) o n t o (t + L~A, 0) ). {0} (sending ( !n ' t) onto (n' Clearly, the identity map i d x is in the closure (in the space C (X, X)) of the set {ht: t E (0, 1)}, and since h t ( X ) C_ X - ( 0 , 0 ) it follows that the origin (0, 0) is a Z-set in X. At the same time (0, 0) is not a strong Z-set in X. To see this, observe that X - G is disconnected for sufficiently small neighborhoods G of the point (0, 0). The following statement provides a characterization of strong Z-sets in Polish A N R-spaces. Its proof is similar to the proof of Proposition 2.2.3 and is therefore omitted. PROPOSITION 2.2.6. Let A be a closed subset of a Polish A N R-space X . Then the following conditions are equivalent: (i) A is a strong Z - s e t in X . (ii) The set { f E C ( N • I ~ , X ) : c l x f ( N • w ) = 0 } is dense in the space c ( g x I "~ , X ) . (iii) For each locally compact polyhedron P, the set { f E C ( P , X ) : c l x f ( P ) M A -- 0} is dense in the space C ( P , X ) } . (iv) The set { f E C ( X , X ) : c l x f ( X ) M A -- 0} is dense in the s p a c e C ( X , X ) . (v) For each Polish space Y , the set { f E C ( Y , X ) : c l x f ( Y ) M A -- 0} is dense in the space C (Y, X ). Condition (v) demonstrates the main advantage of the concept of strong Z-set over the concept of Z-set (in the locally compact case, see Corollary 2.2.4). The proof of the following statement is a useful exercise for the reader. PROPOSITION 2.2.7. Let X and Y be Polish spaces. (i) I f A is a (strong) Z - s e t in X , then A • Y is a (strong) Z - s e t in X • Y . (ii) Let A n , n E w be a Z - s e t in X and suppose the union A = U { A n : n e w } is closed in X .

Then A is a Z - s e t in X .

PROPOSITION 2.2.8. Let A be a closed subset of the product X = 1-I{xn: n E w} of Polish spaces X n . I f zrn(A) 7~ X n for infinitely m a n y indices n, then A is a Z - s e t in X . PROOF. Take an increasing sequence no < n l < --', satisfying the relation rnk (A) ~ Xnk for each k E w. Take a point xk E Xnk - rnk (A). Now, for each

2.2. Z-SETS IN ANR-SPACES

51

m a p f E C ( I ~ , X ) define t h e m a p fk" IW --+ X , k E w as follows:

-~-fk(~) = { ~ j f(~)' xk,

if j # ~k if j -- nk.

It only r e m a i n s to observe t h a t t h e sequence {fk" k E w} converges to t h e m a p f (in t h e space C ( I ~ , X ) ) a n d t h a t f k ( I W ) M A -- q} for each k E w. [] PROPOSITION 2.2.9. Let A be a closed subset of a Polish ANR-space X and G be an open neighborhood of A in X . Then the following conditions are equivalent: (i) A is a (strong) Z-set in X . (ii) A is a (strong) Z-set in G. PROOF. Since t h e proofs of the two cases are similar, we consider only case of s t r o n g Z-sets. First, suppose t h a t A is a s t r o n g Z - s e t in G a n d lg E coy(X). O u r goal is to find a m a p h" X --. X which is /g-close to i d e n t i t y i d x a n d satisfies c l x h ( X ) M A -- 0. Consider open subsets G1 a n d of X such t h a t

the let the G2

A C_ G1 C_ clxG1 C_ G2 C_ clxG2 C_ G. Let V = {G2, X clzG1} and d e n o t e by 142 E coy(X) an open r e f i n e m e n t of b / A V such t h a t t h e pair (142,5/A V) satisfies t h e condition of P r o p o s i t i o n 2.1.22. Consider now t h e open cover W / G E coy(G). Since A is a s t r o n g Zset in G, t h e r e exists a m a p g~" G ~ G such t h a t g~ is 142/G-close to idG a n d cIGg~(G) M A = q}. Let g = g~/cIxG2. T h e n , by the choice of )112, t h e r e exists a m a p h" X ---, X , b / A V-close to idx, such t h a t h/cIxG2 = g. T h e l a t t e r implies t h a t clx(h(cIxG2)) M A -- 0. Let x E X - clxG2. T h e V-closeness of h a n d i d x g u a r a n t e e s t h a t h(x) E X - c l x G ~ and, consequently, c l x h ( X - c l x G 2 ) M A = 0. This shows t h a t c I x h ( X ) M A = q}. T h e r e f o r e A is a s t r o n g Z - s e t in X . Let us show t h e converse. Let A be a s t r o n g Z - s e t in X . As above, take o p e n subsets G1 and G2 of X such t h a t

A C_ G1 C_ clxG1 C_ G2 C_ clxG2 C_ G. L e t / 4 E coy(G) and V = {G2, G - clxG1}. Since G is also a Polish ANR-space, we can, by P r o p o s i t i o n 2.1.22, choose an o p e n refinement W E coy(G) of U A V such t h a t the pair ()/Y,/4(A V) satisfies t h e c o n d i t i o n of t h a t P r o p o s i t i o n . Now let

)4)1 = W U { X - cIxG2}. Clearly, W1 E coy(X). Since, by our a s s u m p t i o n , A is a s t r o n g Z - s e t in X , t h e r e exists a m a p g~" X --~ X , YVl-close to i d x , such t h a t c I x g ' ( X ) M A = 0. Let g = g~/clxG2. T h e n c I x g ( c l x G 2 ) M A = 0. Moreover, t h e 1421-closeness of g and idclxC2 implies t h e inclusion g(clxG2) C_ G. In a d d i t i o n , it is easy to see t h a t g a n d idclxG2 are 142-close and, consequently, by t h e choice of 142, t h e r e exists a m a p h" G --~ G , / g A V-close to idG, such t h a t h / c l x G 2 = g. In particular, clGh(cIxG2)M A = 0. C o n s i d e r now a point x E G - clxG2. T h e V-closedness of t h e m a p s h a n d idG g u a r a n t e e s t h a t h(x) E G - clxG1. Therefore, c l c h ( G - clxG2)M A - - 0 . B u t t h e n clch(G)M A = 0 a n d t h e p r o o f is complete.

E]

52

2. INFINITE-DIMENSIONAL MANIFOLDS

COROLLARY 2.2.10. I f A is a (strong) Z - s e t in a Polish A N R - s p a c e X , then A M U is a (strong) Z - s e t in U f o r each open subspace U of X . PROOF. A M U can be represented as the union A M U = U{An: n E w}, where each of the sets An, n E w, is closed in X. Obviously, A n is a strong Z-set in X. Proposition 2.2.9 allows us to conclude that each of these sets An, n E w, is then a strong Z-set in U. Therefore, by Proposition 2.2.7, the set A M U, as a countable union of strong Z-sets of U, is itself a strong Z-set in U. W1

COROLLARY 2.2.11. Let A be a closed set in a Polish A N R - s p a c e X . I f there exists an open cover IX E c o y ( X ) such that A N U is a strong Z - s e t in U for each U E Lt, then A is a strong Z - s e t in X . PROOF. W i t h o u t loss of generality we can assume that 1X - { U n : n E w} is countable. Each of the intersections A M Un, n E w, can be represented as AMUn -- U{Anm" m E w}, where A m is a strong Z-set in X, m E w. Consequently, by Proposition 2.2.9, the sets A m are strong Z-sets in Un, n , m E w. Since A = U{A m" n, m E w}, Proposition 2.2.7 finishes the proof. I-1

2.2.2. S k e l e t o i d s a n d a b s o r b e r s in P o l i s h s p a c e s . In this Subsection X denotes a Polish space and A u t h ( X ) the set of all autohomeomorphisms of X, endowed with the topology induced by C ( X , X ) . If d is a complete bounded metric on X, then this topology is generated by the following metric

D(f,g)

= d(]', f)

+ d~,(f-l,g-l).

It is well known that A u t h ( X ) is a topological group. Further assume that F is a closed subgroup of A u t h ( X ) and that K: is an additive collection of F-invariant closed subsets of X. By K:a we denote the collection consisting of the countable unions of elements of K~. The subgroup {f E F" f / ( X - M ) - i d x - M } of A u t h ( X ) , where M be a subspace of X, is denoted by FM. Finally, F / M denotes the subgroup { g / M ' g E FM} of the group A u t h ( M ) . The key idea exploited in the following definition is the notion of compact absorption set, due to R. D. Anderson. This notion has been subsequently generalized by different authors (see [35], [31], [31], [158], [294], [295], [313]). DEFINITION 2.2.12. We call a set M C X a (F,K:)-absorbing set (or, shortly, a (F, ~)-absorber), if M E )~a and the following condition is satisfied:

(,) i d x E cl({g E Fu" g(A M U) C M}), whenever A E )~ and U is an open set in X .

The proof of the following s t a t e m e n t is a direct consequence of the above definition. PROPOSITION 2.2.13. Let M be a (F, lC)-absorbing set in X .

2.2. Z-SETS IN ANR-SPACES

53

(i) If A E Ea, then M U A is a (F, M)-absorbing set in X . (ii) If g E F, then the set g ( M ) is a (F, lC)-absorbing set in Z . (iii) I f V is an open set in X , then M M U is a (Fv, lCu)-absorbing set in X , where 2Cu denotes the collection of all closed subsets of X contained in the sets of the form U M A, A E ~ . (iv) I f U is an open subset of X , then M M V is a (r/U,t:v)-absorbing set in u . T h e following s t a t e m e n t also can be easily proved. LEMMA 2.2.14. Let U be an open subset of X , g E F and A E ]C. Then

g E cl({.f E F" f / ( X

- U) -- g / ( X - U) and f ( A M U) C_ M } ) .

THEOREM 2.2.15. Let M1 and M2 be two (F, lC)-absorbing sets in X . the set {g E F" g(M1) = M2} is dense in F.

Then

PROOF. Obviously, it suffices to show t h a t

i d x E clr({g E F" g ( M 1 ) - - M2}). In other words, for each a E C ( X , (0, 1)) we need to show the existence of an element g E F M B d ( i d x , a) such t h a t g(M1) ---- M2 (compare with Subsection 2.1.1). Let Ao -- K1 -- L1 -- 0 and go -- i d x . Represent the sets M1 and M2 as the unions

M1 = U{Kn" n - - 2,3, ...} and M2 = u { n n " n -

2, 3, ...},

where Kn, Ln E ]C. Let us now inductively construct a sequence of h o m e o m o r phisms {gn} C_ F and a sequence {An} of closed sets of X satisfying the following conditions:

(a)n M1 D_ An D_ U{Ki" i _< n} and M2 D gn(An) D_ U{Li" i <_ n}. (b)n gn/ U {Ai" i < n} = - g n - 1 / U {Ai" i < n}. -1 l(x), g n 1 (x)) ~ 2 -(n+l)a (x) for each x E X . (C)n d(gn-1 (x), gn(X)) + d(gn_ It can easily be seen t h a t the h o m e o m o r p h i s m g = lim gn is the desired one. n----~(X:} Letgl=idx andAl=0. Suppose now t h a t the h o m e o m o r p h i s m s go, g l , . . . , g n - 1 and the closed sets Ao, A 1 , . . . , A n _ I , satisfying conditions (a)k, (b)k and (c)k, 0 _< k _ n - 1, have already been constructed. We are going to define a h o m e o m o r p h i s m gn and --1 a closed set An. Applying L e m m a 2.2.14 to the objects M = M1, g = gn-1, A = Ln and U = X - g n - l ( A n - 1 ) , we can conclude t h a t there exists a homeomorphism f E F such t h a t

f / g n - l ( A n - 1 ) = gn-1/gn-l(An-1), -1 f ( L n ) C_ M1 and

d ( g ~ l _ l ( x ) , f ( x ) ) + d ( g n _ z ( x ) , f - l ( x ) ) < 2 - ( n + l ) a ( x ) for each x E X.

54

2. INFINITE-DIMENSIONAL MANIFOLDS

Applying L e m m a 2.2.14 to the objects M = M2, g -- f - l , A = K n and U -- Z - (.f-1 (L,~))U A n - l ) we find a homeomorphism gn E F such that gn/(An-1 U f-l(Ln)) = f /(An-1 U f-l(Ln)),

gn(Kn) C M2

and

d(f-l(x),gn(x)) + d(f(x),gnl(x)) ~

2-(n+l)c~(x) for each x C X.

Straightforward verifications show t h a t the conditions (a)n, (b)n and (c)n are satisfied. This completes the inductive step of our construction and finishes the proof. [::] COROLLARY 2.2.16. Let M be a (r, lC)-absorbing set in X . IS A E ICe and U is an open subset of X , then idu E c l r / u ( { g E F / U : g ( ( i

N U) U (A N U)) -- i

N U}).

PROOF. By Proposition 2.2.13, both M N U and (M N A) U (A n U) are (F/U, K:v)-absorbing sets in U. It only remains to apply Theorem 2.2.15. [-1 COROLLARY 2.2.17. Let M1 and M2 be (F,M)-absorbing sets in X . If g E F and A is a closed subset of X such that g(A) fl M2 = g(A fl M1), then g e cl({h e F: h / A -- g / A and h ( M 1 ) = h(M2). PROOF. Consider an open cover b / e c o y ( X ) and let M3 -- g(M1), U -- X - A . By Proposition 2.2.13, both M2 n U and M 3 n U are (Fv, K:v)-absorbing sets in X. Therefore, by Theorem 2.2.15, there exists a homeomorphism f E Fu, U-close to i d x , such that f ( M 3 N U) -- M2 n U. Then the homeomorphism h - f .g E F is U-close to i d x , h / A -- g / A and h(M1) -- h(M2). I-1 COROLLARY 2.2.18. Let A be a closed subset of X and suppose there exists a (F, IC)-absorbing set Mo in X such that M o n g -- 0. Then for each (F, IC)absorbing set M in X , the complement M - A is a (F, ]C)-absorbing set in X . PROOF. Consider the open set U -- X - A of X. By Proposition 2.2.13, both M n A ---- M - A and M0 n U -- M0 are (Fv, tgv)-absorbing sets in X and, consequently, by Theorem 2.2.15, there exists a homeomorphism h C F u such that h(Mo N U) -- M n U. Then, by Proposition 2.2.13, the set M - A -- h(Mo) is (F, /(:)-absorbing in X. [-1 Let us now consider the closely related concept of skeleton. DEFINITION 2.2.19. By a F - K-skeleton, we mean any increasing sequence ( A i : i E N } of members of )~ such that, for every i E N , for every A , B E ]C and for every F E F with F ( B ) C Ai, we have f E c l ( U j ( g e F: H / B = F / B and H ( A ) C_ Aj}).

2.2. Z-SETS IN ANR-SPACES

55

A subset of X which can be expressed as the union of elements of a F-K:-skeleton is called a F-/C-skeletoid. When F = A u t h ( X ) , it is simply called a K:-skeletoid.

It is easy to see that any F - ]C-skeletoid is a F - K:-absorber. Any two F - K~skeletoids are equivalent in the sense of the next Theorem. In particular, if there exists a F - K:-skeletoid, then it is equivalent to any F - E-absorber. THEOREM 2.2.20. I f A and B are F F" H (A) -- B } is dense in F.

]C-skeletoids in X , then the set { H E

A closed subset A of a complete metric space X is called a thin set, if for each open neighborhood V of A and for each open cover b / o f X, there is an element F e A u t h ( X ) such that F is b/-close to i d x , F / ( X - V) = id and F ( A ) M A -- O. DEFINITION 2.2.21. A n additive A u t h ( X ) - i n v a r i a n t collection ]C consisting of closed subsets of a Polish space X is called a perfect collection if ]C satisfies the following conditions" (i) Each member of IC is a compact thin set in X . (ii) I f B is a compact subset of A E IC, then B E ]C. (iii) (Estimated Extension P r o p e r t y ) For each A E 1C, for each neighborhood V of A and for each open cover bl of X , there exists an open cover V of X refining 34 such that 9 for each B E ]C with B C_ V , and each homeomorphism f " A ~ B which is 1)-close to id, there is an extension F E A u t h ( X ) of f which is Ll-close to id and F / ( X - V ) = id. W h e n the collection K: is perfect, F - / C - s k e l e t o i d s are characterized by the following theorem. THEOREM 2.2.22. Suppose that the collection ]C is perfect and let {Ai" i E N } be an increasing sequence of members of lC. Then {Ai" i E N } is a F-IC-skeleton if and only if, for every K E ]C, for every i E N and for every e > O, there exist an index j > i and an embedding f " K ---+ X such that (,) d ( f , idK) < e, f / ( K M Ai) =- id and f ( K ) C_ A j . Suppose that U Auth(XiIXU)A u t h ( X i i X - V)}. can be regarded as

is an open set in X and let K:(U) - { g E K : ' K C U}, {H E Auth(Z)" H/(XU)----id} and F ( U ) - { H / U " g E A u t h ( X I I X - U) is a closed subgroup of A u t h ( X ) and F(U) a closed subgroup of A u t h ( U ) .

THEOREM 2.2.23. Suppose that A is a 1E-skeletoid in X and U is an open subset of X . Then A M V is a F ( U ) - 1E(U)-skeletoid in V and is also an A u t h ( X i i X - U) - lE(U)-skeletoid in X . PROPOSITION 2.2.24. Let A be a IE-skeletoid in X and f " K1 ~ K2 be a homeomorphism between members of lE such that f ( K 1 M A) -- K2 M A. Then there exists F E A u t h ( X ) such that F / K 1 = f and F ( A ) -- A.

56

2. INFINITE-DIMENSIONAL MANIFOLDS

PROPOSITION 2.2.25. I f A is a lC-skeletoid in X and B is a countable union of members of IC, then A U B is also a lC-skeletoid.

2.2.3. P r o p e r t i e s o f Z - s e t s in I ~ a n d R ~ . In this Subsection the Hilbert cube I ~ is denoted by Q and is represented as the countable infinite power of the closed segment [-1, 1]. Obviously the countable infinite power P - ( - 1 , 1) ~ is a copy of R ~ . We call P the pseudo-interior of the cube Q and the complement Q - P the pseudo-boundary of Q. Evidently, Q can be considered as a subspace of R ~ and, since the latter space has the structure of a topological linear space, we are in position to add elements of Q as well as multiply them by real numbers. Of course, the results of these operations are not necessarily elements of Q. We consider the following metric on Q" I

Yn]

2n n--1

where xn is an n-th coordinate of a point {xn} E Q (i.e.x. = ~ . ( { x . } ) , where Irn" Q --* [-1, 1]n denotes the corresponding projection). We will also denote by A u t h ( X , A ) , where A C X, the subgroup {f E A u t h ( X ) " f ( A ) = A} of the group A u t h ( X ). LEMMA 2.2.26. For each compactum A in P there exists a homeomorphism h E Auth(Q, P) such that ~rnh(A) = {0} for each odd index n. PROOF. Without loss of generality we may assume that the compactum A is contained in the countable infinite product of the segment [- 89 89 First we show that there is a homeomorphism g E A u t h ( Q , P ) such that zqg(A) -- {0}. Consider the homeomorphism f E A u t h ( Q , P ) defined by letting f ( { x n } ) {Yn}, where Yl -- Xl,

1 1 (n-1 and Yn -- --Xnn "[- -~Xl -Jr- )~(Xn) n

Xn

1) -- ~Xl

,

n>2_,

and where 1 A(t)= ~(12t-1 i+I2t+l[-2)

for each t E [-1,1].

Obviously, f ( A ) C P. Now represent the cube Q as the product [-1,111 x Q1, where QI=H{[

-1,1]n'nk2}.

Denote by s: [--1, 1]1 X Q1 -~ {0} x Q1 the map defined by letting s ( t , x ) = (0, x) for each ( t , x ) E [-1,111 x Q1. It follows from the construction of the homeomorphism f that the restriction f l = s / f ( A ) : f ( A ) ~ s f ( A ) is a home0morphism. Straightforward verification shows

2.2. Z-SETS IN ANR-SPACES

57

that f l is the restriction (to f ( A ) ) of some homeomorphism go 9 A u t h ( Q , P ) . Let g = gof. It only remains to observe t h a t g 9 Auth(Q, P) and ~rlg(A) = {0}. Let us now prove our statement in full generality. Represent the set N of all natural numbers as the union U{Nn : n 9 N } , where Nn's are pairwise disjoint infinite subsets of N such that 2 n - 1 9 Nn for each n 9 N. Let Q -,

=

I-[{[-1, 1]k" k 9 Nn}

denote the corresponding projection and An = 7rNn(A). By the case considered above, there exists a homeomorphism gn 9 A u t h ( Q N ~ , P N ~ ) such t h a t ~r2n-lgn(An) ---- {0}, n 9 g . Then the desired homeomorphism g 9 A u t h ( Q , P ) is defined to be the product g x {gn: n 9 N } . [:] The following statement implies, in particular, the topological homogeneity of Rw. LEMMA 2.2.27. Let f " K ~ L be a homeomorphism between compacta contained in P. Then f has an extension h 9 A u t h ( Q , P ) . PROOF. Let

Qi=H{[-1,1ln'n9

and Pi=E{(-1,1)n'n 9

where N1 denotes the set of all odd indices and N2 = N - N1. First consider the special case when K C_ P1 x {0} and L C {0} x /:'2. Extend the map r N 2 f " g --. t'2 to a map A" Q1 x {0} ~ P2. For each e 9 ( - 1 , 1 ) denote by gE 9 A u t h ( [ - 1 , 1]) a non-decreasing function, linear on the intervals [-1, 0] and [0, 1] and such that dE(0) -- e. For each point y 9 P2 denote by ry 9 auth(Q2, P2) the homeomorphism defined by letting r y ( x ) = {gym ( x n ) ' n 9 N2}. Now define the map fl" Q1 x Q2 ~ Q1 • Q2 as follows:

f l ( x , y ) = (x,r~(x)(y)),

whenever (x,y) 9 Qz x Q2.

Obviously, f l 9 Auth(Q1 x Q2, Pz x P2) and f z ( x , y ) =

(x, ~rN2f (x, y)), (x,y) 9 K.

Similarly, we can construct a homeomorphism f2 9 Auth(Q1 x Q2, P1 x P2) such that f2(x, y) = ( l r g l f - l ( x , y), y) whenever ( x , y ) 9 L. It is easy to see t h a t the composition h -- f 2 f l 1 is the desired extension of the homeomorphism f. Let us now consider the general case. By L e m m a 2.2.26, there exist homeomorphisms f~ 9 A u t h ( Q , P ) , i = 1,2, such that I I ( K ) C_ P1 x {0} and I2(L) C {0} • P2. By the case considered above, the homeomorphism g -f2f.f~ -1" f l ( K ) ~ I2(L) has an extension g' 9 a u t h ( Q , P ) . T h e n the composition h = f 2 1 g l f 1 is the desired extension of f F-1 Similar arguments prove the following statement.

58

2. INFINITE-DIMENSIONAL MANIFOLDS

PROPOSITION 2.2.28. Let K be a closed subset of Q. Then K is a Z - s e t in Q if and only if K M P is a Z - s e t in P. In particular, this is the case if K C P or KC_Q-P. LEMMA 2.2.29. Let K be a Z - s e t in Q. f E Auth(Q) such that f ( g ) C P.

Then there is a h o m e o m o r p h i s m

PROOF. Let W2k-l= {xEQ'xk=-l}

and W 2 k - - { x E Q ' x k - - 1 } .

Then Q - P = u { w n " n E N } . Therefore, it suffices to show that the set Bn = {f E Auth(Q)" f ( g ) M

Wn = 0}

is dense in Auth(Q). In turn, by symmetry, it suffices to establish the denseness of B1 in Auth(Q). Consider an arbitrary homeomorphism g E Auth(Q) and take a homeomorphism f E Auth(Q) such that f ( W 1 ) c_ P. Since the set f g ( g ) is a Z-set in f ( W 1 ) ~ Q we can conclude, by Proposition 2.2.3, that the map ids(w1 ) can be arbitrarily closely approximated by embeddings h: f ( W 1 ) --~ P such t h a t h f ( W 1 ) M f g ( g ) -- q). Then the identity map idQ can be approximated by homeomorphisms h E A u t h ( Q , P ) such that h f ( W 1 ) M f g ( g ) -- 0 (we use Lemma 2.2.27). Therefore, as close as we wish to the map idQ there is a homeomorphism h E A u t h ( Q , P ) such that f - l h - l f g ( g ) M W1 -- 0. [:] THEOREM 2.2.30. E v e r y h o m e o m o r p h i s m f " A1 ~ A2 between Z - s e t s of the Hilbert cube Q can be extended to a h o m e o m o r p h i s m h E Auth(Q). PROOF. By Lemma 2.2.29, there are two homeomorphisms gl,g2 E Auth(Q) such that gi(Ai) C P, i -- 1,2. Then, by Lemma 2.2.27, the homeomorphism g -- g 2 f ( g l / A 1 ) -1" g l ( A 1 ) ~ g2(A2) has an extension g' e A u t h ( Q , P ) . It only remains to remark that the composition h -- g21g'gl is the desired extension of f . [::] By Proposition 2.2.28, each point in Q is a Z-set. Consequently, applying Theorem 2.2.30, we obtain the following result.

COROLLARY 2.2.31. The Hilbert cube is topologically homogeneous. More detailed consideration of the above situation allows us to obtain the following "controlled" version of Theorem 2.2.30. THEOREM 2.2.32. For each open cover b[ E coy(Q) there exists an open cover ]) E coy(Q) such that the following condition is satisfied: 9 if a h o m e o m o r p h i s m f : Z1 --~ Z2 between two Z - s e t s of Q is ])-close to the inclusion map i 1 : Z 1 ~-~ Q, then h can be extended to a homeomorp h i s m h: Q ---+ Q which is U-close to the identity map idQ. Let us now consider some properties of Z-sets in the space R ~ .

2.2. Z-SETS IN ANR-SPACES

59

THEOREM 2.2.33. Every h o m e o m o r p h i s m f : A1 --* A2 between Z - s e t s of P can be extended to a h o m e o m o r p h i s m h E A u t h ( P ) . Moreover, f o r each open cover blcov(P) there exists an open cover V E c o y ( P ) such that the following condition is satisfied: 9 if a h o m e o m o r p h i s m f : A1 ~ A2 between two Z - s e t s of P is )P-close to the inclusion map i 1 : A 1 ~ P, then f can be extended to a h o m e o m o r phism h: P ~ P which is U-close to the identity map i d p . PROOF. We prove only the first p a r t of the theorem. Let f : A1 --~ A2 be a h o m e o m o r p h i s m between Z-sets of R ~ (we identify P with R ~ in order to use t h e linear s t r u c t u r e ) . First we consider the case when there exist h o m e o m o r p h i s m s g l , g 2 : Tl w • R w -+ R w such t h a t gl(A1) C_ R w • {0} and g2(A2) C_ {0} • R w . Let K i = gi(Ai), i -- 1,2 and g -- g 2 f ( g l / A 1 ) - l : K1 ~ K2. Let us show t h a t t h e r e is a h o m e o m o r p h i s m g' E A u t h ( R w • R W ) such t h a t g ' / K 1 -- g. Since K1 is closed in R ~ • {0} and since R ~ is an a b s o l u t e extensor, we easily see t h a t t h e r e is a m a p ~ : R ~ x{0} --, R ~ s u c h t h a t ~ ( x ) = ~r2g(x) for each x E g l . Define a h o m e o m o r p h i s m hi E A u t h ( R ~ • R ~ ) by letting h l ( X l , X2) -- (Xl,X2 + (fl(Xl, 0)) whenever (xl, x2) E R ~ • R ~ . It is easy to see t h a t h i ( x ) = (Trl(X),Tr2g(x)) for each x e g l . Similarly, t h e r e exists a h o m e o m o r p h i s m h2 E A u t h ( R ~ • R ~ ) satisfying t h e equality

h2(~) = ( - ~ g - ~ ( ~ ) , ~2(~)) for ~ c h

~ e g2.

T h e n h2g(x) -- h i ( x ) for each x E K1 and, consequently, the h o m e o m o r p h i s m g' - h 2 1 h l E Auth(R~ • R ~ ) e x t e n d s the h o m e o m o r p h i s m g. T h e desired extension h can now be defined by l e t t i n g h -- g21g'g1. Thus, in o r d e r to c o m p l e t e the proof, it suffices to show t h a t for each Z - s e t A in P there is a h o m e o m o r p h i s m g: P ~ P • P such t h a t g ( A ) C P • {0}. D e n o t e by A' t h e closure of A in Q. By P r o p o s i t i o n 2.2.28, A' is a Z - s e t in Q. Observe also that, by P r o p o s i t i o n 2.2.7, A' • {0} is a Z - s e t in t h e p r o d u c t Q • Q. By T h e o r e m 2.2.30, t h e r e is a h o m e o m o r p h i s m h: Q --+ Q • Q such t h a t h(x) -- (x,0) for each x E A'. Now consider the sets M1 -- Q - P and Q • Q - (P • P ) . It is not h a r d to see t h a t b o t h these sets are a b s o r b i n g (with respect to the collection of all Z-sets in Q a n d Q • Q and groups A u t h ( Q ) a n d A u t h ( Q • Q) respectively). T h e r e f o r e t h e set M2 -- h - l ( Q • Q - (P • P ) ) is also an absorbing set in Q. By C o r o l l a r y 2.2.17, there is a h o m e o m o r p h i s m h' E A u t h ( Q ) such t h a t h ' / A ' = id and h ' ( M 1 ) -- M2. It only remains to r e m a r k t h a t the c o m p o s i t i o n g - h h ' / P is a h o m e o m o r p h i s m of P onto P • P a n d g ( A ) C_P x {0}. I--! We also have t h e following s t a t e m e n t .

60

2. INFINITE-DIMENSIONAL MANIFOLDS PROPOSITION 2.2.34. L e t A be a Z a - s e t in R ~ . T h e n the i n c l u s i o n R ~ - A

R ~ is a n e a r - h o m e o m o r p h i s m .

PROOF. We identify R Wwith the p s e u d o - i n t e r i o r P of Q. Let A = UnC~=lAn, where each of the sets A n is a Z-set in P. By Proposition 2.2.28, each of the sets B n -- c I Q A n is a Z-set in Q. As observed in the proof of T h e o r e m 2.2.33, the p s e u d o - b o u n d a r y Q - P of the cube Q is an absorbing set in Q. Therefore, by Proposition 2.2.13, the union (Q - P ) u B, where B = U n = l B n , is also an absorbing set in Q. In this situation T h e o r e m 2.2.15 guarantees the existence of a h o m e o m o r p h i s m (as close to the identity as we wish) g E A u t h ( Q ) such t h a t g ( ( Q - P ) u B ) - Q - P . T h e n g ( P - B ) -- P . It only remains to remark t h a t oo

P-B=P-A.

F]

COROLLARY 2.2.35. let A be a Z a - s e t in an R W - m a n i f o l d X .

Then X -A

is

also an R ~ - m a n i f o l d .

PROOF. Represent X as the countable union X - U n = I X n of open subspaces each of which is h o m e o m o r p h i c to R ~ . Let A -- UkCC=lAk, where Ak is a Z-set in X for each k. By Corollary 2.2.10, each of the sets X n n A k is a Z-set in Xn, k -- 1 , 2 , . . . . Therefore, Xn n A -- Xn n U k = l A k is a Zo-set in Xn for each n. Applying Proposition 2.2.34, we conclude t h a t Xn - A is homeomorphic to R W Now observe t h a t X - A -- U cr ( X n A ) and t h a t each of the s u m m a n d s X n - A is open in X - A. Consequently, X - A is an R "~ -manifold. W1 Xn

oo

9

n--1

--

H i s t o r i c a l a n d bibliographical n o t e s 2.2. The notion of Z-set was first introduced by Anderson in [19] (see also [142]) in order to generalize the concept of infinite codimension. He used the following definition: a closed subset A of a space X has a p r o p e r t y Z in X if for each n o n - e m p t y contractible open set G C X , the c o m p l e m e n t G - A is contractible in X - A. In all cases considered in this book, this definition is equivalent to Definition 2.2.2 (compare with [293]; see also [111]). T h e concept of a stong Z-set was introduced by Henderson [167]. T h e r e are a n u m b e r of papers devoted to these two notions (see, for example, [20], [297], [35], [34]). Corollary 2.2.4, Proposition 2.2.6 and an example of Z-set which is not a strong Z-set a p p e a r e d in [34]. T h e key idea exploited in the definition of the notion of c o m p a c t absorption set (skeletoid) belongs to Anderson. This notion was subsequently generalized by various authors (see [35], [31], [32], [158], [294], [295], [313], [64]). The results of Subsection 2.2.3 are mainly obtained in [19], [20], [24], [26] (see also [69] and [32]). Corollary 2.2.31 was proved in [192].

2.3. R w- AND I ~-MANIFOLDS 2.3. R ~

61

and I w -manifolds

Traditionally, by a Y-manifold (i.e. t h e manifold, m o d e l e d over t h e space Y) we u n d e r s t a n d a space, each point of which has an open n e i g h b o r h o o d h o m e o m o r phic to an open subspace of the space Y. Below we consider t h e two i m p o r t a n t cases in which Y = I ~ and Y = R ~ . 2 . 3 . 1 . S t a b i l i t y o f I ~ - a n d R ~ - m a n i f o l d s . In this Subsection we show t h a t if Y E { I W ,R ~ }, t h e n every Y - m a n i f o l d X is Y-stable, m e a n i n g t h a t t h e p r o d u c t X x Y is h o m e o m o r p h i c to X . For the p r o o f of these i m p o r t a n t s t a t e m e n t s we need some p r e l i m i n a r y results. F i r s t we recall the following topological c o n s t r u c t i o n . For a c o n t i n u o u s m a p f " X --* Y and a closed subset A of in Y, t h e adjunction space X U / Y is defined to be t h e disjoint union of ( X - f - I ( A ) ) a n d A with t h e t o p o l o g y consisting of t h e usual open subsets X - f - I ( A ) t o g e t h e r w i t h sets of t h e form f - I ( U - A ) U ( U M A ) for o p e n subsets U C y . It is easy to see t h a t in this s i t u a t i o n we have two n a t u r a l l y arising m a p s f A" X ---+X Uf A and PA" X Uf A ~ Y. T h e i r definitions are: X,

PA(X) --

.f (x),

if x E X - f - Z ( A ) if x E f - l ( A )

and fA(x)

=

S .f(x), x

[,

ifx E x-f-I(A)

ifxEA.

Obviously, f = PAfA. It is left as a m a n a g e a b l e exercise for t h e reader to verify t h a t if f : X --~ Y is a fine h o m o t o p y equivalence b e t w e e n Polish ANR-spaces a n d A is a (strong) Z - s e t in Y, t h e n X U / A is also a Polish A N R - s p a c e a n d A is a (strong) Z - s e t in X U / A . Additionally, in this s i t u a t i o n , t h e m a p s PA a n d f n also are fine h o m o t o p y equivalences. Consider now an i m p o r t a n t p a r t i c u l a r case of t h e above c o n s t r u c t i o n . S u p p o s e t h a t two spaces X and Y are given and A is closed in X . C o n s i d e r t h e p r o j e c t i o n ~r : X x Y ~ X onto t h e first c o o r d i n a t e a n d form the a d j u n c t i o n space (X x Y)U~ A. This space, for simplicity, is d e n o t e d by (X x Y)A and is called t h e reduced (or partial) product of spaces X and Y over t h e set A. T h e m a p p: (X x Y)A ~ X is d e n o t e d in this p a r t i c u l a r case by p x . Clearly, if A = O, t h e n t h e space (X x Y ) A coincides with t h e usual p r o d u c t X x Y. T h e reader can easily observe t h a t by enlarging set A this c o n s t r u c t i o n " g r a d u a l l y " t u r n s t h e p r o d u c t X x Y into t h e space X . DEFINITION 2.3.1. The space X is strictly Y - s t a b l e if for each closed subset A C_ X , open neighborhood G of A, and neighborhood W of the projection 7rx" X x Y ---, X (in the space C ( X x Y , X ) ) , there exists a homeomorphism f" Z x Y --, (X x Y ) A such that p x f E W and f ( z ) = z for each point zE(X-G) xY.

62

2. INFINITE-DIMENSIONAL MANIFOLDS

PROPOSITION 2.3.2. Let X be a Y-manifold. If each open subspace of Y is strictly Y-stable, then the space X is also strictly Y-stable. PROOF. Consider the collection b / o f open subspaces of the space X satisfying the following condition: (.) If U E b/, V is open in X and V C U, then V is strictly Y-stable. Obviously, the collection/d is hereditary with respect to open subspaces. It is also easy to see that a discrete countable union of elements of/g is an element of /d. Therefore, by the well-known localization principle (see, for example, [32]), it only remains to show that if V1, V2 E b/, then V - V1 tA V2 is strictly Y-stable. Let A be a closed subset of V and G be an open neighborhood of A in V. Take any metric d on V generating the topology of Y and let a E C(V, (0, c~)). Let Ak = A N Vk, k -- 1,2. Take open neighborhoods G1 and G2 of the sets A1 and A2 respectively (in X) so that

Ak C A N G k

C clGk C Vk, k - - 1,2.

Since V1 is strictly Y-stable, there exists a homeomorphism 9" V1 x r ---+ (V1 x Y)A~ such that g ( z ) = z for each point Z E (V1 - G 1 ) x Y and 1

d(pv~g(z),Trv~(z)) < ~a~rv~(z) for each z E 111 x Y. Completing the homeomorphism g by the identity map on (V2 - V1) x Y, we obtain the homeomorphism g " V x Y ---, (V x Y)A~. Since V2 - A1 is strictly Y-stable, we have a homeomorphism h" ( V 2 - A1) x Y --, ( ( V 2 - A1) x Y)A2-A1 such t h a t h ( z ) -

z for each z E ( V 2 - G 2 - A1) x Y and

d(py2-A~h(z), 7ry2-A1 (z)) < ~(Try2-A~ (z)) for each z e ( V 2 - A1) x Y, where

fl(x)-

m i n { 2 ~ ( x ) , d ( x , A1) }.

Completing the homeomorphism h by the identity map on ( ( V 1 - A - V 2 ) x Y ) U A I , we obtain the homeomorphism h " (Y x Y)A~ ~ (V x Y)A. It only remains to note that the composition f - hlg I is a homeomorphism of the product Y x Y onto the space (V x Y)A, coinciding with the identity on ( V - G) x Y and satisfying the desired inequality d ( p v f ( z ) , T r v ( z ) ) < aTrv(z) for each z e

V x Y.

[:3

DEFINITION 2.3.3. A space Y is said to have the reflective isotopy property (or, briefly, R I P ) if there exists a homotopy g" Y x Y x [0, 1] --. Y x Y (called a reflection isotopy) such that the following conditions are satisfied: (i) g(yl, y2, 0 ) - - (yl, y2) for each (yl, Y2) E Y x Y.

2.3. R ~- AND I ~-MANIFOLDS

63

(ii) g(Yl, Y2, 1)---- (Y2, Yl) for each (yl, y2) E Y x Y. (iii) The map .~" Y x Y x [0, 1] ~ Y x [0, 1] defined by letting g(Yl, y2, t) -(g(Yl, Y2), t) is a homeomorphism. LEMMA 2.3.4. The spaces R r and I r have the reflective isotopy property for each T > w. PROOF. Consider a complex linear topological space E and define a m a p g" E • E • [0, 1] --. E x E as follows:

{ g ( x l , x2, t) --

9 (e2~rZtxl,x2), ( - x l sin r t - x2 cos ~rt, - x l cos 7rt + x2 sin ~t),

if 0 _ t < 1 if 89 _ t < 1.

Straightforward verification shows t h a t g is a reflection isotopy. We also r e m a r k t h a t for each linear topological space over the field R of real numbers, its square can be considered as a complex linear topological space. Therefore, the plane R 2 has the reflective isotopy property. Let us now show t h a t the square [-1, 1] 2 also has the reflective isotopy property. Consider the reflection isotopy g" R 2 • R 2 • [0, 1] --+ R 2 • R 2 c o n s t r u c t e d above. Let g -- {(Zl, Z2, za, z 4 ) e R 4" z 2 + z 2 + z 2 + z 2 <_ 1}. Obviously, the 4-dimensional ball K is invariant with respect to g~ ( R 2 x R 2 x {t}) for each t E [0, 1]. Consider now the radial h o m e o m o r p h i s m f" [ - 1, 1] 2 x [ - 1, 1] 2 ---+ K, shrinking the 4-dimensional cube [ - 1 , 1 1 2 x [-1, 1] 2 onto K along the lines passing t h r o u g h the origin. Finally, observe t h a t the m a p h" [-1, 1] 2 x [-1, 1] 2 x [0, 1] --+ [-1, 1] 2 x [-1, 1] 2, defined by letting

h(xl, z2,t) = f - l g ( f ( x l , x 2 ) , t )

for each (Xl, X2,t) e [-1, 1] 2 x [-1, 1] 2 x [0, 1]

is the reflection isotopy for [-1, 1] 2. Thus we have shown t h a t the spaces R 2 and [ - 1 , 1] 2 b o t h have the reflective isotopy property. Taking products of the c o r r e s p o n d i n g reflection isotopies we can easily conclude t h a t the spaces R r and I r , r _ w also have reflective isotopy property. [-1 DEFINITION 2.3.5. A homotopy y N x y x [1, c~) -~ y N is called a displacement isotopy if the following two conditions are satisfied: (i) s n h ( x , y, t) -- Sn(X) for each n <_ t and (x, y, t) E y N x y x [1, oo), where Sn" Y N ._+ y n denotes the projection onto the corresponding subproduct. (ii) the map h" y g x Y • [1, c~) --+ y y x [1, c~) defined by letting h(x, y , t ) = (h((x, y, t, ), t) is a homeomorphism.

64

2. INFINITE-DIMENSIONAL MANIFOLDS

LEMMA 2.3.6. I f a space has the reflective isotopy property, then it admits a displacement isotopy. PROOF. Let g" Y x Y x [0, 1] ~ Y be a reflection isotopy for the space Y. Then the corresponding displacement homotopy can now be defined as follows. For each ({Yk},Y) e y N x Y and for each n e N and t 9 [0, 1], let h ( { y k } , y , n + t ) = {zk}, where zk = Yk if k < n, Zk = Yk-1 if k >_ n + 2, and (Zn, Zn+l) =

g(y, y.t).

c]

Let V be an open subset in y N . We call a function A E C(V, (0, c~)) a stearing f u n c t i o n if for each n E N and for x, x' E V, the conditions

A(x) < _ n + l

and s n ( x ) = s n ( x ' )

imply the equality A(x) = A(x'). LEMMA 2.3.7. The following conditions are satisfied: (a) /fA1 and A2 are stearing f u n c t i o n s on V, then A(x) = max{Al(x), A2(x)} is also a stearing f u n c t i o n on V . (b) Let Y be an open subset in Y g a n d a e C ( V , ( O , c ~ ) ) . Then there is a stearing f u n c t i o n A on V such that A > a. (c) Let V1 and V2 be open subsets of y N, B be a closed subset of V1 and B C V2 C_ 1/'1. Then f o r any two f u n c t i o n s a l E C(Vl,(0, c~)) and ~2 e C(V2, (0, oo)), there exists a stearing f u n c t i o n Ak on Vk, k = 1,2, such that A 1 / B = A 2 / B and Ak > ak, k = 1,2. (d) Let A be a stearing f u n c t i o n on an open subset V C y g . Suppose that h" y g x Y x [1, oo) ~ y N is a displacement isotopy for Y . Suppose also that on y g we have the metric oo

d({yn}, {y~}) = E

min{l'

P(Yn, Ytn)},

n=l

where p is a metric on Y . I f 2 -x(x) < 89 y N - V) for each x E V, then the map f " V x Y --~ y N, defined by letting

f ( ~ , y) = h(~, y, ~(~)),

(~, y) e v • Y

is a h o m e o m o r p h i s m of the product V x Y onto V. In addition, d(f(x,y),x)

< 2 -x(x)+l for each ( x , y ) E V x Y.

PROOF. Conditions (a)-(c) are obvious. Let us verify condition (d). For a point (x, y) e Y x Y take a number n such that A(x) e [n, n + 1]. Then, by the definition of displacement isotopy, we have the following two conditions:

(~) ~ s ( ~ , y) = ~ ( x ) . (ii) A f ( x , y ) = A(x).

2.3. R ~- AND I "~-MANIFOLDS

65

T h e first condition implies t h a t g ( f (x, y), x) < 2 - n < 2-~(z) +1 for all points ( x , y ) 9 Y • Y. In particular, f ( Y • Y ) c Y . T h e second condition implies t h a t the m a p g ( x ) - ( h / ( Y g x Y x {A(x)})) -1, x 9 V, sends the set Y into the p r o d u c t V x Y. Moreover, one can see t h a t g is an inverse of f. W1 PROPOSITION 2.3.8. I f a space Y a d m i t s a d i s p l a c e m e n t isotopy, then every open subspace of Y N is strictly Y - s t a b l e . PROOF. Let U be an open subspace of y N and A and B be disjoint closed subsets of U. Also let a E C ( U , (0, c~)). Our goal is to c o n s t r u c t a homeom o r p h i s m f : V • Y ~ (U • Y ) A such t h a t f ( z ) = z for each z E B • Y, and d ( p u f ( x , y ) , x ) < a ( x ) for each ( x , y ) e Y x Y (here d denotes the metric from condition (d) of L e m m a 2.3.7). Let U1 -- U, U2 -- U - A a n d let i l k ( X ) - - - l o g 2 a k ( x ) for each x e Uk, where a k ( X ) = ~1 m i n { a ( x ) , d ( x , y g

-- Uk)},

k = 1, 2.

By condition (c) of L e m m a 2.3.7, there exist two stearing functions Ak on Uk such t h a t )~I/B = )~2/B and Aa > ilk, k = 1,2. We define m a p s f k : Uk • Y --+ y y by letting

fk(x, y) = h(~, y, ~k(~)),

k = 1, 2

(here h denotes the displacement isotopy for Y). It follows from condition (d) of L e m m a 2.3.7 t h a t f k is a h o m e o m o r p h i s m of the p r o d u c t Uk • Y onto Uk and d ( f k ( x , y ) , x ) < 2 a k ( x ) for each ( x , y ) e Vk • Y, k = 1,2.. Now consider the m a p f3: (U • Y ) A ---+ U coinciding w i t h f2 on U2 • Y and with t h e i d e n t i t y m a p on A. Obviously, f3 is a h o m e o m o r p h i s m . Finally, let f - f 3 1 f l . It is not difficult to see t h a t f sends the p r o d u c t U • Y h o m e o m o r p h i c a l l y onto t h e space (U • Y ) A . It only remains to r e m a r k t h a t f ( z ) = z for each z 9 B • Y a n d t h a t d ( p u f ( x , y ) , x ) < 2 a l ( x ) + 2a2(x) _< c~(x) for each (x, y) 9 U • Y. W1 T h e s t a t e m e n t s proved so far in this section imply the following two i m p o r t a n t stability t h e o r e m s for R "~- and I "~ -manifolds. THEOREM 2.3.9. I f X is an R ~ -manifold, then X • R W ~ X . projection ~ x : X x R W ~ X is a n e a r - h o m e o m o r p h i s m .

Moreover, the

THEOREM 2.3.10. I f X is an R ~ -manifold, then X x I ~ ,~ X . Moreover, the proj e c t i o n ~ x : X • I ~ ~ X is a n e a r - h o m e o m o r p h i s m .

COROLLARY 2.3.11. I f X

is an R W - or I W -manifold,

then the proj ect i on

7r : X • I ~ X is a n e a r - h o m e o m o r p h i s m .

PROOF. Consider only the case of R ~ -manifolds (the r e m a i n i n g case is completely analogous to this one). D e n o t e by ~ x : X • R ~ --~ X and lr1: R W • I ---, R W the projections onto the first coordinates. By T h e o r e m 2.3.9, r x is a nearh o m e o m o r p h i s m . T h e projection 7rl is also a n e a r - h o m e o m o r p h i s m (see [25] for details). Now consider the following c o m m u t a t i v e diagram:

66

2. INFINITE-DIMENSIONAL MANIFOLDS

7rx x id X •

r xI

,.-X •

id x 7rl

X xR"

,,

7rx

,~ X

As observed above, b o t h horizontal arrows, as well as t h e left vertical arrow, are n e a r - h o m e o m o r p h i s m s . In this s i t u a t i o n t h e projection 7r is also a near-hom e o m o r p h i s m . [--1

2 . 3 . 2 . The strong universality o f I ~ - a n d R ~ - m a n i f o l d s . In Subsection 1.1.1 we have already seen ( P r o p o s i t i o n s 1.1.3 and 1.1.8) t h a t t h e spaces R ~ and I W are inuversal with respect to the classes of all Polish spaces a n d all c o m p a c t m e t r i z a b l e spaces respectively. F u r t h e r m o r e , in the first case, it is essential t h a t each Polish space a d m i t s a closed e m b e d d i n g into R ~ . This p r o p e r t y of the space R ~ is non-trivial, because it is easy to see every Polish (and, in general, every separable metrizable) space can be e m b e d d e d into t h e Hilbert cube I W as well. Below, we d e n o t e by Jt~,oo and B~,oo t h e class of all Polish spaces and the class of all m e t r i z a b l e c o m p a c t a respectively. T h e sense of these notations will become clear in Section 6.5. DEFINITION 2.3.12. Let 7~ E {Jtw,oo, Bw,oo}. A Polish space X is said to be strongly P - u n i v e r s a l if f o r each space Y f r o m the class 7~, the set of all closed embeddings of Y in X is dense in the space C (Y, X ) . PROPOSITION 2.3.13. E a c h I ~ - m a n i f o l d is strongly Bw,oo-universat. PROOF. We consider only t h e c o m p a c t case. Let X be a c o m p a c t I~-manifold. By T h e o r e m 2.3.10, it suffices to establish t h e s t r o n g B~,oo-universality of the p r o d u c t X x I ~ . Let f " Y --~ X x I ~ , where Y is a m e t r i z a b l e c o m p a c t u m and Lt E c o v ( X • I ~ ). W i t h o u t loss of generality we can a s s u m e t h a t the cover L/ has the form ]) • W , where • E c o y ( X ) and ]4? E c o v ( I ~ ). Take a finite subset A C w and an open cover ~)A E COV(IA) such t h a t the cover 7rAI()4)A) refines (here 7rA" I ~ --, I A d e n o t e s t h e projection onto the c o r r e s p o n d i n g s u b p r o d u c t ) . Since the space I w - A is h o m e o m o r p h i c to I ~ , we can see, by P r o p o s i t i o n 1.1.8, t h a t there is an (closed) e m b e d d i n g g" Y ~ I w - A . It only r e m a i n s to note t h a t t h e diagonal p r o d u c t 7rlfATrATr2Ag" Y ~ X • I W

2.3. R w- AND I w-MANIFOLDS

67

is an e m b e d d i n g / g - c l o s e to f (here lrl a n d r 2 d e n o t e t h e p r o j e c t i o n of X • I ~ o n t o t h e first a n d t h e second c o o r d i n a t e r e s p e c t i v e l y ) ,

i-1

PROPOSITION 2.3.14. Each R ~ -manifold is strongly ,4~,oo-universal. PROOF. B y T h e o r e m 2.3.9, t h e g e n e r a l case can be r e d u c e d to t h e case X = R ~ . T h e space R ~ is identified w i t h its c o u n t a b l e power X = 1-I{Rn~ : n E w}, w h e r e R ~ = R ~ for each n E w. Let lrn" X + R ~ d e n o t e t h e p r o j e c t i o n o n t o t h e c o r r e s p o n d i n g c o o r d i n a t e . Take a m e t r i c dn on t h e space R ~ b o u n d e d by 1 a n d consider t h e following m e t r i c on t h e p r o d u c t X :

d(x, (x') = m a x { d n ( r n ( x ) , r n ( x ' ) ) " n e w}, x , x ' e X . Let Y be a Polish space. S u p p o s e also t h a t m a p s f : Y - . X a n d a : X --, (0, 1) are given. For our p u r p o s e s it suffices to c o n s t r u c t a closed e m b e d d i n g g: Y - . X such t h a t d ( f ( y ) , g ( y ) ) <_ a f ( y ) for each p o i n t y E Y (see S u b s e c t i o n 2.1.1). Since R ~ is a Polish a b s o l u t e r e t r a c t , t h e r e exists a c o n t i n u o u s m a p

h~.R~ • R~ • [0, ~ ) - ~

R~

such t h a t

h n ( x l , x 2 , t) = x l for each t _< 1 and

h n ( x l , x 2 , t) -- x2 for each t _ 2. B y P r o p o s i t i o n 1.1.3, t h e r e exists a closed e m b e d d i n g ~on" Y -* R n~, n E w. Let t h e m a p gn" Y --* R~, n E w, be defined by l e t t i n g

gn -- hn(~n.f A ~ P n A 2 n a f ) 9 We now define t h e m a p g" Y --~ X to be t h e d i a g o n a l p r o d u c t

If y E Y, t h e n t h e r e exists n E w such t h a t 2 - ( n + l ) <__ a f ( y ) <_ 2 - n . T h e r e f o r e , for k _< n we h a v e ga(y) = ~raf(y). T h i s implies t h a t

d ( f ( y ) , g ( y ) ) -- m a x ( d k ( P k f ( y ) , p k g ( y ) ) "

k -- n + 1 , . . . } <_ 2 - ( n + l ) _< (~f(y).

T h e injectivity of g is obvious. Let us show t h a t g is a closed e m b e d d i n g . Consider an a r b i t r a r y s e q u e n c e (Yk} of p o i n t s in Y such t h a t t h e s e q u e n c e (g(Yk)} converges in X to a p o i n t x -- lim g(Yk). Obviously, it suffices to show t h a t t h e s e q u e n c e (Yk} also converges in Y to s o m e p o i n t y E Y (indeed, this will imply, first of all, t h a t t h e m a p g" Y --+ g ( Y ) is o p e n and, secondly, closedness of t h e set g ( Y ) in X ) . First, a s s u m e t h a t 0 < e = ~ i n f ( a f ( y ~ ) } .

C h o o s e a n u m b e r n E w so t h a t e >_

2 - n + l . T h e n we have rng(Yk) -- gn(Yk) -- ~on(Yk) for each k. C o n t i n u i t y of t h e p r o j e c t i o n ~rn g u a r a n t e e s t h a t r n ( x ) - lim ~n(yk). B u t ~ n is a closed e m b e d d i n g . Ther e fo r e , t h e s e q u e n c e (Yk} converges in Y to t h e p o i n t y -- ~ l ( ~ r n ( x ) ) .

68

2. INFINITE-DIMENSIONAL MANIFOLDS

T h u s it only remains to show t h a t e > 0. Assume to the contrary t h a t e -- 0. T h e n there exists a subsequence {Yk,} of the sequence {Yk} such t h a t lira a f ( y k ~ ) -- x. B u t this equality, coupled with the continuity of a, leads us to a ( x ) -- 0 and this is a contradiction. This completes the proof. [-! LEMMA 2.3.15. Every Z - s e t in a strongly flt~,oo-universal Polish A N R - s p a c e is a strong Z-set. PROOF. Let A be a Z-set in a strongly A~,c~-universal Polish A N R - s p a c e X. According to Proposition 2.2.6, it suffices to show t h a t if P is a locally compact polyhedron then the set of those maps f : P ---, X such t h a t c l x f ( P ) M A = 0 is dense in the space C ( P , X). Represent P as the countable union of its compact subspaces (simplexes) Pn, n E w. D e n o t e by B n the set of those maps f : P ~ X for which f ( P n ) M A -- 0. Obviously, B n is an open and dense subset of C ( P , X ) for each n E w. Recall t h a t the space C(P, X ) contains a dense (and obviously G ~ - ) subspace F , consisting of all closed embeddings (apply L e m m a 2.1.2). By the Baire property in C ( P , X ) (see Proposition 2.1.7), the set OO

F CI Mn= 1Bn

is also dense in C (P, X).

W1

LEMMA 2.3.16. Each compact set in a strongly ,4,~,oo-universal Polish A N R space is a strong Z-set. PROOF. Let K be a compact subset of a strongly A ~ , ~ - u n i v e r s a l Polish A N R - s p a c e X . By L e m m a 2.3.15, it suffices to show t h a t K is a Z-set in X. Let L / E c o y ( X ) and f : I W ---, X be a map. Denote by r : N • X ~ X the n a t u r a l projection. T h e strong A ~ , ~ - u n i v e r s a l i t y of X guarantees the existence of a closed e m b e d d i n g h: N • X ---, X which is U-close to the composition r f . Obviously, the collection {h({k} • I ~ ): k E N } is discrete in Z . Therefore, by compactness of K, there exists an index k0 E g such t h a t h({k0} • I ~ ) M K = 0. Now note t h a t the m a p g: I ~ ---, X defined by letting g(a) - h(ko, a), a E I W , is U-close to f and g ( I ~ ) M K -- 0. W] T h e following two s t a t e m e n t s show that, in fact, the spaces R W and I ~ both satisfy a stronger version of the universality property discussed above. THEOREM 2.3.17. Let X be a R w -manifold and A be a closed subset of a Polish space B . Then each map f : B ---, X , such that f / A is a Z-embedding, can be arbitrarily closely approximated by Z-embeddings coinciding with f on A. In particular, the set of all Z - e m b e d d i n g s B ---, X is dense (and a G b) in the space C (B , X ). THEOREM 2.3.18. Let X be an I ~ -manifold and A be a closed subset of a locally compact space B . Then each proper map f : B --~ X , such that f / A is a Z-embedding, can be arbitrarily closely approximated by Z-embeddings coinciding with f on A.

2.3. R~- AND I ~-MANIFOLDS

69

The following two statements also belong to the same category of results. THEOREM 2.3.19. L e t X be an I w - m a n i f o l d , A be a locally c o m p a c t space a n d F: A x I -, X

be a p r o p e r m a p s u c h t h a t Fo a n d F1 are Z - e m b e d d i n g s .

there exists an i s o t o p y H : X

• I --, X

Then

s u c h t h a t Ho -- id a n d H 1 F o -- F1.

M o r e o v e r , i f F is a b l - h o m o t o p y (Lt E c o y ( X ) ) ,

t h e n we m a y choose H to also

be a L t - i s o t o p y .

THEOREM 2.3.20. L e t X be an R w - m a n i f o l d , A be a P o l i s h space, a n d F : A x I --~ X

be a m a p s u c h t h a t Fo and F1 are Z - e m b e d d i n g s .

isotopy H: X x I ~

T h e n there e x i s t s an

X s u c h t h a t Ho -- id a n d H I F o -- F1. M o r e o v e r , i f F is an

l d - h o m o t o p y (ld E c o v ( X ) ) ,

t h e n we m a y choose H to also be a lA-isotopy.

2.3.3. F u r t h e r p r o p e r t i e s of I ~ - a n d R w - m a n i f o l d s . The following two statements, known as A N R - t h e o r e m s , are among the main results of infinitedimensional manifold theory. THEOREM 2.3.21. T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t f o r each locally com p a c t space X :

(i) X x I ~ is a I ~ - m a n i f o l d . (ii) X is an A N R - s p a c e . A complete proof of this theorem is contained in Chapman's Lectures [69], and for this reason a proof is not presented here. A similar approach proves the following statement. THEOREM 2.3.22. T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t f o r each P o l i s h space X :

(i) X x R W is h o m e o m o r p h i c to R ~ (respectively, X x R W is a R ~ - m a n i f o l d ) . (ii) X is an A ( N ) R - s p a c e . The fact t h a t every compact contractible I ~ -manifold is homeomorphic to I ~ (see [65]), coupled with Theorem 2.3.21, implies the following result. COROLLARY 2.3.23. T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t f o r each c o m p a c t space X :

(i) X • I ~ is h o m e o m o r p h i c to I ~ . (ii) X is an A R - s p a c e . Open embedding theorems are also very important in infinite-dimensional manifold theory. In the case of R ~ -manifolds, the following shows that the class of R ~ -manifolds coincides with the class of open subspaces of the model space Rw .

THEOREM 2.3.24. T h e f o l l o w i n g co'nditions are e q u i v a l e n t f o r a n y space X : (i) X is an R ~ - m a n i f o l d . (ii) X a d m i t s an open e m b e d d i n g i n t o R ~ .

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2. INFINITE-DIMENSIONAL MANIFOLDS

PROOF. Obviously, every open subspace of R W is an R w -manifold. Consider now an a r b i t r a r y R W -manifold X , and d e n o t e by Y the open cone of X , i.e. the reduced p r o d u c t ([0,1) x X){0}. It is not hard to see t h a t Y is a Polish absolute retract. Consequently, by T h e o r e m 2.3.22, the p r o d u c t Y x R ~ can be identified with R ~ . Observe also t h a t the p r o d u c t (0, 1) x Y is open in Y (its c o m p l e m e n t in Y is one point). Consequently, the p r o d u c t (0,1) x X x R W is open in R ~ . It only remains to note that, by T h e o r e m 2.3.9, the indicated p r o d u c t is h o m e o m o r p h i c to X. D Of course, not e v e r y / W - m a n i f o l d can be identified with an open subspace of I W . C o m p a c t n e s s is an obvious obstruction. Nevertheless, in this case we have the following result. THEOREM 2.3.25. L e t X

be an I ~ - m a n i f o l d .

T h e n the p r o d u c t X x [0, 1)

a d m i t s an open e m b e d d i n g into I ~ .

PROOF. First assume t h a t X is compact. T h e n its cone, C o n ( X ) , is an absolute retract. By Corollary 2.3.23, the p r o d u c t C o n ( X ) x I ~ is homeomorphic to I ~ . Since X x [0, 1) is open in C o n ( X ) , T h e o r e m 2.3.10 finishes the proof in this case. Now suppose t h a t X is a n o n - c o m p a c t I w -manifold. Denote by Y the onepoint compactification of the p r o d u c t X x [0, 1). It follows from [178] t h a t Y is an absolute retract. Corollary 2.3.23 implies t h a t the product Y x I W is h o m e o m o r p h i c to I ~. Consequently, the p r o d u c t X x [0, 1) x I ~ is homeomorphic to an open subspace of I "~ . T h e o r e m 2.3.10 completes the proof. [-! A detailed proof of the following s t a t e m e n t can be found in [69]. THEOREM 2.3.26. I ~ m a n i f o l d s X a n d Y are h o m o t o p y equivalent if a n d only i f the p r o d u c t s X x [0, 1) a n d Y x [0, 1) are h o m e o m o r p h i c .

A similar result is valid for R "~ -manifolds as well. Observing, additionally, t h a t every R W -manifold is [0, 1)-stable (this follows from Theorems 2.3.9 and 2.3.22) we obtain the following h o m o t o p y classification of R W -manifolds. THEOREM 2.3.27. H o m o t o p y e q u i v a l e n t R W - m a n i f o l d s are h o m e o m o r p h i c . We conclude this section with triangulation t h e o r e m s for I ~ - and R ~ -manifolds. THEOREM 2.3.28. F o r each I " ~ - m a n i f o l d X there exists a locally c o m p a c t polyhedron K such that X ~ K x I ~ .

THEOREM 2.3.29. F o r each R "~ - m a n i f o l d X

there exists a locally c o m p a c t

polyhedron K such that X ~ K • R W .

Applying T h e o r e m s 2.3.21 and 2.3.28, we obtain the following result (compare with T h e o r e m 2.1.20).

2.3. R"~- AND I "~-MANIFOLDS

71

COROLLARY 2.3.30. Each metrizable A N R - c o m p a c t u m has the h o m o t o p y type of a compact polyhedron.

Finally we mention relative versions of Theorems 2.3.28 and 2.3.29. THEOREM 2.3.31. Let the I W -manifold X o be a Z - s e t in the I W - m a n i f o l d X. Suppose that ho: Xo ---+ Ko • I W is a triangulation of X o (i.e. ho is a h o m e o m o r p h i s m and Ko is a locally compact polyhedron). Then there exists a triangulation h: X ---+ K • I W such that Ko is a subpolyhedron of K and h extends ho.

X.

THEOREM 2.3.32. Let the R W - m a n i f o l d X o be a Z - s e t in the R W - m a n i f o l d Suppose that ho: Xo ---+ Ko • R W is a triangulation of X o (i.e. ho is a

h o m e o m o r p h i s m and Ko is a locally compact polyhedron). triangulation h: X

Then there exists a

---+ K x R W such that Ko is a subpolyhedron of K

and h

extends ho.

Historical and bibliographical notes 2.3. The Stability Theorems 2.3.9 and 2.3.10 for R W- and I W -manifolds were obtained by Anderson and Schori [25]. Lemma 2.3.4 was proved by West in [311]. The strong B~,c~-universality of I Wmanifolds was observed in [298]. For locally compact A N R-spaces, the property of strong B~,c~-universality is equivalent to the existence of the so-called Disj o i n t n-Disks Property ( D D n P ) for every n E w. The last property can be obtained from Definition 2.3.12 by replacing in it the space Y by the disjoint sum (0, 1~ • I n of two n-dimensional cubes (n-disks). The Disjoint n-Disks Property plays a very important role in modern geometric topology. This property was formally introduced by Cannon [60], [61], although it was used earlier in [37]. The strong ~4w,~-universality of R W-manifolds was observed by Torudczyk [299]. Theorems 2.3.17 and 2.3.20 were proved in [24] and Theorems 2.3.18 and 2.3.19 in [23] (see also [69]). Theorems 2.3.21 and 2.3.22 are due to Edwards [139] and Torudczyk [296] respectively. Theorems 2.3.24, 2.3.27 and 2.3.29 were proved by Henderson [165], [166]. Theorems 2.3.25, 2.3.26 and 2.3.28 are due to Chapman [65], [66]. The complete classification of/W-manifolds in terms of (infinite) simple homotopy types (see Whitehead [317]) was obtained by Chapman [68] (see also [283]). Corollary 2.3.30 was obtained by West [314]. It solved the long standing problem of Borsuk concerning the finiteness of homotopy types of AN R-compacta.

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2. INFINITE-DIMENSIONAL MANIFOLDS 2.4. T o p o l o g y of R ~ - a n d I ~ - m a n i f o l d s

2.4.1. T o p o l o g y of R ~ - m a n i f o l d s . In the previous section we have seen that each RW-manifold is strongly A~,~-universal. Our goal in this Subsection is to outline the proof of the converse statement, which in fact gives us a topological characterization of R ~ -manifolds. THEOREM 2.4.1. Let X be a Polish A N R - s p a c e . Then the following conditions are equivalent: (i) Z is homeomorphic to R ~ (respectively, X is an R ~ -manifold). (ii) X is strongly ~4~,oo-universal. The strategy of the proof of this fundamental result consists of two major steps. The first proves the existence of a "good" resolution for a given Polish A N R-space X . This means that there exists a fine homotopy equivalence f : M ~ X, where M is an R "~ -manifold. Of course, formally, such a resolution can be obtained by applying Theorem 2.3.22. Indeed, by this theorem, the product X x R ~ is a R ~-manifold and the projection ~rx: X x R ~ ~ X, b e i n g a soft map is, by Proposition 2.1.27, a fine homotopy equivalence. Since the proof of Theorem 2.3.22 was not presented, we refer the reader to [231] and [139] (see also discussion in [34]). The second step of this strategy proves the following fact: if M --. X is a fine homotopy equivalence, M is an R ~ -manifold and X is a strongly jr,,n-universal Polish A N R - s p a c e , then f is a near-homeomorphism. Obviously these two steps complete the proof of Theorem 2.4.1. First let us introduce some additional definitions. For a map f : X ~ Y between Polish spaces, a point y E Y is called a nondegenerate value of f provided, for a complete metric d on X, there is an e > 0 such that the d-diameter of f - l ( U ) is greater than e for each neighborhood U of y. It is easily seen that different choices of complete metrics for X yield the same non degenerate values. Indeed, if f ( X ) is dense in Y, a purely topological (in other words metric independent) description of a nondegenerate value y E Y is that either 9 f-l(y)

= 0,

9 f - 1 contains at least two points, or

9 f-l(y)_ {x} but f - l ( B ) is not a neighborhood basis for x, where B is a neighborhood basis for y. The set of all non degenerate values of f is denoted by N / . A straightforward verification shows that N / i s an Fa-subset of Y. Observe also that the restriction of f is a homeomorphism from f - 1 ( y _ N / ) to Y - N / . Of course, if f is a proper map, then N / {y E Y : I f - l ( y ) l > 1}. In general, fine homotopy equivalences are not surjective, though they have dense images, and points not in the image are necessarily non degenerate values. We begin with the following lemma.

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73

LEMMA 2.4.2. Let X be a strongly .Aw,oo-universal Polish A N R-space. Then the space C (I W , X ) contains a dense set consisting of Z-embeddings with pairwise disjoint images. PROOF. Let d be a c o m p l e t e m e t r i c on X , and let /dk E c o y ( X ) d e n o t e an open cover of X whose e l e m e n t s have d i a m e t e r less t h a n ~, k E N . Let {an" n E N } be a c o u n t a b l e dense subset of t h e space C ( I W , X ) . Consider the map a" g x I W ~ X defined by letting a (n, a) = an(a) for each (n,a) e g x I W. By L e m m a 2.3.16, each c o m p a c t subset of X is a s t r o n g Z - s e t in X . T h e r e f o r e there exists a m a p #1" N x I W --~ X such t h a t beta l is /dl-close to a a n d # l ( g x I W ) M a ( N x I W ) = O. If m a p s #i" N x I W --~ X , each/d~-close to a a n d satisfying #i(N

x I ~ ) M (a(N

x I ~ ) U Uj
x I W )) -- O,

have a l r e a d y been c o n s t r u c t e d for each i w i t h i _ k, t h e n t h e s t r o n g Aw,oouniversality of X implies the existence of a Z - e m b e d d i n g #k+l" N x I ~ --~ X , /dk+l-close to c~ such t h a t # k + l ( N x I ~' ) M (c~(N x I W ) U Uj<_k/3j(N x I W )) -- O. It only r e m a i n s to note t h a t { f l ~ - n , k e N } , where fl~V(a) = # k ( n , a ) , is t h e desired c o u n t a b l e dense subset of C ( I W , X ) consisting of Z - e m b e d d i n g s . V1 PROPOSITION 2.4.3. I f a Polish A N R - s p a c e X is expressed as the union X = M U A of an RW-manifold M and a strong Z - s e t A, then the inclusion i" M r X is a near-homeomorphism. PROOF. Let /d E c o y ( X ) be an o p e n cover of X a n d 1~ C c o y ( M ) an o p e n cover o f M . S i n c e A is a s t r o n g Z - s e t i n X , t h e r e is a m a p g : X --+ M t h a t is /d-homotopic to i d x . Take open covers/do E c o y ( X ) and 120 E c o v ( M ) such t h a t /d0 refines b o t h / d and g-1(12) and 1~0 refines b o t h 12 a n d / d o / M = {U M M : U E /do}. Also consider a s t a r - r e f i n e m e n t ~21 E c o y ( M ) of ]20 a n d a h o m e o m o r p h i s m H : M x I ~ M t h a t is Vl-close to t h e p r o j e c t i o n 7r: M x I ~ M (see C o r o l l a r y C:1.4.9). Note t h a t any m a p of t h e form H h H - I :

M --~ M , where h: M x I --+

M x I preserves the first coordinate, is l~0-close to idM. In order to prove our s t a t e m e n t we need, by T h e o r e m 2.1.8, a s h r i n k i n g h o m e o m o r p h i s m h: M ~ M . We c o n s t r u c t such a h o m e o m o r p h i s m as a c o m p o s i t i o n h -- h2h3h-~ 1 of h o m e o m o r p h i s m s of M .

Construction of hi. Let /dl C c o y ( X ) be an open cover of X which star refines/do. Take a n / d l - h o m o t o p y a : X x I ~ X such t h a t c l x ( a l ( X ) ) M A = 0. Choose a Z - e m b e d d i n g e: H ( M x {0}) ~ M such t h a t e i s / d l / M - h o m o t o p i c to a l / H ( M x {0}) and such t h a t t h e image of e is disjoint from A (observe t h a t this image is closed in M ) . Since e i s / d 0 / M - h o m o t o p i c to t h e inclusion H ( M ) x {0}) ~ M , an a p p l i c a t i o n of T h e o r e m 2.3.20 gives a h o m e o m o r p h i s m h i : M -+ M e x t e n d i n g e t h a t i s / d 0 / M - h o m o t o p i c to idM.

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2. INFINITE-DIMENSIONAL MANIFOLDS

Construction of h2. Choose a Z - e m b e d d i n g e0" H ( M x {1}) ~ M which is l)0-close to the restriction g i h l / S ( i x {1}), and extend it to a homeomorphism h2" M --, M . Since the composition gihl is st(/go/M,/g/M)-homotopic to the identity idM, we may assume t h a t h2 is st2(/g/M)-closed to idM. Observe t h a t for each a E A, if W is a neighborhood of a in X t h a t is contained in an element of/go and lr" M x I --~ M x { 1} is the projection, then h 2 U ~ S - l h - ~ l ( w - A) is contained in an element of std())). In order to verify this, notice that, as h2 is V0-close to gihl on H ( M x {1}), it suffices to show t h a t g i h l H r H - l h ~ l ( W - A ) is contained in an element of st3(l)). Since/go refines g - l ( V ) , it suffices to show t h a t i h l H r S - l h ~ l ( w A)is contained in an element of st3(/go). T h e last containment is clear since W - A is contained in an element of/go, hi is/go/M-close to idM, 1)o refines/go/M and Hlr H - 1 is Vo-close to idM. Construction of h3. Since A and h l ( H ( i x {0})) are disjoint closed subsets of X , A has a closed neighborhood N disjoint from h l ( H ( i x {0})). Consequently, H - I h ~ ( N - A) is a closed subset of M x I and is contained in i • (0, 1]. Let ho" M x I --, M x I be a h o m e o m o r p h i s m t h a t preserves the first coordinates and "pushes" H-lh-~I(N - A ) near i x {1}. For an appropriately chosen ho, the h o m e o m o r p h i s m h3 = HhoH -1 is "close enough" to H~rH -1 on neighborhoods h ~ l ( W - A ) (where W C_ N) as in the above observation t h a t h 2 h 3 h 1 1 ( W - A ) is contained in an element of std(l)). This finishes the verification of the conditions of T h e o r e m 2.1.8 and completes the proof of the proposition. [] PROPOSITION 2.4.4. If f" M -* X is a fine homotopy equivalence from the R ~ -manifold into the Polish ANR-space X , and c l ( N l ) is a strong Z-set in X , then f is a near-homeomorphism. PROOF. T h e idea of the proof is to approximate f by a fine homotopy equivalence g: M --. X such t h a t N g is contained in cl(Nf) and g-i(cl(Ng)) is a (possibly empty) Z-set. Since M is an R ~ -manifold, we conclude, by Proposition 2.3.14 and L e m m a 2.3.15, t h a t g-l(cl(Ng)) is a strong Z-set in i . Applying Proposition 2.4.3, we see t h a t b o t h inclusions i: g-~(X - c l ( N g ) ) ~ M and j : (X - c l ( N g ) ) ~-~ X are near-homeomorphisms. Therefore, g is approximable by a h o m e o m o r p h i s m of the form h2goh~ 1, where h l approximates i, h2 approximates j and go -- g / g - l ( X cl(Ng)). The fine h o m o t o p y equivalence g a p p r o x i m a t i n g f is the limit of a sequence fo, f l , f2,. 99 of fine homotopy equivalences constructed inductively. First, applying L e m m a 2.4.2, we take a countable dense subset {en : n E N } C_ C(I W , M ) consisting of Z - e m b e d d i n g s with pairwise disjoint images. Let f0 - f. The map f 0 e l : I ~ -* X is homotopic, by a "small" homotopy, to a Z - e m b e d d i n g j l : I ~ --, X - c l ( N l o ). Using an approximate lift of the h o m o t o p y as a guide, T h e o r e m 2.3.20 produces a homeomorphism h i : M ~ M , fixed outside a small neighborhood of f o l f o ( e l ( I ~ )), such t h a t hie1 -- f o l j l and fohl is "sufficiently close" to f0. Let fl - fohl. Observe t h a t f l is one-to-one over the set el(I ~ ). This means t h a t f ~ l f l ( m ) - m for each

2.4. TOPOLOGY OF R ~~ AND I w-MANIFOLDS

75

m E e l ( I ~ ). Continuing this process we obtain a sequence f0, f l , f 2 , . . , such t h a t f i + l - fihi+l, where hi+l" M --+ M is a h o m e o m o r p h i s m fixed outside a small neighborhood of f~-lfi(ei+l(I"~ )) missing Uik=lek(I ~~). Further, letting N i + l denote the non degeneracy set of f i + l , we want hi+lei+l -- f:~lji+l, where j i + l " I ~ ~ X - c l ( N i ) is a Z - e m b e d d i n g a p p r o x i m a t i n g fiei+l, and we also want fihi+l to be "close" to fi. Note t h a t uk=~f~+~ i+1. (ek(ioa )) c_ X - cl(Ni+l ). Since Ni = N$, we can verify directly t h a t if fi+l was chosen sufficiently close to fi, then the limit map g = lim{fi} is a fine h o m o t o p y equivalence a p p r o x i m a t i n g f and satisfying the following two conditions Ng C_ e l ( N / ) and g(ei(I ~ )) C_ X - e l ( N : )

for each i.

It only remains to remark t h a t g-l(cl(Ng)) is a a - s e t in M .

E:]

PROPOSITION 2.4.5. Let f" M --+ X be a fine homotopy equivalence from an R ~ -manifold to a strongly Jtw,oo-universal Polish ANR-space. If N f and f - l ( N f ) are Za-sets in X and M respectively, then f is a near-homeomorphism. PROOF. Let /g E coy(X) be an open cover of X and 1) E coy(M) an open cover of M. Take a map g" X --+ M such t h a t g f is f - l ( / g ) - h o m o t o p i c to idM. Take open covers /go E coy(X) and 1)0 E coy(M) such t h a t /go refines b o t h / g and g-1(1)) and 1)0 refines b o t h 1) and f - l ( / g 0 ) . Take a star-refinement 1)1 E coy(M) of 1)0 and a h o m e o m o r p h i s m H" M x I ---+ M t h a t is 1)l-close to the projection ~r" M x I --+ M (see Corollary C:1.4.9). Note t h a t any m a p of the form H hH -1" M ---, M, where h" M x I ---+ M x I preserves the first coordinate, is 1)0-close to idM. As in the proof of Proposition 2.4.3, the shrinking h o m e o m o r p h i s m will be constructed as the composition h -- h2h3h-~ 1 of h o m e o m o r p h i s m s of M . W i t h minor changes h o m e o m o r p h i s m s h l and h2 can be c o n s t r u c t e d as the corresponding h o m e o m o r p h i s m s in the proof of Proposition 2.4.3. Construction of h 3. This h o m e o m o r p h i s m is constructed as the composition h3 -- HhoH -1, where h0 preserves the first coordinates. Since M x {0} misses the inverse image under (hlH) -1 of an f - s a t u r a t e d neighborhood 2 of f - l ( g f ) , for any given point a E N f , we could choose a neighborhood Wa of a contained in an element of/go with c l ( f - l ( W a ) ) M h l H ( M x {0}) -- q} and specify t h a t h0 "push" ( h l H ) - l ( f - l ( W a ) ) so near M x {1} t h a t h2HhoH-lh-~ 1 "shrinks" f - l ( W a ) . T h e problem is that, presumably, any inverse image u n d e r ( h l H ) -~ of an f - s a t u r a t e d neighborhood of f - l ( N f ) is dense, thereby making it impossible to simultaneously do this for every point of N f . This situation can be rectified by observing t h a t for any point a E N I sufficiently close to f h l H ( M x {0}), the inverse image f - l ( a ) has an f - s a t u r a t e d n e i g h b o r h o o d having d i a m e t e r close to 2By an f-saturated neighborhood of a set f - l ( A ) , for A C_X, we mean a neighborhood of the form f-1 (N) for some open neighborhood N of A.

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2. INFINITE-DIMENSIONAL MANIFOLDS

zero. We now proceed with the construction of h0. As above, h0 is the identity on M • {0} and moves only the first coordinates. For a pair of maps e,5" M ~ [0, 1] with e(m) < 5(m) for each m e M , let

r(~, ~ ) = {(re, t). ~(m) < t < ~(~)) and let pc,6" M x I --~ F(e, 5) be the retraction sending {m} x [0, e(m)] to (m,e(m)) and {m} x [5(m), 1] to (m,$(m)). Take a m a p 6" M ---. (0, 1) such t h a t (i) each a E N / has a n e i g h b o r h o o d Wa such t h a t

h2 g Trl_a,l g - 1 hi l (f-1 (Wa)) is st4(y)-small, and (ii) for each (re,t) e M x I, h2g(B2a(m)(m) x B26(m)(t)) is st4(]2)-small. Next we inductively specify maps e(i)" M ~ (0,1], i -- 0 , 1 , 2 , . . . , with e(O)(m) - 1 and e(i)(m) > e(i + 1)(m) for each m e M so t h a t every point a E N I has a neighborhood Wa such t h a t either

(iii)l H-lh~l(.f-l(Wa)) C_Fc(i+2),~(i) from some i >_ 0 and 7rH-lh~l(f-l(wa)) is contained in B~(m) for each m E M , or (iii)2 H - l h T l ( f - l ( W a ) ) C_ r~(1),~(0 ). Choosing e(1). Consider the set Z0 consisting of those points a E X t h a t do not have a n e i g h b o r h o o d Wa such t h a t 7rH-lhll(f-l(Wa)) is contained in B~(m)(m) for some m E M. This set is closed in X and is contained in N / . Thus, there is a closed f - s a t u r a t e d neighborhood No of f-l(Zo) disjoint from h l H ( U x {0}). Choose e(1) so t h a t F(0, el) is disjoint from H-lh~l(intNo) 3. Observe t h a t points in Z0 have neighborhoods satisfying (iii)2. Choosing e(i + 1). Consider the set Zi consisting of those points a E X t h a t do not have a neighborhood Wa such that H - l h ~ l ( f - l ( W a ) ) C_F(0, e(i)). This set is closed in X, and f - l ( Z i ) is disjoint from hlH(U x {0}). Choose any neighborhood N of h l H ( U x {0}) missing an f - s a t u r a t e d neighborhood of f - l ( z i ) , and any e(i + 1) with e(i + 1)(m) < ~(i)(m) for each m E U and with F(0, e(i + 1)) C_ H-lh~l(intN). Making the further restriction t h a t e(i)(m) < ~, i = 1 , 2 , . . . , we specify a h o m e o m o r p h i s m h0" M • I --~ M • I t h a t is the identity on M • {0, 1} and sends the graph of e(i) onto the graph of ( 1 - 5) ~, i = 1 , 2 , . . . 4. Setting ha -- hoH -1 and h = h2h3h~ 1, we claim t h a t h is the desired shrinking h o m e o m o r p h i s m . For, if a E N / and (iii)l is satisfied, then

hoU-lhll(.f-l(wa)) C

r((1

3here "0" denotes the constant map M --~ (0}. 4(1 - 5)i(m) = (1 - 5(m)) i.

-

~)~+2,(1

-

-

5) i)

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

77

and so h o H - 1 h l l ( f - 1 ( W a ) ) is contained in B25(m)(m) • B2~(m)(t) for some (re.t) E i • I and condition (ii) implies that h ( f - l ( W a ) ) is sta02)-small. If a E N / and (iii)2 is satisfied, then

hoH-lh~l(f-l(Wa))

C_ F(1 - 5, 1)

and condition (i) implies that h ( f - l ( W a ) ) is st4(lZ)-small.

D

PROPOSITION 2.4.6. If f : M ~ X is a fine homotopy equivalence from an R W -manifold to a strongly Jtw,cc-universal Polish A N R - s p a c e and N / is a Zaset in X , then f is a near-homeomorphism. PROOF. The proof parallels closely that of Proposition 2.4.4. An approximating fine homotopy equivalence g is produced using Proposition 2.4.5 in place of Proposition 2.4.3. [-1 PROPOSITION 2.4.7. A fine homotopy equivalence f : M ---+ X from an R Wmanifold to a strongly .Aw,cc-universal Polish A N R-space is a near-homeomorphism. PROOF. By lemma 2.4.2, there exists a countable dense subset {en : n E N } C_ C ( I W , X ) , consisting of Z-embeddings with pairwise disjoint images. The map f : M --, X factors through the adjunction space as f -- plq, where the maps q: M --* M U / e l ( I w ) and P l : M U f e I ( I w ) --~ X are described in Subsection 2.3.1 on page 61. Obviously, Nq C_ e l ( I w ) and e l ( I W ) is a strong Z-set in M U I e l ( I W ). By Proposition 2.4.4, the map q, being clearly a fine homotopy equivalence, is a near-homeomorphism. Take a homeomorphism h approximating q and let f l = plh: M -~ X . Observe t h a t f l approximates f and is one-to-one over the set e l ( I W). Continuing this process, we obtain a sequence f = f0, f l , f 2 , . . , consisting of fine homotopy equivalences (from M to X) such t h a t f n + l a p p r o x i m a t e s fn and is one-to one over the set un+l k=lek(I W ). If f,~+l is sufficiently close to fn (for each n E N), then the map g - lim fn is also a fine homotopy equivalence approximating f. It is not hard oo 1e n(I W ). This fact, coupled with to see that g is o n e - t o - o n e over the set Un= the choice of the collection {en}, guarantees that Ng is a Zo-set in X. It only remains to apply Proposition 2.4.6. E] We are now ready to prove Theorem 2.4.1. Proof of Theorem 2.4.1. One part of the s t a t e m e n t follows from Proposition 2.3.14. Consider now a strongly jtw,~-universal Polish A N R - s p a c e X . By Theorem 2.3.22, there exists a fine homotopy equivalence f : M --, X where M is an R ~ -manifold. Proposition 2.4.7 implies t h a t f is a near-homeomorphism and, consequently, t h a t X is an R~-manifold. If, in addition, X is an absolute retract, then M is contractible (recall that f is a fine homotopy equivalence). It only remains to note t h a t in this case, according to Theorem 2.3.27, M (as well as X ) is homeomorphic to R ~ . This completes the proof of Theorem 2.4.1.

78

2. INFINITE-DIMENSIONAL MANIFOLDS Sometimes it is much more convenient to use the following version of T h e o r e m

2.4.1. PROPOSITION 2.4.8. Let X be a Polish A ( N ) R - s p a c e . Then the following conditions are equivalent: (i) Z is homeomorphic to R W (respectively, Z is an R W -manifold). (ii) The set

{ f e C ( N x I ~ , X ) " the collection { f ( { n } x I r176 ) ' n E N } is discrete} is dense in the space C ( N x I ~ , X ) . (iii) The set { f e C ( $ { I n" n e w } , X ) " the collection { f ( I n ) " n e w} is discrete} is dense in the space C ( ~ { I n " n E w } , X ) . In order to prove this s t a t e m e n t we need some preliminary results. The following l e m m a is geometrically obvious. LEMMA 2.4.9. Let K be an n-dimensional countable locally finite simplicial complex and let G be an open neighborhood of IK(n-1)l in IKI. Then the convex halls Co of the sets Iol- a , o ~ K , forms a discrete collection in IKI, PROOF. Assuming the contrary, there is a point x E c l C - C, where C = U{Co" a E K } . T h e n x E I~1 for some T E K (n-l). By an obvious geometrical argument, dist(la I - G , I~1)- dist(Co, I~1) for all a E s t ( r , K ) . Since I~1 c_ a, we have dist(C Cl Ist(r, K ) , I~1) > 0, contrary to the fact t h a t x e el C fq Irl. v1 LEMMA 2.4.10. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8, K be a finite-dimensional countable locally finite simplicial complex, and Lt e cov(IKI) be the cover of IKI by open stars of the vertices of K . Then the set of all U-maps of the polyhedron IKI into the product X x I ~~ is open and dense in the space C ( I K I , X x I ~ ). PROOF. T h a t the set of all U-maps from IKI to x x I ~ is open follows from L e m m a 2.1.1. Let us show t h a t this set is dense. If dim K = 0, then this follows immediately from condition (iii) of Proposition 2.4.8. Suppose that dim K = n and t h a t the l e m m a has already been proved for complexes of dimension < n - 1. Suppose also t h a t a map f" [K[ ~ X • I ~~ and an open cover 1) E cov(X x I ~ ) are given. Our goal is to show the existence of a / 4 - m a p g" [KI --, X x I "~ t h a t is V-close to f. By our inductive assumption and Corollary 2.1.23, we may assume without loss of generality t h a t the restriction f / l g ( n - 1 ) [ is an U-map. By L e m m a 2.1.3, there is an open neighborhood G of [K(n-1)[ in Igl such t h a t the restriction I / G is still a U-map. Denote by )4; E cov(X x I ~ ) a double star-refinement of 1) such t h a t for each map g" [K[--. X x I ~ t h a t is W-close to f , the restriction g i g is a n / , / - m a p (here we use L e m m a 2.1.1 and Corollary 2.1.23). By L e m m a 2.4.9, the

2.4. TOPOLOGY OF R ~- AND I ~~

79

c o m p l e m e n t I K I - G can be covered by a discrete (in IKI)collection {Ca" a e K } such t h a t Co ~ I n and Ca C_ i n t l a l for each a E g . By condition (iii) of Proposition 2.4.8 and Corollary 2.1.23, there exists a m a p go" IKI --+ X x I ~ such t h a t go is W-close to f and the collection {g0(Ca" a E K } is discrete in X x I W . Let {Da" a E K } be a discrete collection of open subsets of IKI such t h a t Ca C_ Da C_ i n t l a l for e a c h a E g . Take a f u n c t i o n ~ o " I g l - - + [0,1] such that ~o(U{Ca" a e g } ) = l

and ~ o ( [ g [ - U { D a "

aeK})=0.

A straightforward verification shows t h a t the diagonal p r o d u c t g I =g0/kgo" IKI---+X x I ~ x [0,1] is a / g - m a p . Since the projection rr" I " x [0, 1] --+ I ~~ is a n e a r - h o m e o m o r p h i s m (Corollary 2.3.11), there exists a h o m e o m o r p h i s m h ' X x I

~

x[O, 1] ---+XxI'"

t h a t is V-close to id x ft. T h e n the desired m a p g can be defined as the composition g - hg ~. F1 LEMMA 2.4.11. For each n E w and for each n - d i m e n s i o n a l Polish space X , there exists an (n + 1)-dimensional Polish absolute retract containing X as a closed subspace. PROOF. Represent X as the limit space of some inverse sequence S x = { X i ,Pi _i+1 } consisting of n-dimensional locally c o m p a c t p o l y h e d r a and surjective limit projections (see [179]). We construct a new inverse sequence s r = {Yi, q~+1} as follows. As the first element II1 of this s p e c t r u m we take c o n ( X 1 ) the cone over t h e p o l y h e d r o n X1. Obviously, Y1 is an (n + 1)-dimensional Polish absolute retract. We identify X1 with its n a t u r a l l y e m b e d d e d copy in Y1. Suppose that for each i, with i _< k, an (n + 1)-dimensional Polish absolute retract Yi, containing Xi as a closed subspace, and a fine h o m o t o p y equivalence qi-1 i . Yi '"+ Yi-1 , extending Pi-1, have already been constructed. As the i space Yk+l we take the m a p p i n g cylinder M a p ( p ~ +1) of the m a p Pkk+l , considered as a m a p from X k + l into Yk (recall t h a t p k + l ( X k + l ) = X k is closed in Yk). Obviously, Yk+l is an (n + 1)-dimensional Polish absolute retract (see [161] for details). T h e closed e m b e d d i n g of the p o l y h e d r o n X k+l into Yk+l and the k " Yk+l --+ Yk are defined canonically. It is also fine h o m o t o p y equivalence qk+l clear t h a t qkk+l(Xk+ 1) -- X k and q lk + lk/ x k + = Pkk+l . Let Y denote the limit space of the inverse sequence S y . It follows from the construction t h a t Y is an (n + 1)-dimensional Polish space containing X as a closed subspace. Since all projections of this s p e c t r u m are fine h o m o t o p y equivalences and since Y1 is a Polish absolute r e t r a c t we easily conclude t h a t Y is also an absolute r e t r a c t (see [112] for details). El

80

2. INFINITE-DIMENSIONAL MANIFOLDS

LEMMA 2.4.12. Let X be a Polish space satisfying condition (iii) of Proposition 2.4.8. Then for each finite-dimensional Polish space Y , the set of closed embeddings of Y into X x I ~ is dense in the space C ( Y , X x I "~). PROOF. By L e m m a s 2.1.1, 2.1.2 and Proposition 2.1.7, it suffices to show t h a t for each open cover IX E c o y ( Y ) , the set of all IX-maps is dense in the space C ( Y , X x I ~ ). By L e m m a 2.4.11 we may additionally assume t h a t Y is a finite-dimensional Polish absolute retract. But then, by T h e o r e m 2.1.19, we may assume, w i t h o u t loss of generality, t h a t Y = IK], where K is a finite-dimensional countable locally finite simplicial complex and IX is the cover of the polyhedron ]K I by open stars of its vertices. In this situation L e m m a 2.4.10 completes the proof, l1 LEMMA 2.4.13. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8, K be a countable locally finite simplicial complex, and IX be is the cover of the polyhedron ]K] by open stars of its vertices. Then the set of maps f : ]K I ---, X x I ~ such that the collection of fibers of the composition 7 r z f , where lrz : X x I"; ---. X is the projection, refines IX and is a dense and G h-subset of the ~par162c ( I g l , x x I ~ ). PROOF. Given integers m and n, consider for each a E K (n) - K (n-l) the image of ]hi under the (1 - G1 ) - h o m o t h e r y with respect to the barycenter of la]. Obviously, the collection .An,m of all such images is discrete in IK]. By condition (iii) of Proposition 2.4.8 and the compactness of I ~ , the set Fn,m = { f E C ( I K I , X

x I ~ ): the collection

~zf(A~.m)

is discrete in X }

is open and dense in C(IKI, X x I "~ ). It only remains to remark that the intersection MFn,m is the desired set. [Z] LEMMA 2.4.14. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8. Then for each Polish space Y , the set of closed embeddings of X into the p r o d u c t X • is dense in the s p a c e C ( Y , X x I "~). PROOF. As in the proof of L e m m a 2.4.12 it suffices to show t h a t if K is a locally c o m p a c t polyhedron and IX E c o y ( K ) , then the set of all/,/-maps is dense in the space C(Y, X x I ~ ). For simplicity we assume t h a t X is a Polish absolute retract (this simplification has a purely technical n a t u r e and does not affect the general argument). Represent K as the union K = U { K n : n E w} of its finite-dimensional s u b c o m p a c t a in such a way t h a t (a) K n C int K n + l , n E w, and (b) T h e collections { K 2 n + l - i n t K2n : n E w} and { K 2 n - i n t K2n-1 : n E N } are discrete in K. Next, take an arbitrary map f : K --, X x I ~ and introduce the notation A = U{K2n+I - int K2r, : n E w} and B = U { K 2 n - int K2n-1 : n E N}. Obviously, the sets A and B are closed in K. By Proposition 2.1.7, Corollary 2.1.23 and L e m m a 2.4.12, we may assume w i t h o u t loss of generality t h a t the restrictions

2.4. TOPOLOGY OF R w- AND I w -MANIFOLDS

81

S / K n are closed e m b e d d i n g s (we use the finite-dimensionality of K n also). Next, identifying each of the "rings" K2n +1 - i n t K2n, n 9 w, with a subspace of a finitedimensional c u b e I rim, we may also assume t h a t the subspace A is identified with a closed s u b s p a c e of the discrete sum @{In'~: m 9 w}. T h e restriction f / A can be e x t e n d e d to a m a p fl:

@ {in,,,: m 9 w} ---+ X x I ~ .

By condition (iii) of P r o p o s i t i o n 2.4.8, we m a y assume w i t h o u t loss of g e n e r a l i t y t h a t the collection {S#(Xnm): m 9 w} is discrete in t h e p r o d u c t X x I ~ . Therefore, the collection

{f#(K2n+l - int K 2 n ) : n e w} is also discrete in X x I W . As above, we can now conclude t h a t the restriction f / A is a closed e m b e d d i n g of A into X x I ~ . A c o m p l e t e l y analogous a r g u m e n t shows t h a t the restriction f / B is also a closed e m b e d d i n g of B into X x I ~ . Since every closed e m b e d d i n g is a U - m a p , there exists an open n e i g h b o r h o o d G of A in K such t h a t the m a p f / G is still a / g - m a p (here we a p p l y L e m m a 2.1.3; observe also that, by L e m m a 2.1.1 and Corollary 2.1.23, the restriction g / G o f a m a p g: K ~ X x I w t h a t is sufficiently close to f is also a / g - m a p ) . In addition, we assume t h a t G n B -- 0. Now take a function ~ : K ---+ [0, 1] such t h a t ~(A) = 0 and ~ ( K - G) = 1. A s t r a i g h t f o r w a r d verification shows t h a t the diagonal p r o d u c t

g'=fA~:K---+X

•

~ x [0,1]

is a H-map. Take a h o m e o m o r p h i s m h: I ~ • [0, 1] ~ I ~ sufficiently close to t h e projection 7r: I ~ x [0, 1] ~ I ~ . T h e n the c o m p o s i t i o n

( i d x x h)g~: K ---+ X x I ~ is a / g - m a p , sufficiently close to f.

F-1

We are now r e a d y to prove P r o p o s i t i o n 2.4.8. Proof of Proposition 2.4.8. T h e implications (i)===~(ii) and (ii)==~(iii) are obvious. Let us prove the validity of the implication ( i i i ) ~ ( i ) . By T h e o r e m 2.4.1, it suffices to show t h a t for each Polish space Y, the set of closed e m b e d d i n g of Y into X is dense in the space C(Y, X ) . As above, in order to show this last fact, it suffices to prove t h a t for each c o u n t a b l e locally finite simplicial c o m p l e x K the set of all U - m a p s of [g[ into X , w h e r e / d is the cover of [g[ by open stars (with respect to the t r i a n g u l a t i o n K ) of its vertices, is dense in the space C([K[, X ) . By L e m m a s 2.4.13, 2.4.14 and P r o p o s i t i o n 2.1.7, the space CIK[, X x I ~ ) contains a dense set F , consisting of closed e m b e d d i n g s such t h a t for each f E F t h e collection of fibers of the c o m p o s i t i o n ~ z f , where 7rx: X x I ~ ~ X d e n o t e s the projection, refines b/. Each such c o m p o s i t i o n is a p r o p e r map. Therefore, as can be seen in the proof of Corollary 2.1.9, each such c o m p o s i t i o n is a U-map. It only

82

2. INFINITE-DIMENSIONAL MANIFOLDS

remains to remark t h a t the set of maps g: IKI ~ X, t h a t can be represented as a composition g = r x f , f E F is dense in the space C ( I K I , X ). The proof of Proposition 2.4.8 is complete.

2.4.2. T o p o l o g y o f I ~ - m a n i f o l d s . The strategy of the proof of the main result of this Subsection (Theorem 2.4.18 and Proposition 2.4.19) is the same as that of Subsection 2.4.1. As a very useful (and sufficiently difficult, but manageable) exercise we recommend that the reader supply the omitted pieces of the corresponding proofs. PROPOSITION 2.4.15. Let f : M ~ X be a proper C E - m a p of an I ~-manifold onto a strongly Bw,oo-universal locally compact A N R - s p a c e X . If the set c l N i is a Z - s e t in X , then f is a near-homeomorphism. PROPOSITION 2.4.16. Let f : M --~ X be a proper C E - m a p of an I ~ - m a n i f o l d onto a strongly Bw,oo-universal locally compact A N R-space X . If the set N I is a Za-se$ in X , then f is a near-homeomorphism. As in Subsection 2.4.1 these two propositions imply the following statement. PROPOSITION 2.4.17. A proper C E - m a p of an I W -manifold onto a strongly Bw,c~-universal locally compact A N R-space is a near-homeomorphism. As in the case of R "~ -manifolds, the problem of the existence of a resolution for a given locally compact A N R - s p a c e X (i.e. the existence of a proper C E map f : M ~ X, where M is an I W-manifold) arises here as well. The existence of such a resolution follows, for example, from Theorem 2.3.21. Alternatively we can apply Miller's approach (see, for instance, [231] or [139]). Therefore, Proposition 2.4.17 implies the following characterization theorem. THEOREM 2.4.18. The following conditions are equivalent for each (locally) compact A ( N ) R - s p a c e X : (i) z i~ hom~omo~p~i~ to I ~ ( ~ p ~ t i ~ t y , x i~ ~ ~ - m ~ n i / o l d ) . (ii) X is strongly B~,oo-universal. Sometimes it is more convenient to use the following version of Theorem 2.4.18. PROPOSITION 2.4.19. The following conditions are equivalent for each locally compact A N R-space X : (i) X is an I ~ -manifold. (ii) The set {f EC(D •

~,x):

y({0} x I ~ ) A f ( { 1 }

is dense in the space C ( D • I "~ , X ) . 5Recall that D = {0,1) denotes the two-point discrete space

xI W)=0} 5

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

83

(iii) For each n E w, the set

{f e

C ( D x In, X ) : f({0} x I n ) f3 f({1} • I n ) = O}

is dense in the space C ( D x I n, X ) . PROOF. The implications (i)==>(ii) and (ii)==~(iii) are obvious. Let us prove the validity of the implication ( i i i ) ~ ( i ) . According to Theorem 2.4.18 we only have to show the strong Bw,oo-universality of X. First of all let us establish the following fact: 9 For metrizable compacta T1 and T2 the set

{f c C(T1 9 T2, X):.Y(T1)n :(T~) = O} is dense in the space C (T1 (9 T2, X). Indeed, by Theorem 2.1.19, we may assume without loss of generality that T1 -]Ki], where Ki is a finite simplicial complex, i - 1, 2. For each pair of simplexes a l E K1 and a2 E g 2 , let

C(~1, ~2) : {f e C(IKll 9 IK21,X): f ( l ~ l l ) n f(l~2l) - 0}. By condition (iii) and Corollary 2.1.23, the set C ( a l , a2) is dense (and obviously open) in the space C ( I K l l (9 I g 2 ] , X ) . Consequently, according to Proposition 2.1.7, we conclude that the intersection N { C ( a l , a 2 ) : al C K l , a 2 C K2} is also a dense subset of the space C(IK~J 9 JK21,X). It only remains to observe that for each map f from this intersection, we obviously have f ( I g l ] ) N . f ( ] g 2 1 ) =

0. We proceed now to the direct proof of the strong B~,oo-universality of the space X. Consider a metrizable c o m p a c t u m Y and for each pair of disjoint s u b c o m p a c t a T1 and T2 of Y denote by C (T1, T2) the set

{S e c ( Y , X ) : S(T~)n S(T2) = 0}. By the fact established above, the set C(T1, T2) is dense (and obviously open) in the space C (Y, X). Take now a countable open basis 1) for the space Y. T h e n for each pair (V1,1:2) of elements of V, with cl V1 gl cl 1:2 - 0, the set C (cl V1, cl 1/2) is an open and dense subset of C(Y, X ) . Consequently, by Proposition 2.1.7, the intersection of all sets of the form C (cl V~, cl 1/2) is dense in the space C (Y, X). It only remains to observe that each map from this intersection embeds the c o m p a c t u m Y into the space X. I-1

84

2. INFINITE-DIMENSIONAL MANIFOLDS

2.4.3. A p p l i c a t i o n s o f c h a r a c t e r i z a t i o n t h e o r e m s . Theorems 2.4.1 and 2.4.18 (as well as their a l t e r n a t e versions Propositions 2.4.8 and 2.4.19) can be successfully used in m a n y different situations. In this Subsection we present some of them. PROPOSITION 2.4.20. Let f be a map between R W -manifolds. lowing conditions are equivalent: (i) f is approximately soft. (ii) f is a fine h o m o t o p y equivalence. (iii) f is a near-homeomorphism.

Then the fol-

PROOF. T h e implications (i)==~(ii) and ( i i ) - ~ ( i i i ) follow from Propositions 2.1.27 and 2.1.28 respectively. The implication ( i i i ) ~ ( i ) follows from Propositions 2.3.14 and 2.4.7. D PROPOSITION 2.4.21. Let f be a proper map between I ~ -manifolds. Then the following conditions are equivalent: (i) f is approximately soft. (ii) f is a fine h o m o t o p y equivalence. (iii) f is a C E - m a p . (iv) f is a near-homeomorphism. Corollary 2.2.35 states t h a t if A is a Za-set in an R ~ -manifold X, then the complement X - A is also an RW-manifold. Moreover, since the inclusion X - A r X is a (fine) h o m o t o p y equivalence, we can, by virtue of T h e o r e m 2.3.27, conclude t h a t the R"~-manifolds X - A and X are homeomorphic. Proposition 2.4.7 implies an even stronger result. PROPOSITION 2.4.22. Let A be a Z a - s e t in an R w -manifold X . inclusion X - A ~ X is a n e a r - h o m e o m o r p h i s m .

Then the

PROPOSITION 2.4.23. A countable infinite product of non-compact Polish absolute retracts is homeomorphic to R "~ . PROOF. Let X = 1-ln~__lXn, where each X n is a non-compact Polish absolute retract. Obviously, X itself is a Polish absolute retract. Consequently, by Theorem 2.4.1, it suffices to show the strong .A~,~-universality of X. It follows from the proof of Proposition 2.3.14 t h a t we will be done if each of the spaces X n , n C N, contains a closed copy of R "J . Represent the set N of natural numbers as a countable infinite disjoint union of infinite subsets Nk, k E N . T h e n oo

X---- I " [ Y k '

where Y k - - N ( X k ' n E N k } , k E N .

k--1

Therefore it only remains to show t h a t each of the spaces Yk, k E N , contains a closed copy of R ~ . This last condition will be satisfied if we show t h a t the

2.4. TOPOLOGY OF R ~

AND I w-MANIFOLDS

85

real line R admits a closed e m b e d d i n g into the p r o d u c t A x B of any two nonc o m p a c t Polish absolute retracts. Take c o u n t a b l e infinite and discrete subsets {an: n E Z} and {bn: n E Z} of the spaces A and B respectively (use nonc o m p a c t n e s s of these spaces). Let [an, an+I] denote the copy of the closed unit s e g m e n t in A with end points coinciding with a n and an+l. T h e same m e a n i n g shall be given to the symbol [bn, bn+l], n E Z. Let cn = (an, bn) E A x B and Cn+l = (an+I, bn+l) E A x B. Obviously the segment

[Cn, Cn+l] = ([an, an+l] X {bn}) U ({an+l x [bn, bn+l]) connects the points cn and Cn+l in A x B. Moreover, different s e g m e n t s of the form [cn, cn+l] are either disjoint or have exactly one c o m m o n point. It only r e m a i n s to observe t h a t the union U{[cn, Cn+l]: n E Z} is closed in A x B and is h o m e o m o r p h i c to the real line R. D PROPOSITION 2.4.24. A proper retract of an R ~

is an R ~

PROOF. Let r" M ~ X be a p r o p e r r e t r a c t i o n of an R ~ -manifold M into a s u b s p a c e X . First of all we show t h a t each c o m p a c t subset K C X is a Z - s e t i n X . Let 3 / E c o y ( X ) be an open cover of X and suppose a m a p f " I ~ ~ X is given. Define a m a p g" g x I ~ ~ M by letting g ( n , a ) - f ( a ) for each ( n , a ) E N x I ~ . T h e strong Jtoj,oo-universality of the R ~ -manifold M guarantees the existence of a map h" N x I ~ ~ M which is r - l ( U ) - c l o s e to g and such t h a t the collection {h({n} x I ~ )" n E N } is discrete in M . Consider t h e collection { r h ( { n } x I ~ E N}. P r o p e r n e s s of the r e t r a c t i o n r implies t h a t this collection is locally finite in X. Consequently, for each point x E K t h e r e is an open n e i g h b o r h o o d G x of x in X intersecting only finitely m a n y sets of the form r h ( { n } x I ~ ). Selecting a finite subcover of the cover {Gx" x E K } of K , we obtain an o p e n n e i g h b o r h o o d G of the c o m p a c t u m K in X which intersects only finitely m a n y sets having t h e indicated form. T h e r e f o r e t h e r e exists an index n E N such t h a t r h ( { n } x I ~ ) M K -- 0. Obviously, t h e m a p f ' " I ~~ -+ X, defined by letting f ' ( a ) = r h ( n , a) for each a E I ~~, is U-close to f a n d f ' ( I "~ ) M K = 0. Thus, every c o m p a c t subset of X is a Z-set in X. This fact, coupled with P r o p o s i t i o n 2.1.7 and Corollary 2.1.23, shows t h a t t h e set

Al={f

E C(N xI',X)"

f({n} xI')Mf({m}

xI ~)=0;

n#m}

is a dense G~-subset of the space C ( N x I ~ , X ) . On the o t h e r hand, the set

A2 = { f E C ( N x I ~ , X ) "

the collection { f ( { n } x I ~ ) ' n

E N } is locally finite}

is open and dense in the space C ( N x I ~ , X ) . By P r o p o s i t i o n 2.1.7, the intersection A1 M A2 is also dense in the space C ( N x I ~ , X ) . Finally, observe t h a t if f E A1MA2, t h e n the collection { f ( { n } x I ~ ): n E N } is discrete in X. P r o p o s i t i o n 2.4.8 completes the proof. [:]

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2. INFINITE-DIMENSIONAL MANIFOLDS

The following result (compare with Proposition 2.4.23) follows directly from Proposition 2.4.19. PROPOSITION 2.4.25. A countable infinite product of n o n - t r i v i a ~ compact absolute retracts is h o m e o m o r p h i c to I "~ .

Let X be a Polish space. As usual, by e x p X we denote the hyperspace of all non-empty compact subsets of X endowed with the Hausdorff metric (or, equivalently, having the Vietoris topology; see, for example, [145]). The correspondence X ~ e x p X can be naturally extended to maps, i.e. for each map f : X - , Y we can assign a map exp f : exp X --, exp Y. Moreover, it is easy to see that this correspondence is functorial. Obviously the functor exp preserves the class of compacta. The following statement shows that exp significantly improves the properties of spaces. PROPOSITION 2.4.26. I f X is a P e a n o c o n t i n u u m , exp X is h o m e o m o r p h i c to I "~ .

i.e.

X

9 AE(1),

then

PROOF. Take a convex m e t r i c p on X (see [361). This means that each pair of points in X is contained in a subspace isometric (with respect to metric p) to a closed segment on the real line. Consider the following metric on exp X: d ( K 1 , K2) = max{p(x, Ki): i = 1,2, x 9 K1 U K2}.

By the result of Wojdyslawski [321], exp X is an AR-compactum. For each e > 0, the map fe defined by letting re(K) = {x 9 X : p ( x , K )

<_e} K 9

is e-close to the identity map i d e x p X . It is not hard to see that the image fe(exp X) is a Z-set in exp Z (see [116] for details). Proposition 2.4.19 completes the proof. [3 The following is a non-compact counter-part of the previous result. PROPOSITION 2.4.27. Let X be a nowhere locally compact Polish A E ( 1 ) - s p a c e . T h e n exp X is h o m e o m o r p h i c to R ~ .

Throughout the remainder of this Section, X stands for a separable Frgchet space. It is well known (see, for example, [262, Theorem 1.24]) that every Frgchet

space admits an i n v a r i a n t m e t r i c d such that all maps of the half-line (0, c~) into itself of the form t ~ d ( t x , 0), x E X , are non-decreasing. For simplicity we use the notation IIxi] = d(x, 0), x E X (warning: ]I" II is not a n o r m ) . The proofs of the following statements are left to the reader. LEMMA 2.4.28. Let e > 0 and let f : I n ~ X be a map.

T h e n there exists a

m a p g: I n ---, X such that g ( I n) is contained in the convex hall of a finite subset of f ( I n) and I I f ( q ) - g(q)ll < e f o r each q E I n .

6i.e. containing at least two points

2.4. TOPOLOGY OF R ~- AND I "~-MANIFOLDS

87

LEMMA 2.4.29. Let E be a proper closed linear subspace of X and 0 < a ( b < s u p ( i i x - e i l : ( x , e ) E X • E } . Then there exists a map u: [a,b] --, X such that the image u([a, b]) is contained in a finite-dimensional linear subspace of X and the following conditions are also satisfied: IIu(t)I I < 5 t

t and I i u ( t ) - e l I > ~ for each t E [a,b] and e E E.

PROPOSITION 2.4.30. Each infinite-dimensional separable Frgchet space is hom e o m o r p h i c to R ~ . PROOF. By T h e o r e m 2.1.17, X is a Polish absolute retract. Therefore, it suffices to verify condition (iii) of Proposition 2.4.8. If X contains a linear subspace X0 isomorphic to R ~ , then , by T h e o r e m 2.1.18, X is h o m e o m o r p h i c to the product R ~ • Y, where Y is a separable Frgchet space (namely, the factor space X / X o ) . By T h e o r e m s 2.1.17 and 2.3.22, X is h o m e o m o r p h i c to R W . Next, consider the case when X does not contain a linear subspace isomorphic to R ~ . In this case, there exists a n u m b e r E > 0 such t h a t no finite-dimensional linear subspace of X is an e-net in (X, d). Let us verify t h a t condition (iii) of Proposition 2.4.8 holds. Take two maps f : (9 ( I n : n E N } -+ X and (~: X --, (0, c~). W i t h o u t loss of generality we may assume t h a t c~ < c. In addition, by L e m m a 2.4.28, we may assume t h a t for each n E N, the c o m p a c t u m f ( I n) is contained in a finite-dimensional linear subspace of X. We construct a m a p g : ~ ( I n : n E N } --, X by induction. Let g / I 1 = f / i 1 and assume t h a t g has already been defined on the union @ ( I k : k < n } in such a way t h a t g ( I k ) , k < n, is contained in a finite-dimensional linear subspace E of X and the following condition is satisfied:

(,) lAg(q)- f(q)II < 5 ~ f ( q )

and lAg(q)- g(ql)ll > -~af(q)

for e a c h q , ql E I k , w h e r e k < m < n. Apply L e m m a 2.4.29 to the segment [a, b] -- a f ( I n) and proper (recall t h a t X is infinite-dimensional) linear subspace E. Let u: [a, b] --~ X be the m a p whose existence is g u a r a n t e e d by t h a t L e m m a . Let g(q) = f (q) + uc~ f (q), q E I n . This completes the inductive step of our construction. Let us show t h a t the map g: @ { I n : n E N } -~ X constructed above sends the collection {In: n E N } into a discrete (in X ) collection. Assuming the contrary, we can find sequences of indices kl < k2 < . . . and points qm E I k'~ such t h a t { g ( q m ) } converges in X. By (.), lira a f ( q m ) = 0 and the sequence { f ( q m ) } also converges in X . This, in turn, implies the false equality c~(lim f ( q m ) ) -- O. D PROPOSITION 2.4.31. Each infinite-dimensional compact convex subset of a separable Fr~chet space is h o m e o m o r p h i c to I ~ .

88

2. INFINITE-DIMENSIONAL MANIFOLDS

PROOF. Let K be an infinite-dimensional c o m p a c t convex subset of a separable Frgchet space. By T h e o r e m 2.1.17, K is an a b s o l u t e r e t r a c t . Let us verify t h a t condition (iii) of P r o p o s i t i o n 2.4.19 holds. Let e > 0 and let f , g" I n --, K be maps. By L e m m a 2.4.28, we m a y a s s u m e t h a t f ( I n) U g ( I n) is contained in t h e convex hull of a finite set {kl, k2, "., kin} C_ K . Since dim K ---- oc, there exists a point k E K which does not belong to t h e indicated convex hull. T h e comp a c t n e s s of K g u a r a n t e e s t h a t t h e r e is a 5 > 0 such t h a t 5 ( K - K ) is contained in the ball of radius e with c e n t e r at the origin. Let re(q) = (1 - 5 ) f ( q ) + 5k, q E I n. T h e n d(f~(q), f ( q ) ) = d ( 5 ( k - f ( q ) ) , O) < e, q E I n. Let gE = g. It only remains to n o t e t h a t fE(I n) M ge(I n) = 0.

[:]

2 . 4 . 4 . T r i v i a l B u n d l e s . In this Subsection we present p a r a m e t e r i z e d versions of T h e o r e m s 2.4.1 and 2.4.18. We also establish some corollaries which will be essential in C h a p t e r 6. First of all let us i n t r o d u c e some notations. Let a m a p f : X ~ Y be given 9 For any space Z a n d any m a p g: Z ---, X , C g ( Z , X ) shall d e n o t e the subspace of the space C ( Z , X ) consisting of all m a p s h: Z ~ X satisfying t h e equality f h -- f g . T h e following two t h e o r e m s characterize trivial R ~ - and I W-bundles. THEOREM 2.4.32 9 Let f : X ~ Y be a soft m a p between Polish A N R-spaces. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber R "~ . (ii) For each Polish space Z and any m a p g: Z ~ embeddings is dense in the space C g ( Z , X ) .

X,

the set of closed

THEOREM 2.4.33. Let f : X --~ Y be a proper soft m a p between locally compact A N R-spaces. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber I W . (ii) For each locally compact space Z and any proper m a p g: Z ~ set of closed embeddings is dense in the space c g ( z , x ) .

X,

the

T h e following s t a t e m e n t s are direct consequences of T h e o r e m s 2.4.32 and 2.4.33. PROPOSITION 2.4.34. Let f " X ~ Y be a soft m a p between Polish A N R spaces. T h e n the composition f i r x , where 1rx " X • R ~ ---. X is the projection, is a trivial bundle with fiber R ~ . PROPOSITION 2.4.35. Let f " X --~ Y be a proper soft m a p between locally compact A N R - s p a c e s . T h e n the composition f ~ r x , where ~ x " X • I w --. X is the projection, is a trivial bundle with fiber I • .

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

89

PROPOSITION 2.4.36. Let f : X --, Y be a proper soft map between locally compact A N R - s p a e e s . I f f o r each open cover Lt E c o y ( X ) there are m a p s f l , f2: X --+ X , U-close to i d x , such that f l ( X ) n 12(X) -- q} and f f i i = 1, 2, then S is a trivial bundle with fiber I ~ .

-- f ,

PROOF. By T h e o r e m 2.4.33, it suffices to show t h a t if g: Z ~ X is a proper m a p of a locally compact space Z into X, then the set of closed e m b e d d i n g s of Z into X is dense in the space c g ( z , X ) . Take any two disjoint compact subsets K1 and K2 of Z. Obviously the set L ( K 1 , K 2 ) - {h e c g ( z , x ) :

h ( K 1 ) N h ( K 2 ) -- q}}

is open in the space c g ( z , x ) . Let us show t h a t these sets are dense in c g ( z , x ) . Take any p E C g ( Z , X ) and any b / E c o v ( X ) . Softness of the map f guarantees the existence of a refinement )2 E c o y ( X ) of b / s u c h t h a t the following condition is satisfied: (.) If a m a p h': K1 U K2 ~ X is ]2-close to the restriction p / ( K 1 U K2) and f h ~ = f p / ( K 1 U K2), then there exists a map h: Z --~ X t h a t is U-close to p and satisfies h i ( K 1 U K2) -- h ~ and f h -- f p . By our assumption, there are maps f l , f 2 : X ---, X, )2-close to i d x , such t h a t f l ( X ) N . f 2 ( Z ) = 0 a n d . f . f i = f , i = 1,2. Define the map h': K l U K 2 --* X by letting it be equal to f i p / K i on K i , i -- 1,2. Condition (.) guarantees the existence of a m a p h: Z ~ X such t h a t h is U-close to p, h / ( K 1 U K2) -- h l, and f h -- f p . It only remains to r e m a r k t h a t h C L ( K 1 , K2). Therefore, the set L ( K I , K 2 ) is open and dense in the space C g ( Z , X ) . Next, consider (in the space Z) a countable open basis B = {Vi: i E g } , the elements of which have compact closures. Let L denote the intersection of all sets of the form L(cl Yi, cl Yj), where Vi, Vj E B and cl Vi N cl Vj = 0. By Proposition 2.1.7, L is dense in the space C g ( Z , X ) . It only remains to r e m a r k t h a t each map h E L is a closed embedding. F-1 T h e proof of the following s t a t e m e n t is similar to t h a t of Proposition 2.4.19. PROPOSITION 2.4.37. Let f : X ~ Y be a proper soft map between locally compact A N R-spaces. Then the following conditions are equivalent: (i) f is a trivial bundle with fiber I • . (ii) For each map g: D • I ~ --~ X , the set {hecg(n

xI W,X): h({0}xI W)nh({0} xI ~)=0}

is dense in the space C g ( D x I ~ , X ) .

As an i m m e d i a t e consequence of Proposition 2.4.37 we have the following statement. PROPOSITION 2.4.38 " Let 3 = { X n ,~n ~n-F1 , w } be an inverse sequence consisting of locally compact A N R-spaces and proper soft short projections, each

90

2. INFINITE-DIMENSIONAL MANIFOLDS

of which has two sections with disjoint images. lira S ~ X o is a trivial bundle with fiber I ~ .

Then the limit projection p0:

T h e reader can provide the proof of the following useful s t a t e m e n t by analyzing the proof of P r o p o s i t i o n 2.4.8 (keeping in m i n d T h e o r e m 2.4.32). PROPOSITION 2.4.39. Let f : X --~ Y be a soft map between Polish A N R spaces. Then the following conditions are equivalent: (i) f is a trivial bundle with fiber R W . (ii) For each map g: N • I "~ ~ X , the set {h: C g ( N • I ~ , X ) : collection {h({n} • I ~ ): n E N } is discrete in X } is dense in the space C g ( N • I ~ , X ) .

PROPOSITION 2.4.40. Let S = { X n , p n n + l , w } be an inverse sequence consisting of Polish A N R - s p a c e s and soft short projections, each of which has a countable (infinite) f a m i l y of sections, the images of which f o r m discrete collections. Then the limit projection p0: lim S ---, X o is a trivial bundle with fiber R ~ .

PROOF. Obviously the limit projection p0 is a soft m a p and the limit space X - lim S is a Polish A N R-space. Consequently, we only have to verify condition (ii) of P r o p o s i t i o n 2.4.39. Take a metric dn on the space X n , n E w, b o u n d e d by ~ . On X we consider the metric d, defined as follows: d({xn}, {x~}) -- m a x { d g ( x n , x'n)" n e w}. Consider two m a p s f : N • I ~ --, X and a : X ~ (0, 1). Our goal is to construct a m a p g: N x I ~ ~ X satisfying the following conditions: (a) POg -- POf . (b) d ( f ( y ) , g(y)) <_ a f ( y ) for each y E N x I ~ . (c) T h e collection {g({k} x io~): k E w} is discrete in X. For each n E w take a closed e m b e d d i n g i n + l : X n + i --* R W 9 Since R ~ is a Polish absolute r e t r a c t there exists a m a p

h" R w x R w x [0, o o ) - - * R w such t h a t h(a,b,t)=a

for each t_< 1

and h(a, b, t) -- b for each t _ > 2 . Since in+l is a closed embedding, the diagonal p r o d u c t pnn+l ~ k i n + l " X n + l -'-* X n x R w

is also a closed embedding. Consequently, softness of the short projection pn+l n " Xn+l

~

Xn

2.4. TOPOLOGY OF R ~ - AND I "~-MANIFOLDS

91

g u a r a n t e e s t h e existence of a m a p r n + l " X n • R w --* X n ~ - I

such t h a t Pn ..n+l rn+l -- rX~ a n d r n + l / A n + l

U'n+lAin+l)--i -~ ~Pn

where

lrx~" X n x R w -~ X n is the p r o j e c t i o n onto t h e first c o o r d i n a t e a n d A n + l -- (Pnn+l A Z9n + l ) ( X n + l ) . Let go -- p o f . By our a s s u m p t i o n s , t h e p r o j e c t i o n p~" X1 ~ X0 a d m i t s a c o u n t a b l e infinite family of sections, t h e images of which form a discrete collection in X1. This m e a n s t h a t t h e space X1 contains a closed s u b s p a c e h o m e o m o r p h i c to t h e product N x Xo in such a way t h a t p ~ / ( N x Xo) = Axo, where AXo" N x Xo --~ Xo is t h e p r o j e c t i o n onto t h e first c o o r d i n a t e . C o n s i d e r t h e m a p ~1" N x I "~ -~ X1 defined by l e t t i n g ~1 = A A p o f , where A" N x I W --~ N is t h e p r o j e c t i o n onto t h e first c o o r d i n a t e . Obviously, p ~ l = go a n d t h e collection {~1 ({k} x I ~ )" k E N } is discrete in X1. Now define a m a p gl" N x I ~ -~ X1 as follows:

gl = rl ( g o / k h ( i l P l f /kil~Ol /k2~ f )). It is easy to see t h a t p l g I = go. I f y E N x I W a n d o L f ( y ) <_ 89 t h e n , by t h e p r o p e r t i e s of h, we have gl (y) p l f (y). Let us describe t h e c o n s t r u c t i o n of t h e m a p g2" N x I W ~ X2. As above we take a m a p ~2" N x I ~ --. X2 such t h a t p2~2 = gl and for which t h e collection {~o2({k} x I ~ )" k E N } is discrete in X2. Let =

g2 -- r 2 ( g l A h ( i 2 p 2 f

Ai2~2A22af)).

Clearly, p292 - gl. As above, if a f (y) < ~ , t h e n g2(y) - p 2 f (y). If a f (y) > 89 t h e n it follows from t h e c o n s t r u c t i o n t h a t g2(y) -- ~(y). S u p p o s e now t h a t for each i, w i t h 2 _ i _ n, we have a l r e a d y c o n s t r u c t e d m a p s gi" N • I ~ ~ Xi a n d ~oi" N x I ~ ~ X i satisfying t h e following conditions: (1)i

P ii - i g i

---- # i - l .

(2)~ Pi-l~Oi ~ -- g i - 1 . (3)i If a f ( y ) <_ ~ , t h e n g i ( y ) = P i f (y) (4)i If a f ( y ) >_ 2.1-~_~,t h e n gi(y) - ~i(y). (5)i T h e collection { ~ i ( { k } x I W )" k e N } is discrete in Xi. We now c o n s t r u c t t h e m a p

gnH-l" N x I w - * Xn.-F1 a n d ~On-.Fl" g x I w ~ Xn-.F1. Following t h e above description, we o b t a i n a m a p ~ n + l " N x I ~ ~ X n + l such t h a t p n + l ~ n + l = gn a n d t h e collection { ~ n + l ( { k } x I ~ )" k E N } is dis(/rete in

Xn+I. T h e n g n + l -- r n + l ( g n A h ( i n + l P n + l f

/kin+l~n+lA2n+la

f)).

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2. INFINITE-DIMENSIONAL MANIFOLDS

Straightforward verification shows t h a t the maps gn+l and ~ n + l satisfy conditions ( 1 ) n + l - (5)n+1. It follows from the conditions (1)n t h a t the diagonal p r o d u c t g = / k { g n" n E w}" N • I ~ ---. X

satisfies the equalities p n g = gn for each n E w. In particular, we have pod go - p o f . If y E N • I W , then there is an index n E w such t h a t

1 2.+1 < ~ f ( y ) <

1

~.

Therefore, for each i _< n, we conclude, by conditions (3)i, t h a t gi(Y) = p i f ( y ) . This shows t h a t d(f(y),g(y))

-- m a x { d k ( p k f ( y ) , p k g ( y ) ) "

1 k >__n -t- 1} _< 2n+1 a f ( y ) .

This establishes the required closeness of the maps f and g. Next we show t h a t g ( { k } x I " ) N g ( { k ' } x I " ) -- 0 whenever k -fi k'.

Indeed, let yl E {k} x I ~ and y2 E {k'} x I ~ . Take a sufficiently large n to ensure t h a t a f ( y i ) _< 2.1--~_, i = 1,2. Then, by condition (4),~, P n g ( Y i ) = gn(Yi) = ~ n ( y i ) . Consequently, by condition (5)n, yl ~ y2. Finally, let us show t h a t the collection {g({k} x I ~ )" k E} is discrete in X . Since this collection consists (as was just shown) of pairwise disjoint compacta, it suffices to show t h a t for each (infinite) subset g ' of N the set g ( N ' x I ~ ) is closed in X. Consider an arbitrary sequence { y i } of points in N ~ x I ~ so t h a t the sequence { g ( y i ) } converges to a point x E X. Assume that 0 < e -inf{af(yi)}. Take n e w so large t h a t e > 2,1--!:-r_. Then, by condition (4)n, p n g ( y i ) - gn(Yi) -- r for each i. Continuity of the projection pn implies t h a t p n ( x ) - lira ~ n ( y i ) . By condition (5)n, the collection {~n({k} • I ~ )" k e g ' } is discrete in X,~. Therefore, without loss of generality, we may assume t h a t the convergent (in X n ) sequence { ~ n ( y i ) } is completely contained in the c o m p a c t u m {~n({k} x I W) for some k e N'. But then x e g({k} x I W ). Thus, it only remains to see t h a t e > 0. Assume the contrary. T h e n there

exists a subsequence {yi~} of { y i } such t h a t l i m a f ( y i ~ ) = 0. Then, keeping in mind the closedness of the maps f and g, we conclude t h a t lira f ( y i ~ ) = x. This equality implies, by continuity of c~, the contradictory s t a t e m e n t c~(x) = 0. Therefore e > 0. [-7

H i s t o r i c a l and bibliographical n o t e s 2.4. T h e characterization Theorems 2.4.1 and 2.4.18 were obtained by Toruitczyk in [299] and [298] respectively. The proof presented here is taken from [34] (see also [139] and [306]). Proposition 2.4.8 is due to Toruitczyk [299]. L e m m a 2.4.11 is taken from [86] (in the compact

2.5. INCOMPLETE MANIFOLDS

93

case this result had been obtained earlier in [48]). Proposition 2.4.20 appears in [299] and [149]. Proposition 2.4.21 is in fact contained in [139]. Propositions 2.4.23 and 2.4.24 were also proved in [299]. Proposition 2.4.25 is due to West [312]. Propositions 2.4.26 and 2.4.27 can be found in [115], [117] and [113]. Proposition 2.4.30 is the well known theorem of Anderson-Kadec. Proposition 2.4.31 were proved in [192] (see also [195]). The proofs of the last two s t a t e m e n t s are taken from [124]. Theorems 2.4.32 and 2.4.33 parametric versions of Theorems 2.4.1 and 2.4.18 - - are due to Torudczyk and West [300]. Formally this work contains only the proof of Theorem 2.4.33, and the proof of Theorem 2.4.32 is still unpublished. Nevertheless, this result is known to experts and has often been used. [300] contains an example of an n-soft map of the Hilbert cube onto itself, all fibers of which are also copies of the Hilbert cube, but which does not admit two sections with disjoint images. Therefore, a "fibered" version of Theorem 2.4.18 is not generally true. Propositions 2.4.38 and 2.4.40 were proved by S~epin and the author respectively.

2.5. I n c o m p l e t e m a n i f o l d s In Subsection 2.2.3, on page 56, we have already encountered the pseudoboundary E (and the pseudo-interior P of the Hilbert cube I • ). This space can also be described as =

{x~} e 12"

(i~) ~ < ~ i--1

Consider also the space a-

{{xi} E/2: all but finitely many xi---0}.

Alternatively, E can be thought as the span of the standard Hilbert cube in the Hilbert space 12 and a as the span of the usual orthonormal basis of 12. These two spaces, a and E, represent the minimal and maximal topological types of infinite-dimensional a-compact locally convex metrizable linear spaces in the following sense: every infinite-dimensional a-compact locally convex linear metrizable space contains a copy of a and is contained in E. It is not hard to see that both spaces a and E are a-compact absolute retracts. Each of these spaces can be represented as an increasing (countable) union of strong Z-sets. Incidently, every compact subset in either of these spaces is a strong Z-set. The topological characterizations of these spaces also involve a version of the universality property. Namely we have the following two results. THEOREM 2.5.1. Let X be a metrizable A ( N ) R - s p a c e . Then X is homeomorphic to E (respectively, is a E-manifold) if and only if the following conditions are satisfied:

94

2. INFINITE-DIMENSIONAL MANIFOLDS (i) X is a-compact. (ii) X is a countable union of strong Z-sets. (iii) For each compactum B and closed subspace A of B, every map f" B a, such that f / A is an embedding, can be arbitrarily closely approximated by embeddings coinciding with f on A.

THEOREM 2.5.2. Let X be a metrizable A(N)R-space. Then X is homeomorphic to a (respectively, is a a-manifold) if and only if the following conditions are satisfied: (i) X is a countable union of finite-dimensional compacta. (ii) X is a countable union of strong Z-sets. (iii) For each finite-dimensional compactum B and closed subspace A of B, every map f " B ~ a, such that f / A is an embedding, can be arbitrarily closely approximated by embeddings coinciding with f on A. One of the main ingredient of the proofs of the above theorems is the following version of Bing's shrinking criterion (Theorem 2.1.8) for incomplete a-compact spaces. PROPOSITION 2.5.3. Let X and Y be metrizable A N R-spaces which are countable unions of (finite-dimensional) compacta. Suppose that X has the estimated extension property 7 for compacta and Y has the universality property formulated in condition (iii) of Theorem 2.5.1 (of Theorem 2.5.2, respectively). Let f " X ---, Y be a map with the property that for every compactum A in Y and closed subset B of A, the map s

f A / ( X -- f - l ( B ) ) "

X - f-l(B)

---+ (X kJf A ) -

B

is a near-homeomorphism. Then f is a near-homeomorphism. PROOF. We consider only the parenthetical case, the other case being similar. oo oo Let X = kJn=lAn and Y = kJn=lBn, where An and Bn are finite-dimensional c o m p a c t a for each n - 1, 2 , . . . . Let d be any metric on Y. It is sufficient (see Subsection 2.1.1) to show t h a t there is a h o m e o m o r p h i s m h" X ~ Y such t h a t d(h(x) , f ( x ) ) < 1 for each x e X " We inductively construct a sequence {Cn} n~~ - - - - 0 of compact subsets of Y and a sequence {hn}n=0~176 of h o m e o m o r p h i s m s of X onto X Uf Cn = Xn such that, for each n = 1, 2, 9

(a)~ c~ 2 B~ U C~_~. (b)~ h~(A~) C C~.

(c)~ h~/h;~_~(C~_~)=h~_~/h;_~(C~_~). (~)~ p~h~/(X-h;!~(C~_~)is ~-close to p~_lh~_~/(X-h;!~(C~_~), where an" Y -

Cn-1 --'* (0, 1) is defined by 1 an(y) = ~-~ min{1, d(y, Cn-1)}

7See page 55. SSee definition of an adjunction space and induced maps on page 61

2.5. INCOMPLETE MANIFOLDS

95

and pn" X n ~ Y is the m a p defined by P n f c , -- f . We let Co - 0 and ho = id. A s s u m e t h a t hi" X ~ Xi and Xi satisfying conditions (a)i, (b)i, (c)i and (d)i for each i with 0 < i < n have already been constructed. Observe t h a t p n ( X n - Cn) C_ Y - Ca. L e t / 4 E cov(Y - Cn) be an open cover of Y - C n such t h a t

B ( p n / ( X n - Cn, st3/4) C B ( p n / ( X n - Cn, an+l). By the properties of Y, we can find an e m b e d d i n g

v" Dn+l = h n ( A n + l ) U Cn ~ Y such t h a t v / C n = idc~ and v / ( D n + l Take

Cn) is b/-homotopic to P n / ( D n + l -

Cn).

Cn+l = Bn+l U v(Dn+l) U g ( ( D n + l - Cn) x [0, 1]), where H" (Dn+l - On) x [0, 1] -~ Y is a / 4 - h o m o t o p y with U (x, 0) -- p n ( x ) and U ( x , 1) = v(x) for each x E n n + l - C n . Since the restriction f c ~ / ( X - f - l ( c n ) ) is a h o m e o m o r p h i s m of X - f - l ( C n ) onto X n - C n , by the a s s u m p t i o n a b o u t the m a p f , there exists a h o m e o m o r p h i s m gn+l" X n --~ X n + l such t h a t g n + l / C n --1 (/4)-homotopic to the m a p f c . + l , c , ) / ( X n - Cn) , id and g n + l / ( X n - Cn) is Pn+l where fc,+~,c.~) is defined by the equality f c . + ~ , c , ) f c ~ = fc.+x. T h e embed-1 dings gn+l/Dn+l and ( p n + l / C n + l ) - l v are st(pn+l(/4))-homotopic. Since X n + l , being h o m e o m o r p h i c to X , has the e s t i m a t e d extension p r o p e r t y for c o m p a c t a , 1 there exists a h o m e o m o r p h i s m Un+l . X n + l -~ Xn+l which is st 2 (~p -n+l(/'/))-cl~ to the identity m a p and such t h a t

Un+lgn+l/Dn+l -- ( P n + l / C n + l ) - l v . Let hn+l -- Un+lgn+lhn. T h e n Pn+lhn+l is st2(/4)-close to Pn+lgn+lhn and, consequently, pn+lhn+l is st3(/4)-close to pnhn. It is not h a r d to see t h a t the conditions (a)n+l, (b)n+l, (c)n+l and (d)n+l are all satisfied. We leave it as a m a n a g e a b l e exercise for the reader, to verify the fact t h a t the m a p h -- l i m p n h n is a h o m e o m o r p h i s m of X onto Y such t h a t d ( f ( x ) , h ( x ) ) < 1 for all x E X . Fi T h e o r e m s 2.5.1 and 2.5.2 have several corollaries. PROPOSITION 2.5.4. The following conditions are equivalent for each space

X" (i) X • E is homeomorphic to E (respectively, is a E-manifold). (ii) X is a retract of E (respectively, of an open subspace of E). Similarly, we have the following. PROPOSITION 2.5.5. The following conditions are equivalent for each space

X" (i) X • a is homeomorphic to a (respectively, is a a-manifold). (ii) X is a retract of a (respectively, of an open subspace of a).

96

2. INFINITE-DIMENSIONAL MANIFOLDS Triangulation theorems also hold.

PROPOSITION 2.5.6. Each E-manifold is homeomorphic to the product K x E, where K is a locally compact polyhedron. PROPOSITION 2.5.7. Each a-manifold is homeomorphic to the product K x a, where K is a locally compact polyhedron. We also have open embedding and homotopy classification results. PROPOSITION 2.5.8. Each E-manifold is homeomorphic to an open subspace of~. PROPOSITION 2.5.9. Each a-manifold is homeomorphic to an open subspace ofa. PROPOSITION 2.5.10. Homotopy equivalent E-manifolds are homeomorphic. PROPOSITION 2.5.11. Homotopy equivalent a-manifolds are homeomorphic.

Historical and bibliographical notes 2.5. The characterization Theorems 2.5.1 and 2.5.2 were obtained by Mogilski [234]. The triangulation theorems (Propositions 2.5.6 and 2.5.7), open embedding theorems (Propositions 2.5.8 and 2.5.9), homotopy classification theorems (Propositions 2.5.10 and 2.5.11) were proved by Chapman [64]. The spaces E and a are, in some sense, basic a-compact (incomplete) A g R - s p a c e s . In [35] the reader can find constructions of similar objects for higher Borel classes (additive or multiplicative). See also Section 5.6.

CHAPTER 3

Cohomological Dimension

3.1. C o h o m o l o g i c a l D i m e n s i o n For each commutative group G and any nonnegative integer n, we denote by K ( G , n) the corresponding Eilenberg-MacLane complex, i.e. a C)/V-complex, satisfying the following conditions" lri(K (G, n)) - f 0, [ G,

if i ~ n if i = n .

We are especially interested in the spaces g (Z, n). The complex K (Z, n) can be thought as a C)/Y-complex obtained from the n-dimensional sphere S n by attaching cells of dimension >_ n + 2. In this interpretation, the n-dimensional and (n + 1)-dimensional skeleta of K ( Z , n) both coincide with S n. Obviously, every compact subset of K (Z, n) is contained in the union of finitely many cells (recall that C)/V-complexes are endowed with the weak topology). Therefore, if a map f" A --~ K(Z, n) is defined on a closed subspace A of a c o m p a c t u m B, then f can be extended over an open neighborhood of A in B. In other words, the complexes K (Z, n) are absolute extensors with respect to compact spaces. Denote by H i ( X , G) t h e / - d i m e n s i o n a l Cech cohomology group of the compactum X with coefficients in G. It is well known [291], [287] that H i ( X , G) -- [X, K (G, i)],

where the square brackets denotes the set of homotopy classes of maps of the indicated spaces. Now we recall the definition of the cohomological dimension d i m e with respect to a group G. DEFINITION 3.1.1. The cohomological dimension of a compactum X with respect to a group G does not exceed n, i.e. d i m G X ~ n, if for each closed subspace A of X we have H m ( X , A ; G) -- 0 whenever m > n. I f dimG X __ n and dimG X ~ n - l , then dimG X = n. I f for every n, we have d i m G X ~ n, then dimG X -- oc. 97

98

3. COHOMOLOGICAL DIMENSION We begin with the following statement. THEOREM 3.1.2. The following conditions are equivalent for each compactum

X" (1)n d i m a X _ n. (2)n H n + I ( X , A ; G ) = 0 for each closed subspace A of X . (3)n For each closed subspace A of X , any map f" A --~ g ( G , n ) extension F" X ---, K ( G , n ) .

has an

PROOF. The implication (1)n ~ (2)n is obvious. Condition (3)n simply means that for each closed subspaces A of X, the inclusion homomorphism H n ( X ; G ) ~ H n ( A ; G ) is an epimorphism. Consequently, by the exact cohomology sequence of the pair (X, A), condition (2)n implies condition (3)n. Therefore it suffices to establish the implication (3)n ~ (1)n. Consider an additional condition: (4)n For each closed subspace A of X we have H n + I ( A ; G ) = O. Observe that condition (4)n implies condition (3)n+1, and conditions (3)n and (4)n imply condition (2)n. In turn, condition (1)n is equivalent to the collection of conditions (2)k with k > n. Therefore, it suffices to show the implication (3)k ===~ (4)k for k _ n. Consider an arbitrary map f" A ---, K (G, k + 1), defined on a closed subspace A of X. Take a point k0 e g ( G , k + 1), and by E = P ( g ( V , k + 1)), k0) denote the space of paths ,J" I --~ K ( G , k + 1), such that w(0) - k0, endowed with the compact open topology. Consider the natural map a" E --, K ( G , k + 1) assigning to a path w E E the point w(1) E K ( G , k + 1). The map a is a locally trivial bundle with fiber K ( G , k ) . It is also important to remark that the space E is contractible. Consequently, in order to show that the map f is homotopic to a constant map, it suffices to construct a map g" A --~ E such that ag = f . By compactness of the space A, there exists a compact polyhedron P c_ K ( G , k + 1) containing the image f ( n ) . Let P ( 0 d e n o t e the/-dimensional skeleton of P (with respect to some triangulation of P). We construct the map g by induction, first on f - l ( p ( O ) ) , then on f - l ( p ( 1 ) ) , and so forth until we reach f - l ( p ) = A. Choose a section s" p(0) ~ E of the map a over p(0) and let go = s f / f - l ( P ( ~ 9 Suppose that gi" / - l ( p ( O ) ~ E is a lifting of the map f / f - l ( p ( O ) . We are going to construct a lifting gi+l" f - l ( p ( i + l ) ) ___+E of the map f / f - l ( p ( i + l ) ) . Consider an (i + 1)-dimensional simplex a in P. The contractibility of a implies that the space a - l ( a ) is homeomorphic to the product a • K ( G , k ) . Denote by h" OL--I(Gr) --+ ~ X K ( G , k )

the corresponding homeomorphism and by ~" a • K ( G , k ) ~ K ( G , k )

3.1. COHOMOLOGICAL DIMENSION

99

the projection onto the second coordinate. By condition (3)k, the map ~hgi/f-l(Oa):

has an extension r

f-l(a)

/-l(Oa)--.

K(a,k)

~ K ( G , k). But then the map

h-l(f/f-l(a)Ar

f-l(a)

~ a-l(a)

is an extension of the lifting gi onto the whole set f - l ( a ) . Performing the same procedure with all of the (i + 1)-dimensional simplexes, we obtain a lifting gi+l defined on the set f - l ( p ( i + l ) ) . [3 Thus, each of the conditions from T h e o r e m 3.1.2 can be taken as the definition of cohomological dimension. Below, in most cases, we use the definition given by condition (3)n. In this connection, recall that if we replace the complex g (G, n) by the sphere S n in condition (3)n, then we obtain the definition of the usual Lebesgue dimension dim. More formally, we have the following statement. THEOREM 3.1.3. For each compactum X , the inequality d i m X <_ n is equivalent to the condition: 9 For each closed subspace A of X , any map f " A --, S n into the ndimensional sphere S n has an extension F" X --, S n. Theorems 3.1.2 and 3.1.3 imply the following statement. PROPOSITION 3.1.4. I f X is a compactum, then dim Z X _ d i m X . PROOF. If dim X -- cx), then the indicated inequality is trivially true. Assume t h a t dim X _ n and consider a map f" A ---, K ( Z , n ) , defined on a closed subspace A of the c o m p a c t u m X. Since dim X < n, the map f is homotopic to a map g" A --~ K ( Z , n ) (n) = S n. By Theorem 3.1.3, there is an extension G" X --. S n of g. Since the complex K (Z, n) is an absolute extensor with respect to compact spaces, we conclude t h a t the original map f also has an extension F" X --* g ( Z , n ) (homotopic to G). V1 THEOREM 3.1.5. I f X pactum, then

is a f i n i t e - d i m e n s i o n a l (in the sense of dim) com-

dim X -- dim Z X. PROOF. By Proposition 3.1.4, it suffices to show that dim X <_ dim Z X. Assume the contrary. Then there exists a c o m p a c t u m X such t h a t for some integer m we have dimX =m+l

and

dim Z X _ _ _ m .

By the first equality, there exist a closed subspace A of X and a m a p f" A --, S m which cannot be extended to the space X. Identify the sphere S m with the m-dimensional skeleton of the complex K (Z, m). Then, by the inequality c-dimzX < m , there exists an extension G" X ---. K ( Z , m ) of f . Again, by the

100

3. COHOMOLOGICAL DIMENSION

first equality and Theorem 3.1.3, there is a map F" X ~ K ( Z , m) homotopic to G and such that F / G - I ( K ( Z , m ) ( m + I ) ) -- G / G - I ( K ( Z , m ) ( m + I ) ) . In this situation, it can easily be seen that F is an extension of the map f into the sphere S m. This contradiction the choice of f . WI It follows from the universal coefficients formula (see, for example, [2871) that for each group G and c o m p a c t u m X we have dimG X < dim Z X. This fact, coupled with Proposition 3.1.4, shows that the dimension dim Z is "closer" to the dimension dim than is the dimension dim(; for any other group G. Below we consider only the cohomological dimension dim Z with integral coefficients. The main problem of determining whether the dimensions dim and dim Z coincide is known as Alexandrov's problem [5]. Keeping in mind Theorem 3.1.5, this problem can be formulated as follows: 9 Does there exist an infinite-dimensional compactum having finite cohomological dimension dim Z ? In this section we show that the above problem is equivalent to the so-called CE-problem: 9 Do cell-like maps of compacta raise dimension? First we present a spectral characterization of cohomological dimension. For this we need to introduce the following two notions. DEFINITION 3.1.6. The n-dimensional diameter of a map f" X ~ Y is defined as the following number: a n ( f ) = i n f { d i s t ( f , ~o)" ~o e C ( X , Y )

and dim~o(X) _< n}.

DEFINITION 3.1.7. The binary (n, k)-dimensional diameter of a map f" X --~ Y is defined as the following number: a ~ ( f ) = s u p { a k ( f /A)" A is closed in X and dim A _< n}. THEOREM 3.1.8. Let X be the limit space of an inverse sequence 3 -{Xi,p~ +1}, consisting of compact polyhedra. Then the following conditions are equivalent: (i) dim Z x _ n . (ii) limccann+l(p~.) - 0 for each index j. PROOF. (ii) ==, (i). We assume that the sequence S -- {Xi,p~ +1} is realized in the Hilbert cube I ~ , with a given metric on it. Let f" A --. K (Z, n) be a map defined on a closed subspaces A of X. Since the complex K (Z, n) is an absolute extensor with respect to compact spaces, we can extend the map f to a map

3.1. COHOMOLOGICAL DIMENSION

101

f" O A --~ K(Z, n) defined on a compact neighborhood O A of the set A in X. Choose a number e > 0 so small that any two e-close maps of any c o m p a c t u m O A have homotopic compositions with f. Take an index k such that for each i _> k the projection pi moves points not more than e and, additionally, the e-neighborhood Oe(pi(A)) of the set pi(A) in I ~ is contained in O A. By (ii), there is an index i(k) such t h a t a~+l(pk (k) < ~. Take a triangulation of mesh < ~ of the polyhedron Xi(k) and denote by Ai(k) the union of all those simplexes of this triangulation that intersect Pi(k)(A). Obviously, _

into

Pi(k)(A) C_ Ai(k) C OA. X i(k) ( n + l ) --, X k o f t h e ( n + l ) Bythechoiceoftheindexi(k) , thereexistsamapr dimensional skeleton of the polyhedron Xi(k) (with respect to the above indicated

triangulation) with at most n-dimensional image such t h a t dist(r Obviously, r rA(n+l)) i(k) C OA and the restrictions of the maps f r

and f onto

(n+l) (i.e. n-dimensional skeleton of Ai(k) are homotopic. ~(k) ( y (n-t-l)) Since dim Cv'i(k) < n , the map f admits an extension defined on the union (y(n+l)~ OA U C w , i(k) J" -

Consequently, the map f r

--- (n+l)

.y. ( n + l )

can be extended to ~'i(k)

. By the remark

made in the beginning, f liA(n+ i(k) 1) also has an extension to X(n+l)i(k). Now recall t h a t there are no obstructions to extending a map from the m-dimensional skeleton onto the (m + 1)-dimensional one if the range has trivial m-dimensional homotopy group. Therefore there is a map g" Xi(k) K ( Z , n ) extending Jl~i(k) 7,A(n+I) 9 Since the maps g/Ai(k) and f/Ai(k) coincide on the (n + 1)-skeleton, they are homotopic. This shows that f/Ai(k) has an extension to the space Xi(k). It only remains to remark t h a t f is homotopic to fpi(k)/A and, consequently, has an extension to X. By Theorem 3.1.2, dim Z X _ n. (i) ~ (ii). It is not hard to see t h a t the inequality dim Z X < 1 implies the inequality dim X <_ 1. This observation shows that the implication (i) ~ (ii) is true for n = 1. Thus, we may assume below t h a t n >_ 2. Consider a finite simplicial complex L. We now present a description of a connected C14~-complex L associated with L. First, represent L as the union L = L (n) tJ al LJ ... [_J~s, where the simplexes ai are indexed so t h a t n + 1 _ d i m a l < d i m a 2 _< --- _< as. We construct the complex L by induction on s. In the meantime the following conditions will be satisfied: (a) /~ = L (n) U g ( a l ) U - - - U K(as), where g ( a i ) is an Eilenberg-MacLane complex of type g ( r n ( a } n), n). (b) L (n) ~(n+l) L (n) and L (n) N K(ai) _(n) for each i < s. _~_

-~

.~

o

i

102

3. COHOMOLOGICAL DIMENSION

(c) aiNaj, g(aiNaj),

g(ai) Ng(aj)--

if d i m ( a ~ N a j ) < n if d i m ( a i N a j ) > _ n + l .

If s = 0, then we let f, = l (n). If a CW-complex L (n) U g ( a l ) U ... U g ( a k ) , associated to the complex L (n) Uaz U . . . U a k and satisfying the above conditions, has already been constructed, then the complex K(ak+l) can be obtained by killing all homotopy groups in dimensions larger than n of the complex

.~(n) g ( o a k + l ) = "k+l

U

u { g ( a i ) " ai C 0ak+l}.

Since the embedding Oak+ ~ (n)1 ~ K(Oak+l) induces isomorphisms of the homotopy groups of dimension _ n, we conclude that condition (a) is satisfied. The remaining conditions are obvious. We now proceed to the direct proof of the implication (i) ~ (ii). Let e > 0 and an index j be given. Consider a triangulation of the polyhedron Xj of mesh < 5" Consider the complex

f(j

=

X~ n)

U

g (61)

U""

U

g (as)

associated with this triangulation. Define a map i6j" X ~ )Cj so that the restriations of pj and i6j onto p-~Z(X~n)) coincide and such that for each k < s we have

p~(p~ (~k))c g(~k). ^

--1

We construct the desired map 16j by induction, defining it on the sets of the form

pj Z(x~n)U a z a . " U ak), where 0 < k _< s. If the map 16j has already been defined on the set pj Z(X n) U a z a . . . U ak), then, by the inductive assumption,

pj(pyl(Oak+l)) ~

g(ak+l).

Since dim Z X <_ n, we conclude that the map ~j/p~Z(Oak+l) admits an extension to a map of the whole inverse image p~Z(ak+l ) into

g(ak+z) ~-- g ( e [ Z , n) --~ 1~ K(Z, n). 1

This completes the inductive step and shows the existence of the map ~6j. Extend the map 16j onto a neighborhood O X of the limit space X in I ~ and denote by pj^i the restriction of this extension on Xi. For sufficiently large i the following condition will be satisfied: (.) Ifpji(x) e a C_ X a n d p ^i j ( x ) e al C_ X ~n) , then a A al -7(=0; i f p^i j(x) e g ( a m ) , then a A am ~ 0. Next consider an at most (n + 1)-dimensional compactum Y C Xi. Since, by the (n+z (n) such that construction, )~j ) = X~ '~), there exists a map g~: Y ---, X j =

3.1. COHOMOLOGICAL DIMENSION

103

^i and, moreover, gji(y) E a w h e n e v e r pj(y)a, y E Y a n d a E X j . It only r e m a i n s to observe t h a t , by (.), the m a p s pji a n d gji are e-close. T h e p r o o f is c o m p l e t e . V1 Now we are r e a d y to prove the equivalence of A l e x a n d r o v ' s p r o b l e m a n d t h e CE-problem. F i r s t notice t h a t every inverse s e q u e n c e S - {Xi, Pi i + 1 }, consisting of metrizable c o m p a c t , can be realized in t h e H i l b e r t c u b e I ~ (see above). Take any m e t r i c d on I ~ . W i t h o u t loss of g e n e r a l i t y we can a s s u m e t h a t all p r o j e c t i o n s of t h e sequence $ are non-stretching 1 m a p s w i t h r e s p e c t to the m e t r i c d. Next, s u p p o s e t h a t two inverse sequences S -- {Xi, P~+I} a n d S ' - {Yi, qi~+l } are realized in t h e H i l b e r t c u b e I ~ . We say t h a t a sequence { f i : X i ~ Yi} of m a p s converges to a m a p f : X : lim,.q ~ l i m , . q ' : Y if the m a p If[: [5[ --, [,.q'[, defined by r e l a t i o n s [ f [ / X : f a n d [ f [ / X i = fi for each i, is c o n t i n u o u s . Here IS[ d e n o t e s the union of all e l e m e n t s of ,.q a n d the limit space lim 5 (recall t h a t all these spaces are considered as s u b s p a c e s of I w ). LEMMA 3.1.9. Let S ' -- {Yi, qr +1} be an inverse sequence realized in the Hilbert cube I ~ with a given metric d. If all projections of S' are non-stretching, i + l fi+l), imthen the convergence of the series ~i~176 ei, where ei -- dist(fiP~ +1, qi plies the convergence of the sequence {fi}. Furthermore, the limit map f" X ~ Y satisfied the following condition for each i" oo

dist(fipi, qif) <_ E ej. j-'i PROOF. O n e can easily see, using i n d u c t i o n on k, t h a t k-1

/~ i+k qi+ dist[jipi , k'fi+k) <-- E ei+j. j=O T h e n , for each j <_ k we have k-1

dist (qiJ f JPJ, q~k f kPk) ~-- E en. n--j T h e r e f o r e the sequence {qJifjpj}j~176 converges uniformly. It only r e m a i n s to observe t h a t t h e limit m a p f" X --~ Y can be u n i q u e l y d e t e r m i n e d from t h e equalities qif -- limj__.cr q~ f jpj. [~ THEOREM 3.1.10. Let X be a metrizable compactum. Then the following con-

ditions are equivalent: (i) dim Z X < n . (ii) There exists a cell-like map f" Y ~ X such that d i m Y _< n. 1A map f" X --, Y is non-stretching if dist(f(x), f(y)) ~ dist(x, y) whenever x, y e X.

104

3. COHOMOLOGICAL DIMENSION

PROOF. (ii) ~ (i). Let f" Y --+ X be a cell-like map and d i m Y ~ n. Consider a closed subspace A of the compactum X and denote by fA the restriction of the map f to the inverse image . f - l ( A ) . The natural inclusion maps of A into X and of f - l ( A ) into Y are denoted by i and j respectively. By Proposition 3.1.4, we have dim Y _< n. Consequently, by Theorem 3.1.2, the map induced by the inclusion j, i.e. the map

j*" [Y,K(Z,n)]---. [ f - I ( A ) , K ( Z , n ) ] is a surjection (recall that the symbol [A, B] denotes the set of homotopy classes of maps from A into B). Since both maps f : Y ---. X and fA: f - l ( A ) ---* A are cell-like, we conclude, using the Vietoris-Beagle Theorem [287], that the induced maps f * : [X,K(Z,n)] ~ [Y,K(Z,n)] and

f~ : [A,K(Z,n)] ---. [ f - I ( A ) , K ( Z , n ) ] are bijections. The equality f j -- irA implies the commutativity of the following diagram:

i* [X,K(Z,n)]

~ [A,K(Z,n)]

*

[Y,K(Z,n)]

j

*

,.-[f-I(A),K(Z,n)]

It is now easy to see that the map

[A,K(Z,n)] is a surjection. Consequently, by Theorem 3.1.2, dim Z X _< n. (i) ==~ (ii). Represent X as the limit space of an inverse sequence S = {Xi, p~ _ i + 1 } consisting of compact polyhedra and surjective projections. As above, we may assume without loss of generality that the sequence S is realized in the Hilbert cube I W . We may assume also that all projections of our spectra are non-stretching. If a sequence {ei} of positive numbers is given, then, by Theorem 3.1.8, we can define an increasing sequence {k(i)} of natural numbers such that a~+l(pk(i) < ei for each i. On each polyhedron Xk(~), consider a triangulation of mesh < e~ and denote by X k(i) (m) an m-dimensional skeleton of this triangulation. Since a~ +11(pk(i)k(i+l)) < k ( ~ + l ) // X k(~+l) (n+l) map of X k(~+l) (n+l) into Xk(i) with ei, there exists an e~-close to ~'k(i)

3.1. COHOMOLOGICAL DIMENSION

105

y(n) at most n-dimensional image. Pushing this image into ..k(i), we obtain a map k(~+l)/x (n+l) g~+l. v""(k(~+l) n + l ) ~ v""(k(~) n ) which is 2ei-close to the restriction Pk(i) / k(i+l) _i+1 y(n) _i+1 i+2 J and consider an inverse Let q~ = gii + l l/'~k(i+l), q~j -- q~ "qi+l . . . . . qj-1

_

v(n)

A+I

(n)

sequence S ~ {"k(i),qi }. Let ei" Xk(i) ~ Xk(i) denote the inclusion map. If the series ~ ei converges, then, by L e m m a 3.1.9, the limit map f : Y -~ X is well defined (here Y - lira S'). Now we show t h a t if we choose the sequence {e~} in a certain way then the map f will be cell-like. (x) Let ~i - )-]~j=i 2ei. Then, by L e m m a 3.1.9, we have the following inequality for each i:

(.) dist(pk(i)f , qi) <_ ~i. The required condition on the sequence {Ei} is the following:

For each point x E Xk(i+l), its 2~i+l-neighborhood can be contracted to a point within its ei-neighborhood. The construction of such a sequence is trivial, and therefore we assume below that this condition for the sequence {ei} is satisfied. Let x be an arbitrary point in X. We are going to show that the inverse image f - l ( x ) is a cell-like set. Denote by (gEy the closed e-neighborhood of the point y. Also let xi = Pk(i)(x). i+l The map qi sends the s e t 02~i+lxi+1 into the set 02~xi. Denote by q:~ the

(n) Observe t h a t the indicated restriction of the map q~ to the set 0 2 ~ x j N X k(j)" intersection is non-empty (recall that the mesh of the triangulation of Xk(j) is less than ei and ej < 2~j). Now we show t h a t the limit space of the inverse sequence 8 x "- { 0 2 ~ X i A X (k(n~,q-~+l}

coincides with the inverse image f - l ( x ) . If a point y E Y limit space lira Sx, then qi(y) ~ 0 2 ~ x i for some index i. It (,) that dist(qi(y),pk(i)f(y)) <_ gi. Therefore Pk(i)f(Y) ~ y E lim Sx, then limi__.~ dist(qi(y), xi) = 0. By condition

does not belong to the follows from condition xi and y r f - l ( x ) . If (,), we then have

lim dist(qi(y) Pk(i)f(Y))--O. i--~c~ Consequently, limi--.cr dist(xi,pk(i)f(y)) -- 0 and hence y E f - l ( x ) . Since all spaces of the inverse sequence Sx are non-empty and compact, so is its limit. This shows that f is a surjective map. Further, i m ( ~ +1) C_ 02E~xi and, consequently, contracts to a point within v(n+l) Oe~+lxi. Pushing the corresponding homotopy into ~. k(i) w e o b t a i n a h o m o t o p y (n+l) (recall that the mesh of the triangulation of Xk(~) is less in OE,_I+E,(X) NXk(i) than ei). Consider the composition of the final homotopy and the map gi-1. i The image of this composition is contained in the neighborhood of the point xi-1 of =i+1 is radius not greater than ei-1 + ei + 2ei_1 < 4ei-1 < 2gi-1. Therefore, ui-1 homotopic to a constant map.

106

3. COHOMOLOGICAL DIMENSION

Finally, observe that all projections of the subspectrum of the spectrum Sx, indexed by even numbers, are non essential and, consequently, its limit space, i.e. the fiber f - l ( x ) , is cell-like. This completes the proof of the Theorem. V1 Remark 3.1.11. The proof of the implication (ii) ~ (i) in Theorem 3.1.10 shows that cell-like maps cannot raise the cohomological dimension dim Z.

COROLLARY 3.1.12. Alexandrov's problem is equivalent to CE-problem.

Historical and bibliographical notes 3.1. Originally Theorem 3.1.2 was proved in [110], although the proof presented in the text is due to Ferry (see [305], [130]). Theorems 3.1.3 and 3.1.5 were obtained by Alexandrov [4]. Cohomological dimension theory (in particular, the C E - p r o b l e m ) is an important and popular area of investigation; see, for instance, [5], [6], [43], [44], [118], [126],

[12S], [1301, [132], [1361, [2051, [206], [198], [61], [207], [2541, [262], [263], [138]. Theorems 3.1.8 and 3.1.10 are due to Edwards (see [305]).

3.2.

Cell-like mappings raising dimension

In this Section we present a positive solution of Alexandrov's problem (see p.lO0). We begin with the following definition. DEFINITION 3.2.1. Let f" X -~ Y be a map. We write dimG f ~_ n if the following condition is satisfied: 9 For each closed subspace A of Y and map ~" A --, K ( G , n), there is an extension r X --~ K ( G , n ) of the composition ~of / f - l ( A ) . Obviously, d i m a i d x = d i m a X, where i d x denotes the identity map of X. If f" X ~ P is a map into a polyhedron P with triangulation T, then the inequality d i m e ( f , T) stands for the above condition, with A a subpolyhedron of P with respect to the triangulation v. LEMMA 3.2.2. Suppose that an inverse sequence 8 -- {Xi, Pii+l~J, consisting of compact polyhedra, is realized in the Hilbert cube I ~ with a metric d. Assume, in addition, that Ti is a triangulation of X i such that limi--,oo mesh(Ti) -- O. If dimG~p ~ ii+l, Ti) ~_ n for almost all indexes, then dimG lim S ~ n. PROOF. Let A be a closed subspace of the limit space X - lim S and ~" A - . K (G, n) be an arbitrary map. Compactness of A guarantees that there is a finite subcomplex K C_ K ( G , n ) such that ~(A) C_ K. Since K is an ANR-space, there exists an extension ~ " U - . K of ~ to some neighborhood U of the set A in the Hilbert cube I • . Obviously, if i is sufficiently large, then we have pi(A) C_ U. Denote by Ai the star of the set px(A) in Xi with respect to the triangulation

3.2. CELL-LIKE MAPPINGS RAISING DIMENSION

107

Ti, i.e. Ai is the union of those simplexes of Ti t h a t intersect the set pi(A). Since mesh(Ti) ~ O, we conclude t h a t for almost all indexes i we have Ai C U. It is also easy to see t h a t the sequence {~'pi/A: A ~ K } converges to the m a p ~o. Since sufficiently close maps into an A N R-compactum are h o m o t o p i c , t h e r e exists an index i such t h a t ~'pi/A ~- ~. Consequently, it suffices to c o n s t r u c t an extension of the m a p ~'pi/A. By the a s s u m p t i o n s , for sufficiently large indexes i we have dima(pii + 1 , Ti) <_ n. This g u a r a n t e e s the existence of an extension / i + 1 --1 r Xi+l --~ K(G, n) of the m a p w, , t,.i. i + l /(Pi ) (Ai). It only r e m a i n s to observe t h a t the composition r e x t e n d s the m a p ~'pi/A. [3 LEMMA 3.2.3. Let a compactum X be the limit space of an inverse sequence ,S = {Pi ,Pi i + 1 }, consisting of compact polyhedra, T be a triangulation of the polyhedron Po and dimG(P0, T) <_ n, where G is a finitely generated group (G - - Z for instance). Then there exists an index i such that dima(p~, v) < n. PROOF. Let A0 be a s u b p o l y h e d r o n of the p o l y h e d r o n P0 with respect to the t r i a n g u l a t i o n T, and ~" Ao ~ K(G, n) be a m a p defining one of t h e g e n e r a t o r s of the group Hn(Ao;G). As before, we may assume t h a t the sequence S is realized in the Hilbert cube I ~ . By t h e condition dima(Po, T) < n, t h e r e exists an extension ~" X ---, g ( G , n ) of the m a p ~po/pol(Ao). Consider t h e closed subset i -1 (Ao)) 8 = x u (U oo=o(p0) of the Hilbert cube I W . T h e collection of m a p s {@} U {~pio/(p$)-l(Ao)}i~1760 defines a continuous m a p r B ~ K ( G , n ) . By the c o m p a c t n e s s of B, t h e r e is a finite complex g such t h a t r C K C K ( G , n ) . Since g is an AYR-space, t h e r e is an extension r U ~ K of the m a p r to a n e i g h b o r h o o d U of t h e set B in I ~ . Obviously, idx -- limp/. This shows t h a t there is an index k such t h a t Pi c U for each i >_ k. Also observe t h a t the m a p r e x t e n d s the m a p ~p$/(p$)-l(Ao) whenever i >_ k. Since there are only finitely m a n y different s u b p o l y h e d r a A of the p o l y h e d r o n P0, and since the c o r r e s p o n d i n g cohomologies Hn(A; G) are finitely generated, we see t h a t t h e r e exists an index k such t h a t for each s u b p o l y h e d r o n A C P0 and m a p ~" A --+ K(G, n), c o r r e s p o n d i n g to one of the g e n e r a t o r s of the group Hn(A; G), t h e r e exists an extension r Pk --* _

_

g ( G , n ) of the m a p ~pko/(Pko)-l(A ). In t e r m s of cohomology, this j u s t m e a n s t h a t for each s u b p o l y h e d r o n A C_ P0, and for each g e n e r a t o r a E Hn(A;G), t h e r e is an element fl(a) e Hn(pk; G) such t h a t j*(fl(a)) = (pko/...)*(a), where j * denotes the h o m o m o r p h i s m induced by the inclusion j" (pko)-l(A) C Pk. If -), E Hn(A; G) is not a g e n e r a t o r , t h e n -), can be expressed in t e r m s of the g e n e r a t o r s , - y = ~ )~iai, and, consequently,

108

3. COHOMOLOGICAL DIMENSION

Translating this fact back into the language of maps into K (G, n), we obtain the desired conclusion. KI LEMMA 3.2.4. Let L be a compact polyhedron with a triangulation T. Then there exists a map Cr" E(T) ---. L satisfying the following conditions: (i) Fo~ ~ach ~ i m p l ~ ~ ~ ~, th~ i n ~ ~ im~g~ r ha~ th~ homotopy type of the complex K (O17"o" Z, 3), where ra rankTr3(a(3)). (ii) The space E(T) is a Cld]-complex admitting a triangulation and such that its m-dimensional skeleton E ( r ) (m) is a finite subcomplex for each m.

(iii) The map Cr is combinatorial with respect to the triangulation ~- and the C}4~-complex structure of E ( r ) . (iv) d i m z ( r T) _ 3. PROOF. We use induction on the dimension of L. First let us consider one additional condition: (v) For each 3-connected subpolyhedron K C_ L, the space r is 2connected and H3(r = @rlKZ, where r g = rank ~3(K(3)). If dim L _< 3, then E(T) = L and CT = idL. Next we consider the case dim L -- 4. For each 4-dimensional simplex a, consider the inclusion ja" a(3) ~ K(Z, 3) as the 3-dimensional skeleton. We may assume that the finite-dimensional skeleta of the (:W-complex K(Z, 3) are finite complexes. In addition, we may assume that K ( Z , 3) (4) = a (3). For each a we attach the mapping cylinder Mj~ of the inclusion ja to L (3) along the subset a (3). Denote by ca the center of the simplex a. Now we define c r ( g ( Z , 3)) = ca for each a, and Cr(L (3)) = idL(3). Next we extend Cr onto L(3) U (UaMjo) by linearity. Finally we let

E(T) -- L (3)

U

(UaMj~).

Conditions (i)-(v) are satisfied by the construction. Suppose that the lemma has already been proved for polyhedra M with d i m M _< m, m > 3, and consider an (m + 1)-dimensional polyhedron L with triangulation T. By the inductive assumption, there exists a map

Cr(m)" E(T (m)) --+ L(m) satisfying conditions (i)-(v). Take any (m + 1)-dimensional simplex a e T and -I @("9). By condition (v), we have consider the space Cr(m) H 3 ( r --1 ) (a(m)))

-

-

(~rl~Z.

--1 ((7 (m) ). By condition (v) and the Uurewicz theorem, we have Let ]I3 = CT(-,)

~r3(Y3) = @[~Z.

By condition (ii), y(5) is a finite complex.

Since 7r4(Y3) =

7rd(y(5)), and since y(5) is one-connected, we conclude that 7r4(Y3) is a finitely

3.2. CELL-LIKE MAPPINGS RAISING DIMENSION

109

g e n e r a t e d group (see [270]). A t t a c h i n g 5-dimensional cells to Y3, we construct a complex Y4 with 7r4(Y4) - 0. A t t a c h i n g 5-dimensional cells to Y4, we obtain a complex Y5 with ~r3(Yh) = @1rcr Z and 7ri(Yh) = 0 for each i < 3 and i = 4,5 Continuing in this m a n n e r we obtain an increasing sequence II3 c_ Y4 c_ Y5 _ c . . . c_ Y3+~ c _ . . . . Let Y~ = li___.m{Y3+i}. Obviously the complex Yoo is h o m o t o p y equivalent to the space K (@[~Z, 3). --1 (0. (m) ) into Yoo and by Mj,, the D e n o t e by ja the inclusion of the space Cr(,~) m a p p i n g cylinder of this inclusion. For each (m + 1)-dimensional simplex a, --1 (a(m) )" we a t t a c h the space Mj,, to the space E(T (m)) along the subspace Cr(--) T h e complex obtained in this way is the required space E(T). E x t e n d the m a p Cr(m) " E('r (m)) ---+L (m) to the m a p Cr" E(T) ---+L as follows. For each simplex a, let Cr(Yoo) = ca be the center of the simplex ~, and further e x t e n d Cr by linearity. In other words, shrinking Yoo into a point, we obtain from Mj~ the -1 (a(m)), and since a is the cone over a (m) we see t h a t cone over the space Cr(-,) the m a p

,r

,~,r(m )-1 (o.(rn)) ~ (7(rn)

between the bases of these cones can be n a t u r a l l y e x t e n d e d to the m a p between these cones. It only remains to verify conditions (i)-(v). Conditions (i)-(iii) are satisfied by the construction. Let K C L be a 3-connected s u b p o l y h e d r o n of the p o l y h e d r o n L. Consider an a r b i t r a r y (m + 1.)-dimensional simplex a C K. If there is no such a, t h e n condition (v) is satisfied by the inductive hypothesis. In the m a p p i n g cylinder Mj,,, shrink each segment connecting a point x E 'r --1 (G(m) ) with the point ja(x), to a point. This defines a cell-like m a p ~" r --+ Za. Obviously, ~(Mj,,) = Yoo. Observe also t h a t H3(r --1

= H3(r

(K (m)) U

--1

(m)) U (UG'#GMj~ ) U Mj,, ---

U V3) --

--1

U

Consequently, removing single simplexes, we get H3(r

= H 3 ( r --1

= G~KZ.

Let us show now t h a t 7r~(r -- 0 for each i < 3. Obviously, the cells of dimension more t h a n 3 do not affect t h e / - d i m e n s i o n a l h o m o t o p y groups with i < 3. Therefore, for each simplex a C K we can remove the cells of dimension larger t h a n 3 from the space Mj~, C r As a result we o b t a i n a space, deformable onto r Observing t h a t 7ri(r -- 0 for each i < 3, we finish the verification of condition (v).

110

3. COHOMOLOGICAL DIMENSION

Let us verify the remaining condition (iv). Let A C_ L be a subpolyhedron of the polyhedron L and ~" A --~ K ( Z , 3) be an arbitrary map. By the inductive hypothesis there is a map

I9r

r

K ( Z , 3),

extending the map ~ r All we now have to do is extend the map ~ / r to a map ~a" Mj~ --, K ( Z , 3) for each simplex a C L with a ~ A. But by the construction, the inclusion j a induces an epimorphism (even an isomorphism) of 3-dimensional cohomologies. Therefore the indicated extension exists for each a. The proof is complete. [-1 R e m a r k 3.2.5. The 3-dimensional skeleton E(T) (3) of the complex E(T) is, by construction, homeomorphic to the 3-dimensional skeleton L (3) of the polyhedron L. Moreover, the restriction r (3) serves as the corresponding homeomorphism. Let us now recall some facts from the generalized cohomology theories. A comprehensive introduction to the corresponding results can found in several textbooks (for instance, see [291]). Recall that the reduced generalized cohomology theory h* on the category 7)T ~ of pointed spaces with morphisms homotopy classes of maps, preserving base points, is a collection {h n" 7)T ~ ~ A} of contravariant functors into the category of Abelian groups and natural equivalences a n" h n ---+ h n + l ~ *, where E* - is the functor of taking the reduced suspension and n E Z. These objects satisfy the following exactness axiom: 9 For each pair (X, A, x0) the following sequence is exact .

.

.

~" h n ( X ) , J" h n ( X U C *A ) , 5, hn_ 1 (A) ~-.

.

.

Here i" (A, x0) ~-~ (X, x0) and j" ( X , xo) ~ ( X U C ' A , .) stands for the reduced cone, i* = hn([i]) and j* - hn([j]). 5n coincides with the composition k* 9 v* 9 a n - l , where inverse in the H-cogroup and k" X U C * A ~ E*A shrinks Applying the above exact sequence to ({x}, {x},x) we for each n. The following lemma is also obvious.

.

denote inclusions, C* The homomorphism v" E*A --, E* is the X into a point. see that h n ( { x } ) -- 0

LEMMA 3.2.6. I f a map f " X ---. Y is homotopic to a constant map, then the h o m o m o r p h i s m f*" h*(Y)---, h * ( X ) is trivial. One of the major ways of defining the generalized cohomology theories is based on the so called spectra. The spectrum E is a collection {(En, .)" n e Z} of (:W-complexes such that E * E n C_ En+l. The cofunctor h* can then be defined by letting hn(.) : li__,m[)-]k.;enWk]. It is known t h a t under some restrictions, all generalized cohomology theories can be obtained as indicated above 2. 2This fundamental fact was proved by Brown, see [291].

3.2. CELL-LIKE MAPPINGS RAISING DIMENSION

111

The Unreduced generalized cohomology theory k* on the category T 2 of pairs of topological spaces is a sequence (k n" 7"2 --. Jr, n E Z} of contravariant functors and natural equivalences {6 n" k n - 1 . R -+ kn, n e Z} (here R ( X , A ) = (A,O)) satisfying the following axioms: (1) Values of k* are the same for homotopic maps. (2) For each pair ( X , A ) , the following sequence is exact .. k n ( X , 0) ' j. 9" ~- k n + l ( X , A ) ~n+l , kn(A, 0) ~L_

k n ( X , A) ~ . . . 9

(3) For each pair ( X , A ) , and for any set U with V C_ clU C intA, the homomorphism induced by the inclusion ( X - U , A U) r (X, A) is an isomorphism. Conditions (1)-(3) are the well known Eilenberg-Steenrod axioms 4-6 (see [290]). Axioms 1-3 of Eilenberg-Steenrod are contained in the above definition. Generally speaking, it is not required that axiom 7 of Eilenberg-Steenrod be satisfied. This is exactly why these cohomology theories are called generalized. Below we use the notation k* (X) = k* (X, 0). The following statement is well known and can be found in [291]. LEMMA 3.2.7. For a cellular triad ( X ; A , B ) etoris exact sequence 9.. r

k n ( A n B) r

we have the following Mayer-Vi-

k n ( A ) @ k n ( B ) ~-- k n ( X ) ~-- k n - l ( A n B ) ~---'"

and this sequence is functorial.

As usual X -- A U B and (A, B) is an ordered pair. Each reduced cohomology theory h* defines a corresponding unreduced cohomology h* simply by letting hn(X,A)--hn(XUC*AUpt).

Here an isolated point pt is taken as the base point. If A ---- 0, then h n ( X ) -hn(ZUpt). LEMMA 3.2.8. Let f " ( Z , x) ~ (Y, y) be a continuous map. Then the following conditions are equivalent: (i) f*" h*(Y)--~ h * ( X ) is an isomorphism. (ii) S*" h * ( Y ) ~ h * ( X ) is an isomorphism. PROOF. Our statement follows from the five lemma, applied to the diagram generated by the exact sequences of pairs (X U pt, (x} U pt), (Y U pt, (y} U pt) and their transformation f U idpt. Wi Let U be an infinite unitary group and B U its classifying space. It is known [291] t h a t B U is a C~A)-complex. Define a spectrum E by letting E2n -- B U and E2n+l : U for each n E Z. Bott periodicity implies that E is a spectrum.

112

3. COHOMOLOGICAL DIMENSION

The corresponding reduced generalized cohomology theory is known to be the reduced complex K - t h e o r y / < ~ (see [291] for details). Let B 2 be the Moore space, that is B 2 = SI[.J2 B2 is the circle with an attached (via a map of degree 2) 2-dimensional cell. By A we denote the smash product. Recall that x A y = (x • Y ) / ( z • {yo})u ({~o} • Y), where xo and yo are base points in the spaces X and Y respectively and (X •

{y0})tO({x0} x Y) defines the base point in X AY. The K-functor with coefficients in Z2 is defined as follows: / ~ 5 ( X ; Z2) - / ( 5 ( X

A B2).

Observe that K~(.; Z2) is the reduced generalized cohomology theory. The following two statements play a key role. PROPOSITION 3.2.9. Let X = UX n be an arbitrary filtration of a C~4;-complex X by its subcomplexes. Then K~ c* ( X ; Z 2 ) : l i m K~ C* ( X n ; Z2).

PROPOSITION 3.2.10. If r is a natural number, then K-* c ( K ( O r1Z, 3) ; Z2) = 0.

LEMMA 3.2.11. The map Cr" E(T) --~ L from L e m m a 3.2.4 induces an isomorphism (r

gc~*(L ; Z2) ~ k ~ ( E ( r ) ; Z2).

PROOF. Let us show that each continuous map f" E(T) --, L, satisfying conditions (i) and (iii) of Lemma 3.2.4, has the required property. We use induction on the dimension of L. If dim L = 0, then L consists of finitely many points, and for each of these points x, we have f - l ( x ) "~ K(@Z, 3). Therefore the required conclusion follows from Proposition 3.2.10. Suppose that our statement has already been proved for polyhedra M with dim M _< n. Consider an (n + 1)-dimensional compact polyhedron L with triangulation T. We now proceed by induction on the number of (n + 1)-dimensional simplexes of T. If there is no (n + 1)-dimensional simplex, then dim L < n. Let L contains m simplexes of dimension n + 1. Take one of these simplexes a, and denote by M the polyhedron L - int a. Consider the restriction g of the map f" E ( T ) --+ L to the subspace f - l ( M ) and the restriction q of the map f to the subspace f - l ( o a ) . By the inductive assumption, the homomorphisms g* and q* induce isomorphisms of the corresponding K ~ ( . ; Z 2 ) groups. Since f - l ( a ) ~ g (@[Z, 3), we conclude, by Proposition 3.2.10, that the restriction s of the map f to the inverse image f - l ( a ) induces a trivial isomorphism 9 9 kh(~;z2)

~

K- c* ( f

-1(o'); Z2).

3.2. CELL-LIKE MAPPINGS RAISING DIMENSION

113

Lemma 3.2.8 guarantees that the homomorphisms t)*, q* and ~* induce isomorphisms of the corresponding reduced K-groups. Applying L e m m a 3.2.7 to the triads ( L ; a , M ) and ( E ( T ) ; f - l ( a ) , f - l ( M ) ) w e obtain the following commutative diagram: ...+--hi(f-l(O~r))~ ---- h i ( f - l ( c r ) ) (~ h i ( f - l ( M ) )

" +-hi(Oh)~

*-- h i ( E ( T ) ) .-- h i - l ( f - l ( O c r ) ) e - -

h'(L)-,

hi(a) EDh i ( M ) ~

h~-l(0a)e--

Here h*(-) = /r The five lemma implies that f* is an isomorphism. It only remains to remark that, by Lemma 3.2.8, f* is also an isomorphism. U1 The following statement follows immediately from Definition 3.2.1. LEMMA 3.2.12. Let f " X ~ L be a map, d i m z ( f , r) <_ n, and Y C X . d i m z ( . f / Y , T) <_ n.

Then

LEMMA 3.2.13. Let L be a compact polyhedron with triangulation T. I f c~ is a non-trivial element of the group K ~ ( L ; Z 2 ) , then there exists a compact polyhedron xa C E ( T ) such that ( r -7/=0. PROOF. By condition (ii) of Lemma 3.2.4, there is a filtration X 1 C X 2 C X 3 c_ . . . of the complex E(~-), consisting of finite subcomplexes. By L e m m a 3.2.11, r --/: 0. By Proposition 3.2.9, there exists an index i such that 7"(r ~ 0, where 7~ denotes the homomorphism induced by the inclusion 7i" X i r E(T). Let Xa = X i. It only remains to remark that ( r

Now we are ready to prove the main result of this Section. THEOREM 3.2.14. There exists an infinite-dimensional compactum X that dim Z X = 3.

such

PROOF. We construct the c o m p a c t u m X as the limit space of an inverse sequence ,S = {X~ ,Pii+1lj consisting of compact polyhedra. The spectrum 8 itself is constructed by induction and realized in the Hilbert cube I ~ . Represent the Hilbert cube I ~ as an infinite product I~ x I ~

•

•

x...

,

114

3. COHOMOLOGICAL DIMENSION

where I~' -- I ~~ for each n E w. T h e polyhedron X n will be constructed as a subspace of the s u b p r o d u c t

I~ •

•215

• {0} • 2 1 5

{0}

•

where 0 E I ~ is the base point. In addition, the e m b e d d i n g of Xn into I ~' will be chosen so t h a t the restriction of the projection

~+1.

n+l n H IW --'+ H Iw i=0 i--0

to Xn+x coincides with pnn+l. Clearly, in this situation, X = l i m b C_ I ~ and l i m i ~ pi = i d x , i.e. the sequence 8 is realized in the Hilbert cube I ~ . Take any metric d on I w , and define a sequence {Ti} of triangulations of the polyhedra X~ so t h a t limi__,~ mesh(Ti) = 0. Further, by induction, we define a sequence {c~} of non-trivial elements a~ E K ~ ( X ~ ; Z 2 ) . As the polyhedron Xo we take the 4-dimensional sphere S 4. E m b e d Xo into I~' x { 0 } . . - x {0} x - - . , and take a triangulation ro of Xo with mesh(To) < 1. As the element c~o take a generator of the g r o u p / ~ ( $ 4 ; Z 2 ) = Z2. Suppose t h a t the polyhedra Xi, projections Pi-1, i triangulations r~ of X~ and non-trivial elements ai E K ~ ( X i ; Z 2 ) , satisfying the above conditions, have already been constructed for each i < n. By L e m m a 3.2.13, there exists a compact polyhedron Xa,, C_ E(Tn) such t h a t ( r # O. Let Xn+I = Za,,, 6g pn+l = CTn/Xa,~ and c~n+l = (pnn+l)*(an). E m b e d Xn+l into In+ 1 a n d consider the graph of pnn+l in the product I~~ x "'" x In+ 1 x

{0} • 2 1 5

{0}

•

This graph is the desired e m b e d d i n g of Xn+I into I ~' . Finally, we take any 1 triangulation of Xn+I with mesh(Tn+l) < n+l" By condition (iv) of L e m m a 3.2.4 and by L e m m a 3.2 912, d l"m z ( p nn+l , T n ) < _ 3 for each n E w. L e m m a 3.2.2 guarantees t h a t dim Z X < 3. It is also obvious (see R e m a r k 3.2.5) t h a t dim Z > 3. Therefore, dim Z = 3. Let us verify t h a t X is infinite-dimensional. Assume the contrary. Then, by T h e o r e m 3.1.5, dim X = 3. Consequently, the limit projection po" X ~ Xo = S 4 is homotopic to a constant map. T h e n we can find an index i such that the short projection p~" Xi ~ X0 of the s p e c t r u m S is also homotopic to a constant map (see, for example, [290]). By L e m m a 3.2.6, the h o m o m o r p h i s m

(p~)* 9 ~ 5 ( x 0 ; z2) -~gc-* (x~ ; Z~) is trivial. On the other hand, by the above construction, (p~)*(a0) --- c~i =fi 0. This contradiction finishes the proof, i--I T h e specific n a t u r e of K - t h e o r y plays no direct role in the above construction. T h e essential feature is t h a t it is a generalized cohomology theory for which the complex K ( Z , 3) behaves as a point. T h e absence of a generalized cohomology

3.2. CELL-LIKE MAPPINGS RAISING DIMENSION

115

theory for which the complex K ( Z , 2) is known to behave as a point requires an alternate approach in order to produce an infinite-dimensional c o m p a c t u m having integral cohomological dimension equal to 2. Such an approach, based on the validity (see [230]) of the following Sullivan Conjecture, was carried out in [136]: 9 If G is a locally finite group (i.e., each finitely generated subgroup is finite) and L is a finite-dimensional gl/Y-complex, then the space of pointed maps from K ( G , 1) ~ L has trivial homotopy groups. This leads to the following result. THEOREM 3.2.15. There exists an infinite-dimensional c o m p a c t u m X

such

that dim Z X = 2.

Theorems 3.1.10 and 3.2.15 imply the following. COROLLARY 3.2.16. There exists a cell-like map f : X ~ Y between compacta X and Y f o r which d i m X = 2 and d i m Y = c~.

COROLLARY 3.2.17. There exists a cell-like map f : 15 ~ Y , where dim Y = (X3

.

PROOF. E m b e d the c o m p a c t u m X from Corollary 3.2.16 into the cube 15. Then consider the quotient space of 15 with respect to the decomposition, whose non-trivial elements are the fibers f - l ( y ) , where f : X ~ Y denotes the map from Corollary 3.2.16. f-1 It is known (see [307]) t h a t cell-like maps defined on the 3-dimensional cube cannot raise dimension. Thus, the following problem remains open. PROBLEM 3.2.18. Can cell-like maps defined on the 4 - d i m e n s i o n a l cube raise dimension? We conclude this Chapter by showing t h a t cell-like images of A N R - c o m p a c t a are not necessarily A N R-spaces. First we need the following concept. DEFINITION 3.2.19. We say that a map f : X --, Y is approximately invertible if f o r each open c o v e r / 4 E c o v ( Y ) there is a map gu: Y ~ X such that the composition f gu is ~4-close to the identity map i d y . LEMMA 3.2.20. I f f : X dim Y.

--~ Y

is approximately invertible, then dim X

PROOF. Consider an arbitrary open cover/4 E c o v ( Y ) . Denote by ~) a starrefinement of Y, and take a map g: Y --. X such that the composition f g is ])-close to i d y . Consider the open cover /-~(v)

= {/-~(w): w e v}

116

3. COHOMOLOGICAL DIMENSION

of the space X. Take an open refinement )/Y E c o y ( X ) of f - l ( V ) m o s t dim X + 1. Obviously, the order of the cover g-l(~v)--

{g-l(w):

of order at

W e ~)}

is also at most dim X + 1. Consequently, it suffices to show t h a t g-10/Y ) refines L/. In turn, in order to show this fact it is sufficient to show t h a t g - l f - l ( v ) refines b/. Thus, let V E V and y E g - l f - l ( v ) . Since the maps f g and i d y are V-close, there exists an element Pry E V such t h a t b o t h points f g ( y ) and y are c o n t a i n e d in Vy. Since y E g - l f - l ( v ) , we see t h a t f g ( y ) E V and, consequently, Vy N Y ~ q}. This implies t h a t g - l f - l ( Y ) c_ St(V, V). WI COROLLARY 3.2.21. There are non cell-like images of the 5-dimensional cube that are not A N R's. PROOF. Let f" 15 ~ Y be the m a p from Corollary 3.2.17. Assuming t h a t X is an A N R - c o m p a c t u m , by L e m m a 3.2.20 and P r o p o s i t i o n 2.1.29 we conclude t h a t d i m Y < 5. This is a contradiction. [:] m

Historical and bibliographical notes 3.2. This Section contains the solution of the C E - p r o b l e m given by Dranishnikov [130]. P r o p o s i t i o n s 3.2.9 and 3.2.10 a p p e a r e d in [59]. T h e o r e m 3.2.15, as well as Corollaries 3.2.16, 3.2.17, were proved by D y d a k and Walsh [136].

3.3. U n i v e r s a l s p a c e for c o h o m o l o g i c a l d i m e n s i o n In this Section we show t h a t there exist universal spaces with respect to integral cohomological dimension.

3 . 3 . 1 . (Z, n ) - i n v e r t i b l e a n d ( Z , n ) - s o f t m a p s . THEOREM 3.3.1. There exists a Polish subspace Y C_ D ~ • I ~ satisfying the

following conditions: (i) dim Z Y = n ; (ii) For every G~-subset Y~ C I W with dim Z Y ~ n, there exists c E D "~ such t h a t ( I ~ • { c } ) N Y = Y ' • {c}. PROOF. Let c~: M ---. I ~ be an n-invertible m a p of an n-dimensional comp a c t u m M onto the Hilbert cube I ~ . T h e results of C h a p t e r 4 show t h a t c~ has the following property: 9 For any m a p g: Z ~ I W of an at most n - d i m e n s i o n a l separable metrizable space Z, there is an e m b e d d i n g i: Z ~ M such t h a t c~i - g.

3.3. UNIVERSAL SPACES Let ~" D W - ~ e x p M

be a surjection, where e x p M

117

denotes t h e h y p e r s p a c e of M .

Consider the s u b s p a c e

A=U{{c} xfl(c)'cEDW } of the p r o d u c t D W x M

and define t h e m a p # "

It(C, x ) = (c, a ( x ) )

A - - ~ D ~ x I W by l e t t i n g

for each (c, x) e A.

Let Y =- { y E D W x I W " S h ( # - l ( y ) ) = . } .

It can easily be seen t h a t Y is a G~-subspace of t h e p r o d u c t D ~ x I W . Clearly dim#-l(Y)

_< d i m A _< d i m ( D ~ x M ) = n.

Therefore, by t h e n o n - c o m p a c t version of P r o p o s i t i o n 3.1.4, dim Z Y = n (observe t h a t by the definition of Y, t h e restriction # / # - l ( y ) . # - l ( y ) __. y is a p r o p e r CE-map). Let us now show t h a t if Y~ is a G s - s u b s e t of I W t h e n t h e r e is a point c E D W such t h a t ( I W • { c } ) A Y = y ' • {c}. Using once again the n o n - c o m p a c t version of P r o p o s i t i o n 3.1.4, we have an at m o s t n - d i m e n s i o n a l Polish space X ~ a d m i t t i n g a p r o p e r C E - m a p f~" X ~ -~ Y~ onto Y~. Choose an e m b e d d i n g i ~" X ~ -~ M such t h a t a i ~ = f~. W i t h o u t loss of generality, we m a y a s s u m e t h a t i ~ = i d , i.e. X ~ C_ M . Finally, choose c E D ~ w i t h c l X ~ = f~(c). A m a p f 9 X -~ Y is called ( Z , n ) - i n v e r t i b l e

[-1

provided t h a t for each m a p

g 9 Z -~ Y with d i m Z Z _ n, t h e r e exists a m a p h 9 Z --~ X such t h a t f = g h . COROLLARY 3.3.2. T h e r e e x i s t s a ( Z , n ) - i n v e r t i b l e Y

map

f

9Y

---+ I W , w h e r e

is a P o l i s h s p a c e w i t h dim Z Y _< n.

PROOF. Let Y C I W x D W be t h e space from T h e o r e m 3.3.1 a n d f " Y --. I W be the restriction of t h e p r o j e c t i o n m a p ~1 " IW x D W --. I W . Split I W into t h e p r o d u c t , I W ~- I W x I W , and let F be t h e c o m p o s i t i o n Y

]

......

~I W

--

,.- I W x I W

7rl

~

IW

.

['-'],,

T h e following definition is a n a t u r a l c o u n t e r p a r t of t h a t of n-soft m a p (Definition 2.1.33). DEFINITION 3.3.3. A m a p f " commutative

diagram

X

~

Y

is c a l l e d (Z, n)-soft p r o v i d e d

for

each

118

3. COHOMOLOGICAL DIMENSION

A

Z

~X

r

,-Y

where A is a closed subset of a space Z , dim Z _ n, there exists a m a p (h 9 Z -~ X such that f ~ = r and O]A = r

DEFINITION 3.3.4. A space X is called an absolute extensor in integral cohomological dimension n (briefly, X E A E ( Z , n ) ) provided f o r every map f " A --, X defined on a closed subset of a space Y , d i m z <_ n, there exists an extension of f to the space Y . Obviously, X E A E ( Z , n) if the constant map X --~ {x} is (Z, n)-soft. PROPOSITION 3.3.5. Let f " X - , Y be a (Z, n ) - s o f t map, dim Z X Y CAE. ThenX EAE(Z,n).

<__ n and

LEMMA 3.3.6. For any closed subset A, dim Z A <_ n, of a Polish space B there exists a ( Z , n ) - i n v e r t i b l e map g" K - , B of a Polish space K , dim Z K <_ n, such that g / g - l ( A ) " g - l ( A ) - ~ A is an embedding.

PROOF. Let f 9 X --~ B be a (Z, n)-invertible map of a Polish space X with dim Z X ~ n. E m b e d f into a map f" .~ - , B of compact spaces, and let G be the decomposition of )( into singletons and sets of the form ] - l ( a ) , a E A ( the closure of A in /~). Let q ' ) ( --, ) ( / G be the quotient map and f = g'q a factorization of f . We can take K ---- q ( X ) and g -- g ' / K . E] PROPOSITION 3.3.7. Let n E w. There exists a (Z, n ) - s o f t m a p f " X --~ I ~ , where dim Z X < n.

PROOF. Let s denote the pseudo-interior of the Hilbert cube I ~ . Let ql " )~1 --~ I W = Z0 be a (Z,n)-invertible map of a Polish space -~1 with dim Z ) ( 1 = n (apply Corollary 3.3.2). T h e r e exists a closed embedding )(1 ~-~ I ~ • s = Zx such t h a t ql = 7r01/)(1 (r~ denotes the projection of I W • s onto the first factor). By L e m m a 3.3.6 there exists a (Z, n)-invertible map q2" )(2 -* Z1 of a Polish space -X1 with dim Z ) ( 2 = n such t h a t the restriction q2/q21(f(1)

" q2--1(.,~'1) --+

.z~"1

is a h o m e o m o r p h i s m . Let el" )~1 --* -~2 be a m a p such t h a t q2el = i d f c 1.

3.3. UNIVERSAL SPACES

119

P r o c e e d i n g as a b o v e we o b t a i n a c o m m u t a t i v e d i a g r a m

Z0 -~

Z1 -q

X1 -4

el

X2 -~ -

7r21

Z2 -,

e2

X3 -4

e3

...

w h e r e Zi - I ~ • s x .-- • s (i + 1 factors), ~r~+1" Z i + l -* Zi are t h e p r o j e c t i o n s o n t o t h e initial i factors, )(i is a closed s u b s p a c e of Zi w i t h d i m Z )~i = n, a n d qi are (Z, n ) - i n v e r t i b l e m a p s such t h a t t h e r e s t r i c t i o n s

qi/qi -1(-~i-1) " q~-1(.~i_1) --+ ) ( i - 1 are m a p s such t h a t q i + l e i + l = i d a . C o n s i d e r t h e inverse s y s t e m S = {Zi, ~ + 1 } a n d let ~0 " l i m S --~ Zo = I ~ be t h e limit p r o j e c t i o n . Let Xi={(xj)~=

1" x i e ) ~ j

and xj+l=ei(xj)

for e v e r y j k i } C l i m S .

Also let X=U~=IXi

and f==0/X'X---.Z0=I

~.

Obviously, t h e X i are closed s u b s p a c e s of X , d i m Z X i = n, a n d hence, by t h e C o u n t a b l e S u m T h e o r e m , d i m Z X = n. To s h o w t h a t f is a ( Z , n ) - s o f t m a p , c o n s i d e r a m a p ~/ 9 D ~

Z0, w i t h

d i m z D _< n, a n d a m a p a - C --~ X defined on a closed s u b s e t C of D such t h a t

fa = 7/D. Ci = a - l ( x i )

L e t a = (ai)~__l, w h e r e a i " C --~ Zi are t h e c o o r d i n a t e m a p s . L e t a n d let D1 C_ D2 C . . . be a s e q u e n c e of closed s u b s e t s of D such

t h a t Di N C -- Ci a n d D = U oio= l D i . W e a r e g o i n g to c o n s t r u c t t h e m a p f~" D --~ X in t h e f o r m ~ = (f~J)~=0, w h e r e f~j 9 D --. Z j a r e t h e c o o r d i n a t e m a p s . L e t f~0 = 7. Since s E A E , t h e r e exists a m a p ~1" D -+ Z1 such t h a t ~ 1 / C = a l a n d ~r~f~l = f~0 = 7. A s s u m e t h a t for e v e r y j _ k, a m a p f~j" D --~ Z j is defined in such a w a y t h a t ~ j / C = a j a n d ~j_lf~jJ = f~j-1.

Since t h e m a p qk+l is ( Z , n ) - i n v e r t i b l e ,

t h e r e exists a m a p gk" Dk --+ X k + l such t h a t qk+lgk -- ;3k/Dk.

gk/Ck-

ek/Ck-

OLk+l/Ck and, t h e r e f o r e , t h e m a p aa+l U gk " C U Dk

--~

)(k+l

Note that

120

3. COHOMOLOGICAL DIMENSION

is well-defined. Since the m a p rr~ +1 is soft, there exists a m a p ilk+l" D -+ Z k + l _k+l,~ such t h a t "k pk+l = /~k and ~ k + I / ( C U D k ) = ak+lUgk. The map /~ = /~ c~ . ( j)j=o D ~ X is well-defined,/~/C = a a n d / / ~ = -y. F-! We need a slightly stronger version of the above s t a t e m e n t . PROPOSITION 3.3.8. For any closed subset A of a separable metrizable space Y such that dim Z A <_ n, there exists a ( Z , n ) - s o f t m a p f " X -+ Y such that dim Z X < n and the restriction f / f - l ( A ) "

f - 1 (A) --+ A is a h o m e o m o r p h i s m .

PROOF. Let / " X ' --+ Y be a ( Z , n ) - s o f t m a p with dim X ' < n. E m b e d f ' into a m a p /" )( --+ I7" of c o m p a c t a and let G be a decomposition of )( into singletons and the inverse images of the form ] - l ( a ) , a E A (here ,a denotes the closure of A in 17"). Let q" )( --+ ) ( / 6 denote the quotient m a p and ] = rq the factorization of f . Set X = q ( X ' ) and f = r / X " X -+ Y . Note t h a t dim Z X < n and f / f - l ( A ) " f - l ( A ) --+ A is a homeomorphism. Consider a c o m m u t a t i v e diagram

B

-

*-Z

Z

r

,-Y

where B is a closed subset of a separable metrizable space Z with dimTz Z < n. We add the set r and its derivatives to the topology of Z. The space thus o b t a o n e d is denoted by Z r and its underlying set is naturally identified with Z. Since f ' is (Z, n)-soft, there exists a m a p r

B Nr

such t h a t f'~0 - f ~ o / ( B N r

Obviously, there exists a map ~" B -

such t h a t q~3 = ~o/(B - r map

(A) --+ X '

r

X'

T h e maps ~o and ~ uniquely determine the ~ = ~o tO ~ " B r -+ X '

(here B r denotes the set B with the topology inherited from Zr).

3.3. UNIVERSAL SPACES

121

Since f ' ~ = r by t h e (7/,, n)-softness of f ' there exists a m a p ~," Z r --+ X ' such t h a t ~ / B r -- ~ and f ' ~ = r T h e n the m a p 9 = q~" Z --+ X is continuous and =q%a=%o a n d f r 1 6 2

9 /B=q~/B

Hence, the m a p f is (Z, n)-soft.

FI

PROPOSITION 3.3.9. L e t f " X ~ n. f:

There

exists a complete

)(--~ I ~

space f(

such that f /X=

I W be a (Z, n ) - s o f t D X,

m a p , w h e r e dim Z X _ dim~, X <_ n a n d a ( Z , n ) - s o f t m a p

f.

PROOF. We m a y assume t h a t X C I W x I ~ and f = l r l / X :

X

--+ I ~~ where

rrl : I ~~ x I W --+ I ~ denotes the projection onto the first coordinate. We identity I W x I W with the s u b s e t I W x I W x {0} c I Claim.

Let K

W xI W xI W.

C I W x I W C I W x I W x I W a n d L K -- ( I w x I w x I w ) --

(I w x I W x { 0 } - - g ) . T h e n t h e m a p l r l / L K : n K + I W is s o f t . P r o o f of Claim. Consider a c o m m u t a t i v e d i a g r a m

A

'-LK

7rl/LK

B

42

,..i W

Let a = ( c ~ i ) ~ l " B --+ I W be a m a p such t h a t o l - l ( { 0 } )

-- A (here, I W -

oo

1--[i=1 Ii, Ii -- [0, 1], and hi" B ~ Ii is a c o o r d i n a t e map, i E N ) . Let ~'--

( ~ ' 1 , 0 ' 2 , 0 ' 3 ) " B --+ I W x I W • I W

be a m a p such t h a t O ' / A = ~o a n d 7rl (I)! = r Let ~ = ( O ~ i ) ~ l , where O~i" B --+ Ii is a c o o r d i n a t e m a p for each i e N . Put r

= (r

oo

where ~3i(b) = m i n { ~ i ( b ) + c~(b), 1}.

Now we can take r - ( ~ , ~ , r T h e claim has been established. Let X C X ' C I W x I W C I W x I W x I W , where X ' is a Polish space, dim Z X ' = n , a n d X D X ' (the c l o s u r e X is taken i n I W x I W x I W ) . Let g" Y ' --+ L x , be a (7/,, n)-soft map, where d i m z Y' - n a n d

gl~ - * (x'). G -~ (x')

-+

x'

is a h o m e o m o r p h i s m . Let Y = g - l ( L x ) . Consider the c o m m u t a t i v e d i a g r a m

122

3. COHOMOLOGICAL DIMENSION

g/g-l(x) g-~(x)

Y

~ x

(lrl/Lx)(g/Y)

.io. ,

B y the (Z, n)-softness of f , there exists a m a p r : Y ~ X such t h a t r / g - l ( X ) g / g - l ( x ) and f r (rl/ix)(g/Y). By t h e Lavrentieff theorem, there exist Ga-subsets )( C X ~ and 1~ c Y~ and a m a p ~: 1) - , )( such t h a t ~ / Y - r. Obviously, we may assume t h a t Y D g - l ( . ~ ) (if not, take I7" -- Y U g - l ( ) ~ ) ) . We shall show t h a t the m a p ] - r l / ) ( : )( --* I ~ is (Z, n)-soft. Let ~o A

Z

~X

r

,-Y

be a c o m m u t a t i v e diagram, where A is a closed subset of a space B and dim Z B < n. By the Claim, there exists a m a p fl" B ~ L 2 such t h a t f l / A = (p and ffl -- r F u r t h e r , by the ( Z , n ) - s o f t n e s s of g t h e r e exists a map 7" B ~ Y~ such t h a t 7 / A = g - l ( p and g7 = f~. Obviously, 7(A) C Y and the m a p (I)= ~7 satisfies the conditions r = ~ and ] ( I ) = r K]

COROLLARY 3.3.10. For every Polish space Y , there exist a Polish space K with dim Z K <_ n and a (Z, n)-soft map g" K --, Z . PROOF. Consider Z as a subset of I ~ a n d let K -- ] - I ( Z ) Z.

and g - ] / K " K ---.

[3

3.3.2. Universal space for c o h o m o l o g i c a l d i m e n s i o n . Let A41(Z,n) denote the class of Polish spaces X with dim Z X < n.

3.3. UNIVERSAL SPACES

123

LEMMA 3.3.11. Let X -- lim{Xi,p{+l}, where the maps p{+l are surjections and dim Z X i <_ n. Then dim Z X _ < n . PROOF. Let ~., be an equivalence relation on X x [0, 1] defined as follows: 1 (x,t) ~ (y,t') if t -- t' and either x - y or 71 < t - t t -< /--4-Y a n d p i i-+-I (x) --

p{+l(y). Let q" X x [0, 1]--, (X x [0, 1])/.~ = Y denote the quotient map. with q(Xi x { 1}).

We identify X with the subset q(X x {0}) and Xi

Let f " A --+ K ( Z , n) be m a p defined on a closed subset A of X . Assuming K ( Z , n) to be an absolute neighborhood extensor, we can obtain an extension f" U --+ K ( Z , n) of f to a neighborhood U of A in Y (recall t h a t X C Y). Let V be a neighborhood of A with c l V C U. There exists a continuous function ~" A --+ (0, 1] such t h a t _

1

d(q(cp(a)), Y - V) < -~d(a, Y - V) for every a E A (d denotes a compatible metric on Y) and such t h a t the m a p

g = fq(idA x ~o) is homotopic to f . Let ~" X -~ (0, 1] be an extension of ~o. T h e n the space q(idx x ~ ) ( X ) is a countable union of closed subspaces which are h o m e o m o r p h i c to closed subsets of spaces Xi, i E N. Thus, dim Z q(idx x ~ ) ( X ) < n and the m a p _

f /(idA x ~)(A)" (idA x ~)(A)---+ K ( Z , n ) has an extension f" (idx x ~ ) ( X ) ---+K ( Z , n ) . Therefore, the m a p

f q(idA X ~)" A ---+K ( Z , n) has an extension, namely,

gq(idx • ~) " X -+ K (Z, n). By the H o m o t o p y Extension Theorem, the m a p f" A --~ K ( Z , n), which is homotopic to f q ( i d A • ~0), can be e x t e n d e d over X. [:] THEOREM 3.3.12. There exists a space Y(Z, n) satisfying the following prop-

erties:

(i) Y(Z,~) e M~(z,~). (ii) Y ( Z , n) E A S ( Z , n). (iii) Y (~'., n) is strongly .h/[ 1 (Z, n)-universal, i.e. for any Polish space C with d i m z C <_ n, the set of closed (even Z-) embeddings is dense in the space

c(c, Y(Z,n)).

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3. COHOMOLOGICAL DIMENSION

PROOF. We shall define an inverse sequence ,9 - (Xi, ~'i-i+l} by induction. Let X0 be a singleton. Assuming that the spaces X~ and maps Pi-1 i have already been defined, let f 9 X~+I --* Xi • s (here s denotes the pseudo-interior of I ~ ) be a (Z, n)-soft map of a space X~+l E A/II(Z, n) (see Corollary 3.3.10) and let pi+l i lrlf. Now let Y ( Z , n ) - l i r a S . By L e m m a 3 . 3 . ~ , d i m Y ( Z , n ) - n. Since all the bonding projections are (Z, n)-soft maps, we easily conclude that Y(Z, n) E A S (Z, n) z (compare with Lemma 6.2.6). Note t h a t the bonding projections satisfy the following property" 9 For each Z E A~II(Z, n) and each map g" Z -~ Xi+I there exists a closed embeddingg'" Z~Xi+I such t h a t p ~i + 1 g = P i i + 1 g t. Following the argument of Proposition 5.1.11, it can now be shown that the limit projection pl " Y (Z, n) satisfies the following fiberwise universality property: (.) For every space Z 6 A/II(Z,n), every map g ' Z ~ Y ( Z , n ) and every open cover U 6 c o v ( Y ( Z , n ) ) there exists a closed embedding g'" Z --~ Y (Z, n) which is U-close to g and such that p i g - p i g ' . In fact, slightly modifying the argument we may assume that g' is even a Zembedding. [3 R e m a r k 3.3.13. Comparing the properties of the space Y (Z, n) from Theorem 3.3.12 with the properties of the universal n-dimensional Nhbeling space -'7% ]V 2 n + l we can see t h a t the space Y ( Z , n ) plays the similar role with respect to the cohomological dimension as that o f /--7% v 27%+1 with respect to the usual dimension.

It is still unknown whether there exists a compact space universal in the class of spaces with dim Z < n. The space Y in Theorem 3.3.2 cannot be made compact. Indeed, we have the following statement. PROPOSITION 3.3.14. T h e r e is no (Z, n ) - i n v e r t i b l e m a p f " Y ~ I ~ where Y is a c o m p a c t u m with dim Z Y < n. I n o t h e r words, f o r a n y m a p f " Y --~ I ~ of a c o m p a c t u m Y to the Hilbert cube I W s u c h t h a t dim Z Y < n, there is a c o m p a c t u m X C I "~ s u c h t h a t d i m Z X < n

a n d the i n c l u s i o n X

~I

~ does n o t lift t o Y .

PROOF. Choose a C)4;-complex K which is a K (Z, n) and has infinite skeleta. Let X k C_ I ~ be compacta with the following properties" (a) dim Z X k <_ n for each n E w. (b) Each X k contains the same copy of the n-dimensional sphere S7%. (c) The inclusion S n ~ K cannot be extended over Xk so that the image of the extension is contained in the k-dimensional skeleton K (k) of K. Suppose now t h a t there is a (Z, n)-invertible map f" Y --* I ~ where Y is a comp a c t u m with dim Z Y <__n. Choose an extension g" Y --, K of f - l ( s n ) --~ S7% K. Then there exists an integer k such that g ( Y ) c_ K ( k ) , which demonstrates t h a t the inclusion X k "-* I ~ cannot be lifted to Y. This contradicts with the choice of f. [3

3.3. UNIVERSAL SPACES

125

Historical and bibliographical notes 3.3. Theorem 3.3.1 was in fact obtained in [i34]. All of the other results of this Section appear in [326]. The existence of a Polish space universal in the class of Polish spaces with dim Z _ n (a weaker version of Theorem 3.3.12) was shown earlier in [134] (see also [96]). The compacta X k used in the proof of Proposition 3.3.14 were constructed in [i37].

CHAPTER 4

Menger Manifolds

In this Chapter we present a survey of Menger manifold theory.

4.1. G e n e r a l T h e o r y Before we start a detailed discussion, let us outline the construction of the kdimensional universal Menger compactum tt k, k > 0. We partition the standard unit cube I2k+l, lying in (2k + 1)-dimensional Euclidean space R 2k§ into 32k§ congruent cubes of the "first rank" by hyperplanes drawn perpendicular to the edges of the cube 12k+1 at points dividing the edges into three equal parts, and we choose from these 32k+l cubes those which intersect the k-dimensional skeleton of the cube 12k+l. The union of the selected cubes of the "first rank" is denoted by I(k, 1). In an analogous way, we divide every cube entering as a term in I(k, 1) into 3 2k+1 congruent cubes of the "second rank" and the union of all analogously selected cubes of the "second rank" is denoted by I(k, 2). If we continue the process we get a decreasing sequence of compacta

I(k, 1) D I(k,2) D - - . . The compactum #k = N{I(k,i)" i E N } is called the k-dimensional universal Menger compactum. Note that #k - - IV1 . k2k+l , where Mk, n 0 ~ k < n, denotes the "k-dimensional Menger compactum constructed in the n-dimensional cube I n,' (a precise definition of M~ is given below). Obviously, #0 coincides with the Cantor discontinuum D ~ and, consequently, is the only zero-dimensional compactum with no isolated points (Theorem 1.1.10). It was shown by Sierpinski [285] that #0 is a universal space for the class of all zero-dimensional metrizable compacta. Positive dimensional Menger compacta M~ were originally defined within the classical dimension theory by Menger in [221]. They are generalizations of the Cantor discontinuum and Sierpinski's universal curve M 2 [284]. Let us recall that the space M12 is universal for the class of all at most 1-dimensional planar compacta [284]. Further, it was shown by Menger [221] that the 1dimensional Menger compactum #1 = M 3 is universal for the class of all at 127

128

4. MENGER MANIFOLDS

most 1-dimensional compacta. Generally, it was conjectured [221] that M ~ is a universal space for the class of all at most k-dimensional compacta embeddable in R n (Menger's problem). As was already mentioned, this problem was known to have a positive solution for (k, n) = (1, 2) and (k, n) = (1, 3). A positive solution in the case k -- n - 1 was also given by Menger [221]. Results of Lefschetz [209] and Bothe [48] produced a positive solution in the case n = 2k + 1. The ultimate affirmative solution of Menger's was obtained by Stanko [289]. The topological characterization of #0 has already been mentioned. Note also that #0, being a topological group, is homogeneous. The 1-dimensional Menger compacta M 2 and #1 = M 3 were also characterized topologically by W h y b u r n [318] and Anderson [14] (see also [15] ), respectively. Anderson's theorem characterizes #1 as a 1-dimensional locally connected continuum with no local separating points and with no non-empty open subspaces embeddable into the plane. Comparing the characterizations of #0 and #1, it is very hard to see what is at the root of these results, what is common between them and, finally, how they can be generalized to higher dimensions in order to (at least) make a reasonable conjecture concerning the characteristics of M~. In full generality, some of these questions still remain open. To the best of our knowledge, there are no conjectures concerning the characteristic properties of the compacta M~ when l
4.1.1. C o n s t r u c t i o n o f M e n d e r c o m p a c t a . In this Subsection we describe the constructions of Mender compacta given by Mender, Lefschetz and Bestvina. Although the Mender construction has already been described above, we restate it in a slightly different (but equivalent) form for later use. Throughout this section we fix integers 0 < k _ n. I. M e n g e r ' s

construction

[221]

As a metric on R n, we use the maximum metric, i.e. d ( { x i } , {yi}) -- max{I x i -

for each { x i ) , { y i ) e R n. If A e-neighborhood of A with respect Let I n be the n-cell in R n with i >_ O, L i denotes the cell complex II{

me m t 3i +l 3i ,

yi I" 1 _< i _< n )

is a subset of R n, then g ( A , e ) denotes the to the above metric. the standard linear structure. For each integer structure of I n whose n-cells are of the form ] " m r = 0, 1 , . . . , 3 i - 1 } .

We define the Mender c o m p a c t u m M ~ as follows: Let M o = I n and (by induction) for each integer i _ 1 let M i + l -- s t ( [ L ~ k ) [ , L i + l ) f3 M i -- s t ( [ L ~ k ) [ , L i + l [ M i )

-- s t ( [ L ~ k ) [ M i [ , L i + l [ M i ) .

4.1. GENERAL THEORY

129

Clearly, {Mi} is a decreasing sequence of compacta and M ~ = AMi is called the Menger compactum of type (k,n). W h e n n = 2k 4- 1, we use the symbol #k - - " zaz2k+l to denote the k dimensional universal Menger compactum. v.L~ We will also use another description of M ~ [120, Chap.2]. Let V~ = {(2t4- 1 ) / 2 - 3 ~ : t = 0 , 1 , . . . , 3 ~ - 1} and Y = t2V~. Note t h a t Bi = Vi • .-. • 1// (n factors) is the set of centers of n-cells of Li. Let :P be the finite collection of homeomorphisms of R n defined by p e r m u t a t i o n s of coordinates of R n. For each i we define Di--= N{a({c} • I n - k - 1 ) : a E 7) and c E Vik+l} and N~ = N ( D i , 1(2-3i)). Then Di can be regarded as the "dual (n - k - 1)skeleton"of Li and Ni as the regular neighborhood of Di. It is easy to see that U~= In-U{int(Ni) : i= 1,...,n},In= U i U N i and OUi = U i N ( N i U O I n ) . If we perform the above construction starting with R n (instead of I n) we get a closed subspace U~ of R n which is a countable union of copies of M ~ (in this case L0 is the partition of R n into unit cubes).

II. Lefschetz's c o n s t r u c t i o n [209] Replacing the cell complexes in (I) by simplicial complexes, we obtain Lefschetz's construction. We describe it in slightly general form. Let M be a PL n-manifold with a (combinatorial) triangulation L. Inductively, we define a sequence {Mi} of PL n-manifolds and their triangulations Li as follows. Let M0 = M and L0 = L. Let M1 = st(L(k),~2Lo),L1 = ~2LolM1 and suppose that Mi and Li have already been defined. Consider ~2Li and let Mi+l -- st(L~k),~2Li) and Li+l -- ~2L~IMi+I. Then {Mi} is a decreasing sequence and NMi ~- 0. If M is the n-simplex with the s t a n d a r d simplicial complex structure, then the resulting c o m p a c t u m NM~ is denoted by L~. In particular, L2kk+l -- #k (We use the same symbol as in (I). This notation is justified by the Characterization Theorem 2.4.1). Notice that Mi+l may be regarded as a regular neighborhood of the k-skeleton of Mi (with respect to Li).

III. B e s t v i n a ' s c o n s t r u c t i o n [33] In Bestvina's construction, the k-skeleta in (II) are replaced by the dual kskeleta. Suppose t h a t M is a PL n-manifold with a (combinatorial) triangulation L. As in (II), we define a sequence {Mi} of PL n-manifolds and their triangulations {Li} as follows: Let M0 = M, L0 = L and suppose t h a t we already have defined Mi and Li. Then Mi+l = U{st(ba,~2Li) 9 ba is the barycenter o f a E L~ with d i m a > n - k } and Li+l = fl2Li[Mi+l.

130

4. MENGER MANIFOLDS

If M is the n-simplex with the standard simplicial complex structure, the resulting compactum DMi is denoted by T~. In particular, T 2k+1 is denoted by ~k. (Again this is justified by the Characterization Theorem). Observe that Mi+l is regarded as a regular neighborhood of the dual k-skeleton of Mi (with respect to Li). It might be worth noting the differences among these constructions. Consider the properties of the partitions which are naturally induced by each of the above constructions. For simplicity, we formulate these properties only for M~, L~ and

PROPOSITION 4.1.1. There are sequences {Pi}, { Q i } and { R i } of partitions of M ~ , L~ and T ~ , respectively, satisfying the following conditions: (a) Pi+l, Qi+l and Ri+l are refinements of Pi, Qi and Ri, respectively. (b) lim mesh Pi = lim mesh Qi = lim mesh Ri -- 0. (c) ord Pi = n ~- 1,1imordQi = c~ and ordRi = k + 1. (d) For each p l , p 2 , . . . , p t E Pi, D{pj : j -- 1 , . . . , t } is an at most k-dimensional L C k-1 D C k - l - c o m p a c t u m o r (e) For each q l , q 2 , . . . , a t E Qi, D{qj : j sional L C k-1 gl C k - l - c o m p a c t u m o r (f) For each r l , r 2 , . . . , r t 9 Ri, D{rj : j sional L C k - t CI C k - t -compactum.

an at m o s t -- 1 , . . . , t } an at m o s t -- 1 , . . . , t }

k - d i m e n s i o n a l cell. is an at most k-dimenk - d i m e n s i o n a l simplex. is a ( k - t + 1)-dimen-

PROOF. The partitions defined below satisfy the desired conditions: Pi-- {eD M~ : e E Li},Qi--

{ s D L~ : s E L i } a n d R i -

{ s D T~ : s E Li}.

[:] R e m a r k 4.1.2. (1) In the last case, if ( k , n ) -- (1,3) we have a partition of #1 with 0-dimensional intersections of all adjacent elements. In this sense, the partition determined by Bestvina's construction can be regarded as a generalization of the partition of the Menger curve considered in [14, 15] and [235].

(2) We may obtain characterizations of (compact) Menger manifolds as well as (compact) Q-manifolds in terms of the existence of certain types of partitions

[18s]. 4.1.2. n - h o m o t o p y . We are going to describe an adequate homotopy language for #n+l-manifold theory. This is the so called n-homotopy theory. The related notion of #n+l-homotopy was first exploited in [33]. DEFINITION 4.1.3. Two maps f , g : X - , Y are said to be n - h o m o t o p i c (writn

ten f "~ g) if the compositions f a and ga are homotopic in the usual sense f o r any map ~ : Z --, X of an at m o s t n - d i m e n s i o n a l space Z into X .

4.1. GENERAL THEORY

131

It can easily be seen [87, Proposition 2.3] that if dim X _ n + 1 and Y E L C n, then maps f , g 9 X - , Y are #n+l-homotopic in the sense of Bestvina [33, Definition 2.1.7] if and only if they are n-homotopic. Note also that if, in the above definition, we consider, instead of compact, only polyhedral Z, then we get Fox's definition of n-homotopy [154]. In practice it is convenient to use the following statement. PROPOSITION 4.1.4. M a p s f , g 9 X ~ Y are n - h o m o t o p i c if and only if f o r s o m e (or, equivalently, any) n - i n v e r t i b l e m a p ~ 9 Z ~ X with dim Z ~_ n, the c o m p o s i t i o n s f (~ and g(~ are h o m o t o p i c .

DEFINITION 4.1.5. A m a p ~ 9 A --. X is said to be n - i n v e r t i b l e if f o r any m a p 9 B --. X with d i m B ~_ n, there is a m a p ~/9 B --~ A such that ~/~ - ~.

Note that 0-invertible maps between metrizable compacta are precisely surjections with a regular averaging operator (see Definition 6.1.24). Note also that each compactum is an n-invertible image of an n-dimensional compactum. Of course, homotopic maps are n-homotopic for each n _ 0, but not conversely. Indeed, consider the identity map and the constant map of an arbitrary non-contractible L C ~ 1 7 6 C~176 Nevertheless, n-homotopic maps have several useful properties. PROPOSITION 4.1.6. For each Y

E L C n, there exists an open cover Lt E

c o y ( Y ) such that any two U - c l o s e m a p s of any space into Y are n - h o m o t o p i c .

PROPOSITION 4.1.7. ( n - H o m o t o p y E x t e n s i o n T h e o r e m ) . Let Y E L C n. T h e n f o r each 34 E c o y ( Y ) , there exists 1) E c o y ( Y ) refining lg such that the following condition holds: (*)n For any at m o s t (n -t- 1 ) - d i m e n s i o n a l space B , any closed subspace A of B , and any two V - c l o s e m a p s f , g : A --. Y such that f has an e x t e n s i o n F : B --+ Y , it follows that g also has an e x t e n s i o n G : B --~ Y which is U-close to F .

PROPOSITION 4.1.8. Let Y E L C n. Suppose that A is closed in B and dim B n + 1.

If maps f,g

9 A --, Y

are n - h o m o t o p i c and f

a d m i t s an e x t e n s i o n

F 9 B - . Y , then g also a d m i t s an e x t e n s i o n G 9 B --+ Y , and it m a y be a s s u m e d that F ~ G.

A map f : X -~ Y is an n - h o m o t o p y equivalence if there is a map g : Y --~ X such that g f ~ i d x and f g ~ i d y [87]. The spaces X and Y in this case are said to be n - h o m o t o p y equivalent. For example, any map between arcwise connected spaces is a 0-homotopy equivalence. Note also t h a t the (n + 1)-dimensional sphere S n+l is n-homotopy equivalent to the one-point space. In general, we have the following algebraic characterization of n-homotopy equivalences [316, Theorem 2].

132

4. MENGER MANIFOLDS

PROPOSITION 4.1.9. A map f " X -~ Y between at most (n + 1)-dimensional locally finite polyhedra is an n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n, i.e., f induces a bijection between the components of X and Y and the h o m o m o r p h i s m ~ k ( f ' ) " ~rk(Cx) --~ ~ k ( C y ) is an isomorphism for each k ~_ n and each pair of components C x C_ X and C y C_ Y with f ( C x ) C_ C y , where f l " C X - , C y denotes the restriction of

f. Recall that each A N R - c o m p a c t u m is homotopy equivalent to a finite polyhedron (Corollary 2.3.30). The following statement is an "n-homotopy version" of West's result. PROPOSITION 4.1.10. Every at most ( n T 1 ) - d i m e n s i o n a l locally compact L C nspace is properly n - h o m o t o p y equivalent to an at most (n + 1)-dimensional locally finite polyhedron. Therefore, 4.1.9 holds even for maps between at most (n + 1)-dimensional locally compact LCn-spaces. Proper n-homotopies, and all associated notions, are defined in the natural way and we do not repeat them here. In order to state an algebraic characterization of proper n-homotopy equivalences similar to Proposition 4.1.9, we need some preliminary definitions. We say that a proper map f 9 X --, Y between locally compact spaces induces an epimorphism of i-th homotopy groups of ends (i >__ 0) if for every compactum C C_ Y there exists a compactum K C_ Y such that for each point x E X - f - l ( K ) and every map ~ " (S i, ,) --, (Y - K, f ( x ) ) there exists a map ~ " (S i, ,) --, ( X - f - l ( C ) , x ) and a homotopy f ~ "~ a(rel ,) in Y - C . We say that f 9 X - , Y induces a m o n o m o r p h i s m of i-th homotopy groups of ends if for every compactum C C_ Y there exists a compactum K C_ Y such that for every map ~ 9 S i --, X - f - l ( K ) with the property that f ~ is null-homotopic in Y - K it follows that a is null-homotopic in X - f - l ( C ) . As usual, f is said to induce an isomorphism of i-th homotopy groups of ends if it simultaneously induces an epimorphism and a monomorphism. PROPOSITION 4.1.11. A proper map f 9 X - , Y between at most (n ~- 1)dimensional locally compact LCn-spaces is a proper n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n and isomorphisms of homotopy groups of ends of dimension ~_ n. Note that proper n-homotopies have also been studied from the categorical point of view [169]. The following proposition will be used below and indicates a difference between the n-homotopy and usual homotopy theories (compare with [303]). PROPOSITION 4.1.12. Let M be an at m o s t (n -t- 1)-dimensional locally finite polyhedron. Suppose that there exists an at m o s t (n + 1)-dimensional finite polyn

hedron K and two maps f " M --~ K and g " K -+ M such that g f ~_ idM (i.e.

4.1. GENERAL THEORY

133

g is an n - h o m o t o p y domination). Then there exist an (n + 1)-dimensional finite polyhedron T, containing K as a subpolyhedron, and an n-homotopy equivalence h 9 T ~ M extending g such that f is a n-homotopy inverse of h. PROOF. Obviously, it suffices to consider only connected polyhedra. Consequently, the case n -- 0 is trivial. If n -- I, then, by the assumption, lrl(g)" l r l ( g ) --* ~rl(M) is an epimorphism and g e r ( l r l ( g ) ) is a finitely generated group. Select finitely m a n y generators of Ker(~rl(g)) and use t h e m to a t t a c h 2-cells to K and to e x t e n d g over these cells. In this way we obtain a 2-dimensional finite polyhedron T, containing K as a subpolyhedron, and a map h" T --, M , extending g, which induces an isomorphism of f u n d a m e n t a l groups. By Proposition 4.1.9, h is a 1-homotopy equivalence. Assume, by way of induction, t h a t the proposition is already proved in the cases n ~_ m, m _ 1, and consider the case n - m + 1. W i t h o u t loss of generality we can suppose t h a t f ( M (i)) C_ K(~) and g ( K (~)) C_ M (~) for each i _< m + 1. Since g f m+l idM it follows easily t h a t g f / M (re+l) m idM(m+l)" By the inductive hypothesis, there are an (m + 1)-dimensional finite polyhedron R, containing K (re+l) as a subpolyhedron, and an m - h o m o t o p y equivalence r" R - . M (m+l) e x t e n d i n g g / K (re+l). Sewing together the polyhedra K and R along naturally e m b e d d e d copies of K (re+l), we obtain the (m + 2)-dimensional finite polyhedron L, containing K and R as subpolyhedra, and the map s" L --. M which coincides with g on K and with r on R, whence s f = g f

m-l-1

~_ idM and f s / L (re+l) =

m

f r ~_ idL(m+~). By these conditions, we conclude t h a t ~i(s)" ~ri(L) --, ~ i ( i ) is an isomorphism for each i ___ m and an epimorphism for i -- m + 1. One can easily verify t h a t in this situation Ker(lrm+l(S)) is a finitely g e n e r a t e d Z(lrl(L))module. Select Z ( ~ l ( L ) ) - g e n e r a t o r s for Ker(lrm+l(S)) and use t h e m to a t t a c h (m + 2)-cells to L and to extend s over these cells. Let T denote the resulting (m + 2)-dimensional finite polyhedron, containing L as a subpolyhedron, and h" T --~ M the corresponding extension of s. T h e n lri(h) is an isomorphism for each i _< m + 1. Again, by Proposition 4.1.9, h is an (m + 1)-homotopy equivalence. This completes the inductive step and finishes the proof. [-1 T h e analogous s t a t e m e n t for proper n - h o m o t o p y dominations (near c~) will be discussed in Subsection 4.4.4

4.1.3. Z-set unknotting

and topological homogeneity.

PROPOSITION 4.1.13. Let A be a closed subset of a Polish A N E ( n ) - s p a c e X . Then the following conditions are equivalent: (i) A is a Z n - s e t . (ii) For each at most n-dimensional locally finite polyhedron P, the set { f E C ( P , X ) : f ( P ) A A - - 0 } is dense in C ( P , X ) . (iii) For each at most n-dimensional Polish space Y , the set { f e C (Y, X ) : f (Y) N A = 0} is dense in C (Y, X ) .

134

4. MENGER MANIFOLDS

Note that each Z n - s e t in any at m o s t n - d i m e n s i o n a l L C n - l - s p a c e is a Z-set. If, in Proposition 4.1.13, X is locally compact, then the listed conditions are equivalent to the following: (iv) For each at most n - d i m e n s i o n a l Polish space Y , the set { f 9 C ( Y , X ) : c l ( f ( Y ) ) N A - 0} is dense in C ( Y , X ) . Closed subsets satisfying t h e p r o p e r t y (4), as in the case of infinite-dimensional manifolds, are called strong Zn-sets. These sets are especially i m p o r t a n t in the non-locally c o m p a c t setting. PROPOSITION 4.1.14. One-point subsets of M e n g e r manifolds are Z-sets. T h e following s t a t e m e n t s are versions of the powerful Z-set u n k n o t t i n g theorem. THEOREM 4.1.15. Let Z1 and Z2 be two Z - s e t s in a ~ n + l - m a n i f o l d M , and let h : Z1 ---* Z2 be a h o m e o m o r p h i s m . Denote by ij : Z j --, M the inclusion map (j = 1,2). I f il and i2h are properly n-homotopic, then h extends to a h o m e o m o r p h i s m H : M ---, M which is properly n - h o m o t o p i c to idM.

COROLLARY 4.1.16. Every h o m e o m o r p h i s m between Z - s e t s of #n can be extended to an a u t o h o m e o m o r p h i s m of #n. For n - 0 this result is well-known. A closed subset of ~1 is a Z-set if and only if it does not locally s e p a r a t e ~1. T h e result for n - 1 originally a p p e a r e d

in [235]. A c o m p a c t u m X is called strongly locally homogeneous if for each point x E X and each n e i g h b o r h o o d U of x , there is a n e i g h b o r h o o d V of x contained in U such t h a t the following condition holds: for each point y E V, there is a h o m e o m o r p h i s m h : X ---, X such t h a t h(x) -- y and h / ( X - U) -- id.

COROLLARY 4.1.17. #n is topologically homogeneous. Moreover, it is strongly locally homogeneous. THEOREM 4.1.18. Let M be a #n-manifold. For each open c o v e r U E c o y ( M ) , there is an open cover )2 E c o y ( M ) with the following property: (,) if a h o m e o m o r p h i s m h : Z1 ---* Z2 between two Z - s e t s of M is )?-close to il (see notations in 2.3.5), then h can be extended to a h o m e o m o r p h i s m H : M ~ M which is l~-close to idM.

4.1. GENERAL THEORY

135

4.1.4. Topological characterization. The following characterization theorem [33, T h e o r e m 5.2.1] is central to the whole theory. We recall t h a t a space X has D D n P (Disjoint n-Disks P r o p e r t y ) if for each open cover L / 9 c o y ( X ) and any two m a p s a : I n ---. X and fl : I n ---+ X , there are maps a l : I n ---* X and ~1 : I n --. X such t h a t a l is U-close to a, ~1 is U-close to ~ and a l (I n) Aft1 (I n) = 0.

THEOREM 4.1.19. The following conditions are equivalent for any n - d i m e n sional locally compact A N E ( n ) - s p a c e X : (i) X is a #n-manifold. (ii) X is strongly B•,n-universal, i.e. has D D n p . (iii) Each map of the discrete union I n @ I n into X can be approximated arbitrarily closely by embeddings. (iv) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by closed embeddings. (v) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by Z-embeddings. (vi) Each proper map f : Y --. X of any at m o s t n - d i m e n s i o n a l locally compact space Y into X such that the restriction f /Yo onto a closed subset Iio is a Z-embedding can be arbitrarily closely approximated by Z-embeddings coinciding with f on Yo. Additionally, if X is c o m p a c t and ( n - 1)-connected (i.e. X 9 A E ( n ) ) , then conditions (ii)-(vi) give a topological characterization of the c o m p a c t u m #n. Note t h a t the t h e o r e m remains true even in the case n - c~ (see C h a p t e r 2). If n -- 0 and X is compact, condition (ii) trivially implies t h a t X has no isolated points. Therefore, in this case, X is h o m e o m o r p h i c to the C a n t o r disc o n t i n u u m as has already been noted above. A p p l y i n g the above characterization in the case k -- 1, we see t h a t a comp a c t u m is h o m e o m o r p h i c to #1 if and only if it is a 1-dimensional, locally connected c o n t i n u u m with D D 1 P . It is known (see [15] or [235]) t h a t a locally connected c o n t i n u u m has D D 1 P if and only if it has no local s e p a r a t i n g points and has no open subspaces e m b e d d a b l e in the plane. In this sense, Bestvina's characterization of #1 reduces to Anderson's. In Subsection 4.1.1, three m a j o r geometric constructions of the universal Menger c o m p a c t u m have been presented. Let us indicate another, spectral construction, given in [248]. Let {G~} be a basis of open sets of S 1, such t h a t G~ is an open cell with the p r o p e r t y t h a t diamG~ --. O. Let us construct an inverse sequence {Xi,p~ +1} as follows. We set X0 = S 1, and we get X~+I from X~ by "bubbling over G~" , i.e. X i + l is the quotient space o b t a i n e d from the disjoint union Xi ~ X~ by identifying the two copies of x 9 Xi precisely when p~(x) ~ Gi (here p*o" Z i ~ X o denotes the corresponding projection). T h e projection ~i ~ + 1 " X i + l ~ Xi is defined in the obvious way. Note t h a t p~+ 1is a retraction. One can check directly t h a t X = lim{Xi, Pi -i+1 } is a 1-dimensional 9

136

4. MENGER MANIFOLDS

A E ( 1 ) - c o m p a c t u m with D D 1 P and hence, by 4.1.19, X is a copy of #1. More careful consideration shows [33] that we get #n if we start with S '~ = X0 (and proceed as above). There are several other constructions of #n. To the best of our knowledge, all of them are defined as the limit spaces of inverse sequences(see, for example,

[155], [ls6]). 4.1.5. Approximation by Homeomorphisms. THEOREM 4.1.20. Proper U V n - m a p s between #n+l-manifolds are near-homeomorphisms. Another important result of I ~-manifold theory states that an infinite simple homotopy equivalence between/W-manifolds is homotopic to a homeomorphism. Let us note that this is not the case for homotopy equivalences (i.e. there exist non-homeomorphic but homotopy equivalent compact I ~ -manifolds). In #n-manifold theory we do not have simple homotopy obstructions and this significantly simplifies the corresponding result. THEOREM 4.1.21. Each proper n-homotopy equivalence between #n+l-manifolds is properly n-homotopic to a homeomorphism. The following result, due to Ferry [150, Proposition 1.7] and improved slightly in [188], also illustrates this situation. PROPOSITION 4.1.22. Let f 9 P ~ L be a map between compact polyhedra which induces an isomorphism between the i-th homotopy groups for each i < n. Then there is a compact polyhedron Z and U V n - m a p s ~ 9 Z ---. P and ~ 9 Z ~ L such that f a n ~.

Historical and bibliographical notes 4.1. Most of the results of this Section were obtained by Bestvina in his fundamental work [33]. The notion of nhomotopy is due to the author [87]. Proposition 4.1.12 was proved by the author

[100]. 4.2. n-soft m a p p i n g s of c o m p a c t a , raising d i m e n s i o n The following statement is the first resolution theorem for locally compact L C n - l - s p a c e s , in particular for #n-manifolds. THEOREM 4.2.1. Every locally compact L C n - l - s p a c e is an (n-invertibte) proper u v n - l - i m a g e of a #n-manifold. We also will be using the following result, which states that the u v n - r e s o l u tions from Theorem 4.2.1 can be improved over Z-sets.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION PROPOSITION 4.2.2. Let f " M ~

X

be a proper U V n - s u r j e c t i o n of a #n+l_

m a n i f o l d onto a locally c o m p a c t L C n - s p a c e and let Z be a Z - s e t in M . X

is any proper m a p properly n - h o m o t o p i c to f / Z ,

uvn-surjection

137

I f g" Z ---.

then there exists a proper

h" M ---. X such that h / Z -- g.

Below, in this subsection, we prove much s t r o n g e r results. We begin by fixing a n a t u r a l n u m b e r n. Let B n+l -- { x E R n ' i i x i l < 1} d e n o t e t h e unit closed ball in R n+l. Consider a m a n y - v a l u e d r e t r a c t i o n l:ln" B n+l "--+ S n of the ball B n+l onto its boundary, the sphere S n - OB n+l, defined as follows" R n ( x ) - {y E S n" ( x , y ) >_ 4[[xI[ 2 - 3[[x[[} for each point x E B n+l.

Here (x, y) denotes the s t a n d a r d scalar p r o d u c t in the E u c l i d e a n space R n+l and Ilxll = v / ( x , x ) denotes the induced norm. T h e above to each point of the ball B~'+1 = {x E R n+l" IIxlI < If x E B n+~ - (B~ +1 U Sn), t h e n R n ( x ) is a ball on x E S n, then R n ( x ) = x, i.e. R n is a retraction. It continuous. Consider the g r a p h F Rn of this map, i.e.

m a n y - v a l u e d m a p assigns 89 the whole sphere S n. the sphere S n. Finally, if is easy to see t h a t R n is

FRo = { ( ( x , y) e B ~+1 • S ~ : y e R ~ ( ~ ) } . By Pn: FRn ---+ B n+l and qn: FRn --'+ S n we d e n o t e the restrictions to FR., of the n a t u r a l projections of the p r o d u c t B n+l x S n onto its coordinates. LEMMA 4.2.3. T h e m a p Pn" FRn -'+ B n+l is ( n -

1)-soft.

PROOF. C o n t i n u i t y of the r e t r a c t i o n R n implies openness of Pn. Therefore, by T h e o r e m 2.1.15, it suffices to show t h a t the collection { R n ( x ) " x E B n + l } is c o n n e c t e d and uniformly locally c o n n e c t e d in all dimensions less t h a n n - 1. As was m e n t i o n e d above, topologically t h e r e are only three t y p e s of elements of the collection { R n ( x ) " x e B n + l } . Indeed, R n ( x ) is the sphere S n (for x E B~'+I), X a closed ball B n with center at i[~-][ in the sphere S n (for x E B n+l - B~'+I), or a point (for x E Sn). C o n s e q u e n t l y the i n d i c a t e d collection is c o n n e c t e d in all dimensions not exceeding n - 1. Let us show the uniform local c o n n e c t i v i t y of this collection in all dimensions less t h a n n - 1. Let k _ n - 2, e > 0 and e Consider a point x E B n+l and an a r b i t r a r y m a p a" S k ---, R n ( x ) 5 - - ~. such t h a t the d i a m e t e r of its image a ( S k) is less t h a n 5. E v i d e n t l y t h e r e is a ball B n on the sphere S n, containing this image a ( S k ) . Let &" B k+l ---+ B n be an extension of a from the b o u n d a r y S k = OB k+l onto the whole B k+l. Additionally, we can assume t h a t if R n ( x ) = x (which occurs when x E S n ) , t h e n a = a. Obviously, in this case as well as in the case w h e n R n ( x ) = S n, ~ is the desired extension. Suppose now t h a t x E B n+l - (B n+l U s n ) . T h e n R n ( x ) is a ball on S n of non-zero diameter. If S '~ - R , ( x ) ~= B n, t h e n the intersection B n M R n ( x ) is h o m e o m o r p h i c to the n - d i m e n s i o n a l ball and, consequently, t h e r e exists a r e t r a c t i o n r" B n ~ B n M R n ( x ) . Therefore, in this case, the c o m p o s i t i o n a r is the desired extension of a with d i a m e t e r of the image less t h a n e. In the r e m a i n i n g case, when S ~ - R n ( x ) C B n, we first observe t h a t the b o u n d a r y

138

4. MENGER

MANIFOLDS

O R n ( x ) of the ball n n ( x ) is contained in B n. Since k 4- 1 < n, there is a h o m o t o p y (in B n) connecting the m a p 5 with a m a p / 5 " B k+l ~ B n N R n ( x ) fixed on ~ - l ( R n ( x ) ) . Then fl does not move points of OB k+l -- S k as well. Clearly, ~ is the required extension of c~. [7

LEMMA 4.2.4. The map Pn" FRn ~ B n + l is n-invertible. PROOF. Let us consider a lower semi-continuous many-valued retraction Fn" B n+l ---* S n defined as follows:

= ( S

I1 11 > ifll ll <

89 89

Obviously, F(x) C R ( x ) for each point x E B n+l. It is also easy to see t h a t the collection {Fn(x)" x e B n+l} is connected and uniformly locally connected in all dimensions less t h a n n. Let FEn C B n+l • S n denote the graph of the map Fn and let PEn -- ~rBn+l/FFn, where 7rBn+~" B n+l • n ~ B n§ is the projection onto the first coordinate. It follows from T h e o r e m 2.1.15 t h a t PEn is n-soft. In particular, PEn is n~invertible. Now observe t h a t FEn C FRn and PEn -- Pn/FFn. This is obviously enough to conclude t h a t pn is also n-invertible. D LEMMA 4.2.5. Let Y be a closed subset of the sphere S n and f " Y --, B n§ be a map into the ball B n+l. Then there exists a h o m o t o p y H" Y • [0, 1] ---+ B n+l, fixed on f - l ( O B n + l ) , connecting f with a map whose image is contained in OB n+l. Moreover, g ( Y x [0, 1]) does not contain the center of B n+l.

PROOF. T h e case when 0 r f ( Y ) is trivial. Consider the case when 0 E f ( Y ) and let U -- f - l ( B ~ + l - {0}). Obviously, V is a proper subset of the sphere S n and, consequently, has trivial n-dimensional cohomology. Let us show t h a t f / U " U ~ B ~ + 1 - {0} is null-homotopic. Obviously, B~ + 1 - {0} is homeomorphic to the p r o d u c t S n x (0, 89 Therefore we can fix a canonical h o m e o m o r p h i s m sending S n x {t} onto the set {x e Rn+l"]]x]] -- t), t e (0, 89 Suppose now t h a t the m a p f / V is not null-homotopic. Since the projection r l " S n • (0, 89 ~ S n is a h o m o t o p y equivalence (notice t h a t the half-interval is an AR-space) we conclude t h a t the composition 7rf/U" U ~ S n is also not null-homotopic. In this situation, this composition generates a non-trivial element in the n-dimensional cohomology, c o n t r a r y to the r e m a r k above. Let lr2" S n • (0, 89 ~ (0, 89 denote the projection onto the second coordinate. 1 We assume t h a t the half-interval (0, ~] is isometrically e m b e d d e d into B~ +1 as a subspace {x e R n + l " x = te, t e (0,1]}, where e = ( 1 , 0 , . . . , o ) e n n+l. Since (0, 1] is contractible, the projection 7r2 is null-homotopic. Consequently, the composition 7r2f/U is also null-homotopic. Thus, keeping in mind the above facts, we can conclude t h a t the maps f / U and ~r2f/U, considered as maps into B~ +1 - {0}, are homotopic. Take the corresponding h o m o t o p y G" U x [0, 1] B~ + 1 - {0} such t h a t Go = f / U and GI = l r 2 f / U ) . Observe t h a t ~ r 2 G o ( y ) = 7r2Gl(y ) for each y E U. Therefore we may assume without loss of generality

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

139

that ~r2G0(y) -- 7r2Gl(y) for each y E U and each t E [0, 1]. The homotopy H can be extended to a homotopy F" (U U f - l ( 0 ) ) x [0, 1] ~ B~ +1. Further, the homotopy F can be extended to a homotopy H ~" Y x [0, 1] so that H'(f-I(B

n+l - B~ +1) x [0, 1])C_ B n+l - I n t B ~ +1.

We also can assume that H ' is fixed on the set f - l ( O B n + l ) . The intersection of the image of H~ with B~ +z is a segment, in fact a radius of the ball B~ +1. Let us now, after applying the homotopy H', shrink the indicated segment into its end which belongs to the boundary OB~ +1. Next push the ring B n + l - I n t B ~ +1 onto the sphere OB ~+1. As a result, we obtain a h o m o t o p y / ~ " Y x [0, 1] + B n+l. Observe t h a t / ~ - 1 ( 0 ) = / ~ o 1 (0) x [0, a], where a < 1. The decomposition of the product Y x [0, 1] into the intervals {{y} x [0, a ] ' y e / ~ o 1 ( 0 ) } and single points generates a C E - m a p It" Y x [0, 1] + Y x [0, 1] which is divisor of the homotopy H. In other words, there is a map H" Y x [0, 1] + B n+l such that /~ = HIt. The homotopy H satisfies our requirements. D LEMMA 4.2.6. There exists 5 > 0 such that for each m a p r B n --. B n+l, with d i a m ( i m ( r < 5, and for each map 7~" OB n" FRn, satisfying the equality Pn~ = r n, there is a map ~" B n ~ FR~ such that p n ~ -- r and 7~ -- ~ / 0 B n . PROOF. Let 5 = ~ and consider a maps r satisfying the following two conditions: diamr

B n --~ B n+l and ~" OB n --. FRn,

1 n) < ~ and pnT~ -- ~b/OB n.

First assume that r n) N OB~ +1 = 0. Then, by the connectedness of r either r n) C_ B~ +1 or r n) C B n + I - B ~ +1. It follows from the construction of the many-valued map R n that the map

p~! = p ~ / p ; ~ (B~+I ). ;~-1 (B3 + 1) ---+B~+I is a locally trivial bundle with fiber the sphere S n and, consequently, Pin is n-soft. Thus, if r n) C_ B~ +1, then the conclusion of our lemma is true. Observe that the map Pn = P n / p n l (

Bn+l

-- B ~ + I ) " P n l ( B n + I -- B ~ +1) --* B n+l -

B~ +1

is also n-soft (it is even soft). Therefore the lemma is true in the case r n) C B n+l - B~ +1. Next we consider the situation when r n) A OB~ +1 ~ 0. Since diam(im(r < ~, we see that the set r n) does not contain the center of the ball B n+l. Denote by J : B n+l - ( 0 } - . OB n+l the central projection of B n+l - {0} onto the boundary OB n+l. Then the set J ( r is contained in the ball (on the sphere S n) of an angular radius ~. Let z denote the center of this ball. Straightforward calculations show that for each point x E r the

140

4. MENGER MANIFOLDS

71" set R n ( x ) is a ball on the sphere S n with center at i ~z and of radius at least ~. Define a section r/: B n --~ B n • S n of the projection B n • S n ~ B n by letting

r i1r

r/(x) = ( x , - ~ )

for each point x E B n.

Since for each x E B '~ angle between - z and - IIr r is less t h a n ~, ~ we see t h a t there is an i s o m e t r y h: B n x S n --+ B n • S n, with 7rBn -- 7rB,~h, transforming rl into the trivial section 0: B n --~ B n x S n of the projection 71"B,~ (i.e. hrl = 0) d e t e r m i n e d by - z : O ( x ) = ( x , - z ) for each x e B n. D e n o t e by O n the ball (on the sphere S n) with center at - z and of radius ~. T h e m a p ~o9 O B n - - - , F R~ induces a section c~: O B n ~ O B n x S n of the projection O B n x S n ~ O B n. One can see t h a t c~(x) E R n ( r whenever x e O B n = S n - 1 . By L e m m a 4.2.5, we can connect the composition l r s , , h a : O B n --* S n with a m a p f : S n - 1 S n - I n t D n via a h o m o t o p y H t such t h a t H t ( x ) ~ - z w h e n e v e r t > 0. F u r t h e r , let G t : S n - 1 x [0, 1] ~ S n be a h o m o t o p y connecting ~rsnhc~ with a c o n s t a n t m a p (GI (S n - l ) = d), which can be o b t a i n e d from H by adding the contraction of the set S n - I n t D n. For each x E B n denote by rz the push (fixed on R n ( x ) ) in S '~ - R n ( r

with the center at -

:~ = {y e s~: (r

r IIr

onto the sphere

y ) = 411r

311r

if the sphere Ex is defined. If not, then rx denotes the identity map of S n. Observe t h a t if r E O B ~ +1, then Ez consists of the single point, and if r E I n t B ~ +1, t h e n E~ = 0. Define an extension ~ : B n ~ O B n x S n of c~ by letting ifx#O

.1

a(~) = ( ~h-l(x'e)'

if• =0.

Note t h a t the m a p & is well-defined and t h a t 5 ( x )

for each point

E Rn(r

x E B n. It only remains to note t h a t the desired m a p ~5 can now be defined by

letting ~5 = (r • i d s n ) ~ .

Vl

DEFINITION 4.2.7. A m a p f : X ---+ Y is c a l l e d p o l y h e d r a l l y n-soft i f f o r e a c h at m o s t

n-dimensional

g: A ---, X

polyhedron B,

a n d h : B ---, Y

that f k = h and k/A

subpolyhedron

with f g = h/A,

A

of B,

and any two maps

there exists a map

k : B -+ X

such

= g.

T h e following s t a t e m e n t expresses one of the most i m p o r t a n t properties of P n . LEMMA 4.2.8. T h e m a p P n : F R n --+ B n + l is p o l y h e d r a l l y n - s o f t . PROOF. It suffices to show t h a t for every pair of maps h: B n ~ B n + l and FR,~ with P n g - h / O B n, there exists a m a p k: B n --~ FR,~ such t h a t p n k -- h and k / O B n -- g. Take a t r i a n g u l a t i o n of the ball B n small enough to ensure t h a t the d i a m e t e r s of images (under h) of simplexes of this t r i a n g u l a t i o n g: O B n ~

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

141

are less than ~. Denote by B(nn_l) the ( n - 1)-dimensional skeleton of this triangulation. Since, by L e m m a 4.2.3, the map Pn is ( n - 1)-soft, we conclude that there exists a map k ~" B(n-1) ~ FR, such that pn kl -- h / B (n-l) n and k~/OB n = g. Apply L e m m a 4.2.6 to each of the n-dimensional simplexes of the indicated triangulation B n to obtain the desired extension k. El LEMMA 4.2.9. The m a p qn" F R~, ---+ S n is a trivial bundle with fiber homeomorphic to the ball B n+l. PROOF. It follows from the definition of the retraction Rn that the fiber qnl(y) of each point y E S n can be written as qn--1 (y) __~ {X e B n+l 9 (x, y) >_ 4llxII 2 - 3llxII}.

Let B~ +1 = {z"

IIzll

< 89 and define a map fy" q ~ l ( y ) ~ B ~ + I as follows

S~(~) =

4x

3 + ~/9 + 16(~, y)

Straightforward verification shows that the map fy is a homeomorphism. Additionally, the collection of homeomorphisms {fy" y E S n} continuously depends on y, i.e. qn is homeomorphic to the trivial bundle B~ +1 • S n ~ S n. El Let K be a finite simplicial complex. By K (n) we denote the n-dimensional skeleton of K and by IKI the underlying polyhedron of K. If for each at most (n + 1)-dimensional simplex a E K we take a many-valued retraction Rn" lal --* ]a(n+l)l, homeomorphic to the retraction Rn" B n+l --~ S n constructed above, and then consider the union of these retractions, we obtain a retraction

nK(.)" IK(~+~)I ~ IK(~)I. Generally there is no canonical homeomorphism of the ball B n+l onto the ( n + l ) dimensional simplex. Therefore the retractions RK(. ) are not uniquely defined. For this reason RK(,~ ) denotes any retraction constructed in the above indicated way. Further, by FRK(.) _C [K(n+I) I x IK(")] we denote the graph of the manyvalued retraction R K ( , ) 9 IK(n+l)l

• IK(") I.

As above, PRK(,)" FRK(,~) ---+ IK(n+l) ]

and qRK(,) o FRK(.) ~ IK(n)l shall denote the restrictions of the projections of the product IK(n+l) I • IK (n) ] onto the first and second coordinates respectively. LEMMA 4.2.10. The m a p PRK(,)" FRK(,) ---~1 K ( n + l ) l is n-invertible, ( n - 1 ) soft and polyhedrally n-soft. PROOF. Apply Lemmas 4.2.3, 4.2.4 and 4.2.8.

El

For each pair (n, k) of natural numbers (n > k) and for each finite simplicial k,n(K) as follows. Concomplex K, let us now assign a commutative diagram $i,j sider the many-valued retractions R K ( ~ - I ) " [K(n-i+l)[ ---+ [K(n-i)[ constructed

142

4. MENGER MANIFOLDS

above, i = 1 , . . . , k. First, by induction on i + j (beginning with i + j - k), we define spaces Ek, i,jn (K) and maps

k n (K)---+ Eki'nl,j (K) Ei,j

fik'n(K)

n (g) , gik? ( K ) 9 E ki,j' n ( g ) _.+ Ek, i,j-1 t h a t will be part of the diagram 9 For each non-negative integer i _ k, we d e f n e k,n our space as Ei,k_i(K) =1 K (n-l) I. For each non-negative integer i _< k - 1, let

Z ik,. + l , k - i (K) = FR~ (n-~-l)

'

k~n

f i+l,k_i(K) -- pRg(,.,_,_l) and k~?l

gi+l,k_i(K) -- qRK(,,_,_I ) . Suppose now t h a t the spaces and maps, required for these diagrams have already k'n been constructed for all i,j with i + j < m 9 Let us define the space E i,m_i(K) as a fibered p r o d u c t (i.e. with respect to the maps

k,n

pullback) of spaces Eki'~l,m_i(K) and Ei,m_i_l(K )

fk,ni,m_i_l(K)"

k,n k,n Ei,m_i_l(K)--+ Ei_l,m_i_ 1(K)

and ]g~n

k~n

gi--l,m--i--1 ( g ) " E i _ l , m _ i ( g ) ~ Eki'l,m_i_l ( g ) . T h e maps

k,n (K) ---+Ei_l,m_i(K k,n fk,n ) i,m--i (K)" Ei,m_ i and

k,n kn " E ~,.,_~ k'n (K)---+ Ei,m_i_ 1(K) a,~_~(g) are defined to be the canonical projections of the corresponding fibered product. In these notations, we have a c o m m u t a t i v e square diagram

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

fi,~-i(K)

E~ m--i 's (K)

143

k

9.- Ei, ~ - i ( K )

k~S

kn a,;i_~ (g )

gi-l,m-i(K)

ks

f i , ~ - i - 1(K Eki'l,m_i_l (K )

,~Eki'l,m_i(K) k~rt

]g~s

After performing k steps we get a single space Ek, k (K) and two maps fk,k (K) k,n

and gk,k(K) 9 This finishes the construction of the diagram s i , j Below, " when there is no confusion, we omit upper indices in the notations of spaces and ks maps of the diagram s ~ (K). We formulate some elementary properties of these diagrams. 9 The map f = fl,kf2,k'''fk,k: Ek,k ~ Eo,k is ( n - k - 1)-soft, ( n - k)invertible and polyhedrally ( n - k)-soft. 9 Let g be a projection of the space Ek,k onto the space Ek,o. Then

g f - 1 = RK(,_k)RK(n_k+I) 9 The part of the diagram s i,j

... RK(n_I).

consisting of spaces Ei,k n~ (K) with k--m,n--m

indices i _> m is naturally isomorphic to the diagram C~,j

(K). In

kn k-re,n-re(K). particular, Ei,'j (K) ~ Ei_m, j 9 Let K1 be a subcomplex of K.

Then the diagram Ek?(K1) can be k,n

naturally embedded into the diagram Ei, j (K) in the sense that for all indices i , j there exist embeddings ~oi,j" Ei,k,n j (K1) ~ E ki,j, n (K) such that the maps f.k'.n(K1)~,~ and gik,?(gl) coincide with the restrictions of the k,n

kn

maps fi,j (K) and gi,~ ( g ) respectively 9 Observe also that for each i > 0 and j > 0, we have

E ki,j, n ( K 1 ) - (fi,~n(K))-l(Eik,?(K1)) kn

LEMMA 4.2.11. If K is a finite simplicial complex, then Ei, ~ (K) is an A N R -

compactum. PROOF. We prove our statement by induction on k. First consider the case 1,n k = 1. Represent the space E0,1 ( g ) =1 g(n) I as the union a l U a2 U . . . U ar of

144

4. MENGER MANIFOLDS

its simplexes. Then 1 (al) U - - - U f -1,1 1 (fir). El:I n1 (K) = f -1,1

Since the map f1,1 is a projection of the graph of the many-valued retraction RK(,,_~), it can easily be seen that for each i _~ r we have Sl, l ( a i ) n Uj
Ek+l,k+l

fk+l,k+l

~''"

* Ekk+l

fl,k+l

* E0,k+l

gl,k+l

gk,k+l

gk+l,k+l

*El,k+l

fk+~,k Ek+l,k

.

Ek,k

,.-''"

,.- E l , k

Denote by A: K (n) ~ w the map assigning to each simplex a E K (n) the number of n-dimensional simplexes of K containing a as a subcomplex. Applying Lemma 4.2.9, we can see that for each simplex a E K (n) the map gl,k+l restricted to --1 ( I n t I a [) is a locally trivial bundle with base I n t I a [ the inverse image gl,k+l and fiber V,x(a)B n, where Vx(a)B n is a wedge of A(a) n-dimensional spheres.. Contractibility of I n t [ a [ guarantees that this bundle is trivial. Similarly, it --1 can be seen that the space c l ( g--1 l , k + l ( I n t [ a [)) C_ gl,k+l([ a I) (closure is taken in El,k+l) is also the space of a trivial bundle over [ a [ with the same fiber. Let a l , . . . , a m o be the vertices of K, and in general denote by am~_l,...,am~ idimensional simplexes of K. Let Mz = Ui
4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

145

M is a s u b c o m p l e x of K (n-l). It follows from the above listed properties of the d i a g r a m s s k )n (K) t h a t k+l'n(M) ~ ~ k'n-l(M). f{--l(i M ]) = ~ ~k+l,k ~k,k Therefore, by our inductive assumption, f l 1(I M ]) C A N R. Observe t h a t --1 gl,k+l([ a l) ~ V,k(a) B n

for each vertex a E K (~ 9 It is not hard to see t h a t g k-1+ l , k + l f l l ( a ) "~ f l l ( a ) X V)~(a)B n and, consequently, g -k+l,k+lfl 1 -1 (a) is an ANR-space for each vertex a E K (0). --1 Suppose now t h a t the inverse image gk+l,k+lfll(Ml) is an ANR-space for some 1 (1 >_ m0). Observe t h a t [ a l + l I - - M l - I n t l a l + l I. Therefore g-1

1

g-1

-1

1

k+l,k+lfl (M/+I) ---- k + l , k + l f l l ( I n t [ a / + l l) U gk+l,k+lfl (Ml) --- f o l ( c l ( g ~ , ~ + l ( X n t

[ hi+ 1 [)) U g-1 k+l,k+l f 1 1 ( M l ) .

Notice also t h a t

f o l ( d ( g l-1 , k + l ( I n t lal+l I)) ~ f l l ( I al+l [) • VX(al+l)B n. In this situation it suffices to show t h a t

fol(cl(g~,lk+l(IntIal+l I)) n g-lk+l,k+lfll (Ml) is an A N R-space. Note t h a t f o 1 ( c l ( g l-1 ,k+l(Int

f o l ( C l ( g l-1 ,k+l(Int

l al+l I)) N g k-1+ l , k + l f l l ( M l ) l al+l I)) N g l-1 ,k+lfll(Ml)

--

--

--1 --1 f o l ( c l ( g 1,k+1 -1 ( I n t [ hi_t_1 [)) Pl g l , k + l f I (0 ] al+ 1 [) Consider the projection

onto fll(Olaz+~l) which, as shown above, is an ANR-space. Let

r

fll(Olal+l[) • VXal+,)B n --+ E l , k + l

be the composition of f l x idvB n and the trivialization --1 (0lal+ll) 7I" OIal+ll • VB n ---+ gl,k+l

-1 (0 ] az+l I) --1 1( I n t ] al+l ]))I~l gl,k+l of the bundle gl,k+l/cl(gl,k+ flTr = g l , k + l r Consequently there exists a m a p 4: f11(Olal+11) • VX(a,+I) Bn --* Ek+l,k+l,

Notice t h a t

146

4. MENGER MANIFOLDS

lifting b o t h lr and r Since the collection {Tr, r separates points of the compactum f~l(Olal+ll) x VX(a~+I)B n, we conclude that r is injective. On the other hand,

r

x vx(o,+,)B ~)

-

fol(r

n g l-1 , k + l f -1l ( M t ) "

( g l-1 ,k+l(Intlal+ll))

Therefore

f o l ( d (o~,k+~(x~tl"z+~l)) -1 n g-1 k+~,k+~f~-1 (M~) is an A N R-space. --1 1(MI) is an ANR-space for each I. It only suffices to note Thus gk+l,k+lfl that Ek+1,k+1 = g-1 k+1,k+l f -11 (El,k). This finishes the proof.

D

LEMMA 4.2.12. For each finite simplicial complex K and each natural number n, there exist an ANR-compactum X and two maps f : X --*1 K I and g: X --~ IK(n)l satisfying the following conditions: (a) The map f is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

(b) g / f - l ( [ g ( n ) l ) - f / f - l ( [ g ( n ) [ ) . (c) gf-l(lal)c_ Io(")1. n n+k

PROOF. Let d i m K = n + k. Consider the diagram ~r

X

(K) and let

~k,n+k(K )

= ~~k,k

k n+k

k n+k

f = f l , k " " fk,k: Ek' k (g)---* E0'~r (g) = Igl, k:.+k k:.+k g = gl,k gk,k: Zk, k ( g ) ---, Ek, o ( g ) - - I g ( n ) l

for each a e g .

By L e m m a 4.2.11, X is an ANR-compactum. By the properties of the di1)agrams s i , j ' the map f is n-invertible, polyhedrally n-soft and ( n soft. These properties also imply that g f - 1 = RK . . . . RK(,+k-1), and consequently f and g coincide on the inverse image f-l([K(n)[). For the same reason,

g.f-l(lal) c_ Io(n)l. D Remark 4.2.13. All statements proved in this section so far are valid for countable locally finite simplicial complexes as well. In such cases, the resulting spaces are locally compact A N R-spaces and the resulting maps are proper. Remark 4.2.14. More careful consideration shows that the compactum X from L e m m a 4.2.12 is a finite polyhedron. Respectively, if K is countable and locally finite then the polyhedron X is also countable and locally finite. We are ready to prove the main result of this section. THEOREM 4.2.15. For each natural number n, there exists a map fn: #n I ~ of the universal n-dimensional Menger compactum onto the Hilbert cube satisfying the following properties: (i) The map fn is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

147

(ii) For each at most n-dimensional compactum Y, each map f" Y --+ #n and each open cover lg E cov(#n), there is an embedding g" Y --~ #n that is lg-close to f and such that f ng = f n f .

PROOF. We construct the map fn as the limit projection of an quence 8 - {M{, c~ +1} consisting of the Hilbert cube manifolds (in M1 -- Io2 ). The construction is carried out by induction. The limit verse sequence is contained in the product I~' • I~' • --. • I~ • -.the metric

~(~, y) = r ~

inverse separticular, of this inwhich has

d~(x~, y~) 2(~+1)

where di is a (bounded by 1) metric on the Hilbert cube I.~ = ( y l , . . . , y i , . . . ) and xi, yi E I~ for each i. Let ai E Io2 for each i. Then each of the products I~' x --- x I m can be naturally identified with the subspace oo I~' x - - . x I m x {am+l} x - - - x {ai} x - - - of the product l I i = l i ~ . We construct an Io2-manifold as a subspace of the product I~ x --- x I~n and consider the O2 restriction (to Mi+l) of the projection ~ + 1 . 1-Ij=l I~ x Ii+1 ~ 1-I~=11~ as a projection a~+l. Mi+l ---+ Mi of the spectrum 8. In this situation, as can easily be seen, the limit space X = lim 8 is naturally embedded into Io2 - 1--Ii=l~176 i~'. Let M1 = I~ and suppose that the Io2-manifold Mj C YIi=lJI~ c Io2 has already been constructed. By Theorem 2.3.28, the Io2-manifold Mj is homeomorphic to the product of the Hilbert cube jIo2 and a finite polyhedron Kj. Obviously, we may assume that the composition of the projection wj" Kj x j I W ---+ Kj with a certain section sj" Kj" K j x j Io2 is a 2-J-move, i.e. for each point x E K j • Io2, we have d(x, sjwj(x)) < 2 - j . We also assume t h a t Kj is given together with a triangulation such t h a t diamsj(a) < 2 - j for each simplex a of this triangulation. By L e m m a 4.2.12, there exist an A N Rc o m p a c t u m X j + I and maps f j + l " X j + I ~ K j and gj+!" Xj+I ---+ K~ n) such t h a t f j + l is n-invertible, polyhedrally n-soft, ( n - 1)-soft and, in addition, -1 (n) for each simplex a of the triangulation given on Kj. Let Mj+I : Xj+I • Io2. By Theorem 2.3.21, Mj+I is a Io2-manifold. Next, represent the cube j+lIo2 as the product jIo2 x j + 1 I~' of two Hilbert cubes, and denote by yj+l" X j + I • Io2 • I~ --+ Xj+I x jIo2 the product of the identity map idx~+l and the projection ~ " jIo2 • I~ --+j Io2. Define the map ~+~ j 9 Mj+I ~ Mj by letting ~j+l

= (f3+~ • i d ~ ) ~ j + l .

In this situation we have the following diagram:

148

4. MENGER MANIFOLDS

Mj+I

= Xj+l

~ j + l -- i d x

x j I ~~ x j + 1 I~'

7r31

Xj+I xj I ~

Xj+I

f j+l x id

fj+l

,~ K j x j I ~~ = M j

,- K j

Let A" M j + I ~ IS+1 be an embedding. T h e n the desired e m b e d d i n g of the /"a-manifold M j + I into the p r o d u c t I i a x - - - x I~a+l can be defined as the diagonal p r o d u c t of the m a p ~ j + 1 and the e m b e d d i n g A. Let a jj + l ._ 7rJ+lj / M j + I , and j+l observe t h a t this map a j " M j + I ~ M j is n-invertible, polyhedrally n-soft and ( n - 1)-soft (because b o t h f j + l and 7r~ have the c o r r e s p o n d i n g properties). T h u s the c o n s t r u c t i o n of s p e c t r u m 8 = {Mi, a~ +1} is complete. Let ai" X = lira,5' --, M i d e n o t e the i-th limit projection of this s p e c t r u m . As the desired m a p fn" #n __~ i ~ we take the first limit projection a l " X ---+ M1 = I W 9 Of course, we still have to show t h a t X ~ #n. Let us investigate the properties of the c o m p a c t u m X and the m a p fn. Since each of the short projections a~ +1 of the s p e c t r u m S is n-invertible, p o l y h e d r a l l y n-soft and ( n - 1)-soft, we easily see t h a t fn (as well as all other limit projections of the s p e c t r u m S, see L e m m a 6.2.6) has the same properties. This proves one p a r t of the theorem. Let us now show t h a t dim X = n. Since M1 -- I w , M1 contains a topological copy of the n - d i m e n s i o n a l cube I n. T h e n-invertibility of fn then g u a r a n t e e s t h a t X also contains a copy of I n. Therefore dim X _> n. In order to show t h a t dim X _< n we proceed as follows. Obviously, all we need is an existence, for each i, of a ~ - m o v e of X into an n - d i m e n s i o n a l polyhedron. Such a move is i n d i c a t e d below" siwi(gi+l x idii~)~ii+lai+ 1 9 X ---+ K} n). Consequently, dim X - n . Next we show t h a t fn has the second p r o p e r t y formulated in the theorem. Let Y be an at most n - d i m e n s i o n a l c o m p a c t u m , f" Y -4 X be a m a p and

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

149

bl E c o v ( X ) . Choose an index i and an open cover 12 E cov(Mi) such t h a t a~-l(Y) refines/~/. It follows from the construction of the m a p a~ +1 t h a t a~ +1 can represented as the composition 7i+lr/i+l, where ~i+l is a trivial bundle with fiber the Hilbert cube. Consequently, there exists an e m b e d d i n g hi+l" Y ~ M i + l such t h a t a~+lhi+l -- a i f . Since d i m Y < n, and since the limit projection a i + l " X ---+ M i + l in n-invertible, we can find a map h" Y ---+ X such t h a t a i + l h -- hi+l. It is not hard to see t h a t h is an embedding. Moreover, h is U-close to f and f n h = f n f . Finally, let us show t h a t X is homeomorphic to #n. By the property of f n established above, X satisfies the condition from T h e o r e m 4.1.19., i.e. every map of an at most n-dimensional c o m p a c t u m into X can be arbitrarily closely a p p r o x i m a t e d by embeddings. T h e equality dim X -- n was also established above. Therefore, by T h e o r e m 4.1.19, it suffices to show t h a t X E L C n - I N C n - 1 . First we show t h a t X E L C n-1. Let x be an a r b i t r a r y point of X and U be a neighborhood of x. Take an index i and a n e i g h b o r h o o d Ui of a i ( x ) in Mi such t h a t a~-l(Ui) C U. Since, by our construction, Mi is an A N R - c o m p a c t u m (even an I ~ manifold), M i E L C n-1. Consequently, there exists a neighborhood Vi of a i ( x ) in Mi such t h a t Vi C Ui and the following condition is satisfied: 9 for each k < n - 1 and each map ~i" S k -+ V~, there is a m a p r B k+l --+ U~ such t h a t ~i = r k. Let VaT, l ( v i ) . Clearly, V is a neighborhood of x contained in U. Take any map ~o" S k ---+ V, k _< n - 1. By the choice of the n e i g h b o r h o o d Vi, there is a map r B k+l --+ Ui such t h a t ai~o = r k. Since (Sk, B k+l) is a polyhedral pair and the map a i is polyhedrally n-soft (recall t h a t k < n - 1), we conclude t h a t there is a map r B k+l --+ X such t h a t the following d i a g r a m commutes:

X

Sk t

.

~Y

.Bk+ 1

In other words, r k = ~ and a i r = r This obviously implies t h a t k+l) C_C_U. Thus, X E L C n-1. A similar (but simpler) a r g u m e n t shows t h a t X C C n-1. Therefore, X ~ #n. T h e proof is finished. [2] r

In order to o b t a i n other i m p o r t a n t properties of the m a p fn we need some preliminary statements. T h e proof of the following one is, in fact, contained in

150

4. MENGER MANIFOLDS

the proof of L e m m a 4.2.6. LEMMA 4.2.16. For each e > O, there is a 5 > 0 such that for any map ~o: S n-1 --~ F R , , with diam(im(~o)) < 5, and any 5-homotopy H : S n-1 x[0, 1]---, B n+l, with Ho = pn~o, there exists a h o m o t o p y G: S n-1 x [0, 1] ~ FR, such that Go = ~o, pnG = H and d i a m G ( S n-1 x [0, 1]) < e. LEMMA 4.2.17. Let Y be an L C n - l - c o m p a c t u m , K be a finite simplicial complex and ~o: Y --~ Ig(n+l)l be a map. Then the fibered product (pullback) X of the spaces FRK(,~) = 1-'n+l and Y with respect to the maps Pn+l -- PRK(n ) : Pn+l --+

IK(r'+l)l and ~o: Y --, Ig(n+l)l is an L C n - l - c o m p a c t u m . i i c~ l:~ n PROOF. Let c~ (~,~=1~.i ) be the one-point compactification of a discrete collection of n-dimensional disks B.n and let x be the compactifying point. Let S~'-1 denote the boundary of the disk B~'. Assume the contrary. Then there exists a map r (U~IS~ '-1) ---, X such that for each i and for any extension of r -1 (to the disk B~) the diameter of its image is more than some positive number a. Let a be an (n + 1)-dimensional simplex in K (n+l) containing the point y = pn+l~O~r Here ~o~:X --+ Fr,+l denotes the canonical projection (parallel to ~o) of the fibered product X. The map q: X ~ Y has similar meaning (see the following diagram):

a (u~,S~ '-I)

r

, X

,- Fn+l

Pn+l

y

r

,._

IK(n+l) I

We identify the simplex a with the unit (n + 1)-dimensional ball B n+l. Also, we identify the restriction of the many-valued retraction R g ( n ) : l K ( n + l ) l --~ Ig(n)l to lal with the many-valued retraction R n : B n+l ~ OB n+l (see the beginning of this section). We consider two cases. C a s e 1. Assume first that y f~ OBg +1, and take a neighborhood V of y in IK(n+l)l disjoint from OB~ +1. The restriction of Pn+l to this neighborhood is n-soft by construction. But then the restriction of q to q-l(~o-l(Y)) is also nsoft. It only remains to observe that the latter fact, coupled with the condition Y E L C n - l , contradicts the assumption made in the beginning. C a s e 2. Now assume that y E OB~ +1. Since Y c L C n-1 we can conclude that there is an extension r (U~__lBn) ~ Y of the composition qr Take

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

151

an e > 0 such that for each set F C X, the inequalities diam((t'(F)) < e and diam(q(F)) < e imply the inequality d i a m F < a. Next, choose 5 as in Lemma 4.2.16. Let b~ be the center of the ball B~. Connect the points 4(b~) and qr by paths ~i" [0, 1] ~ Y so that limi--.oo diam~i([O, 1]) = 0. Then there is a number k such that

max{diam(rlk([O, 1]) U r

diam((trlk([O , 1]) U (t(~(B~))} < 5.

Let Ht" S~ -1 x [0,1] --~ Y be a homotopy which c o n n e c t s r -1 with the constant map to the point 4(bk) inside of the image 4(B~), and then sends this constant map along the path ~k to another constant map to the point qr By L e m m a 4.2.16, the homotopy (tilt can be lifted to a homotopy Gt . S kn - 1 x [0, 11 --. Fn+l so that Go = (t'r G I ( S ~ -1) C_ Pn~I(Y) and d i a m ( G t ( S ~ -1 x [0, 1]) < e. Since Rn+I(Y) -1 "~ S n and the convex hall of G I ( S ~ -1) in the sphere S n has diameter less than e, we may conclude that, also shrinking c o n v ( G l ( S ~ -1) to a point, the homotopy Gt can be extended to an e-homotopy Lt" S~ -1 x [0, 1] --* Fn+l. Extending the homotopy Ht by adding the identity map, we obtain a homotopy Nt" S~ -1 • [0, 1] ~ Y such that (tiNt -- Pn+lLt. But then (recall that X is the fibered product in the above indicated diagram) we get a well-defined a-homotopy Mr" S~ -1 x [0,1] ~ X such that M0 = r -1 and M I ( S ~ -1) = r This contradiction finishes the proof. V1 LEMMA 4.2.18. Let K be an m-dimensional finite simplicial complex and f" X --+ [K[ be a map, constructed in Lemma 4.2.12. Then for each L C n - l - c o m p a c t u m Y , m > n, and for each map (t" Y ~ [K[, the fibered product Z of X and Y with respect to f and (t is an L C n - l - c o m p a c t u m . PROOF. Denote by X1 the inverse limit of the following diagram Fm

Fm-1

IKI

IK(m-1) [

Fn+2

...

[K(n+2) [

IK(n+l)[

and assume that p~" X1 ~ IK(i)l and q~" X1 ~ F~ denote the corresponding projections, n-softness of the projection p~ implies t h a t the fibered product y I of X1 and Y with respect to Pm I and (t is an L C n - l - c o m p a c t u m . Let (t~ Y~ ~ X1 be the projection of this fibered product, parallel to (t. It is easy to see that X is also the fibered product of spaces X1 and Fn+l with respect to maps ! . X l '"+ IK ( n + l ) [ and pn+l . Fn+l ---+ [ K ( n + l ) [. Then Z itself is the fibered Pn+l product of spaces Fn+l and Y~ with respect to Pn+l: Fn+l --~ IK(n+I)I and

152

Pn+l

4. MENGER MANIFOLDS

~ . y~

"-+ Ig

lemma is proved.

(n+l)

I" By L e m m a 4.2.17, Z is an L C n - l - c o m p a c t u m .

The

F'I

The following statement expresses one of the most important properties of the map fn: I~n --+ I~ constructed above. THEOREM 4.2.19. For each LCn-l-compactum Y contained in the Hilbert cube I ~, the inverse image f ~ l ( y ) is a #n-manifold. Additionally, i f Y E C n-l, then f ~ - l ( y ) is homeomorphic to #n. PROOF. We use the same notations as in the proof of Theorem 4.2.15. Recall t h a t the map f n : # n --~ I • was constructed as the limit projection of the inverse sequence S = {Mi, c~ +1} consisting of I ~ -manifolds (M1 -- I W) and n-invertible, polyhedrally n-soft and ( n - 1)-soft short projections. If Y is an LCn-l-compactum contained in I 0~ , then the inverse image fnZ(Y) is the limit space of the induced inverse sequence S ' - {Yi, r~+l}, where Y1 = Y, Yi+l = (c~+l)-l(Yi) and r~+1 = ~+l/Yi+l. Since the restriction of an ninvertible, polyhedrally n-soft and ( n - 1)-soft map onto the inverse image still has all these properties, we see that all short projections of the spectrum ,~' = {Yi, r~ +1} are n-invertible, polyhedrally n-soft and (n - 1)-soft. While proving Theorem 4.2.15 we have already seen t h a t the inverse limit of an inverse sequence consisting of L C n - l - c o m p a c t a and polyhedrally n-soft short projections is an LCn-l-compactum. Therefore it suffices to show that Yi E LC n-1 for each i. We prove this fact by induction. By assumption, Y1 E LC n-1. Assume t h a t Yi E LC n-1. The projection a~ +1 can be represented as the composition ")'i+1~i+1, where ~i+1 is a trivial bundle with fiber the Hilbert cube and "yi+l is homeomorphic to the map f~+l x i d i ~ . Here f~+l: X~+I ~ Ki is the map from Lemma 4.2.12 (see the diagram in the proof of Lemma 4.2.12). Consequently, by Lemma 4.2.18, ~i+l(Yi+l) is an LCn-l-compactum. But then Yi+l ~ ~i+l(Yi+l) x I ~ is also an LCn-l-compactum. Thus f ~ l ( y ) c LC n-1. Polyhedral n-softness of the short projections guarantees that if, in addition, Y E C n - l , then f n l ( Y ) e C n-1. Obviously, dim f ~ l ( y ) < dim #n = n. Theorem 4.2.15 shows that dim f n l ( Y ) ~n. Theorem 4.2.15 also guarantees that any map of an at most n-dimensional c o m p a c t u m into f~-I ( y ) can be arbitrarily closely approximated by embeddings. Therefore, by Theorem 4.1.19, f ~ Z ( y ) is a #n-manifold (and is homeomorphic to #n i f Y E Cn-1). [:]

Remark 4.2.20. In fact a stronger result can be proved: for any LC n-1c o m p a c t u m Y and for any map ~: Y ~ I W (not only for embeddings, as in Theorem 4.2.19) the fibered product of Y and #n with respect to ~ and fn is an L C n - 1_co m pact um. Summarizing the results proved above, we have.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

153

THEOREM 4.2.21. There are maps fn: #n ~ i W and gn: ttn ---* #n satisfying the following properties: (i) The maps fn and gn are n-invertible, ( n - 1 ) - s o f t and polyhedraUy n-soft. (ii) All the fibers of the maps fn and gn are homeomorphic to #n. (iii) The inverse images of L C n - l - c o m p a c t a under the maps fn and gn are #n-manifolds. (iv) The maps fn and gn both satisfy the parametric version of D D n P , that is, any two maps ~, ~ : I n ~ #n can be arbitrarily closely approximated by maps a I, ~l: I n ~ #n such that f n a l = f n ~ , f n ~ I = f n ~ , g n d = gnu, g n f l ' - - gnZ and i m ( a ' ) A im(Z') = O. Using the map fn as a guide, additional considerations allow us to obtain the following result. THEOREM 4.2.22. Any metrizable A ( N ) E ( n + 1)-compactum is an U V n - I image of: (i) An (n + 1)-dimensional A ( N ) E - c o m p a c t u m . (ii) The Hilbert cube I ~ (an I W -manifold, respectively). (iii) The (2n + 1)-dimensional cell ( a (2n + 1)-dimensional topological manifold, respectively). A simple comparison of the major ingredients of the Hilbert cube manifold theory (see Chapter 2) with the corresponding results of Menger manifold theory presented so far, shows that from a certain point of view the n-dimensional analog of the Hilbert cube Q should be considered to be, not the usual ndimensional cube I n, but the n-dimensional universal Menger compactum #n (moreover, the Hilbert cube itself may be viewed as the "infinite-dimensional Menger compactum"). In addition, one can observe a fairly deep analogy between the theories of #n-manifolds and Q-manifolds themselves. On the other hand, at first glance it is not clear what is the analog of the operation of "taking the product by Q" in #n-manifold theory - the operation which is involved in the formulations of triangulation (Theorem 2.3.28) and stability (Theorem 2.3.10) theorems for Q-manifolds. A decisive step in finding a "full" analog of this operation in #n-manifold theory is based on Theorems 4.2.15 and 4.2.19. First observe that taking the product X • Q of a space X and the Hilbert cube Q may be interpreted as taking the inverse image lr~-l(X) of a space X c Q, where r l " Q • Q ~ Q denotes the natural projection onto the first coordinate. It turns out that the map gn : f n / . f n l ( # n ) " #n ___+ttn in Theorem 4.2.15 may be thought of as the analog of the projection ~1 in the theory of #'~-manifolds. If this is agreed, everything then falls in place. The following statement is a triangulation theorem in #n-manifold theory. THEOREM 4.2.23. For any #n-manifold M , there is an n-dimensional polyhedron P such that for any embedding of P into #n the inverse image g~-l(p) is homeomorphic to M .

154

4. MENGER MANIFOLDS

PROOF. We consider the compact case. The locally compact case can be proved similarly. Take an n-dimensional finite polyhedron P and a map ~" P --~ M which induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i _< n - 1. E m b e d P into #n. It is easy to see that the composition

~gn/g~l(P) 9gnl(P)~

P ~ M

also induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n 1. Observe also t h a t the inverse image g ~ l ( p ) is a #n-manifold (we use Theorem 4.2.19). Therefore, by Theorem 4.1.21, M and g ~ l ( p ) are homeomorphic. [7 Here is the promised stability theorem for #"-manifolds. THEOREM 4.2.24. For any #n-manifold M in tt n, the inverse image g~-l(M) is homeomorphic to M . PROOF. If M is compact, then, by Theorem 4.2.19, the inverse image g n l ( M ) is a tin-manifold. Since the restriction g n / g n l ( M ) " g n l ( M ) ~ M is polyhedrally n-soft, we see t h a t it induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n - 1. By Theorem 4.1.21, g ~ l ( M ) is homeomorphic to M. The proof of the non-compact case is similar. Kl We conclude this section with the following statement which shows, in particular, that the map fn cannot be made n-soft. THEOREM 4.2.25. There is no n-soft map of an n-dimensional compactum onto a higher dimensional cube (the Hilbert cube in particular). PROOF. If n -- 0, the validity of our statement is obvious: the open image of a zero-dimensional c o m p a c t u m is zero-dimensional. Thus we may assume that n _> 1. If such a map g" Z --* I m existed, then Z must be connected (notice that 1-soft maps are monotone). Recall that cell-like maps cannot raise cohomological dimension (see Remark 3.1.11) and that the cohomological dimension of any cube coincides with its Lebesgue dimension: Therefore, g cannot be cell-like. It is clear now t h a t it suffices to prove the following claim. C l a i m . A n y non-constant n-soft map of a connected n-dimensional compactum is cell-like. Proof of Claim. Let f" X ---, Y be an n-soft map, where X is connected, d i m X = n and I Y I> 1. Since the fiber f - l ( y ) 6 L C n - I N C n - 1 for each y e Y we see t h a t g k ( f - l ( y ) ) = [-Ik(fkl(y)) = 0 for all k < n. Here g k denotes the singular homology with respect to the group of integers a n d / ~ k the (;'ech-homology. Then it follows that I ~ k ( f - l ( y ) ) = 0 for all k < n. Let us show that [-In(f-~(y)) = 0. If so, using the standard criterion that a finite-dimensional Peano continuum with trivial cohomology has trivial shape, we obtain the desired conclusion. Assume the contrary. Namely, t h a t / ~ n ( f - l ( y ) ) __/=0 for some y E Y. Take a map ~" f - l ( y ) ~ g ( Z , n ) (the Eilenberg-Maclane complex) which is not

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

155

homotopic to a constant map. Consider a point x E Y different from y. W i t h o u t loss of generality we may assume t h a t I f - l ( x ) I 1 (otherwise shrink the fiber f - l ( x ) into a point). Since dimX - n, there is an extension ~: Z ~ K(Z,n). Next dengte by Y0 the set of all those points z E Y for which the restriction ~ / f - l ( z ) is homotopic to a constant map. Obviously, Yo is not empty. Indeed, x E Yo. Let us show t h a t Y0 is an open set in Y. Take a point z E Yo, and let h: f - l ( z ) x [0,1] ~ g ( Z , n ) be a h o m o t o p y connecting ~ / f - l ( z ) with a constant map to some point c E K ( Z , n), i.e.

h / ( f - l ( z ) x {0}) = Cp/f-l(z) and h / ( f - l ( z ) x {1}) = c. Consider the closed subset A -- (X x {0, 1} U ( f - l ( z ) x [0, 1]) of the product X x [0, 1] and define the m a p hi" A ~ K ( Z , n) by letting

hl(X

X

{ 1 } ) = c, h l / ( X x { 0 } ) = ~ a n d

hl/(f-l(z)

x

[0, 1 ] ) = h.

Since the CW-complex K (Z, n) is an absolute extensor with respect to the class of compact spaces, we can extend hi to a map h" U ---, K(Z, n), where U is a neighborhood of A in the p r o d u c t X x [0, 1]. T h e n U contains an open set of the form G x [0, 1], where G is a neighborhood of the point z in Y. Clearly G is contained in Y0 (see the definition of Y0). Since z was an arbitrarily chosen point of Y0, we conclude t h a t Y0 is open in Y. Next observe t h a t the complement Y - Y0 is also a n o n - e m p t y set, since it contains y. Let us now show t h a t this complement is also open in Y. Assume the contrary. T h e n there exists a sequence {zk} of points of Y0 such t h a t z = limzk E Y - ]I0. We construct a sequence {gk" f - l ( z ) --+ f--l(zk)} of maps, which converges to the identity map idy-~(z) in the space C ( f - l ( z ) , X ) . Since the space F -- f - l ( z ) x ({zk" k e N } U { z } ) is at most n-dimensional, n-softness of the m a p f guarantees the existence of a map r F --. X such t h a t

r

x

{z}) = idy-l(z)

and r

x

{zk}) C f - l ( z k )

for each k E g .

Let gk -- e l ( f - l ( z ) X {Zk}). T h u s we have a sequence {~gk}of maps each of which is homotopic to a constant map and which converges to the m a p ~ / f - 1 (z). This contradicts the fact t h a t sufficiently close maps into A N R-space are homotopic. Consequently, Y - Y0 is open in Y. Connectedness of Y shows t h a t this is impossible F-1

Historical and bibliographical notes 4.2. T h e o r e m 4.2.1 and Proposition 4.2.2 (a weaker version of it - w i t h o u t stating an n-invertibility) were proved in [33]. T h e construction of n-invertible, ( n - 1)-soft maps presented in this Section, as well as T h e o r e m s 4.2.23 and 4.2.24 in the compact case, are taken from [127](see [90] for the non-compact case). This result completes a circle of works of various

156

4. MENGER MANIFOLDS

authors concerning the existence of dimension raising maps. Historically, the first example of an open map (of the one-dimensional compactum onto the twodimensional "Pontryagin surface") of this sort was constructed by Kolmogorov in [196] in 1937. Further examples, with some additional properties, have been constructed in [11], [12], [15], [319], [320], [189], [190], [191], [200], [211], [245], [248], [304] etc. Theorems 4.2.22 and 4.2.25 are taken from [129] and [125]. Theorem 4.2.22 extends an earlier result from [71].

4.3. n-soft m a p p i n g s of P o l i s h spaces~ r a i s i n g d i m e n s i o n It has already been remarked in Section 4.2 that the maps fn and gn, constructed in that section, cannot be made n-soft. Also, both of them fail to satisfy the property of preservation of Z-sets in the inverse direction. In this section we construct a map with the last property which is "almost" n-soft. We begin with the following technical statement. LEMMA 4.3.1. If f" X ---. Y is an n-soft map, then the inverse image of each Zn-set in Y is a Zn-set in X . For each simplex a, denote the first and second barycentric subdivisions of a by fla and ~2a respectively. Ma denotes the closed star of the barycenter va of a in the triangulation ~2a" Ma = St(Va,~2a). We put No = Int(Ma). Finally let ra "(I a I - { v a } ) ~ ] Oa ]denote the canonical deformation retraction. LEMMA 4.3.2. For each n > 0 and each countable locally-finite simplicial complex K , there exist a countable locally finite simplicial complex B ~ and proper simplicial maps f~" IBm( I--*1 g I and g~" IBm( I---*1 g ( n + l ) I satisfying the following conditions: (i) f ~ is an (n + 1)-invertible u y n - m a p . (ii) If a is a simplex of K, then g ~ ( ( f ~ : ) - l ( ] a ])) C_I a (n+l) ]. (iii) f ~ c / ( f ~ ) - l ( [ K (n+l) I ) = g ~ / ( f ~ : ) - l ( ] K(n+l)I). (iv) There exists a subspace Ang of ] B ~ l such that the restriction f ~ / A ~ " A ~ --*1K[ is an (n + 1)-soft map and the complement l B ~ l - A ~ is a a Z-set in ] B ~ ]. (v) If Z is a Zn+l-set in I g I, then ( f ~ c ) - l ( z ) is a Zn+l-set in ] B ~ ]. PROOF. Let us consider an arbitrary simplex a and define two compact-valued retractions ~ , r l a I---*1 a(n+l) I. Definitions are given by induction on the (n + 1 +/)-dimensional skeleta of a. If x el a (n+l) I, then we put ~ ( x ) = x = r If T is an (n + 2)-dimensional face of a, then

n(x) __ { 107"1, ~~

rr(x),

if x e Nr, i f x e ITI--NT

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

157

and

[OT[, r

--

r~-(x),

i f x 9 M~, if x 9 t~-1- M~

If T is an (n + 3)-dimensional face of a, then

~(~)

=

f I(0~)(-+1) I, t ~2(~(~)),

if x 9 Nr, if x 9 [ v [ - - N r

{ I(0T)(~+I)[, Can(rr(x)),

if x 9 Mr, if x 9 I~1- Mr.

and Can(x) =

Continuing this process, we obtain the desired retractions of [a] onto [a(n+l)[. Finally, if K is an arbitrary locally-finite simplicial complex, then the retractions ~ : , ~b~-IK} ~ ]K(n+l) 1 are defined as the unions of ~ and ~pa n, a 9 g . Now denote by B ~ the standard triangulation (induced by K) of the polyhedron {(x, y) 9 [g[ x ]g(n+l)[ 9 r ~ y}. We put also f~: = 7rl/[B~;[ and g~: = ~2/[B~I, where lr1" IKI • [K(n+l)l--* IKI and 7r2" IKI • IK("+~)I--' IK(n+l)[ denote the natural projections. Straightforward verification shows that: (a) the compact-valued retraction ~ : " [K[ ~ [K("+I)[ is lower semi-continuous. (b) the compact-valued retraction r IKI ---, IK(n+I)I is upper semi-continuous. (c) ~ - ( x ) C_ r for each point x 9 IKI. (d) the collection {r : x 9 IKI} is connected and uniformly locally connected in all dimensions less than n + 1 (we consider the standard metric

on

IKI).

Therefore, properties (i)-(iii) of the lemma are satisfied. Let us verify condition (iv). First of all, consider the subspace A~r = {(x,y) e ] K I • tK(n+l)l" y 9 qo~:(x)}

of IB~I and note that the complement I B ~ I - A~: is an Fa-subset of IB~I. By conditions (a), (c) and Theorem 2.1.15, the restriction f ~ / A ~ is an (n + 1)-soft map. Let us show that the complement IB~:I- A~. is a aZ-set in IB~I. Clearly it is sufficient to show that the last fact is true for the (n + 2)-dimensional simplex a. It follows from the construction that IB~:I- A~c = OMo • Io(-+a)l- T, where Z = {(~, y) e Iol • Io(~+1)1" ~ ( ~ ) = y}. Since OMa is a Z-set in Ma, we can conclude that OMo x [a(n+l)[ is a Z-set in M a x [a(n+l)[. Consequently, [B'~[-A'~ is a aZ-set in M a x [a(n+l)[. Consider now an open subspace U = [ B ~ [ - T of the polyhedron [B~[ Evidently, [ B ~ [ - A n C U C Mo x [a(n+l)[ Then the complement tBant- Aan is a aZ-set in U. Finally, for the same reason, we can conclude that [ B ~ ] - A~ is a aZ-set in [Ban[.

158

4. MENGER MANIFOLDS

The last condition is an easy consequence of condition (iv). Indeed, let Z be a Zn+~-set in [g[. Then, by (iv) and Lemma 4.3.1, the set

C 1 -=

{h e C ( I n + I , A ~ ) " ira(h) n ((S~7)-l(z) n A~:) -- 0}

is a dense G~-subset of C ( I n + I , A ~ ) . At the same time, by (iv), the set

6 2 - - {h e C ( I nq-1, [ B~z [)" ira(h) C Ang}

is also a dense G6-subset of C ( I n+l, IB~[). A Baire category argument finishes the proof, fl

THEOREM 4.3.3. Let n >_ 0 and let X be the limit space of an inverse sequence ~qx ---- {[ Xi [, p~+l} all spaces of which are locally finite polyhedra and all bonding maps of which are proper, simplicial and (n-b 1)-soft. Then there exists an (n-b 1)invertible proper UVn-surjection f ~ " M x ---* X of some #n+l-manifold M x onto X satisfying the following conditions: (i) For each (n -b 1)-dimensional locally compact space Z, closed subset Zo of Z, open cover Lt of M x , and proper map h" Z ~ M x , such that h/Zo is a fibered Z-embedding (with respect to f ~ ) , there exists a fibered Z-embedding (with respect to f ~ ) g" Z ---. M x which is U-close to h and such that f ~ g - f ~ h and g / Z o - - h/Zo. (ii) There exists a subspace A x of M x such that the restriction f ~ / A z " A x ---* X is an ( n + 1)-soft map and the complement M x - A x is a a Z-set in M x . (iii) If Z is a Zn+l-set in X , then ( f ~ ) - l ( Z ) is a Z-set in M x .

PROOF. For simplicity we consider only the compact case. The general case can be handled similarly. A standard Baire category argument reduces the proof to the case when Z0 - 0. After making these assumptions, we proceed as follows. We construct another polyhedral inverse sequences SM -- {[Ki[ , qi _i+1~~, all bonding maps of which are simplicial UVn-surjections. Further, we shall construct a family of ( n + 1)-invertible simplicial UYn-surjections fi: [Ki[ ---* [Xi[ which forms a strictly commutative (n+l)-invertible UYn-morphism {f~: [K~] [Xi[}: SM ~ 8 x . This means that if we fix an index i and consider the naturally arising diagram

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

159

IKi+ll

i+1

r~

iKil

p~+l

........

.fi

~ [Xi[

~ , i + l A .e then the characteristic map li = ~i ~ J i + l of the diagram is an ( n + 1)-invertible UVn-surjection (here ITil denotes the fibered product of IKi{ and IXi+ll with _i+1 respect to maps fi and ~'i ; si and ri denote the corresponding projections of this fibered product). Then we shall obtain the space M x as the limit of t.~M and the map f ~ as the limit map of the morphism {f i}. Let K1 = Bnxl and f l = f ~ l " IKll --~ IXll (see L e m m a 4.3.2). We can suppose, of course, that mesh(Kx) < 1 (if not, then we consider a sufficiently fine subdivision of K1 and denote it again by K1). Suppose now that we have already 1 and simplicial constructed finite simplicial complexes Km with m e s h ( K m ) < ~-~ n UV -surjections fm" [Kml ~ IXml and qm-lm . IKml __~ IKm_ll, m _< i, in such a way that the characteristic maps of all the naturally arising rectangular diagrams are (n + 1)-invertible UVn-surjections. In order to perform the inductive step, consider the fibered product ITil of IKil and IX~+ll with respect to the maps f~ and p~+l and denote by ri" ITil--* Igil and si" ITi[--~ IXi+ll the corresponding projections. Now consider the map f ~ , ' l B L,I n ~ ILil where Li denotes the natural triangulation of the polyhedron ITil x i2n+3 (we again use L e m m a 4.3.2). 1 Let K i + l -- BnL~, assuming at the same time t h a t mesh(K{+l) < 2~--Tf" Also _i+l define f~+l = sili and qi = ril +i, where li = ~lf~, and ~1" ITil x i2n+3 __~ iTil denotes the projection onto the first coordinate. One can easily verify that all our requirements are satisfied and consequently the inductive step is complete. As already remarked, we let M x = limSM and f ~ = lim{fi}. Since all mentioned rectangular diagrams strictly commute (because, by the construction, their characteristic m a p s - li's - are surjective) and all fi's are UVn-surjections, we can conclude that their limit map f } " M x --~ X is also a VVn-surjection. The standard argument (see C h a p t e r 6) shows that (n + 1)invertibility of all fi's and a l l / i ' s implies (n + 1)-invertibility of f ~ . Let us now verify condition (i). Fix an index m and an open cover L/m of IKml such that qml(Llm) refines b/(recall that qi: M x --~ Igil and pi: X ~ IX~I denote the limit projections of the spectra SM and S x respectively). We are going to

160

4. MENGER MANIFOLDS

construct maps gi" Z ~ [Ki[ in such a way t h a t the following conditions are satisfied: (a) If i > m, then gi is an embedding. (b) q i~+1g i + l - - g i .

(c) f~g~ = p ~ / ] h . (d) If i > m and c~i" i n + l __~ IK~I is an arbitrary map, then there exists a m a p 13i . I n+l ~ IKil such t h a t f i ~ __. f i a i , q i - l ~ i ~ q ii_ l a i and im(Z~) n i~(g~) = O. Let gi = qih for each i _< m. We now indicate how the map gi+l can be constructed. First of all fix an e m b e d d i n g u" Z ~ I2n+3 (we use the inequality d i m Z _< n + 1). Now consider the diagonal p r o d u c t g m A p m + l f ~ h , which maps Z into ITm] 9 T h e n the diagonal product (gin • P m + l f xnh ) A u will be an embedding of Z into the product ILml = ITml • I2n+3. By L e m m a 4.3.2, the map f~,," [Kin+l[ ~ [Lml is (n + 1)-invertible. Consequently, there exists an embedding gm+l" Z ---* I g m + l l such t h a t f~,.,,gm+l = ( g m A P m + l f ~ h ) A u . A simple verification shows t h a t conditions (b) and (c) are satisfied. Now consider any m a p a m + l " i n + l ~ IKm+ll" Since u is an e m b e d d i n g and dim Z _ n + 1, there is a point a E i2n+3 such t h a t a qf ira(u). Let the same letter a denote the constant map which sends the whole cube I n+l to the point a. Consider the m a p (lmo~m+l/ka) . I n+l ----* ILml and observe t h a t the image of I n+l under this m a p does not intersect the image of Z under the composition f L m g m + l . Consider now any lifting ~m+l" I n+l ~ IKm+l[ of the product (lmoLm+lAa). Again, straightforward verification shows t h a t condition (d) is also satisfied. Therefore, continuing this process we obtain the maps g~ for each i. By (b), the diagonal p r o d u c t g of all g~'s maps Z into M x . By (a), g is an embedding. By (c), the desired equality f ~ g - f ~ h also holds. T h e choice of an index m and the equalities qmg = gm "-- qmh (which are true by our construction) show t h a t g and h are/g-close. Finally, let us show t h a t ira(g) is a fibered Z-set with respect to the map f ~ . Fix an open cover l; of M x and any map c~" I n+l --+ M x . Clearly we can assume t h a t there exist an index j _> m and an open cover l)j of ]Kj[ such t h a t q~-l(1)j) refines "g. As above we shall inductively construct maps ~i" I n+l ---* [Ki[ in such a way t h a t the following conditions are satisfied" - i + l ~ i + 1 = ~i. (e) qi (f) f ij3i = pif~ca.

(g) im ( ~ + ~) n im (g~+ ~) = r (h) Zj = q ~ . We let ~i ---- qia for each i < j (consequently, the last condition is automatically satisfied). Let us construct the m a p ~3j+1. By (d) (assuming t h a t i = j + 1), we obtain a map/3j+1" I n+l -~[ g y + l [ such t h a t i m ( Z j + l ) ~ i m ( g j + l ) = 0 (i.e. j+l condition (g) is satisfied), qj j3j+l = ~j and fj+lJ3j+l = p j + l f ~ a . For i > j + 2 , we can construct maps ~i" I n+l ~ [ Ki [ in a similar way to the construction of /~y+l. It only remains to r e m a r k t h a t if ~ is the diagonal p r o d u c t of all ~i's, then one can easily verify t h a t ira(Z) ~ ira(g) = O, f ~ Z = f ~ a and t h a t fl and a are

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

161

])-close. This finishes the verification of condition (i). Let us now show that M x is an #n+l-manifold. Since mesh(Ki) ~ 0 we can conclude (conditions (ii) and (iii) of Lemma 4.3.2) that M x admits small maps into (n + 1)-dimensional polyhedra. Consequently, d i m M x _< n + 1. The inverse inequality is obvious, because, by the above verified condition (i), M x contains a copy of every (n + 1)-dimensional compactum. Since M x is, by the construction, the limit space of the polyhedral inverse sequence SM all bonding maps of which are UVn-surjections we conclude that M x is an LCn-compactum. Again, by (i), it follows immediately that M x satisfies the disjoint (n + 1)-disk property and hence, by Theorem 4.1.19, M x is an #n+l-manifold. Next, we verify condition (ii). By Lemma 4.3.2, there exists a subspace A1 of [KI[ such that the restriction f l / n l : A1 ~ [XI[ is an (n + 1)-soft map and the complement [ K I [ - A1 is a aZn+l-set of ]KI[. It follows from our construction that the fibered product R1 of A1 and [ X2 [ w i t h respect to the maps f l / A 1 and p2 is a subspace of [TI[. Clearly the natural projections of R1 onto A1 and IX2] coincide with the restrictions rl/R1 and Sl/R1 respectively. Since f l / A 1 is (n + 1)-soft, we conclude (see Lemma 6.2.5) that sl/R1 is also (n + 1)-soft. Consider the set R1 x i2n+3 and denote by A2 its inverse image under the (n + 1)-soft map f~l (see Lemma 4.3.2). Then the restriction f 2 / A 2 : A 2 ~ IX2[ is (n + 1)-soft. Since the bonding map p2 is (n + 1)-soft, using again Lemma 6.2.5, we see that the map rl is also ( n + l ) - s o f t . Consequently, by Lemma 4.3.1, the set ]TI[-R1 is a aZn+l-set in ]T1]. In this situation one can easily observe that, again by Lemma 4.3.1, the complement [ K 2 [ - A2 is a a Z n + l - s e t in [K2[. Continuing i + l / A i+ 1 ) and an in such a manner we obtain an inverse sequence SA = {Ai, ~i (n + 1)-soft morphism (in the sense of Chapter 6) morphism { f i / A i } of Sn to S x . Clearly, the limit space A x of the spectrum $A is a subspace of M x and the restriction f ~ / A x " A x ---* X , which coincides with the limit map of the morphism {f~/Ai}, is an (n + 1)-soft map (here we use L e m m a 6.2.7). It only remains to remark that the complement M x - A x is a aZ-set in M x . Condition (iii) is a direct consequence of condition (ii) and Lemma 4.3.1 (compare with the proof pf L e m m a 4.3.2). The proof is complete. [::] We also need the following statement. PROPOSITION 4.3.4. Each #n+lmanifold M can be represented as the limit space of an inverse sequence SM -- {Mi ,Pi i+1 } consisting of locally compact poly-

h~d~a a~d p~op~ ~imptic~al (n + 1)-~~t~bt~, n-~oft a~d polyh~d~aUy (,~ + 1)-~oft bonding maps. PROOF. By Proposition 4.1.10, there exists a proper n-homotopy equivalence a : ] K ]---~ M, where g is an at most (n + 1)-dimensional countable locally finite simplex. Let K0 = K. Suppose that the countable locally finite simplicial simplexes K~ and proper simplicial maps Pi-1 ~ 9 ]K~[ ~ [K~-I[, satisfying conditions

162

4. MENGER MANIFOLDS

(i) - (iii) from L e m m a 4.3.2, have already been c o n s t r u c t e d for each i _< m. We m a y also assume t h a t the m a p Pi-1 i 9 IKil - . IKi_ll has the following property: 9 For any m a p a" I n+l --, IKi_ll there exist two maps ill, f~2" I n+l --~ IKil such t h a t P~-lflj i ---- a, j -- 1,2, and im(f~l) N im(fl2) -- 0. In order to construct these objects for the next step, we apply L e m m a 4.3.2 to a sufficiently fine triangulation of the p o l y h e d r o n IKml x [0, 1]. In this way we get the next simplicial complex g m + l ---- B gmx[0,1] n m+l -- 7rlfg,,~ n x [0,1], 9 We let pm where ~rl"lgml x [0, 1] ~ Igml is the projection. As in the proof of Theorem 4.3.3, we can verify t h a t the limit space M ~ of the inverse sequence 8M -{Mi, Pi i + 1 } is a #n+l-manifold. Clearly, the limit projection p0" M ' --, IKI, being a p r o p e r U V n - m a p between at most (n + 1)-dimensional locally finite L C nspaces, is a proper n - h o m o t o p y equivalence. Therefore M and M ~ are properly n - h o m o t o p y equivalent. T h e o r e m 4.1.21 finishes the proof. V-1 Applying T h e o r e m 4.3.3 and P r o p o s i t i o n 4.3.4 we get the following s t a t e m e n t . THEOREM 4.3.5. Let n > 0 and X E ( # k . k >_ n + 2} U {I ~ }. Then there exists an (n + 1)-invertible UYn-surjection f ~ " #n+l __~ X , satisfying conditions ( i ) - (iii) of Theorem 4.3.3. We conclude this section with the following s t a t e m e n t . THEOREM 4.3.6. Let n ~_ O. For each locally compact polyhedron K , there exists a proper (n + 1)-invertible UYn-surjection h~" M ~ +1 ---. g of some #n+l_ manifold M ~ +1 onto K satisfying the following conditions: (i) xf L i~ ~ ~lo~d ~bpoly~d~o~ of K , t ~ it~ i ~ ~ im~g~ (h~)-I(L) is a #n+l-manifold. (ii) If L is a closed subpolyhedron of K and Z is a Z-set in L, then the

i~,~

im~g~ (h~)-l(z)

i~ ~ z - ~ t

i~

(h~)-~(L).

PROOF. R e p e a t the proof of T h e o r e m 4.3.3, first observing that if L is a s u b c o m p l e x of a countable locally finite simplicial complex K, then the simplicial complex B~ from L e m m a 4.3.2 is a s u b c o m p l e x of the complex B ~ and the map f ~ coincides with the restriction f ~ / I B ~ ]. []

Historical and bibliographical notes 4.3. L e m m a 4.3.1 was proved in [279]. All o t h e r results of this Section were o b t a i n e d by the a u t h o r [79], [84], [88], [97], [98]. The non-separable case was considered in [108].

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

163

4.4. Further properties of M e n g e r manifolds Using the existence of the dimension raising maps constructed in Sections 4.2 and 4.3, we now discuss other major ingredients of Menger manifold theory. 4.4.1. n - h o m o t o p y kernel and Open E m b e d d i n g T h e o r e m . The open embedding theorem for I ~ -manifolds states (see C h a p t e r 2) that for each I ~ manifold X, the product X x [0, 1) can be embedded into I ~ as an open subspace. Observe that identifying X with X x [0, 1] (stability of I"~-manifolds), the product X x [0, 1) may be viewed as the complement of the image of an appropriately chosen Z-embedding of X into itself. Using this remark as a guide, we introduce the following notion. Consider a #n+l-manifold M and two Z-embeddings f , g : M --~ M each of which is properly n-homotopic to the identity map idM. Then the homeomorphism g f - 1 . f ( M ) --. g(M) is properly n-homotopic to idf(M) and, consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending g f - 1 . Clearly the restriction h / ( M - f ( M ) ) is a homeomorphism between the complements M - f ( M ) and M - g(M). This shows that the following definition does not depend on the choice of a Z-embedding.

DEFINITION 4.4.1. The n-homotopy kernel K e r n ( M ) of a #n+l-manifold M is defined to be the complement M - f ( M ) , where f : M ~ M is an arbitrary Z-embedding properly n-homotopic to idM.

PROPOSITION 4.4.2. Let M and N be #n+l-manifolds. Then the following conditions are equivalent: (i) N admits a proper UVn-surjection onto the product M x [0, 1). (ii) N is homeomorphic to K e r n ( M ) . PROOF. It suffices to show that K e r n ( M ) also admits a proper UVn-surjecti on onto the product M x [0, 1). Take a proper v v n - s u r j e c t i o n f : M1 : M x [0, 1], where M1 is also a #n+l-manifold (see Theorem 4.2.1). Consider the quotient space M2 of M1 with respect to the partition whose nontrivial elements are fibers f - l ( m , 1) over the Z-set M x {1} of the product M • [0, 1]. Clearly, by Theorem 4.1.19, M2 is a #n+l-manifold. Moreover, if we consider the naturally induced proper UVn-surjection of M1 onto M2, then we conclude, by Theorem 4.1.21, that M1 and M2 are even homeomorphic. Next, it is easy to see that the set g - l ( M • {1}) is a Z-set in M2 and the restriction g / g - l ( M • {1}) is a homeomorphism. By Theorem 4.1.20, the composition 7rMg, where ~rM: M x [0, 1] ~ M is the natural projection, can be arbitrarily closely approximated by homeomorphisms. In particular, there exists a homeomorphism h: M2 ---+ M that is properly n-homotopic to 7rMg. T h e n the map r -- h g - l i , where

164

4. MENGER MANIFOLDS

i: M --* M x {1} is a n a t u r a l h o m e o m o r p h i s m , is a Z - e m b e d d i n g properly nh o m o t o p i c to idM. Indeed, r - h g - l i ~n --p 7rMgg-1 i -- zrMi -- idM. Thus, by Definition 4.4.1, we conclude t h a t the c o m p l e m e n t M - r ( M ) is h o m e o m o r p h i c to K e r n ( M ) . Consequently, the space M 2 - g - l ( M • { 1 } ) - h - l ( M - r ( M ) ) i s also h o m e o m o r p h i c to K e r n ( M ) . It only remains to note t h a t the space M 2 g - l ( M x {1}) a d m i t s a p r o p e r UVn-surjection onto the p r o d u c t M x [0, 1). V1

Now we are ready to prove the open e m b e d d i n g t h e o r e m for Itn+l-manifolds.

THEOREM 4.4.3. The n-homotopy kernel of each #n+l-manifold admits an open embedding into Itn+l.

PROOF. First of all let us show t h a t every Itn+l-manifold M admits a p r o p e r U V n - m a p onto a certain I w -manifold X. For this we take a proper UVn-map ~" M ~ P , where P is a locally c o m p a c t polyhedron (see T h e o r e m 4.2.23). Clearly the p r o d u c t P • I w --- X is a IW-manifold (see C h a p t e r 2). By T h e o r e m 4.2.1, there is a p r o p e r UVn-surjection r M ' ~ M • I w of some #n+l-manifold onto the locally c o m p a c t L C n - s p a c e M x I w . Since the composition r M r M ' M is also a proper U V n - m a p , we conclude, by T h e o r e m 4.1.20, that M ' and M are homeomorphic. It only remains to observe t h a t the required proper UV nsurjection h" M --* X is given by the composition (~o • idlw )r Now we proceed with the direct proof of our theorem. Take a proper UV nsurjection h" M ---+ X, where X is a I w -manifold. By the open e m b e d d i n g t h e o r e m f o r / W - m a n i f o l d s (see T h e o r e m 2.3.25), one m a y suppose that the product Z • [0, 1) lies in I w as an open subspace. Consider a p r o p e r (n + 1)-invertible UVn-map g" M1 ---* M x [0, 1], where M1 is a # n + l - m a n i f o l d (we use T h e o r e m 4.2.1). Since the m a p g is (n q- 1)-invertible and dim M -- n q- 1, we may assume, w i t h o u t loss of generality, t h a t the restriction g / g - l ( M x {1}) is a homeomorphism and A = g - l ( M • {1}) is a Z-set in M1. As above, we conclude, using T h e o r e m 4.1.20, t h a t the # n + l - m a n i f o l d s M and M1 are homeomorphic. Therefore, it only remains to show t h a t the c o m p l e m e n t M1 - A (which is obviously h o m e o m o r p h i c to K e r n ( M ) ) admits an open e m b e d d i n g into #n+l. Consider the m a p f" # n + l ___, i w of T h e o r e m 4.3.5 and let M2 denote the # n + l - m a n i f o l d f - l ( X x [0, 1)), which is open in # n + l . All t h a t remains to be shown is t h a t the # n + l - m a n i f o l d s M2 and M1 - A are homeomorphic. The last fact can be observed in the following way: b o t h # n + l - m a n i f o l d s MI - A and M2 admit proper U Y n - m a p s onto X • [0, 1) (consider the maps (h • id)g and f ) . Therefore, by T h e o r e m 4.1.21, they are homeomorphic. T h e following d i a g r a m helps to reconstruct the complete argument.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS M2

f

M1-A

M1

g

~ M x [0, 1)

g

~M

x

[0,1]

~t n + l

f

hxid

•

h

x [0,1]

•

165

,~ I w

71"

M This finishes the proof.

~X

K]

4.4.2. n - h o m o t o p y C l a s s i f i c a t i o n T h e o r e m . Theorem 4.1.21 completely describes the proper n-homotopy classification of #n+l-manifolds. In particular, n-homotopy equivalent compact #n+l-manifolds are homeomorphic. Obviously, the last fact is incorrect in the non-compact case: compare #n+l and # n + l - { p t } . The main result of this subsection solves the n-homotopy classification problem of arbitrary ttn+lmanifolds (compare with Theorem 2.3.26). PROPOSITION 4.4.4. For each #n+l-manifold M, the spaces K e r n ( M ) and Kern(Kern(M)) are homeomorphic. PROOF. As above, take a proper UVn-surjection g: M --~ X of M onto a I w -manifold X. By Proposition 4.4.2, there exists a proper UVn-surjection f : Kern(M) ~ M x [0, 1). For the same reason, there exists a proper UV nsurjection h: K e r n ( K e r n ( M ) ) --~ Kern(M) x [0, 1). Consequently, we have two proper UVn-surjections:

p = (g x id)f: Kern(M) ~ M x [0, 1) and

q = (p x id)h: K e r n ( K e r n ( M ) ) ~ K e r n ( M ) x [0, 1),

166

4. MENGER MANIFOLDS

where id denotes the identity map of [0, 1). Since X is a Hilbert cube manifold, the product X x [0, 1] is homeomorphic to X (see Chapter 2). Remarking that the spaces [0, 1) x [0, 1) and [0, 1] x [0, 1) are homeomorphic, we have X x [0,1) x [ 0 , 1 ) ~ X x [0,1] x [0,1) ..~ X x [0, 1). Consequently, the #n+l-manifolds K e r n ( M ) and K e r n ( K e r n ( M ) ) admit proper UVn-surjections onto the same I ~ -manifold X x [0, 1). Therefore they are properly n-homotopy equivalent. Theorem 4.1.21 finishes the proof. V1 PROPOSITION 4.4.5. Let M be a #n+l-manifold. I r A is a Z-set in K e r n ( M ) , then the spaces K e r n ( M ) and K e r n ( K e r n ( M ) - A) are homeomorphic. PROOF. As in the proof of Proposition 4.4.4 consider three proper UV nsurjections:

g: M ~ X, f : K e r n ( M ) - - * M x[0, 1 ) a n d p =

(gxid)f: Kern(M)~

Xx[0,1),

where X is a I ~ -manifold and id denotes the identity map of [0, 1). Now we redefine the map p in such a way that the set p(A) will be a Z-set in X x [0, 1). For this, consider any Z-embedding r: A ~ X x [0, 1) properly n-homotopic to the restriction p/A. By Proposition 4.2.2, there exists a proper UVn-surjection q: K e r n ( M ) ---* X x [0,1) such that q/A = r. Moreover, as in the proof of Proposition 4.4.2, we can additionally suppose that A = q-lq(A). Consequently, the restriction

q / ( K e r n ( M ) - A): K e r n ( M ) -

A ---. X x [0, 1 ) - q(A)

is a proper UVn-surjection. As above (compare with the proof of Proposition 4.4.4) this implies that K e r n ( K e r n ( M ) - A) admits a proper UYn-surjection onto the product (X x [0, 1 ) - q(A)) x [0, 1). By the same argument, there exists a proper VYn-surjection of K e r n ( K e r n ( M ) ) onto the product X x [0, 1) x [0, 1). By Proposition 4.4.4, K e r n ( M ) and K e r n ( K e r n ( M ) ) are homeomorphic. Therefore, by Theorem 4.1.21, it suffices to show that the spaces (X • [ 0 , 1 ) - q ( A ) )

x [0,1) and X x [ 0 , 1 ) x [0,1)

are homeomorphic. Indeed, since q(A) is a Z-set in X x [0, 1), we conclude that the I~-manifolds X x [0, 1) and X x [0, 1 ) - q ( A ) are homotopy equivalent. Then, by the homotopy classification theorem for I W-manifolds (Theorem 2.3.26), the products X x [0, 1) x [0, 1) and (X x [0, 1 ) - q(A)) x [0, 1) are homeomorphic. This finishes the proof. E] PROPOSITION 4.4.6. Let a #n+l-manifold M be a Z-set of a #n+l-manifold N , and suppose the inclusion i: M --+ N is an n-homotopy equivalence. Then there exists a Z-set A in N such that the complement N - A is homeomorphic to K e r n ( M ) .

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

167

PROOF9 By Theorem 4.2.23, there exist (n + 1)-invertible proper UVn-sur jections ~: M ---. L and r N ---. K1, where L and K1 are (n + 1)-dimensional locally compact polyhedra. The (n + 1)-invertibility of ~ implies the existence of a proper map s: L ~ M with ~s - idL. Then the composition r L --. K1 is properly homotopic to some proper piecewise linear (PL) map p: L ~ K1. Form the mapping cylinder M(p) -- K of the map p. Recall that this is the space obtained from the disjoint union (L • [0, 1])@ K1, by identifying (/, 1) with p(l) for each l E L. At the same time we identify L with L x {0}. Clearly L x [0, 1) can be considered as an open subspace of K. Since p is a proper PLmap, L • {0} and K1 are subpolyhedra of the polyhedron K. Let c: K ~ K1 be the collapse to the base, i.e. the natural retraction defined by sending (/, t) to p(l). Clearly, c is a proper CE-surjection that is a homotopy equivalence. Now we consider an (n + 1)-invertible proper UYn-surjection f~(" M~(+1 --. K , satisfying the conditions of Theorem 4.3.6. Clearly, the composition cf~" M~(+1 ---* K1 is a proper UVn-surjection, and hence, by Theorem 4.1.21, the #n+l-manifolds M ~ +1 and N are homeomorphic. By Theorem 4.3.6, the inverse image ( f ~ ) - l ( L x {0}) is a #n+l-manifold that, again by Theorem 4.1.21, is homeomorphic to M. One can easily verify, using the assumption and the specifics of the above construction, that the natural inclusion of ( f ~ ) - l ( L x {0}) into M ~ +1 is an n-homotopy equivalence. Moreover, by Theorem 4.3.6, the above inverse image is a Z-set in M ~ +1 (since L • {0} is a Z-set in K). Now redefining the above objects for simplicity, we have the following situation. A proper UVn-surjection f : N ~ K, satisfying the conditions of Theorem 4.3.6, is given, M -- f - l ( L • {0}) is a Z-set in N and the inclusion M ~ N is an n-homotopy equivalence. Clearly, K - K1 -- L x [0, 1), and hence the inverse i m a g e / - I ( L • [0, 1)) admits a proper UYn-surjection onto L • [0, 1). On the other hand, K e r n ( M ) admits a proper vVn-surjection onto M • [0, 1), and hence onto the product L • [0, 1) as well. Thus, by Theorem 4.1.21, the inverse image f - l ( L • [0, 1)) and K e r n ( M ) are homeomorphic. Consequently, to finish the proof it remains to construct an open embedding h: f - l ( L x [0, 1)) ---. N such t h a t the complement A -- g - h ( f - l ( L • [0, 1)) is a Z-set in N. Since N is a #n+l-manifold, by Theorem 4.1.19, there is a countable dense subset {~k: k -- 1 , 2 , . . . } of c ( I n + l , g ) consisting of Z-embeddings. As observed above, f - l ( L • {0}) is a Z-set in N and the inclusion f - l ( L x {0}) ~ N is an n-homotopy equivalence. It easily follows from Proposition 4.2.2 that in this case there exists a retraction rl: N --. f - l ( L x {0}) that is n-homotopic to idN. Consider the restriction 81 = r l / 9

9 f-l(n

x {0}-)U~I(/n+l)

---~ f - l ( n

1

x [0, ~)).

Clearly 81 is a proper map. By Theorem 4.1.19, Sl is properly n-homotopic to a Z-embedding

gl 9 I - I ( L x { 0 } . ) U ~ I

(/n+l

)-"> .f --1 (L x [0, 1 )) Z

168

4. MENGER MANIFOLDS

that coincides with the identity map on f - l ( L x {0}). By Theorem 4.1.15, there exists a homeomorphism GI" N --~ N extending gl. Put hi = G~-1. Then hi is a homeomorphism such t h a t (a) h l / f - l ( L x { 0 } ) - id. (b) (ill(/n+l) C_ h l f - l ( L x [0, 1)). 1 Consider now the polyhedron K - (L x [0, ~)). Since the set L x { 89 is a Z-set in K (L x [0, ~)), 1 we can conclude, by Theorem 4.3.6, that the set h l f - l ( L x { 1}) is a Z-set in a #n+l-manifold N - h l f - l ( L • [0, 89 Moreover, since the inclusion -

1 L x {~}~K-L

1 x [0,~))

is a homotopy equivalence, we conclude that the inclusion

hlf-l(L x { })~

N-hl.f-l(L

x [0,~))

is an n-homotopy equivalence. Again, using the above construction, we see that there is a homeomorphism

h ,2. N - h l f

-1

1

(L x

))---+ N

which is the identity on h l f - l ( L x {89 ~o2(I n+l) C'I ( N -

hlf-l(L

x [0,

-

h 1 f - l ( L x [0, 1

and for which ))) C_ h ~ h l f - l ( L x [~,2-

)).

Extend h~ to a homeomorphism h2 defined on N by defining h2 = id on

h l f - l ( L x [0, 89 Then we have 1

~o1(In+l) U ~o2(I~+1) C_ h 2 h l f - l ( L x [0,2 9 g)). Inductively continuing this process, we construct homeomorphisms hk" N ~ N in such a way that hk+l = id on h k f - l ( L x [O,k. k--~]) and 1

V~l(In+l)IJ...iJ(Pk(In+l ) C_ h k h k _ l . . . h l f - l ( L x [O,k. k + 1) ). Define an open embedding h" f - l ( L x [0, 1)) ~ N by h(x) = limk--.oo h k ' " hi(x) for each x E f - l ( L x [0, 1)). Clearly U{~k ( i n + l )" k = 1 , 2 , . . . } _C h f - l ( L x [0, 1)) and consequently, by the choice of the family {~k" k = 1, 2 , . . . }, the complement [-1

N - h f - l ( L x [0, 1)) is a Z-set in N.

Now we are ready to prove the n-homotopy classification theorem for #n+l_ manifolds. THEOREM 4.4.7. #n+l-manifolds are n-homotopy equivalent if and only if their n-homotopy kernels are homeomorphic.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

169

PROOF. Let M and N be n - h o m o t o p y equivalent tin+l-manifolds. Take maps n n a" M ~ N and/3" N ~ M such t h a t ~ a " ~ i d M and a ~ " ~ i d N . As above, we can find (n + 1)-invertible proper UYn-surjections f " K e r n ( M ) ~ M • [0, 1) and g" K e r n ( N ) ~ N • [0, 1). Let ~" M x [0, 1) ~ N x [0, 1) and r g x [0, 1) --, M • [0, 1) be a proper maps such t h a t ~ is homotopic to a x id and r is homotopic to fl x id, where id denotes the identity map of [0, 1) (compare with [69, L e m m a 21.1]) . Since g is (n + 1)-invertible and dim K e r n ( M ) = n + 1, there exists a proper m a p r" K e r n ( M ) ~ K e r n ( N ) such t h a t gr = ~o.f. Similarly we have a proper map s" K e r n ( N ) ~ K e r n ( M ) such t h a t f s = Cg. In this situation one n

can verify directly t h a t sr "~ idKer,~(g). Moreover, by T h e o r e m 4.1.19, we can additionally suppose t h a t r and s are Z-embeddings. Now consider the Z-set r ( g e r n ( M ) ) in a # n + l - m a n i f o l d K e r n ( N ) . It follows immediately from the above construction t h a t the inclusion map

r(Kern(M)) ~ Kern(N) is an n - h o m o t o p y equivalence. By Proposition 4.4.6, there is a Z-set A in K e r n ( N ) such t h a t K e r n ( N ) - A is h o m e o m o r p h i c to K e r n ( r ( g e r n ( M ) ) ) -K e r n ( K e r n ( M ) ) (recall t h a t r is an embedding). Then, by Propositions 4.4.4 and 4.4.5,

K e r n ( M ) .~ K e r n ( K e r n ( K e r n ( M ) ) ) ..~ K e r n ( K e r n ( N ) as desired. T h e second part of the t h e o r e m is trivial.

A) ..~ K e r n ( N )

V1

4.4.3. n - s h a p e a n d t h e C o m p l e m e n t T h e o r e m . T h e famous Complement T h e o r e m for I • -manifolds [69] states t h a t if X and Y are Z-sets in I ~ , then their complements I ~ - X and I ~ - Y are h o m e o m o r p h i c if and only if the shapes of X and Y coincide, i.e. S h ( Z ) = S h ( Y ) . T h e obvious form of T h e C o m p l e m e n t T h e o r e m fails for #n+l. T h e equality of shapes of two Z-sets X and Y in # n + l is sufficient, but far from necessary, for the c o m p l e m e n t s # n + l - X and # n + l _ y to be homeomorphic. Indeed, it can be easily seen t h a t if the (n + 1)dimensional sphere S n+l is e m b e d d e d into # n + l as a Z-set, t h e n # n + l _ s n + l is h o m e o m o r p h i c to tt n+l - {pt}. At the same time S h ( S n+l) ~ Sh(pt). T h e problem was solved in [87] (see also [90], [92]) where the notion of n - s h a p e was introduced. T h e relation between n - S H A P E and n - H O M O T O P Y categories is of the same n a t u r e as t h a t between the categories of S H A P E and H O M O T O P Y . Roughly, n - S H A P E is a "spectral completion" of n - H O M O T O P Y . T h e main result in this direction is the following. THEOREM 4.4.8. Let X and Y be Z-sets in #n+l. The complements tt n+l - X and # n + l _ y are homeomorphic if and only if n - S h ( X ) -- n - S h ( Y ) .

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We would like to mention some corollaries of this theorem and the definition of n-shape itself. COROLLARY 4.4.9. I f S h ( X ) -- S h ( Y ) , then n -

S h ( X ) -- n - S h ( Y ) .

COROLLARY 4.4.10. I f X and Y are at most n-dimensional, then S h ( X ) S h ( Y ) if and only if n - S h ( X ) -- n - S h ( Y ) .

=

COROLLARY 4.4.11. I f Z - s e t s X and Y in #n+l are #n+l-manifolds, then the complements #n+l _ X and #n+l _ y are homeomorphic if and only if the compacta X and Y are homeomorphic. Let us emphasize that the notion of n-equivalence, introduced by Ferry [150] as a generalization of Whitehead's notion of n-type, coincides in several important cases with the notion of n-shape. Relations between these two concepts have been studied in [97].We conclude this section by noting that Theorem 4.4.8 was extended [280] to a larger class of subspaces than Z-sets. These are the so-called weak Z-sets.

4.4.4. M e n d e r m a n i f o l d s w i t h b o u n d a r i e s . The problem of putting a boundary on various types of manifolds were considered in [57] (PL manifolds), [282] (smooth manifolds) and [70] (I ~ - manifolds). It was proved in [70] that if an I W-manifold M satisfies certain minimal necessary homotopy-theoretical conditions (finite type and tameness at oo), then there are two obstructions a ~ ( M ) and Tc~(M) to M having a boundary. The first one is an element of the group li+___m{/C0rl(M - A ) " A C M A is compact}, where ]C0~rl is the projective class group functor. If c o o ( M ) -- O, then the second obstruction can be defined as an element of the first derived limit of the inverse system li.___m{YVhrl( M - A ) " A C M A is compact}, where ~Vh~rl is the Whitehead group functor. It was shown in [70] t h a t the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms o f t h e group l i m { Y ~ h ~ l ( M - A ) " A C M A is compact}. It should be emphasized that the above mentioned obstructions essentially involve the Wall's finiteness obstruction [303]. The natural analog of Wall's obstruction vanishes in the n-homotopy category. This is exactly what was stated in 4.1.12. We will see that this observation significantly simplifies the situation for #n+l-manifolds. First of all we need the following corollary of Proposition 4.1.12. PROPOSITION 4.4.12. I f a # n + l - m a n i f o l d M is n - h o m o t o p y dominated by an at m o s t (n + 1)-dimensional L C n - c o m p a c t u m , then M is n - h o m o t o p y equivalent to a compact #n+l-manifold.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

171

We say t h a t a #'~+l-manifold M admits a boundary if there exists a compact # n + l manifold N such t h a t M = N - Z, where Z is a Z-set in N. In this case we shall say t h a t N is a compactification of M corresponding to the b o u n d a r y Z, and conversely, Z is a b o u n d a r y of M corresponding to the compactification N. We also need the following definition [100]. DEFINITION 4.4.13. A space X is said to be n - t a m e at c~ if for each compactum A C X there exists a larger compactum B C X such that the inclusion X - B ~-~ X - A factors up to n-homotopy through an at most (n + 1)-dimensional finite polyhedron. PROPOSITION 4.4.14. I f a #n+l-manifold M is n-tame at c~, then M is nhomotopy equivalent to a compact #n+l-manifold. PROOF. Take a proper U V n - r e t r a c t i o n r: M ~ P of the given # n + l - m a n i f o l d M onto some (n + 1)-dimensional locally compact polyhedron P. It follows from elementary properties of proper U V n - m a p s t h a t P is n - t a m e at c~ as well. Using Proposition 4.1.8, one can easily see t h a t P is n - h o m o t o p y d o m i n a t e d by an at most (n + 1)-dimensional compact polyhedron. Proposition 4.4.12 finishes the proof. [::] Let us recall t h a t an I ~ -manifold M lying in a larger I ~ -manifold N is said to be clean if M is closed in N and the topological frontier of M in N is collared b o t h in M and N - I n t M . For obvious dimensional reasons we cannot directly define the corresponding notion for #n+l-manifolds. Nevertheless, the following notion is sufficient for us. DEFINITION 4.4.15. A #n+l-manifold M lying in a #n+l-manifold N is said to be n-clean in N provided that M is closed in N and there exists a closed subspace 5 ( M ) of M such that the following conditions are satisfied:

(i) 6(M) /~ ~ (ii) (iii) (iv) (v)

,~+~-m~ifold.

( N - M ) t2 5 ( M ) is a #n+l-manifold. 5 ( M ) is a Z - s e t in M . 5(M) is a Z - s e t in ( N - M ) U S(M). M - 5 ( M ) is open in g .

Sometimes we say t h a t M is n-clean with respect to 5 ( M ) . Let us indicate the s t a n d a r d situation in which n-clean submanifolds arise naturally. Suppose t h a t L is submanifold of a combinatorial PL-manifold P . Consider a proper UVn-surjection f : N ~ P of a # n + l - m a n i f o l d N from Theorem 4.3.6. Using the properties of f , it is easy to see t h a t M = f - l ( L ) is an n-clean submanifold of N with 5 ( M ) = f - l ( O L ) . Generally speaking, .f is not an open map and consequently 5 ( M ) does not necessarily coincide with the topological frontier of M in N.

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LEMMA 4.4.16. Let N be a #n+l-manifold which is n - t a m e at oo. Suppose that M is a compact and n-clean submanifold of N . Then the #n+l-manifold (N - M ) U 5 ( M ) is n-homotopy equivalent to a compact #n+l-manifold. PROOF. By Proposition 4.4.14, it suffices to show t h a t the #n+l-manifold ( N - M ) U S ( M ) is n - t a m e at 00. Let A be a compact subspace of ( N - M ) U 6 ( M ) . Clearly, K1 = A U M is compact. Since N is n - t a m e at 00, there exists a c o m p a c t u m K2 such t h a t Kz C_ K2 C_ N and the inclusion N - K 2 ~ NK1 factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Let B = ((N - M ) U 6 ( M ) ) gl K2. Clearly B is a c o m p a c t u m and A c B. Note t h a t D

((N-M)U6(M))-B=N-K2

and N - K I C _

((N-M)U6(M))-A.

Consequently, the inclusion ((N-M)U6(M))-B

~

((N-M)U6(M))-A

factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Hence ( g - M ) U 6 ( M ) is n - t a m e at c~. FI LEMMA 4.4.17. A n y #n+l-manifold M can be written as a union M = Ui=IM~ such that all M i ' s are compact and n-clean and M~ C M i + l - 6 ( M i + 1 ) for each i= 1,2,.... oo

PROOF. It suffices to show t h a t for each c o m p a c t u m K C_ M, there exists a compact and n-clean M1 C_ M such t h a t K C_ M1 - 6 ( M 1 ) . As before, take a proper UVn-surjection g" M ~ X, where X is a I ~ -manifold. There is a compact and clean Y _c X such t h a t g ( K ) C_ I n t ( x ( Y ) (see [70]). By the relative triangulation theorem for I~-manifolds (see T h e o r e m 2.3.31), there exists a polyhedron P which can be w r i t t e n as a union of two subpolyhedra P1 and /)2 such t h a t X = P x I ~ , Y = P1 x I ~ , X - I n t x ( Y ) = P2 x I W and B d x ( Y ) = (P1 CI P2) x I W . Note also t h a t the subpolyhedron P1 F1P2 is a Z-set b o t h in P1 and P2. Consider now a proper UVn-surjection f" N ~ P of a #n+l-manifold N onto the polyhedron P satisfying the conditions of T h e o r e m 4.3.6. Consequently, we have two proper u v n - s u r j e c t i o n s f" N ~ P and 7rpg" M --+ P (here ~p" P x I ~ --, P denotes the n a t u r a l projection) of two #n+l-manifolds onto the polyhedron P. Consider an open cover b / - {P - 7 r g g ( K ) , I n t p ( P 1 ) } of P. By T h e o r e m 4.1.20, there exists a h o m e o m o r p h i s m h" M --+ N such t h a t the compositions 7rgg and f h are b/-close. Let M1 -- h - l f - l ( p 1 ) and 6(M1) = h - l f - l ( p 1 rl P2). By the properties of the map f, M1 is compact and n-clean. It only remains to note t h a t K C_ M1 - 6 ( M 1 ) . This finishes the proof. F1 We also need the following s t a t e m e n t , which is a direct consequence of the characterization T h e o r e m 4.1.19.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

173

PROPOSITION 4.4.18. Let a space M be the union of two closed subspaces M1 and M2. If M1, M2 and Mo = M1 A M2 are #n+l-manifolds and Mo is a Z-set both in M1 and M2, then M is a #n+l-manifold. PROOF. It suffices to show t h a t for any m a p f : X ~ M of an at most (n + 1)dimensional c o m p a c t u m X into M , and any open cover U E cov(M), t h e r e exists an e m b e d d i n g g: X ~ M , U-close to f . Let us consider the case when f ( X ) N Mi 7~ 0 for each i = 0, 1, 2. All o t h e r cases are trivial. By P r o p o s i t i o n 4.1.7, t h e r e exists an open cover 12 E coy(M) refining b / s u c h t h a t the following condition is satisfied: (.)~ for any at most (n + 1)-dimensional c o m p a c t u m B, closed s u b s p a c e A of B, and any two P-close m a p s c~1,c~2: A --. M such t h a t c~1 has an extension r B --. M , it follows t h a t C~2 also a d m i t s an extension r B ~ M which is/,/-close to ~1. Let Xi = f - l ( M i ) , i = 0, 1,2. Since M0 is a # n + l - m a n i f o l d , there is a Ze m b e d d i n g go: X0 --~ M0 such t h a t go and f / X o are ]2-close. By (*)n, t h e r e is an extension h: X --. M of G - ) such t h a t h and f are V-close. Since Mo is a Z-set in b o t h M1 and M2 we conclude t h a t go(Xo) is a Z - s e t b o t h in M1 and M2. Consequently, by T h e o r e m 4.1.19, for each i = 1, 2 there is a Z - e m b e d d i n g gi: Xi ~ Mi such t h a t gi/Xo = go and gi is U-close to h / X i . At the same time, w i t h o u t loss of generality we can assume t h a t one of these maps, say gl, has the following p r o p e r t y : g l ( X 1 - X0) N M0 = 0 (we once again use the fact t h a t M0 is a Z-set in M1). T h e n the map g, coinciding with gi on Xi (i = 1,2), is an embedding. It only remains to note t h a t g and .f are N-close. [] LEMMA 4.4.19. If a #n+l-manifold M is n-tame at oo, then we can write M = U~__IMi such that all Mi's are compact and n-clean, Mi C M i + l - ~ ( M i + l ) and the inclusion 5(Mi) r ( M i + l - Mi) U 5(Mi) is n-homotopy equivalence for each i = 1 , 2 , . . . . PROOF. Choose any c o m p a c t and n-clean submanifold A of M . By L e m m a 4.4.17, it suffices to find a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B 5(B) and the inclusion 8(B) ~-~ ( M - B ) U 5(B) is an nh o m o t o p y equivalence. By L e m m a 4.4.16, the # n + l - m a n i f o l d ( M - A)8(A) is n - h o m o t o p y equivalent to some c o m p a c t # n + l - m a n i f o l d X. Fix the corresponding n - h o m o t o p y equivalence r ( M - a ) U S ( A ) + X and its n - h o m o t o p y inverse ~o1" X ~ (M - A ) U 5(A). Obviously there is a m a p ~92" (M - A ) 5 ( A ) ~ X such t h a t r 5(A) --~ X is a Z - e m b e d d i n g a n d r is as close to r as we wish. Similarly, there is a Z - e m b e d d i n g ~o2" X ~ ( M - A ) U 5(A) which is as close to ~Ol as we wish. In particular, we can assume t h a t r and ~o2 are n - h o m o t o p y equivalences. If r and ~o2 were chosen sufficiently close to r and ~ol respectively, then, by T h e o r e m 4.1.18, t h e r e exists a h o m e o m o r phism h" ( M - A ) U 5(A) ~ ( M - A ) U 5(A) which e x t e n d s the h o m e o m o r phism qo2r

5(A) ~

qo2r

and which is sufficiently close to the

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4. MENGER MANIFOLDS

identity map of (M - A ) U 5(A). In particular, we can assume that h is nhomotopic to id(M_A)oh(A ). T h e n the n - h o m o t o p y equivalence ~p - h -1~2" X --~ (iA ) U 6(A) is a Z - e m b e d d i n g and 6(A) C ~ ( X ) -- Y . Since Y is a compact #n+l-manifold, there exists a U V n - r e t r a c t i o n s" Y ~ K onto a finite (n + 1)-dimensional polyhedron K (see T h e o r e m 4.2.23). Similarly, take a proper V Y n - r e t r a c t i o n r" ( M - A) U 6 ( A ) --, T, where T is a polyhedron. Let i" Y "--. ( M - A ) U 6 ( A ) denote the inclusion map and j" K ~ Y be a section of s (i.e. s j - i d g ) . Note t h a t i is an n - h o m o t o p y equivalence. Let p" K --, T be a P L - m a p homotopic to the composition r i j . Form the mapping cylinder M ( p ) -- P of the map p. For the reader's convenience, we again recall t h a t P is the space obtained from the disjoint union ( g x [0, 1]) @ T, by identifying (k, 1) with p ( k ) , k E g . At the same time we identify g with g x {0}. Since p is a P L - m a p , K x {0} and T are s u b p o l y h e d r a of the polyhedron P. Let c" P --. T be the collapse to the base, i.e. the natural retraction defined by sending (k, t) to p ( k ) for each (k, t) E K • [0, 1]. Obviously, c is a proper cell-like map t h a t is a proper h o m o t o p y equivalence. Now consider a proper U V n - s u r j e c t i o n f" N --. P of some # n + l - m a n i f o l d N onto P, satisfying the conditions of Theorem 4.3.6. T h e compact #n+l-manifolds Y and N1 = f - l ( K • {0}) admit U V n - s u r j e c t i o n s s" Y ~ K x {0} and f / N l " N1 ---* K • {0} onto the same polyhedron. Consequently, by T h e o r e m 4.1.21, there exists a homeomorphism n gl" Y ---* N1 such t h a t f g l ~ - s . Similarly, we have two proper U V n - s u r j e c t i o n s r" ( M - A ) U 5 ( A ) --. T and c f " N --~ T. As above, there is a homeomorphism g2" ( M - A ) U 5 ( A ) --. N such t h a t c f g2 ~ p r. By the construction and the corresponding properties of proper U V n - s u r j e c t i o n s , we have n

n

n

c f gl ~-- cs -- ps ~ r i j s ~_ ri ~ c f g2i.

Since c f is a proper n - h o m o t o p y equivalence, we conclude t h a t gl" Y ~ N and g 2 / Y " Y ~ N are n-homotopic. Consider the h o m e o m o r p h i s m

= ~g2--1 /g2(Y)" g 2 ( Y ) ~ N~. Clearly n

--1

~-- g2g2 / g 2 ( Y ) -- idg2(y ).

By the properties of the map f, N1 is a Z-set in N. Note also that, by our construction, g2(Y) is a Z-set in N as well. By T h e o r e m 4.1.15, we can find a h o m e o m o r p h i s m G: N ~ N extending a. Let H -- G g2. Note that H ( Y ) -G g 2 ( Y ) = a g 2 ( Y ) = g l . Finally, let B -- A U H - I ( f - I ( K

• [0, ~])) and 5(B) -- H - I ( f - I ( K

• { })).

It follows from the properties of the map f and Proposition 4.4.18 t h a t B is a compact and n-clean submanifold of M , A C B - 5(B), and the inclusion 5(B) ~ ( M - B ) U S ( B ) is an n - h o m o t o p y equivalence. To see this, observe t h a t

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS 1 the m a p p, and consequently the inclusion K x {~} ~ n - h o m o t o p y equivalence. [-1

175

P - ( K x [0, ~]), 1 is an

LEMMA 4.4.20. Let a #n+l-manifold M be a Z - s e t in a compact # n + l - m a n i fold N . I f the inclusion i" M --~ N is an n-homotopy equivalence, then there exists a UVn-retraction of N onto M . PROOF. Let j" N --~ M be an n - h o m o t o p y inverse of i. By T h e o r e m 4.1.21, ?1 there is a h o m e o m o r p h i s m h" N ~ M such t h a t h n J. T h e n hi rn~ j i ~_ idM. Consequently, by P r o p o s i t i o n 4.2.2, there is a UV'~-surjection r" N ~ M such t h a t ri = idM. U] T h e following result gives us a c h a r a c t e r i z a t i o n of t i n + l - m a n i f o l d s with b o u n d aries. THEOREM 4.4.21. A #n+l-manifold admits a boundary if and only if it is n-tame at oo. PROOF. Let M be a # n + l - m a n i f o l d which is n - t a m e at cx~. By L e m m a 4.4.19, we can represent M as a union M -- U ~ I M i such t h a t all t h e M i ' s are c o m p a c t a n d n-clean, Mi C M i + l - 5 ( M i + I ) and the inclusion 5(Mi) ~ ( M i + l - Mi) U 5(Mi) is an n - h o m o t o p y equivalence for each i = 1 , 2 , . . . . By L e m m a 4.4.20, for each i there exists a u y n - r e t r a c t i o n fi" ( M i + l - Mi) U 5(Mi) ~ 3(Mi). Let the u v n - r e t r a c t i o n ri" M i + l --* Mi coincide with fi on M i + l - Mi and with t h e identity on Mi. T h e n we have an inverse sequence S = {Mi, ri} consisting of c o m p a c t # n + l - m a n i f o l d s and U V n -retractions. By T h e o r e m 4.1.20, ri is a nearh o m e o m o r p h i s m for each i. By [58], each limit p r o j e c t i o n of the s p e c t r u m S is a n e a r - h o m e o m o r p h i s m as well. Consequently, N = lim S, being h o m e o m o r p h i c to M1, is a c o m p a c t # n + l - m a n i f o l d . Since 5(Mi) is a Z-set in Mi for each i, we conclude t h a t the subset Z = l i m { 5 ( M i + l ) , r i / 5 ( M i + l ) } is a Z - s e t in N. It only remains to note t h a t N - Z is n a t u r a l l y h o m e o m o r p h i c to M . Conversely, suppose t h a t the # n + l - m a n i f o l d M a d m i t s a b o u n d a r y . This m e a n s t h a t there are a c o m p a c t # n + l - m a n i f o l d N and a Z - s e t Z in N such that M = N-Z. Let us show t h a t M is n - t a m e at cx~. Let A be a c o m p a c t subspace of M . As in the proof of L e m m a 4.4.17, t h e r e exists a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B - 5 ( B ) . It suffices to show t h a t ( M - B ) U 5(B) is n - h o m o t o p y equivalent to an at m o s t (n + 1)-dimensional finite p o l y h e d r o n . Indeed, it is easy to see t h a t ( M - B ) U 5 ( B ) is n - h o m o t o p y equivalent to a c o m p a c t # n + l - m a n i f o l d ( N - B ) U S ( B ) . It only remains to a p p l y P r o p o s i t i o n 4.1.10. El Not all # n + l - m a n i f o l d s a d m i t b o u n d a r i e s in the above sense. To see this, consider the 3-dimensional (topological) manifold W ( c o n s t r u c t e d by W h i t e h e a d ) which is defined as the c o m p l e m e n t in S 3 of a c o n t i n u u m W h which, in t u r n , is the intersection of a nested sequence of tori in S 3. T h e manifold W has an

176

4. MENGER MANIFOLDS

infinitely generated f u n d a m e n t a l group at c~. Let n >_ 1, and consider a #n+l_ manifold M and a proper U V n - s u r j e c t i o n f " M ---. W . Since n + 1 >_ 2, we see t h a t f induces an isomorphism of f u n d a m e n t a l groups of ends. T h e n it is easy to see t h a t M is not 1-tame at c~ and, therefore, cannot have a boundary. On the other hand, it can shown t h a t the Freudenthal compactification of any connected #l-manifold contains its end as a Z-set. Consequently, the Freudenthal compactification of any connected #1manifold is homeomorphic to #1. In other words, any connected # l - m a n i f o l d has a boundary. We also mention two related results. PROPOSITION 4.4.22. I f the compactum X is a boundary for a #n+l-manifold M , then the compactum Y is also a boundary for M if and only if dim Y < n + 1 andn-Sh(Y)=n-Sh(X). Two compactifications N and T of the same space M are said to be equivalent if for every c o m p a c t u m A C M there is a h o m e o m o r p h i s m of N onto T fixing A point wise. Of course, if the #n+l-manifolds N and T are compactifications of a #n+l_ manifold M , then the inclusions M ~ N and M --~ T are n-homotopy equivalences (because, N - M and T - M are Z-sets in N and T respectively). Consequently, N and T are h o m e o m o r p h i c as n - h o m o t o p y equivalent compact #n+l-manifolds (Theorem 4.1.21). A stronger result can be obtained. PROPOSITION 4.4.23. Every two # n + l - m a n i f o l d compactifications of a given # n + l - m a n i f o l d are equivalent in the above sense. T h e problem of w h e t h e r a # n + l - m a n i f o l d has a b o u n d a r y which is itself a # n + l - m a n i f o l d was also considered in [104]. DEFINITION 4.4.24. A proper map f " Y ---, X between at most (n + 1)-dimensional locally compact spaces is an n - d o m i n a t i o n near c~ provided that there exists a cofinite subspace X1 of X ( i.e. X1 is closed and X - X1 has compact closure) and a proper map g" X1 ---* Y such that f g is properly n-homotopic to the inclusion map X1 ~-* X . If, in addition, for some cofinite subspace Y1 of Y the composition g f /Y1 is properly n - h o m o t o p i c to the inclusion map ]I1 ~ Y , then we say that f is an equivalence near c<). DEFINITION 4.4.25. A space X is finitely n - d o m i n a t e d near 0o if there exists a finite polyhedron P and an n - d o m i n a t i o n near c~, f" (P • [0, 1)) (n+l) ---, X . If, in addition, f is an equivalence near oc, then we say that X has finite n-type near c<). T h e following s t a t e m e n t corresponds to Proposition 4.1.12 in proper n-homotopy category. PROPOSITION 4.4.26. Each finitely n - d o m i n a t e d near oc, at most (n + 1)dimensional, locally compact L C n - s p a c e has finite n-type near c~.

4.5. HOMEOMORPHISM GROUPS

177

Using Proposition 4.4.26, it is possible to prove [104] the following theorem. THEOREM 4.4.27. A ttn+l-manifold M has a boundary which itself is manifold if and only if M is finitely n - d o m i n a t e d near c<).

a

#n+l_

It follows from Proposition 4.4.22 and Theorem 4.1.21 that if such a boundary exists, then it is uniquely determined.

Historical and bibliographical notes 4.4. Theorem 4.4.27 (as well as Proposition 4.4.26) was proved in [104]. All other results of this Section are due to the author. More precisely, the concept of the n-homotopy kernel of a #n+l-manifold was introduced in [98]. Theorem 4.4.3 and 4.4.7 also appear in [98]. A weaker version of Theorem 4.4.3 was proved earlier in [90]. The complement theorem for Z-sets in Mender compacta (Theorem 4.4.8 and its corollaries) were proved in [87] (also, see the related papers [90] and [92]). The results of Subsection 4.4.4 appear in [100].

4.5. Homeomorphism Groups Let M be a Mender manifold and A u t h ( M ) its group of autohomeomorphisms. The study of A u t h ( M ) is well-developed. It is in their spaces of autohomeomorphisms that we see one of the major differences between I ~ -manifolds on the one hand, and Mender manifolds on the other. Homeomorphism groups of I ~ manifolds are /2-manifolds (see Chapter 2), while homeomorphism groups of #n-manifolds M, as it will be shown below, are totally disconnected. If n -- 0, then A u t h ( M ) is 0-dimensional, and if n >_ 1, then A u t h ( M ) is 1-dimensional. Hence, the only compact Lie groups to act effectively on a #n-manifold are finite groups. On the other hand, all compact 0-dimensional metric groups act effectively on all Mender manifolds. The algebraic properties of homeomorphism groups of Mender manifolds are rather similar to the properties of homeomorphism groups of n- and I "~-manifolds. For an autohomeomorphism H" X ~ X , s u p p H ---- cl{x E X" H(x) -~ x}. I f s u p p H is contained in a s u b s e t A, we say that H is supported on A. 4.5.1. D i m e n s i o n of A u t h ( # n ) . We begin with the following Definition. DEFINITION 4.5.1. A c o n t i n u u m X is locally setwise homogeneous if there exists a basis 34 of connected open subsets of X and a dense subset B C_ X such that for each E E Lt and a, b E B M E there exists h E A u t h ( X ) supported on E such that h(a) = b. Clearly, a locally setwise homogeneous continuum is locally connected. The Sierpinski curve M 2 is locally setwise homogeneous but not homogeneous. The solenoids are homogeneous but not locally setwise homogeneous.

178

4. MENGER MANIFOLDS

Suppose t h a t X is a locally setwise h o m o g e n e o u s c o n t i n u u m and e > 0 is a sufficiently small number. Let U be a n e i g h b o r h o o d of i d x in A u t h ( X ) with d i a m U < e. Let x , y E X with d(x, y) -- e, and let A be a small d i a m e t e r arc from x to y in X. One can define, using local setwise homogeneity, a convergent sequence {h~} in A u t h ( X ) "sliding x towards y along A" and such t h a t h -limh~ E A u t h ( X ) and h E Bd(U). Thus, each e - n e i g h b o r h o o d of i d x has none m p t y b o u n d a r y , and so dim A u t h ( X ) >_ 1. Therefore we have the following statement. THEOREM 4.5.2. Let X

be a locally setwise homogeneous continuum. A u t h ( X ) is at least 1-dimensional.

Then

S t r o n g local h o m o g e n e i t y of # " - m a n i f o l d s (Corollary 4.1.17), implies their local setwise homogeneity. Consequently, we have the following. COROLLARY 4.5.3. If M is a compact #"-manifold, then dim A u t h ( M ) >_ 1. Recall t h a t a space X is almost O-dimensional if it has a basis B of open sets such t h a t for each B E B, X - c l B -- U(Ui 9 i E N } where each Ui is b o t h open and closed. Clearly, each 0-dimensional space is almost 0-dimensional. Also, it is easy to see t h a t every almost 0-dimensional space is totally disconnected. T h e c o m p l e t e Erdhs space ~ = (x E 12" xi is irrational for each i} is a 1-dimensional space which is almost 0-dimensional. PROPOSITION 4.5.4. Each almost O-dimensional space is at most 1-dimensi-

onal. SKETCH OF PROOF. Let X be an almost 0-dimensional space and let B be a c o u n t a b l e basis witnessing this fact. Let ~ = ( f i " i E N } be a collection of continuous functions f i " X ~ (0, 1} such t h a t if B , B ' E 13 with cl(B) Mcl(B') -0, there is a f i E ~ ' w i t h f~(B) = 0 and f i ( B ' ) = 1. Let p be the metric o n X given by p(x, y) = ~-~i 2 - i I f i ( x ) - f i ( y ) ]. Let d be a t o t a l l y bounded metric on X. It suffices to show t h a t the metric dimension # d i m ( X , d ) < 1. Now, d ~ -- d + p is also a t o t a l l y b o u n d e d metric on X. Let Y be the completion of X with respect to d ~. It suffices to show t h a t for each t > 0, there is an open set U of Y containing X such t h a t each c o n t i n u u m in U has d i a m e t e r less t h a n t. Let 34 - {U open in Y 9 diam U < t / 3 and U MX E B}. Let C be any continu u m in W - U/d. I f B , B ' E B with B M C ~ 0 ~ B ' M C , then cl(B) N c l ( B ' ) =fi 0, for otherwise there is fi E 9r with f~(B) = 0 and f i ( B ' ) = 1. But f~(X) = {0, 1} and so, by the definition of d', Y = c l y ( f ~ - l ( O ) ) U c l y ( f ~ - l ( 1 ) ) , where cly(f/--l(0)) and c l y ( f ~ - l ( 1 ) ) are disjoint closed sets. Hence C, being a continuum, c a n n o t meet b o t h c l y ( f / - l ( 0 ) ) and c l y ( f ~ - l ( 1 ) ) which is a contradiction. It follows t h a t d i a m C < t. V1 PROPOSITION 4.5.5. If M is an Mkn-manifold with 0 ~_ n < k < oe, then A u t h ( M ) is almost O-dimensional.

4.5. HOMEOMORPHISM GROUPS

179

SKETCH OF PROOF. Let g E A u t h ( M ) and ~ > 0. Let h E A u t h ( M ) with d(g,h) > e. We shall show that there is an open and closed set U containing h such that d ( g , j ) > e for each j E U. Now, d(g, h) = c + 45 for some 5 > 0, and there is x E M so that d(g, h) = d(g(x), h(x)). Choose an n-sphere S in M such that g(S) c_ N ( g ( x ) , 5 ) and h(S) C_ N ( h ( x ) , 5). Since dim M = n, there is a retraction r : M ~ h(S) such that r ( M - N ( h ( x ) , 2 5 ) ) is constant. Let U -- {f E A u t h ( M ) : r f / S ~ .}. Then U is both open and closed in A u t h ( M ) because close maps into S are homotopic. Also h E U. Let f E N(g, e). T h e n f ( S ) C N ( g ( x ) , c + 5). Hence, r . f ( S ) is a point and f ~ U. D COROLLARY 4.5.6. Let 1 < n < cx~. If M is a compact #n-manifold, then d i m A u t h ( M ) - - 1. PROOF. Apply Corollary 4.5.3 and Propositions 4.5.4 and 4.5.5.

[:]

4.5.2. S i m p l i c i t y . Anderson [16], [17] originated a technique for identifying minimal, non-trivial normal subgroups of A u t h ( X ) for spaces with certain dilation and homogeneity properties. DEFINITION 4.5.7. Let X be a space. A subset A of X is deformable if for every non-empty open set U in X , there is h E A u t h ( X ) with h(A) C_ U. Let V be an open set. A collection ({Bi : i E N } , h) is called a dilation system in U if {B~} is a sequence of disjoint non-empty open sets in U with lim Bi -- {p} for some p E U and h E A u t h ( X ) supported on U such that h(B~+l) ---- B~ for each i. PROPOSITION 4.5.8. Let X be a metrizable space in which each non-empty open set contains a dilation system. Let G be a subgroup of A u t h ( X ) generated by all homeomorphisms which are supported by deformable subsets of X . I f G :fi {e}, then G is the smallest non-trivial normal subgroup of A u t h ( X ) . If X is a finite-dimensional manifold without boundary, then A u t h 0 ( X ) , the subgroup of homeomorphisms isotopic to the identity, is simple (see [151] and [141]). It is also known (see [220] and [323]) that Auth(/2) and A u t h ( I ~ ) are simple. DEFINITION 4.5.9. Let M be a #n+l-manifold. A pair (W, 5 ( W ) ) is an nclean pair if W is n-clean with respect to 5 ( W ) in the sense of Definition 4.4.15 and if, in addition, both W and 5 ( W ) are homeomorphic to #n+l. By Theorem 4.1.15 and the existence of n-clean pairs in #n+l, (compare with [184]), it follows that every open set in #n+l has a dilation system. Also, every proper closed set in #n+l is deformable. Since every element of A u t h ( # n+l) is stable (see Theorem 4.5.13 below) we have the following. THEOREM 4.5.10. A u t h ( # n+l) is simple.

180

4. MENGER MANIFOLDS

4.5.3. Stability of homeomorphisms. An a u t o h o m e o m o r p h i s m of a space X is said to be stable [32] if it can be expressed as the composition of finitely m a n y a u t o h o m e o m o r p h i s m s each of which is the identity on some n o n - e m p t y open subspace of X. It is well-known t h a t all a u t o h o m e o m o r p h i s m s of the Hilbert cube I ~ and the Hilbert space 12 are stable (see, for example, [32]). Every orientation-preserving h o m e o m o r p h i s m of R n is stable [194]. LEMMA 4.5.11. For each #n+l-manifold M , there is a Z-embedding a" M ---+ M which is properly n-homotopic to idM and which satisfies the following condition: (.) If F e A u t h ( M ) and F / a ( M ) = id, then F can be expressed as the composition of two autohomeomorphisms of M each of which is the identity on some open subspace of M . PROOF. Take a proper UVnsurjection g" M ---+ K , where K is an at most (n + 1)-dimensional locally compact p o l y h e d r o n ( T h e o r e m 4.2.23). Consider also a p r o p e r UVn-surjection f" MI" K x [-1, 2], satisfying conditions of T h e o r e m 4.3.6. It can easily be checked t h a t the inverse image M -- f - l ( g x [0,2]) is also a copy of the # n + l - m a n i f o l d M. Moreover, since the composition 7 r l f / M --* g x {0}, where ~rl" g x [0, 2] --. g x {0} denotes the projection, is a proper U V nsurjection, we conclude t h a t there exists a h o m e o m o r p h i s m a" M --, f - l ( K x {0}) such t h a t f a ..~n 7 r l f / M o T h e n a is properly n - h o m o t o p i c to idM. By the mp properties of f , we see t h a t a" M --~ M is a Z-embedding. Consequently, it only remains to show t h a t if F E A u t h ( M ) and F / f - l ( g x {0}) -- id, then F can be expressed as the composition of two a u t o h o m e o m o r p h i s m s of M each of which is the identity on some open subspace of M . Let U - F - l ( f - l ( g x (1,2])). Since F / a ( M ) - id we conclude t h a t U n f - l ( K x {0}) = 0. Consequently, there exists a sufficiently small tl > 0 such t h a t f - l ( K x [O, t l ] ) N ( U U F ( U ) ) = O. Let V : f - l ( K x [0, t0)), where to = l t l l + t1l . Consider the h o m e o m o r p h i s m ~" [0, tl] --* [ - 1 , t l ] defined as v~(x) = t-LL--x--1 1+tl Consider also an open cover ZX=

{ [ K x

--1,--

,K x

--~,tl

]}

of the p r o d u c t K x [-1, tl]. Again, by the properties of f , the inverse images N1 = f - l ( K x [0, tl]) and N2 = f - l ( K x [-1, tl]) are #n+l-manifolds. Clearly these a d m i t proper u v n - s u r j e c t i o n s f l = (id x t p ) f / N l " N1 ---+ K x [-1, tl] and f2 = f / N 2 " K x [-1, tl] onto the polyhedron K x [-1, tl]. T h e n we see t h a t the inverse image f - l ( K x {tl}) is a Z-set b o t h in N1 and N2. Moreover, since, v~(tl) = tl, we conclude t h a t the identity h o m e o m o r p h i s m h o f . f - l ( K x {tl}) satisfies the equality f2h = fl/f-l(K x {tl}). Consequently, there exists a h o m e o m o r p h i s m T" N1 ---+ N2 such t h a t T / f - I ( K x {tl}) = id and the composition f 2 T is U-close to f l . Now

4.5. HOMEOMORPHISM GROUPS

181

consider the homeomorphism H" M ~ M1 such that H / f - I ( K x [0, tl]) = T and H / f - I ( K x [tl,2]) = id. It is easy to verify that H ( V ) N M = 0 and H/(UUF(U))=id. Let f

a(x) = ~F(x),

[ x,

ifx eM ifxEM1-M.

Since F / f - I ( K x {0}) = id, we conclude t h a t G E Auth(M1). Now let F2 -H - 1 G H e A u t h ( M ) . If x e U, then H ( x ) - x. Hence, G H ( x ) - F ( x ) e F ( V ) . Consequently, H - 1 G H ( x ) -- F ( x ) . In other words, F2/U -- F l U . Similarly, if x e V, then H ( x ) e M1 - M . Hence G H ( x ) = H ( x ) and H - 1 G H ( x ) = H - 1 H ( x ) = x. This means that F 2 / V = id. Obviously, F -- F2F1, where F1 = F 2 1 F . It only remains to note t h a t F 1 / U -- id and F 2 / V -- id. 0 LEMMA 4.5.12. Let M be an #n+l-manifold and ~" M ~ M be a Z-embedding properly n-homotopic to idM. I f G E A u t h ( M ) and G / ~ ( M ) -- id, then G can be expressed as the composition of two autohomeomorphisms each of which is the identity on some open subspace of M . PROOF. Take a Z-embedding a satisfying condition (.) of L e m m a 4.5.11. Then the homeomorphism / 3 a - l : a ( M ) ~ ~ ( M ) is properly n-homotopic to the inclusion a ( M ) ~ M. Consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending ~ a -1. Consider the homeomorphism h - l G h E A u t h ( M ) . Evidently, h - l G h / a ( M ) -- id. By the choice of a, there exist F1, F2 E A u t h ( M ) and open subspaces 1/1 and 1/2 of M such that F i / V i = id, i -- 1, 2, and h - l G h -- F2F1. Then G -- G2G1, where Gi -- hFih -1, i -- 1, 2. It only remains to note t h a t G i / h ( V i ) -- id, i -- 1, 2. V] THEOREM 4.5.13. Let M be a #n+l-manifold and F E A u t h ( M ) . properly n-homotopic to idM, then F is stable.

I f F is

PROOF. Let ~" M --~ M be a Z-embedding properly n-homotopic to idM. Let a" M ---+ M be another Z-embedding properly n-homotopic to idM and such that a ( M ) N ( ~ ( M ) U F ( ~ ( M ) ) ) = 0. Consider the homeomorphism S" a ( M ) U f l ( M ) ~ a ( M ) U F ( ~ ( M ) )

which coincides with the identity on a ( M ) and with F on /~(M). Clearly, f is properly n-homotopic to the inclusion a ( M ) U ~ ( M ) ~ M . Therefore there exists an extension F2 E A u t h ( M ) of f . Let F1 = F 2 1 F . By L e m m a 4.5.12, F = F2F1 can be expressed as the composition of four a u t o h o m e o m o r p h i s m s each of which is the identity on some open subspace of M. V1 Since each map of #n+l into itself is (properly) n-homotopic to the identity, we obtain the following statement.

COROLLARY 4.5.14. Every autohomeomorphism of#n+1 is stable.

182

4. MENGER MANIFOLDS

4 . 5 . 4 . G r o u p a c t i o n s o n M e n d e r m a n i f o l d s . T h e well-known HilbertSmith conjecture asks w h e t h e r every c o m p a c t group acting effectively on a manifold is a Lie group. This is equivalent to asking w h e t h e r the group A n of p-adic integers acts effectively on a manifold. This long s t a n d i n g problem is still open. T h e situation is r a t h e r different for Mender manifolds. For instance, it is known [13] t h a t any c o m p a c t metrizable zero-dimensional topological group G acts freely on #1 so t h a t the orbit space # I / G is h o m e o m o r p h i c to tt 1. There are several constructions of group actions on #n-manifolds. THEOREM 4.5.15. Let M be a # " - m a n i f o l d . T h e n : (i) E v e r y c o m p a c t z e r o - d i m e n s i o n a l metrizable group G acts on M so that the orbit space M / G

is h o m e o m o r p h i c to M

(ii) Ap acts freely on M so that d i m M / A p -- n + 1 (iii) Ap acts on M so that dim M / A p = n + 2 T h e r e are u n e x p e c t e d ties between group actions on Mender c o m p a c t a and the H i l b e r t - S m i t h problem mentioned above. Namely, a positive solution to the following conjecture would prove t h a t there is no free Ap-action on a connected (topological) manifold M with dim M / A p < 0r

CONJECTURE 4.5.16. Let m and n be positive integers and G be a zero-dim e n s i o n a l c o m p a c t m e t r i c group. I f #mWn and #n are free G-spaces, then there is no equivariant map #mTn ~ #n.

Historical and bibliographical notes 4.5. T h e o r e m 4.5.2 was proved in [54]. Corollary 4.5.6 a p p e a r s in [244]. Corollary 4.5.14 is due to the author [94]. It was shown later t h a t all a u t o h o m e o m o r p h i s m s of all connected Mender manifolds are stable [266]. Item (i) of T h e o r e m 4.5.15 was proved in [131] (see also [265]). Items (ii) and (iii) a p p e a r in [219] (the last s t a t e m e n t is based on the work [257] ). Conjecture 4.5.16 appears in [1].

4.6. w-soft map

of a onto

E

In this Section, using T h e o r e m 4.2.21, we show t h a t there exists an "almost soft" m a p from a onto E. DEFINITION 4.6.1. A m a p f : X c o u n t a b l e - d i m e n s i o n a l I space B , g: A ---, X

~

Y

is called w - s o f t if f o r each strongly

closed subspace A

and h: B ---, Y with f g -- h / A ,

of B

and any two m a p s

there exists a m a p k ~

B ---, X

such that k / A -- g and f k -- h, i.e. if the following diagram

1Recall that a space is strongly countable-dimensional if it can be represented as the countable union of finite-dimensional closed subspaces.

4.6. w-SOFT MAP OF a ONTO E

X

A

183

~Y

~

,.-B

commutes.

Obviously, e v e r y w-soft m a p is n-soft for each n E w. In p a r t i c u l a r , e v e r y wsoft m a p is s u r j e c t i v e a n d o p e n ( c o m p a r e w i t h t h e p r o o f of i m p l i c a t i o n (ii) ~

(i)

in P r o p o s i t i o n 2.1.34).

LEMMA 4.6.2. Let

T

,-X

i

,~ X I

,~Y

be a c o m m u t a t i v e diagram, consisting of compact spaces, where i is an embedding and d i m T < n. T h e n there exists a c o m m u t a t i v e n - s o f t diagram 2

2This means that the diagonal product k~Ah ~, considered as the map of T ~ onto its image, is n-soft (see page 159).

184

4. M E N G E R M A N I F O L D S

TI

\

h!

~

T

,

~

XI

h

f

Z

g

~Y

where j is an embedding and d i m T ~ <_ n + 1.

PROOF. L e t K be t h e fibered p r o d u c t of Z a n d X I w i t h r e s p e c t to t h e m a p s g" Z --~ Y a n d f~" X ~ ---, Z.

Let ~" K

---, Z a n d r

K

--, X ~ d e n o t e t h e

r e s t r i c t i o n s of t h e n a t u r a l p r o j e c t i o n s Z x X ~ --, Z a n d Z x X ~ ~

X ~ onto K

r e s p e c t i v e l y . B y T h e o r e m 4.2.21, t h e r e exists an n - s o f t m a p m" T ~ ~ K , w h e r e d i m T ~ < n + 1. Since d i m T < n we c a n find a n e m b e d d i n g j " T ~ T I such t h a t

m j = k A i h . It o n l y r e m a i n s to o b s e r v e t h a t t h e d e s i r e d m a p s are h ~ = C m a n d k l = ~m. [:]

PROPOSITION 4.6.3. There exists an w-soft map f" X ~ I ~ , where X is a a-compact strongly countable-dimensional AR-space.

PROOF. W e c o n s i d e r t h e H i l b e r t c u b e I w to be t h e limit s p a c e of an inverse _i+1 }, w h e r e Yi = i i a n d qii+1 " i i + l ~ i i d e n o t e s t h e projecs e q u e n c e , ~ = {Yi, ~i t i o n o n t o t h e c o r r e s p o n d i n g s u b p r o d u c t . W e n o w c o n s t r u c t a n inverse s e q u e n c e ,~ -- { X i , p_i+1 i }, a c o l l e c t i o n of c o m p a c t s u b s p a c e s Zi,z C_ Zi,2 C_ . . . C Zi,i -- X i for e a c h i a n d a m o r p h i s m

{ a i " X i ~ Yi}" 8 ~ 8 '

such t h a t t h e following c o n d i t i o n s are satisfied: (i) d i m Zi,k <_ k -t- 1, w h e n e v e r k < i.

4.6. w-SOFT MAP OF a ONTO IE

185

(ii) T h e d i a g r a m

a~+~/z~+~.k Zi+l,k

~)i,k =

i+l i

k i pi+l/Zi+l,

a~/zi,k

Z~,k is k-soft. (iii) T h e d i a g r a m

o~i+1

Xi+l

~)i "--

~ Yi+l

i+l i

i+l i

Xi

,.

'-- Yi

is soft.

Let X1 = I x I and a l = r l " I x I ~ I. Assume t h a t for each i _< j, the m a p s a i and the spaces Zi,k, k = 1, 2 , . . . , i, have already been defined. Let Zj+I, 1 be ' the fibered p r o d u c t of the spaces Zj,1 and Yj+I with respect to the maps aj/Zj,1 and ~j A + I , and ~o" Zj+I, , 1 --, Zj,1 and r Zj+I, 1 ~ Yj+I be the corresponding projections of the fibered product. By T h e o r e m 4.2.21, there exists a i-soft m a p m " Z j + I , 1 --. Z j'+ I , I ~ where dim Z j + l , 1 = 2. We define the maps ~,j ~.j+l and a j + l by means of their restrictions to the subspaces Zj+l,i.

Let pj j+ l / Z j + I , 1 - - ~ o m

and o z j + l / Z j + l , 1 = Cm. Using L e m m a 4.6.2, we get the sequence of spaces Zj+I,1 c_ Zj+I,2 c _ . . . c Z j + l d with dim Zj+l,k _< k + 1, and maps _j+l -

rk--pj

/Zj+l,k" Zj+l,k ~ Zj,k, ~k = aj+l/Zj+l,k" Zj+l,k ~ Yj+I

such t h a t r k / Z k - 1 = rk-1, ~k/Zk-1 = r

a n d the diagrams T~j,k are k-soft.

186

4. MENGER MANIFOLDS

If k = j , let l be an e m b e d d i n g of t h e (j + 1 ) - d i m e n s i o n a l space Z j + I , j into t h e c u b e i2j+3. L e t

X j + l -- X j x Yj+I • i 2 j + 3

Zj+I,j is e m b e d d e d into Xj+I by t h e m a p l' = ~ X j be t h e r e s t r i c t i o n of t h e p r o j e c t i o n onto t h e first c o o r d i n a t e a n d a j + l " X j + I ~ Yj+I be t h e r e s t r i c t i o n of t h e p r o j e c t i o n onto t h e second c o o r d i n a t e . It is easy to see t h a t c o n d i t i o n s (i)-(iii) are satisfied. For each n a t u r a l n u m b e r k E N , let a n d a s s u m e t h a t t h e space

r j / k ~ j / k l 9 L e t t,j ,.j+l " X j + I

Zk = lim{Zi,k,pi + l i / Z i + l , k , i},

X = U~=IZk

and

f = lim{cri}/X" X ~ lim,9' = I ~ . Obviously, t h e space X is a s t r o n g l y c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t space. N e x t we show t h a t t h e m a p f is w-soft. C o n s i d e r t h e c o m m u t a t i v e d i a g r a m

X

~I

A t

.B

w

w h e r e A is a closed s u b s e t of a s t r o n g l y c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t space B. L e t r -- { r a n d r - {r where r A -+ X i a n d r B -+ Yi are t h e coord i n a t e m a p s . W e also let B -- Uj= l c ~ B j , w h e r e t h e B j are at m o s t j - d i m e n s i o n a l s u b s e t s of B a n d B1 C_ B2 C_ . . . . Let A j -- A A B j . T h e n A - U~=IA j. N o t e t h a t , w i t h o u t loss of generality, we can a s s u m e t h a t r C_ Zj. Now we are going to c o n s t r u c t a m a p (I)" B -+ X as (I) -- {q)i}, w h e r e (I)i" B --+ Xi are c o o r d i n a t e m a p s . We c o n s t r u c t s t h e (Ih's by i n d u c t i o n . L e t (I)l" B - , X I be a m a p such t h a t ~1(I)1 -- r a n d ~ 1 / A 1 -- ~1. Now a s s u m e t h a t for every i < j m a p q)i has a l r e a d y b e e n c o n s t r u c t e d so t h a t r C_ Zi,k, k < i. Using t h e 1-softness of t h e s q u a r e d i a g r a m T)j,1 we choose a m a p (I)j(1) + l " B1 --+ Zj+I,1 such t h a t t h e following c o n d i t i o n s are satisfied:

= Cj/B 9 Pjj+1~(1) =j+l _

m(1) 9 ~j+l~j+l 9 r

j+l/r

=

1.

Cj+I/B1.

(A N Zj+l,1) = Cj+l/r

(A N Zj+l,1).

4.6. w-SOFT MAP OF a ONTO E Using 2-softness of the square d i a g r a m T)j , 2 we choose a m a p (I)(2) j+l" such t h a t 9 p- Jj + l d ~~(j2+)l = (~j/B2.

187 B2 ~

Zj+l,2

d~(2) 9 ~j+l~j+l

-- Cj+I/B2. ~(2) /.,.-1 9 "~j+I/Wj+I (A A Zj+I,2) = C j + l / r -1 (A N Zj+l,2) 9 ~(2) = j + l / B 1 = ~(1) =j+l" C o n t i n u i n g this process we obtain a m a p ~ j + l " B j --, Z j + I j conditions" 9 Pjj + l m~ j(+j )l = ~ j / B j 9 aj+l~j+l -- C j + I / B j . ~(j) -1 9 xj+l/~)j+l(A A ZjTl,j) = Cj+I/~);_~I(A

satisfying t h e

N Zj+l,j).

9 ff~" 1 / B j -1 - - ~ j + l " Finally, using softness of t h e d i a g r a m 7)j we can choose a m a p (I)j+l : B ~ X j + I such t h a t _j+l 9 ~j

(I)j+l = (I)j.

9 aj+l(I)j+l

= r

9 ( ~ j + I / B j - - d~(J) =j+l" 9 Cj+I/A = r It follows easily from the c o n s t r u c t i o n t h a t the m a p (I) = ((I)j} is well defined, r = r and f(I) = r It only remains to observe t h a t t h e w-softness of f g u a r a n t e e s t h a t X is an AR-space. O THEOREM 4.6.4. There exists an w-soft map r

a --+ E.

PROOF. Let f" X ~ I ~ be the m a p from P r o p o s i t i o n 4.6.3 a n d let X ~ = f - l ( E ) . Clearly, X ~ is a strongly c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t AR-space. B y P r o p o s i t i o n 2.5.5, the p r o d u c t X ~ x a is h o m e o m o r p h i c to a. T h e desired m a p is r - fir- a --, E, where ~" X ~ x a ~ X ~ denotes the p r o j e c t i o n onto t h e first coordinate. [l

Historical and bibliographical notes 4.6. T h e o r e m 4.6.4 was proved in [325]. A s o m e w h a t weaker version of it was o b t a i n e d earlier in [229].

CHAPTER 4

Menger Manifolds

In this Chapter we present a survey of Menger manifold theory.

4.1. G e n e r a l T h e o r y Before we start a detailed discussion, let us outline the construction of the kdimensional universal Menger compactum tt k, k > 0. We partition the standard unit cube I2k+l, lying in (2k + 1)-dimensional Euclidean space R 2k§ into 32k§ congruent cubes of the "first rank" by hyperplanes drawn perpendicular to the edges of the cube 12k+1 at points dividing the edges into three equal parts, and we choose from these 32k+l cubes those which intersect the k-dimensional skeleton of the cube 12k+l. The union of the selected cubes of the "first rank" is denoted by I(k, 1). In an analogous way, we divide every cube entering as a term in I(k, 1) into 3 2k+1 congruent cubes of the "second rank" and the union of all analogously selected cubes of the "second rank" is denoted by I(k, 2). If we continue the process we get a decreasing sequence of compacta

I(k, 1) D I(k,2) D - - . . The compactum #k = N{I(k,i)" i E N } is called the k-dimensional universal Menger compactum. Note that #k - - IV1 . k2k+l , where Mk, n 0 ~ k < n, denotes the "k-dimensional Menger compactum constructed in the n-dimensional cube I n,' (a precise definition of M~ is given below). Obviously, #0 coincides with the Cantor discontinuum D ~ and, consequently, is the only zero-dimensional compactum with no isolated points (Theorem 1.1.10). It was shown by Sierpinski [285] that #0 is a universal space for the class of all zero-dimensional metrizable compacta. Positive dimensional Menger compacta M~ were originally defined within the classical dimension theory by Menger in [221]. They are generalizations of the Cantor discontinuum and Sierpinski's universal curve M 2 [284]. Let us recall that the space M12 is universal for the class of all at most 1-dimensional planar compacta [284]. Further, it was shown by Menger [221] that the 1dimensional Menger compactum #1 = M 3 is universal for the class of all at 127

128

4. MENGER MANIFOLDS

most 1-dimensional compacta. Generally, it was conjectured [221] that M ~ is a universal space for the class of all at most k-dimensional compacta embeddable in R n (Menger's problem). As was already mentioned, this problem was known to have a positive solution for (k, n) = (1, 2) and (k, n) = (1, 3). A positive solution in the case k -- n - 1 was also given by Menger [221]. Results of Lefschetz [209] and Bothe [48] produced a positive solution in the case n = 2k + 1. The ultimate affirmative solution of Menger's was obtained by Stanko [289]. The topological characterization of #0 has already been mentioned. Note also that #0, being a topological group, is homogeneous. The 1-dimensional Menger compacta M 2 and #1 = M 3 were also characterized topologically by W h y b u r n [318] and Anderson [14] (see also [15] ), respectively. Anderson's theorem characterizes #1 as a 1-dimensional locally connected continuum with no local separating points and with no non-empty open subspaces embeddable into the plane. Comparing the characterizations of #0 and #1, it is very hard to see what is at the root of these results, what is common between them and, finally, how they can be generalized to higher dimensions in order to (at least) make a reasonable conjecture concerning the characteristics of M~. In full generality, some of these questions still remain open. To the best of our knowledge, there are no conjectures concerning the characteristic properties of the compacta M~ when l
4.1.1. C o n s t r u c t i o n o f M e n d e r c o m p a c t a . In this Subsection we describe the constructions of Mender compacta given by Mender, Lefschetz and Bestvina. Although the Mender construction has already been described above, we restate it in a slightly different (but equivalent) form for later use. Throughout this section we fix integers 0 < k _ n. I. M e n g e r ' s

construction

[221]

As a metric on R n, we use the maximum metric, i.e. d ( { x i } , {yi}) -- max{I x i -

for each { x i ) , { y i ) e R n. If A e-neighborhood of A with respect Let I n be the n-cell in R n with i >_ O, L i denotes the cell complex II{

me m t 3i +l 3i ,

yi I" 1 _< i _< n )

is a subset of R n, then g ( A , e ) denotes the to the above metric. the standard linear structure. For each integer structure of I n whose n-cells are of the form ] " m r = 0, 1 , . . . , 3 i - 1 } .

We define the Mender c o m p a c t u m M ~ as follows: Let M o = I n and (by induction) for each integer i _ 1 let M i + l -- s t ( [ L ~ k ) [ , L i + l ) f3 M i -- s t ( [ L ~ k ) [ , L i + l [ M i )

-- s t ( [ L ~ k ) [ M i [ , L i + l [ M i ) .

4.1. GENERAL THEORY

129

Clearly, {Mi} is a decreasing sequence of compacta and M ~ = AMi is called the Menger compactum of type (k,n). W h e n n = 2k 4- 1, we use the symbol #k - - " zaz2k+l to denote the k dimensional universal Menger compactum. v.L~ We will also use another description of M ~ [120, Chap.2]. Let V~ = {(2t4- 1 ) / 2 - 3 ~ : t = 0 , 1 , . . . , 3 ~ - 1} and Y = t2V~. Note t h a t Bi = Vi • .-. • 1// (n factors) is the set of centers of n-cells of Li. Let :P be the finite collection of homeomorphisms of R n defined by p e r m u t a t i o n s of coordinates of R n. For each i we define Di--= N{a({c} • I n - k - 1 ) : a E 7) and c E Vik+l} and N~ = N ( D i , 1(2-3i)). Then Di can be regarded as the "dual (n - k - 1)skeleton"of Li and Ni as the regular neighborhood of Di. It is easy to see that U~= In-U{int(Ni) : i= 1,...,n},In= U i U N i and OUi = U i N ( N i U O I n ) . If we perform the above construction starting with R n (instead of I n) we get a closed subspace U~ of R n which is a countable union of copies of M ~ (in this case L0 is the partition of R n into unit cubes).

II. Lefschetz's c o n s t r u c t i o n [209] Replacing the cell complexes in (I) by simplicial complexes, we obtain Lefschetz's construction. We describe it in slightly general form. Let M be a PL n-manifold with a (combinatorial) triangulation L. Inductively, we define a sequence {Mi} of PL n-manifolds and their triangulations Li as follows. Let M0 = M and L0 = L. Let M1 = st(L(k),~2Lo),L1 = ~2LolM1 and suppose that Mi and Li have already been defined. Consider ~2Li and let Mi+l -- st(L~k),~2Li) and Li+l -- ~2L~IMi+I. Then {Mi} is a decreasing sequence and NMi ~- 0. If M is the n-simplex with the s t a n d a r d simplicial complex structure, then the resulting c o m p a c t u m NM~ is denoted by L~. In particular, L2kk+l -- #k (We use the same symbol as in (I). This notation is justified by the Characterization Theorem 2.4.1). Notice that Mi+l may be regarded as a regular neighborhood of the k-skeleton of Mi (with respect to Li).

III. B e s t v i n a ' s c o n s t r u c t i o n [33] In Bestvina's construction, the k-skeleta in (II) are replaced by the dual kskeleta. Suppose t h a t M is a PL n-manifold with a (combinatorial) triangulation L. As in (II), we define a sequence {Mi} of PL n-manifolds and their triangulations {Li} as follows: Let M0 = M, L0 = L and suppose t h a t we already have defined Mi and Li. Then Mi+l = U{st(ba,~2Li) 9 ba is the barycenter o f a E L~ with d i m a > n - k } and Li+l = fl2Li[Mi+l.

130

4. MENGER MANIFOLDS

If M is the n-simplex with the standard simplicial complex structure, the resulting compactum DMi is denoted by T~. In particular, T 2k+1 is denoted by ~k. (Again this is justified by the Characterization Theorem). Observe that Mi+l is regarded as a regular neighborhood of the dual k-skeleton of Mi (with respect to Li). It might be worth noting the differences among these constructions. Consider the properties of the partitions which are naturally induced by each of the above constructions. For simplicity, we formulate these properties only for M~, L~ and

PROPOSITION 4.1.1. There are sequences {Pi}, { Q i } and { R i } of partitions of M ~ , L~ and T ~ , respectively, satisfying the following conditions: (a) Pi+l, Qi+l and Ri+l are refinements of Pi, Qi and Ri, respectively. (b) lim mesh Pi = lim mesh Qi = lim mesh Ri -- 0. (c) ord Pi = n ~- 1,1imordQi = c~ and ordRi = k + 1. (d) For each p l , p 2 , . . . , p t E Pi, D{pj : j -- 1 , . . . , t } is an at most k-dimensional L C k-1 D C k - l - c o m p a c t u m o r (e) For each q l , q 2 , . . . , a t E Qi, D{qj : j sional L C k-1 gl C k - l - c o m p a c t u m o r (f) For each r l , r 2 , . . . , r t 9 Ri, D{rj : j sional L C k - t CI C k - t -compactum.

an at m o s t -- 1 , . . . , t } an at m o s t -- 1 , . . . , t }

k - d i m e n s i o n a l cell. is an at most k-dimenk - d i m e n s i o n a l simplex. is a ( k - t + 1)-dimen-

PROOF. The partitions defined below satisfy the desired conditions: Pi-- {eD M~ : e E Li},Qi--

{ s D L~ : s E L i } a n d R i -

{ s D T~ : s E Li}.

[:] R e m a r k 4.1.2. (1) In the last case, if ( k , n ) -- (1,3) we have a partition of #1 with 0-dimensional intersections of all adjacent elements. In this sense, the partition determined by Bestvina's construction can be regarded as a generalization of the partition of the Menger curve considered in [14, 15] and [235].

(2) We may obtain characterizations of (compact) Menger manifolds as well as (compact) Q-manifolds in terms of the existence of certain types of partitions

[18s]. 4.1.2. n - h o m o t o p y . We are going to describe an adequate homotopy language for #n+l-manifold theory. This is the so called n-homotopy theory. The related notion of #n+l-homotopy was first exploited in [33]. DEFINITION 4.1.3. Two maps f , g : X - , Y are said to be n - h o m o t o p i c (writn

ten f "~ g) if the compositions f a and ga are homotopic in the usual sense f o r any map ~ : Z --, X of an at m o s t n - d i m e n s i o n a l space Z into X .

4.1. GENERAL THEORY

131

It can easily be seen [87, Proposition 2.3] that if dim X _ n + 1 and Y E L C n, then maps f , g 9 X - , Y are #n+l-homotopic in the sense of Bestvina [33, Definition 2.1.7] if and only if they are n-homotopic. Note also that if, in the above definition, we consider, instead of compact, only polyhedral Z, then we get Fox's definition of n-homotopy [154]. In practice it is convenient to use the following statement. PROPOSITION 4.1.4. M a p s f , g 9 X ~ Y are n - h o m o t o p i c if and only if f o r s o m e (or, equivalently, any) n - i n v e r t i b l e m a p ~ 9 Z ~ X with dim Z ~_ n, the c o m p o s i t i o n s f (~ and g(~ are h o m o t o p i c .

DEFINITION 4.1.5. A m a p ~ 9 A --. X is said to be n - i n v e r t i b l e if f o r any m a p 9 B --. X with d i m B ~_ n, there is a m a p ~/9 B --~ A such that ~/~ - ~.

Note that 0-invertible maps between metrizable compacta are precisely surjections with a regular averaging operator (see Definition 6.1.24). Note also that each compactum is an n-invertible image of an n-dimensional compactum. Of course, homotopic maps are n-homotopic for each n _ 0, but not conversely. Indeed, consider the identity map and the constant map of an arbitrary non-contractible L C ~ 1 7 6 C~176 Nevertheless, n-homotopic maps have several useful properties. PROPOSITION 4.1.6. For each Y

E L C n, there exists an open cover Lt E

c o y ( Y ) such that any two U - c l o s e m a p s of any space into Y are n - h o m o t o p i c .

PROPOSITION 4.1.7. ( n - H o m o t o p y E x t e n s i o n T h e o r e m ) . Let Y E L C n. T h e n f o r each 34 E c o y ( Y ) , there exists 1) E c o y ( Y ) refining lg such that the following condition holds: (*)n For any at m o s t (n -t- 1 ) - d i m e n s i o n a l space B , any closed subspace A of B , and any two V - c l o s e m a p s f , g : A --. Y such that f has an e x t e n s i o n F : B --+ Y , it follows that g also has an e x t e n s i o n G : B --~ Y which is U-close to F .

PROPOSITION 4.1.8. Let Y E L C n. Suppose that A is closed in B and dim B n + 1.

If maps f,g

9 A --, Y

are n - h o m o t o p i c and f

a d m i t s an e x t e n s i o n

F 9 B - . Y , then g also a d m i t s an e x t e n s i o n G 9 B --+ Y , and it m a y be a s s u m e d that F ~ G.

A map f : X -~ Y is an n - h o m o t o p y equivalence if there is a map g : Y --~ X such that g f ~ i d x and f g ~ i d y [87]. The spaces X and Y in this case are said to be n - h o m o t o p y equivalent. For example, any map between arcwise connected spaces is a 0-homotopy equivalence. Note also t h a t the (n + 1)-dimensional sphere S n+l is n-homotopy equivalent to the one-point space. In general, we have the following algebraic characterization of n-homotopy equivalences [316, Theorem 2].

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4. MENGER MANIFOLDS

PROPOSITION 4.1.9. A map f " X -~ Y between at most (n + 1)-dimensional locally finite polyhedra is an n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n, i.e., f induces a bijection between the components of X and Y and the h o m o m o r p h i s m ~ k ( f ' ) " ~rk(Cx) --~ ~ k ( C y ) is an isomorphism for each k ~_ n and each pair of components C x C_ X and C y C_ Y with f ( C x ) C_ C y , where f l " C X - , C y denotes the restriction of

f. Recall that each A N R - c o m p a c t u m is homotopy equivalent to a finite polyhedron (Corollary 2.3.30). The following statement is an "n-homotopy version" of West's result. PROPOSITION 4.1.10. Every at most ( n T 1 ) - d i m e n s i o n a l locally compact L C nspace is properly n - h o m o t o p y equivalent to an at most (n + 1)-dimensional locally finite polyhedron. Therefore, 4.1.9 holds even for maps between at most (n + 1)-dimensional locally compact LCn-spaces. Proper n-homotopies, and all associated notions, are defined in the natural way and we do not repeat them here. In order to state an algebraic characterization of proper n-homotopy equivalences similar to Proposition 4.1.9, we need some preliminary definitions. We say that a proper map f 9 X --, Y between locally compact spaces induces an epimorphism of i-th homotopy groups of ends (i >__ 0) if for every compactum C C_ Y there exists a compactum K C_ Y such that for each point x E X - f - l ( K ) and every map ~ " (S i, ,) --, (Y - K, f ( x ) ) there exists a map ~ " (S i, ,) --, ( X - f - l ( C ) , x ) and a homotopy f ~ "~ a(rel ,) in Y - C . We say that f 9 X - , Y induces a m o n o m o r p h i s m of i-th homotopy groups of ends if for every compactum C C_ Y there exists a compactum K C_ Y such that for every map ~ 9 S i --, X - f - l ( K ) with the property that f ~ is null-homotopic in Y - K it follows that a is null-homotopic in X - f - l ( C ) . As usual, f is said to induce an isomorphism of i-th homotopy groups of ends if it simultaneously induces an epimorphism and a monomorphism. PROPOSITION 4.1.11. A proper map f 9 X - , Y between at most (n ~- 1)dimensional locally compact LCn-spaces is a proper n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n and isomorphisms of homotopy groups of ends of dimension ~_ n. Note that proper n-homotopies have also been studied from the categorical point of view [169]. The following proposition will be used below and indicates a difference between the n-homotopy and usual homotopy theories (compare with [303]). PROPOSITION 4.1.12. Let M be an at m o s t (n -t- 1)-dimensional locally finite polyhedron. Suppose that there exists an at m o s t (n + 1)-dimensional finite polyn

hedron K and two maps f " M --~ K and g " K -+ M such that g f ~_ idM (i.e.

4.1. GENERAL THEORY

133

g is an n - h o m o t o p y domination). Then there exist an (n + 1)-dimensional finite polyhedron T, containing K as a subpolyhedron, and an n-homotopy equivalence h 9 T ~ M extending g such that f is a n-homotopy inverse of h. PROOF. Obviously, it suffices to consider only connected polyhedra. Consequently, the case n -- 0 is trivial. If n -- I, then, by the assumption, lrl(g)" l r l ( g ) --* ~rl(M) is an epimorphism and g e r ( l r l ( g ) ) is a finitely generated group. Select finitely m a n y generators of Ker(~rl(g)) and use t h e m to a t t a c h 2-cells to K and to e x t e n d g over these cells. In this way we obtain a 2-dimensional finite polyhedron T, containing K as a subpolyhedron, and a map h" T --, M , extending g, which induces an isomorphism of f u n d a m e n t a l groups. By Proposition 4.1.9, h is a 1-homotopy equivalence. Assume, by way of induction, t h a t the proposition is already proved in the cases n ~_ m, m _ 1, and consider the case n - m + 1. W i t h o u t loss of generality we can suppose t h a t f ( M (i)) C_ K(~) and g ( K (~)) C_ M (~) for each i _< m + 1. Since g f m+l idM it follows easily t h a t g f / M (re+l) m idM(m+l)" By the inductive hypothesis, there are an (m + 1)-dimensional finite polyhedron R, containing K (re+l) as a subpolyhedron, and an m - h o m o t o p y equivalence r" R - . M (m+l) e x t e n d i n g g / K (re+l). Sewing together the polyhedra K and R along naturally e m b e d d e d copies of K (re+l), we obtain the (m + 2)-dimensional finite polyhedron L, containing K and R as subpolyhedra, and the map s" L --. M which coincides with g on K and with r on R, whence s f = g f

m-l-1

~_ idM and f s / L (re+l) =

m

f r ~_ idL(m+~). By these conditions, we conclude t h a t ~i(s)" ~ri(L) --, ~ i ( i ) is an isomorphism for each i ___ m and an epimorphism for i -- m + 1. One can easily verify t h a t in this situation Ker(lrm+l(S)) is a finitely g e n e r a t e d Z(lrl(L))module. Select Z ( ~ l ( L ) ) - g e n e r a t o r s for Ker(lrm+l(S)) and use t h e m to a t t a c h (m + 2)-cells to L and to extend s over these cells. Let T denote the resulting (m + 2)-dimensional finite polyhedron, containing L as a subpolyhedron, and h" T --~ M the corresponding extension of s. T h e n lri(h) is an isomorphism for each i _< m + 1. Again, by Proposition 4.1.9, h is an (m + 1)-homotopy equivalence. This completes the inductive step and finishes the proof. [-1 T h e analogous s t a t e m e n t for proper n - h o m o t o p y dominations (near c~) will be discussed in Subsection 4.4.4

4.1.3. Z-set unknotting

and topological homogeneity.

PROPOSITION 4.1.13. Let A be a closed subset of a Polish A N E ( n ) - s p a c e X . Then the following conditions are equivalent: (i) A is a Z n - s e t . (ii) For each at most n-dimensional locally finite polyhedron P, the set { f E C ( P , X ) : f ( P ) A A - - 0 } is dense in C ( P , X ) . (iii) For each at most n-dimensional Polish space Y , the set { f e C (Y, X ) : f (Y) N A = 0} is dense in C (Y, X ) .

134

4. MENGER MANIFOLDS

Note that each Z n - s e t in any at m o s t n - d i m e n s i o n a l L C n - l - s p a c e is a Z-set. If, in Proposition 4.1.13, X is locally compact, then the listed conditions are equivalent to the following: (iv) For each at most n - d i m e n s i o n a l Polish space Y , the set { f 9 C ( Y , X ) : c l ( f ( Y ) ) N A - 0} is dense in C ( Y , X ) . Closed subsets satisfying t h e p r o p e r t y (4), as in the case of infinite-dimensional manifolds, are called strong Zn-sets. These sets are especially i m p o r t a n t in the non-locally c o m p a c t setting. PROPOSITION 4.1.14. One-point subsets of M e n g e r manifolds are Z-sets. T h e following s t a t e m e n t s are versions of the powerful Z-set u n k n o t t i n g theorem. THEOREM 4.1.15. Let Z1 and Z2 be two Z - s e t s in a ~ n + l - m a n i f o l d M , and let h : Z1 ---* Z2 be a h o m e o m o r p h i s m . Denote by ij : Z j --, M the inclusion map (j = 1,2). I f il and i2h are properly n-homotopic, then h extends to a h o m e o m o r p h i s m H : M ---, M which is properly n - h o m o t o p i c to idM.

COROLLARY 4.1.16. Every h o m e o m o r p h i s m between Z - s e t s of #n can be extended to an a u t o h o m e o m o r p h i s m of #n. For n - 0 this result is well-known. A closed subset of ~1 is a Z-set if and only if it does not locally s e p a r a t e ~1. T h e result for n - 1 originally a p p e a r e d

in [235]. A c o m p a c t u m X is called strongly locally homogeneous if for each point x E X and each n e i g h b o r h o o d U of x , there is a n e i g h b o r h o o d V of x contained in U such t h a t the following condition holds: for each point y E V, there is a h o m e o m o r p h i s m h : X ---, X such t h a t h(x) -- y and h / ( X - U) -- id.

COROLLARY 4.1.17. #n is topologically homogeneous. Moreover, it is strongly locally homogeneous. THEOREM 4.1.18. Let M be a #n-manifold. For each open c o v e r U E c o y ( M ) , there is an open cover )2 E c o y ( M ) with the following property: (,) if a h o m e o m o r p h i s m h : Z1 ---* Z2 between two Z - s e t s of M is )?-close to il (see notations in 2.3.5), then h can be extended to a h o m e o m o r p h i s m H : M ~ M which is l~-close to idM.

4.1. GENERAL THEORY

135

4.1.4. Topological characterization. The following characterization theorem [33, T h e o r e m 5.2.1] is central to the whole theory. We recall t h a t a space X has D D n P (Disjoint n-Disks P r o p e r t y ) if for each open cover L / 9 c o y ( X ) and any two m a p s a : I n ---. X and fl : I n ---+ X , there are maps a l : I n ---* X and ~1 : I n --. X such t h a t a l is U-close to a, ~1 is U-close to ~ and a l (I n) Aft1 (I n) = 0.

THEOREM 4.1.19. The following conditions are equivalent for any n - d i m e n sional locally compact A N E ( n ) - s p a c e X : (i) X is a #n-manifold. (ii) X is strongly B•,n-universal, i.e. has D D n p . (iii) Each map of the discrete union I n @ I n into X can be approximated arbitrarily closely by embeddings. (iv) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by closed embeddings. (v) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by Z-embeddings. (vi) Each proper map f : Y --. X of any at m o s t n - d i m e n s i o n a l locally compact space Y into X such that the restriction f /Yo onto a closed subset Iio is a Z-embedding can be arbitrarily closely approximated by Z-embeddings coinciding with f on Yo. Additionally, if X is c o m p a c t and ( n - 1)-connected (i.e. X 9 A E ( n ) ) , then conditions (ii)-(vi) give a topological characterization of the c o m p a c t u m #n. Note t h a t the t h e o r e m remains true even in the case n - c~ (see C h a p t e r 2). If n -- 0 and X is compact, condition (ii) trivially implies t h a t X has no isolated points. Therefore, in this case, X is h o m e o m o r p h i c to the C a n t o r disc o n t i n u u m as has already been noted above. A p p l y i n g the above characterization in the case k -- 1, we see t h a t a comp a c t u m is h o m e o m o r p h i c to #1 if and only if it is a 1-dimensional, locally connected c o n t i n u u m with D D 1 P . It is known (see [15] or [235]) t h a t a locally connected c o n t i n u u m has D D 1 P if and only if it has no local s e p a r a t i n g points and has no open subspaces e m b e d d a b l e in the plane. In this sense, Bestvina's characterization of #1 reduces to Anderson's. In Subsection 4.1.1, three m a j o r geometric constructions of the universal Menger c o m p a c t u m have been presented. Let us indicate another, spectral construction, given in [248]. Let {G~} be a basis of open sets of S 1, such t h a t G~ is an open cell with the p r o p e r t y t h a t diamG~ --. O. Let us construct an inverse sequence {Xi,p~ +1} as follows. We set X0 = S 1, and we get X~+I from X~ by "bubbling over G~" , i.e. X i + l is the quotient space o b t a i n e d from the disjoint union Xi ~ X~ by identifying the two copies of x 9 Xi precisely when p~(x) ~ Gi (here p*o" Z i ~ X o denotes the corresponding projection). T h e projection ~i ~ + 1 " X i + l ~ Xi is defined in the obvious way. Note t h a t p~+ 1is a retraction. One can check directly t h a t X = lim{Xi, Pi -i+1 } is a 1-dimensional 9

136

4. MENGER MANIFOLDS

A E ( 1 ) - c o m p a c t u m with D D 1 P and hence, by 4.1.19, X is a copy of #1. More careful consideration shows [33] that we get #n if we start with S '~ = X0 (and proceed as above). There are several other constructions of #n. To the best of our knowledge, all of them are defined as the limit spaces of inverse sequences(see, for example,

[155], [ls6]). 4.1.5. Approximation by Homeomorphisms. THEOREM 4.1.20. Proper U V n - m a p s between #n+l-manifolds are near-homeomorphisms. Another important result of I ~-manifold theory states that an infinite simple homotopy equivalence between/W-manifolds is homotopic to a homeomorphism. Let us note that this is not the case for homotopy equivalences (i.e. there exist non-homeomorphic but homotopy equivalent compact I ~ -manifolds). In #n-manifold theory we do not have simple homotopy obstructions and this significantly simplifies the corresponding result. THEOREM 4.1.21. Each proper n-homotopy equivalence between #n+l-manifolds is properly n-homotopic to a homeomorphism. The following result, due to Ferry [150, Proposition 1.7] and improved slightly in [188], also illustrates this situation. PROPOSITION 4.1.22. Let f 9 P ~ L be a map between compact polyhedra which induces an isomorphism between the i-th homotopy groups for each i < n. Then there is a compact polyhedron Z and U V n - m a p s ~ 9 Z ---. P and ~ 9 Z ~ L such that f a n ~.

Historical and bibliographical notes 4.1. Most of the results of this Section were obtained by Bestvina in his fundamental work [33]. The notion of nhomotopy is due to the author [87]. Proposition 4.1.12 was proved by the author

[100]. 4.2. n-soft m a p p i n g s of c o m p a c t a , raising d i m e n s i o n The following statement is the first resolution theorem for locally compact L C n - l - s p a c e s , in particular for #n-manifolds. THEOREM 4.2.1. Every locally compact L C n - l - s p a c e is an (n-invertibte) proper u v n - l - i m a g e of a #n-manifold. We also will be using the following result, which states that the u v n - r e s o l u tions from Theorem 4.2.1 can be improved over Z-sets.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION PROPOSITION 4.2.2. Let f " M ~

X

be a proper U V n - s u r j e c t i o n of a #n+l_

m a n i f o l d onto a locally c o m p a c t L C n - s p a c e and let Z be a Z - s e t in M . X

is any proper m a p properly n - h o m o t o p i c to f / Z ,

uvn-surjection

137

I f g" Z ---.

then there exists a proper

h" M ---. X such that h / Z -- g.

Below, in this subsection, we prove much s t r o n g e r results. We begin by fixing a n a t u r a l n u m b e r n. Let B n+l -- { x E R n ' i i x i l < 1} d e n o t e t h e unit closed ball in R n+l. Consider a m a n y - v a l u e d r e t r a c t i o n l:ln" B n+l "--+ S n of the ball B n+l onto its boundary, the sphere S n - OB n+l, defined as follows" R n ( x ) - {y E S n" ( x , y ) >_ 4[[xI[ 2 - 3[[x[[} for each point x E B n+l.

Here (x, y) denotes the s t a n d a r d scalar p r o d u c t in the E u c l i d e a n space R n+l and Ilxll = v / ( x , x ) denotes the induced norm. T h e above to each point of the ball B~'+1 = {x E R n+l" IIxlI < If x E B n+~ - (B~ +1 U Sn), t h e n R n ( x ) is a ball on x E S n, then R n ( x ) = x, i.e. R n is a retraction. It continuous. Consider the g r a p h F Rn of this map, i.e.

m a n y - v a l u e d m a p assigns 89 the whole sphere S n. the sphere S n. Finally, if is easy to see t h a t R n is

FRo = { ( ( x , y) e B ~+1 • S ~ : y e R ~ ( ~ ) } . By Pn: FRn ---+ B n+l and qn: FRn --'+ S n we d e n o t e the restrictions to FR., of the n a t u r a l projections of the p r o d u c t B n+l x S n onto its coordinates. LEMMA 4.2.3. T h e m a p Pn" FRn -'+ B n+l is ( n -

1)-soft.

PROOF. C o n t i n u i t y of the r e t r a c t i o n R n implies openness of Pn. Therefore, by T h e o r e m 2.1.15, it suffices to show t h a t the collection { R n ( x ) " x E B n + l } is c o n n e c t e d and uniformly locally c o n n e c t e d in all dimensions less t h a n n - 1. As was m e n t i o n e d above, topologically t h e r e are only three t y p e s of elements of the collection { R n ( x ) " x e B n + l } . Indeed, R n ( x ) is the sphere S n (for x E B~'+I), X a closed ball B n with center at i[~-][ in the sphere S n (for x E B n+l - B~'+I), or a point (for x E Sn). C o n s e q u e n t l y the i n d i c a t e d collection is c o n n e c t e d in all dimensions not exceeding n - 1. Let us show the uniform local c o n n e c t i v i t y of this collection in all dimensions less t h a n n - 1. Let k _ n - 2, e > 0 and e Consider a point x E B n+l and an a r b i t r a r y m a p a" S k ---, R n ( x ) 5 - - ~. such t h a t the d i a m e t e r of its image a ( S k) is less t h a n 5. E v i d e n t l y t h e r e is a ball B n on the sphere S n, containing this image a ( S k ) . Let &" B k+l ---+ B n be an extension of a from the b o u n d a r y S k = OB k+l onto the whole B k+l. Additionally, we can assume t h a t if R n ( x ) = x (which occurs when x E S n ) , t h e n a = a. Obviously, in this case as well as in the case w h e n R n ( x ) = S n, ~ is the desired extension. Suppose now t h a t x E B n+l - (B n+l U s n ) . T h e n R n ( x ) is a ball on S n of non-zero diameter. If S '~ - R , ( x ) ~= B n, t h e n the intersection B n M R n ( x ) is h o m e o m o r p h i c to the n - d i m e n s i o n a l ball and, consequently, t h e r e exists a r e t r a c t i o n r" B n ~ B n M R n ( x ) . Therefore, in this case, the c o m p o s i t i o n a r is the desired extension of a with d i a m e t e r of the image less t h a n e. In the r e m a i n i n g case, when S ~ - R n ( x ) C B n, we first observe t h a t the b o u n d a r y

138

4. MENGER

MANIFOLDS

O R n ( x ) of the ball n n ( x ) is contained in B n. Since k 4- 1 < n, there is a h o m o t o p y (in B n) connecting the m a p 5 with a m a p / 5 " B k+l ~ B n N R n ( x ) fixed on ~ - l ( R n ( x ) ) . Then fl does not move points of OB k+l -- S k as well. Clearly, ~ is the required extension of c~. [7

LEMMA 4.2.4. The map Pn" FRn ~ B n + l is n-invertible. PROOF. Let us consider a lower semi-continuous many-valued retraction Fn" B n+l ---* S n defined as follows:

= ( S

I1 11 > ifll ll <

89 89

Obviously, F(x) C R ( x ) for each point x E B n+l. It is also easy to see t h a t the collection {Fn(x)" x e B n+l} is connected and uniformly locally connected in all dimensions less t h a n n. Let FEn C B n+l • S n denote the graph of the map Fn and let PEn -- ~rBn+l/FFn, where 7rBn+~" B n+l • n ~ B n§ is the projection onto the first coordinate. It follows from T h e o r e m 2.1.15 t h a t PEn is n-soft. In particular, PEn is n~invertible. Now observe t h a t FEn C FRn and PEn -- Pn/FFn. This is obviously enough to conclude t h a t pn is also n-invertible. D LEMMA 4.2.5. Let Y be a closed subset of the sphere S n and f " Y --, B n§ be a map into the ball B n+l. Then there exists a h o m o t o p y H" Y • [0, 1] ---+ B n+l, fixed on f - l ( O B n + l ) , connecting f with a map whose image is contained in OB n+l. Moreover, g ( Y x [0, 1]) does not contain the center of B n+l.

PROOF. T h e case when 0 r f ( Y ) is trivial. Consider the case when 0 E f ( Y ) and let U -- f - l ( B ~ + l - {0}). Obviously, V is a proper subset of the sphere S n and, consequently, has trivial n-dimensional cohomology. Let us show t h a t f / U " U ~ B ~ + 1 - {0} is null-homotopic. Obviously, B~ + 1 - {0} is homeomorphic to the p r o d u c t S n x (0, 89 Therefore we can fix a canonical h o m e o m o r p h i s m sending S n x {t} onto the set {x e Rn+l"]]x]] -- t), t e (0, 89 Suppose now t h a t the m a p f / V is not null-homotopic. Since the projection r l " S n • (0, 89 ~ S n is a h o m o t o p y equivalence (notice t h a t the half-interval is an AR-space) we conclude t h a t the composition 7rf/U" U ~ S n is also not null-homotopic. In this situation, this composition generates a non-trivial element in the n-dimensional cohomology, c o n t r a r y to the r e m a r k above. Let lr2" S n • (0, 89 ~ (0, 89 denote the projection onto the second coordinate. 1 We assume t h a t the half-interval (0, ~] is isometrically e m b e d d e d into B~ +1 as a subspace {x e R n + l " x = te, t e (0,1]}, where e = ( 1 , 0 , . . . , o ) e n n+l. Since (0, 1] is contractible, the projection 7r2 is null-homotopic. Consequently, the composition 7r2f/U is also null-homotopic. Thus, keeping in mind the above facts, we can conclude t h a t the maps f / U and ~r2f/U, considered as maps into B~ +1 - {0}, are homotopic. Take the corresponding h o m o t o p y G" U x [0, 1] B~ + 1 - {0} such t h a t Go = f / U and GI = l r 2 f / U ) . Observe t h a t ~ r 2 G o ( y ) = 7r2Gl(y ) for each y E U. Therefore we may assume without loss of generality

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

139

that ~r2G0(y) -- 7r2Gl(y) for each y E U and each t E [0, 1]. The homotopy H can be extended to a homotopy F" (U U f - l ( 0 ) ) x [0, 1] ~ B~ +1. Further, the homotopy F can be extended to a homotopy H ~" Y x [0, 1] so that H'(f-I(B

n+l - B~ +1) x [0, 1])C_ B n+l - I n t B ~ +1.

We also can assume that H ' is fixed on the set f - l ( O B n + l ) . The intersection of the image of H~ with B~ +z is a segment, in fact a radius of the ball B~ +1. Let us now, after applying the homotopy H', shrink the indicated segment into its end which belongs to the boundary OB~ +1. Next push the ring B n + l - I n t B ~ +1 onto the sphere OB ~+1. As a result, we obtain a h o m o t o p y / ~ " Y x [0, 1] + B n+l. Observe t h a t / ~ - 1 ( 0 ) = / ~ o 1 (0) x [0, a], where a < 1. The decomposition of the product Y x [0, 1] into the intervals {{y} x [0, a ] ' y e / ~ o 1 ( 0 ) } and single points generates a C E - m a p It" Y x [0, 1] + Y x [0, 1] which is divisor of the homotopy H. In other words, there is a map H" Y x [0, 1] + B n+l such that /~ = HIt. The homotopy H satisfies our requirements. D LEMMA 4.2.6. There exists 5 > 0 such that for each m a p r B n --. B n+l, with d i a m ( i m ( r < 5, and for each map 7~" OB n" FRn, satisfying the equality Pn~ = r n, there is a map ~" B n ~ FR~ such that p n ~ -- r and 7~ -- ~ / 0 B n . PROOF. Let 5 = ~ and consider a maps r satisfying the following two conditions: diamr

B n --~ B n+l and ~" OB n --. FRn,

1 n) < ~ and pnT~ -- ~b/OB n.

First assume that r n) N OB~ +1 = 0. Then, by the connectedness of r either r n) C_ B~ +1 or r n) C B n + I - B ~ +1. It follows from the construction of the many-valued map R n that the map

p~! = p ~ / p ; ~ (B~+I ). ;~-1 (B3 + 1) ---+B~+I is a locally trivial bundle with fiber the sphere S n and, consequently, Pin is n-soft. Thus, if r n) C_ B~ +1, then the conclusion of our lemma is true. Observe that the map Pn = P n / p n l (

Bn+l

-- B ~ + I ) " P n l ( B n + I -- B ~ +1) --* B n+l -

B~ +1

is also n-soft (it is even soft). Therefore the lemma is true in the case r n) C B n+l - B~ +1. Next we consider the situation when r n) A OB~ +1 ~ 0. Since diam(im(r < ~, we see that the set r n) does not contain the center of the ball B n+l. Denote by J : B n+l - ( 0 } - . OB n+l the central projection of B n+l - {0} onto the boundary OB n+l. Then the set J ( r is contained in the ball (on the sphere S n) of an angular radius ~. Let z denote the center of this ball. Straightforward calculations show that for each point x E r the

140

4. MENGER MANIFOLDS

71" set R n ( x ) is a ball on the sphere S n with center at i ~z and of radius at least ~. Define a section r/: B n --~ B n • S n of the projection B n • S n ~ B n by letting

r i1r

r/(x) = ( x , - ~ )

for each point x E B n.

Since for each x E B '~ angle between - z and - IIr r is less t h a n ~, ~ we see t h a t there is an i s o m e t r y h: B n x S n --+ B n • S n, with 7rBn -- 7rB,~h, transforming rl into the trivial section 0: B n --~ B n x S n of the projection 71"B,~ (i.e. hrl = 0) d e t e r m i n e d by - z : O ( x ) = ( x , - z ) for each x e B n. D e n o t e by O n the ball (on the sphere S n) with center at - z and of radius ~. T h e m a p ~o9 O B n - - - , F R~ induces a section c~: O B n ~ O B n x S n of the projection O B n x S n ~ O B n. One can see t h a t c~(x) E R n ( r whenever x e O B n = S n - 1 . By L e m m a 4.2.5, we can connect the composition l r s , , h a : O B n --* S n with a m a p f : S n - 1 S n - I n t D n via a h o m o t o p y H t such t h a t H t ( x ) ~ - z w h e n e v e r t > 0. F u r t h e r , let G t : S n - 1 x [0, 1] ~ S n be a h o m o t o p y connecting ~rsnhc~ with a c o n s t a n t m a p (GI (S n - l ) = d), which can be o b t a i n e d from H by adding the contraction of the set S n - I n t D n. For each x E B n denote by rz the push (fixed on R n ( x ) ) in S '~ - R n ( r

with the center at -

:~ = {y e s~: (r

r IIr

onto the sphere

y ) = 411r

311r

if the sphere Ex is defined. If not, then rx denotes the identity map of S n. Observe t h a t if r E O B ~ +1, then Ez consists of the single point, and if r E I n t B ~ +1, t h e n E~ = 0. Define an extension ~ : B n ~ O B n x S n of c~ by letting ifx#O

.1

a(~) = ( ~h-l(x'e)'

if• =0.

Note t h a t the m a p & is well-defined and t h a t 5 ( x )

for each point

E Rn(r

x E B n. It only remains to note t h a t the desired m a p ~5 can now be defined by

letting ~5 = (r • i d s n ) ~ .

Vl

DEFINITION 4.2.7. A m a p f : X ---+ Y is c a l l e d p o l y h e d r a l l y n-soft i f f o r e a c h at m o s t

n-dimensional

g: A ---, X

polyhedron B,

a n d h : B ---, Y

that f k = h and k/A

subpolyhedron

with f g = h/A,

A

of B,

and any two maps

there exists a map

k : B -+ X

such

= g.

T h e following s t a t e m e n t expresses one of the most i m p o r t a n t properties of P n . LEMMA 4.2.8. T h e m a p P n : F R n --+ B n + l is p o l y h e d r a l l y n - s o f t . PROOF. It suffices to show t h a t for every pair of maps h: B n ~ B n + l and FR,~ with P n g - h / O B n, there exists a m a p k: B n --~ FR,~ such t h a t p n k -- h and k / O B n -- g. Take a t r i a n g u l a t i o n of the ball B n small enough to ensure t h a t the d i a m e t e r s of images (under h) of simplexes of this t r i a n g u l a t i o n g: O B n ~

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

141

are less than ~. Denote by B(nn_l) the ( n - 1)-dimensional skeleton of this triangulation. Since, by L e m m a 4.2.3, the map Pn is ( n - 1)-soft, we conclude that there exists a map k ~" B(n-1) ~ FR, such that pn kl -- h / B (n-l) n and k~/OB n = g. Apply L e m m a 4.2.6 to each of the n-dimensional simplexes of the indicated triangulation B n to obtain the desired extension k. El LEMMA 4.2.9. The m a p qn" F R~, ---+ S n is a trivial bundle with fiber homeomorphic to the ball B n+l. PROOF. It follows from the definition of the retraction Rn that the fiber qnl(y) of each point y E S n can be written as qn--1 (y) __~ {X e B n+l 9 (x, y) >_ 4llxII 2 - 3llxII}.

Let B~ +1 = {z"

IIzll

< 89 and define a map fy" q ~ l ( y ) ~ B ~ + I as follows

S~(~) =

4x

3 + ~/9 + 16(~, y)

Straightforward verification shows that the map fy is a homeomorphism. Additionally, the collection of homeomorphisms {fy" y E S n} continuously depends on y, i.e. qn is homeomorphic to the trivial bundle B~ +1 • S n ~ S n. El Let K be a finite simplicial complex. By K (n) we denote the n-dimensional skeleton of K and by IKI the underlying polyhedron of K. If for each at most (n + 1)-dimensional simplex a E K we take a many-valued retraction Rn" lal --* ]a(n+l)l, homeomorphic to the retraction Rn" B n+l --~ S n constructed above, and then consider the union of these retractions, we obtain a retraction

nK(.)" IK(~+~)I ~ IK(~)I. Generally there is no canonical homeomorphism of the ball B n+l onto the ( n + l ) dimensional simplex. Therefore the retractions RK(. ) are not uniquely defined. For this reason RK(,~ ) denotes any retraction constructed in the above indicated way. Further, by FRK(.) _C [K(n+I) I x IK(")] we denote the graph of the manyvalued retraction R K ( , ) 9 IK(n+l)l

• IK(") I.

As above, PRK(,)" FRK(,~) ---+ IK(n+l) ]

and qRK(,) o FRK(.) ~ IK(n)l shall denote the restrictions of the projections of the product IK(n+l) I • IK (n) ] onto the first and second coordinates respectively. LEMMA 4.2.10. The m a p PRK(,)" FRK(,) ---~1 K ( n + l ) l is n-invertible, ( n - 1 ) soft and polyhedrally n-soft. PROOF. Apply Lemmas 4.2.3, 4.2.4 and 4.2.8.

El

For each pair (n, k) of natural numbers (n > k) and for each finite simplicial k,n(K) as follows. Concomplex K, let us now assign a commutative diagram $i,j sider the many-valued retractions R K ( ~ - I ) " [K(n-i+l)[ ---+ [K(n-i)[ constructed

142

4. MENGER MANIFOLDS

above, i = 1 , . . . , k. First, by induction on i + j (beginning with i + j - k), we define spaces Ek, i,jn (K) and maps

k n (K)---+ Eki'nl,j (K) Ei,j

fik'n(K)

n (g) , gik? ( K ) 9 E ki,j' n ( g ) _.+ Ek, i,j-1 t h a t will be part of the diagram 9 For each non-negative integer i _ k, we d e f n e k,n our space as Ei,k_i(K) =1 K (n-l) I. For each non-negative integer i _< k - 1, let

Z ik,. + l , k - i (K) = FR~ (n-~-l)

'

k~n

f i+l,k_i(K) -- pRg(,.,_,_l) and k~?l

gi+l,k_i(K) -- qRK(,,_,_I ) . Suppose now t h a t the spaces and maps, required for these diagrams have already k'n been constructed for all i,j with i + j < m 9 Let us define the space E i,m_i(K) as a fibered p r o d u c t (i.e. with respect to the maps

k,n

pullback) of spaces Eki'~l,m_i(K) and Ei,m_i_l(K )

fk,ni,m_i_l(K)"

k,n k,n Ei,m_i_l(K)--+ Ei_l,m_i_ 1(K)

and ]g~n

k~n

gi--l,m--i--1 ( g ) " E i _ l , m _ i ( g ) ~ Eki'l,m_i_l ( g ) . T h e maps

k,n (K) ---+Ei_l,m_i(K k,n fk,n ) i,m--i (K)" Ei,m_ i and

k,n kn " E ~,.,_~ k'n (K)---+ Ei,m_i_ 1(K) a,~_~(g) are defined to be the canonical projections of the corresponding fibered product. In these notations, we have a c o m m u t a t i v e square diagram

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

fi,~-i(K)

E~ m--i 's (K)

143

k

9.- Ei, ~ - i ( K )

k~S

kn a,;i_~ (g )

gi-l,m-i(K)

ks

f i , ~ - i - 1(K Eki'l,m_i_l (K )

,~Eki'l,m_i(K) k~rt

]g~s

After performing k steps we get a single space Ek, k (K) and two maps fk,k (K) k,n

and gk,k(K) 9 This finishes the construction of the diagram s i , j Below, " when there is no confusion, we omit upper indices in the notations of spaces and ks maps of the diagram s ~ (K). We formulate some elementary properties of these diagrams. 9 The map f = fl,kf2,k'''fk,k: Ek,k ~ Eo,k is ( n - k - 1)-soft, ( n - k)invertible and polyhedrally ( n - k)-soft. 9 Let g be a projection of the space Ek,k onto the space Ek,o. Then

g f - 1 = RK(,_k)RK(n_k+I) 9 The part of the diagram s i,j

... RK(n_I).

consisting of spaces Ei,k n~ (K) with k--m,n--m

indices i _> m is naturally isomorphic to the diagram C~,j

(K). In

kn k-re,n-re(K). particular, Ei,'j (K) ~ Ei_m, j 9 Let K1 be a subcomplex of K.

Then the diagram Ek?(K1) can be k,n

naturally embedded into the diagram Ei, j (K) in the sense that for all indices i , j there exist embeddings ~oi,j" Ei,k,n j (K1) ~ E ki,j, n (K) such that the maps f.k'.n(K1)~,~ and gik,?(gl) coincide with the restrictions of the k,n

kn

maps fi,j (K) and gi,~ ( g ) respectively 9 Observe also that for each i > 0 and j > 0, we have

E ki,j, n ( K 1 ) - (fi,~n(K))-l(Eik,?(K1)) kn

LEMMA 4.2.11. If K is a finite simplicial complex, then Ei, ~ (K) is an A N R -

compactum. PROOF. We prove our statement by induction on k. First consider the case 1,n k = 1. Represent the space E0,1 ( g ) =1 g(n) I as the union a l U a2 U . . . U ar of

144

4. MENGER MANIFOLDS

its simplexes. Then 1 (al) U - - - U f -1,1 1 (fir). El:I n1 (K) = f -1,1

Since the map f1,1 is a projection of the graph of the many-valued retraction RK(,,_~), it can easily be seen that for each i _~ r we have Sl, l ( a i ) n Uj
Ek+l,k+l

fk+l,k+l

~''"

* Ekk+l

fl,k+l

* E0,k+l

gl,k+l

gk,k+l

gk+l,k+l

*El,k+l

fk+~,k Ek+l,k

.

Ek,k

,.-''"

,.- E l , k

Denote by A: K (n) ~ w the map assigning to each simplex a E K (n) the number of n-dimensional simplexes of K containing a as a subcomplex. Applying Lemma 4.2.9, we can see that for each simplex a E K (n) the map gl,k+l restricted to --1 ( I n t I a [) is a locally trivial bundle with base I n t I a [ the inverse image gl,k+l and fiber V,x(a)B n, where Vx(a)B n is a wedge of A(a) n-dimensional spheres.. Contractibility of I n t [ a [ guarantees that this bundle is trivial. Similarly, it --1 can be seen that the space c l ( g--1 l , k + l ( I n t [ a [)) C_ gl,k+l([ a I) (closure is taken in El,k+l) is also the space of a trivial bundle over [ a [ with the same fiber. Let a l , . . . , a m o be the vertices of K, and in general denote by am~_l,...,am~ idimensional simplexes of K. Let Mz = Ui
4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

145

M is a s u b c o m p l e x of K (n-l). It follows from the above listed properties of the d i a g r a m s s k )n (K) t h a t k+l'n(M) ~ ~ k'n-l(M). f{--l(i M ]) = ~ ~k+l,k ~k,k Therefore, by our inductive assumption, f l 1(I M ]) C A N R. Observe t h a t --1 gl,k+l([ a l) ~ V,k(a) B n

for each vertex a E K (~ 9 It is not hard to see t h a t g k-1+ l , k + l f l l ( a ) "~ f l l ( a ) X V)~(a)B n and, consequently, g -k+l,k+lfl 1 -1 (a) is an ANR-space for each vertex a E K (0). --1 Suppose now t h a t the inverse image gk+l,k+lfll(Ml) is an ANR-space for some 1 (1 >_ m0). Observe t h a t [ a l + l I - - M l - I n t l a l + l I. Therefore g-1

1

g-1

-1

1

k+l,k+lfl (M/+I) ---- k + l , k + l f l l ( I n t [ a / + l l) U gk+l,k+lfl (Ml) --- f o l ( c l ( g ~ , ~ + l ( X n t

[ hi+ 1 [)) U g-1 k+l,k+l f 1 1 ( M l ) .

Notice also t h a t

f o l ( d ( g l-1 , k + l ( I n t lal+l I)) ~ f l l ( I al+l [) • VX(al+l)B n. In this situation it suffices to show t h a t

fol(cl(g~,lk+l(IntIal+l I)) n g-lk+l,k+lfll (Ml) is an A N R-space. Note t h a t f o 1 ( c l ( g l-1 ,k+l(Int

f o l ( C l ( g l-1 ,k+l(Int

l al+l I)) N g k-1+ l , k + l f l l ( M l ) l al+l I)) N g l-1 ,k+lfll(Ml)

--

--

--1 --1 f o l ( c l ( g 1,k+1 -1 ( I n t [ hi_t_1 [)) Pl g l , k + l f I (0 ] al+ 1 [) Consider the projection

onto fll(Olaz+~l) which, as shown above, is an ANR-space. Let

r

fll(Olal+l[) • VXal+,)B n --+ E l , k + l

be the composition of f l x idvB n and the trivialization --1 (0lal+ll) 7I" OIal+ll • VB n ---+ gl,k+l

-1 (0 ] az+l I) --1 1( I n t ] al+l ]))I~l gl,k+l of the bundle gl,k+l/cl(gl,k+ flTr = g l , k + l r Consequently there exists a m a p 4: f11(Olal+11) • VX(a,+I) Bn --* Ek+l,k+l,

Notice t h a t

146

4. MENGER MANIFOLDS

lifting b o t h lr and r Since the collection {Tr, r separates points of the compactum f~l(Olal+ll) x VX(a~+I)B n, we conclude that r is injective. On the other hand,

r

x vx(o,+,)B ~)

-

fol(r

n g l-1 , k + l f -1l ( M t ) "

( g l-1 ,k+l(Intlal+ll))

Therefore

f o l ( d (o~,k+~(x~tl"z+~l)) -1 n g-1 k+~,k+~f~-1 (M~) is an A N R-space. --1 1(MI) is an ANR-space for each I. It only suffices to note Thus gk+l,k+lfl that Ek+1,k+1 = g-1 k+1,k+l f -11 (El,k). This finishes the proof.

D

LEMMA 4.2.12. For each finite simplicial complex K and each natural number n, there exist an ANR-compactum X and two maps f : X --*1 K I and g: X --~ IK(n)l satisfying the following conditions: (a) The map f is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

(b) g / f - l ( [ g ( n ) l ) - f / f - l ( [ g ( n ) [ ) . (c) gf-l(lal)c_ Io(")1. n n+k

PROOF. Let d i m K = n + k. Consider the diagram ~r

X

(K) and let

~k,n+k(K )

= ~~k,k

k n+k

k n+k

f = f l , k " " fk,k: Ek' k (g)---* E0'~r (g) = Igl, k:.+k k:.+k g = gl,k gk,k: Zk, k ( g ) ---, Ek, o ( g ) - - I g ( n ) l

for each a e g .

By L e m m a 4.2.11, X is an ANR-compactum. By the properties of the di1)agrams s i , j ' the map f is n-invertible, polyhedrally n-soft and ( n soft. These properties also imply that g f - 1 = RK . . . . RK(,+k-1), and consequently f and g coincide on the inverse image f-l([K(n)[). For the same reason,

g.f-l(lal) c_ Io(n)l. D Remark 4.2.13. All statements proved in this section so far are valid for countable locally finite simplicial complexes as well. In such cases, the resulting spaces are locally compact A N R-spaces and the resulting maps are proper. Remark 4.2.14. More careful consideration shows that the compactum X from L e m m a 4.2.12 is a finite polyhedron. Respectively, if K is countable and locally finite then the polyhedron X is also countable and locally finite. We are ready to prove the main result of this section. THEOREM 4.2.15. For each natural number n, there exists a map fn: #n I ~ of the universal n-dimensional Menger compactum onto the Hilbert cube satisfying the following properties: (i) The map fn is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

147

(ii) For each at most n-dimensional compactum Y, each map f" Y --+ #n and each open cover lg E cov(#n), there is an embedding g" Y --~ #n that is lg-close to f and such that f ng = f n f .

PROOF. We construct the map fn as the limit projection of an quence 8 - {M{, c~ +1} consisting of the Hilbert cube manifolds (in M1 -- Io2 ). The construction is carried out by induction. The limit verse sequence is contained in the product I~' • I~' • --. • I~ • -.the metric

~(~, y) = r ~

inverse separticular, of this inwhich has

d~(x~, y~) 2(~+1)

where di is a (bounded by 1) metric on the Hilbert cube I.~ = ( y l , . . . , y i , . . . ) and xi, yi E I~ for each i. Let ai E Io2 for each i. Then each of the products I~' x --- x I m can be naturally identified with the subspace oo I~' x - - . x I m x {am+l} x - - - x {ai} x - - - of the product l I i = l i ~ . We construct an Io2-manifold as a subspace of the product I~ x --- x I~n and consider the O2 restriction (to Mi+l) of the projection ~ + 1 . 1-Ij=l I~ x Ii+1 ~ 1-I~=11~ as a projection a~+l. Mi+l ---+ Mi of the spectrum 8. In this situation, as can easily be seen, the limit space X = lim 8 is naturally embedded into Io2 - 1--Ii=l~176 i~'. Let M1 = I~ and suppose that the Io2-manifold Mj C YIi=lJI~ c Io2 has already been constructed. By Theorem 2.3.28, the Io2-manifold Mj is homeomorphic to the product of the Hilbert cube jIo2 and a finite polyhedron Kj. Obviously, we may assume that the composition of the projection wj" Kj x j I W ---+ Kj with a certain section sj" Kj" K j x j Io2 is a 2-J-move, i.e. for each point x E K j • Io2, we have d(x, sjwj(x)) < 2 - j . We also assume t h a t Kj is given together with a triangulation such t h a t diamsj(a) < 2 - j for each simplex a of this triangulation. By L e m m a 4.2.12, there exist an A N Rc o m p a c t u m X j + I and maps f j + l " X j + I ~ K j and gj+!" Xj+I ---+ K~ n) such t h a t f j + l is n-invertible, polyhedrally n-soft, ( n - 1)-soft and, in addition, -1 (n) for each simplex a of the triangulation given on Kj. Let Mj+I : Xj+I • Io2. By Theorem 2.3.21, Mj+I is a Io2-manifold. Next, represent the cube j+lIo2 as the product jIo2 x j + 1 I~' of two Hilbert cubes, and denote by yj+l" X j + I • Io2 • I~ --+ Xj+I x jIo2 the product of the identity map idx~+l and the projection ~ " jIo2 • I~ --+j Io2. Define the map ~+~ j 9 Mj+I ~ Mj by letting ~j+l

= (f3+~ • i d ~ ) ~ j + l .

In this situation we have the following diagram:

148

4. MENGER MANIFOLDS

Mj+I

= Xj+l

~ j + l -- i d x

x j I ~~ x j + 1 I~'

7r31

Xj+I xj I ~

Xj+I

f j+l x id

fj+l

,~ K j x j I ~~ = M j

,- K j

Let A" M j + I ~ IS+1 be an embedding. T h e n the desired e m b e d d i n g of the /"a-manifold M j + I into the p r o d u c t I i a x - - - x I~a+l can be defined as the diagonal p r o d u c t of the m a p ~ j + 1 and the e m b e d d i n g A. Let a jj + l ._ 7rJ+lj / M j + I , and j+l observe t h a t this map a j " M j + I ~ M j is n-invertible, polyhedrally n-soft and ( n - 1)-soft (because b o t h f j + l and 7r~ have the c o r r e s p o n d i n g properties). T h u s the c o n s t r u c t i o n of s p e c t r u m 8 = {Mi, a~ +1} is complete. Let ai" X = lira,5' --, M i d e n o t e the i-th limit projection of this s p e c t r u m . As the desired m a p fn" #n __~ i ~ we take the first limit projection a l " X ---+ M1 = I W 9 Of course, we still have to show t h a t X ~ #n. Let us investigate the properties of the c o m p a c t u m X and the m a p fn. Since each of the short projections a~ +1 of the s p e c t r u m S is n-invertible, p o l y h e d r a l l y n-soft and ( n - 1)-soft, we easily see t h a t fn (as well as all other limit projections of the s p e c t r u m S, see L e m m a 6.2.6) has the same properties. This proves one p a r t of the theorem. Let us now show t h a t dim X = n. Since M1 -- I w , M1 contains a topological copy of the n - d i m e n s i o n a l cube I n. T h e n-invertibility of fn then g u a r a n t e e s t h a t X also contains a copy of I n. Therefore dim X _> n. In order to show t h a t dim X _< n we proceed as follows. Obviously, all we need is an existence, for each i, of a ~ - m o v e of X into an n - d i m e n s i o n a l polyhedron. Such a move is i n d i c a t e d below" siwi(gi+l x idii~)~ii+lai+ 1 9 X ---+ K} n). Consequently, dim X - n . Next we show t h a t fn has the second p r o p e r t y formulated in the theorem. Let Y be an at most n - d i m e n s i o n a l c o m p a c t u m , f" Y -4 X be a m a p and

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

149

bl E c o v ( X ) . Choose an index i and an open cover 12 E cov(Mi) such t h a t a~-l(Y) refines/~/. It follows from the construction of the m a p a~ +1 t h a t a~ +1 can represented as the composition 7i+lr/i+l, where ~i+l is a trivial bundle with fiber the Hilbert cube. Consequently, there exists an e m b e d d i n g hi+l" Y ~ M i + l such t h a t a~+lhi+l -- a i f . Since d i m Y < n, and since the limit projection a i + l " X ---+ M i + l in n-invertible, we can find a map h" Y ---+ X such t h a t a i + l h -- hi+l. It is not hard to see t h a t h is an embedding. Moreover, h is U-close to f and f n h = f n f . Finally, let us show t h a t X is homeomorphic to #n. By the property of f n established above, X satisfies the condition from T h e o r e m 4.1.19., i.e. every map of an at most n-dimensional c o m p a c t u m into X can be arbitrarily closely a p p r o x i m a t e d by embeddings. T h e equality dim X -- n was also established above. Therefore, by T h e o r e m 4.1.19, it suffices to show t h a t X E L C n - I N C n - 1 . First we show t h a t X E L C n-1. Let x be an a r b i t r a r y point of X and U be a neighborhood of x. Take an index i and a n e i g h b o r h o o d Ui of a i ( x ) in Mi such t h a t a~-l(Ui) C U. Since, by our construction, Mi is an A N R - c o m p a c t u m (even an I ~ manifold), M i E L C n-1. Consequently, there exists a neighborhood Vi of a i ( x ) in Mi such t h a t Vi C Ui and the following condition is satisfied: 9 for each k < n - 1 and each map ~i" S k -+ V~, there is a m a p r B k+l --+ U~ such t h a t ~i = r k. Let VaT, l ( v i ) . Clearly, V is a neighborhood of x contained in U. Take any map ~o" S k ---+ V, k _< n - 1. By the choice of the n e i g h b o r h o o d Vi, there is a map r B k+l --+ Ui such t h a t ai~o = r k. Since (Sk, B k+l) is a polyhedral pair and the map a i is polyhedrally n-soft (recall t h a t k < n - 1), we conclude t h a t there is a map r B k+l --+ X such t h a t the following d i a g r a m commutes:

X

Sk t

.

~Y

.Bk+ 1

In other words, r k = ~ and a i r = r This obviously implies t h a t k+l) C_C_U. Thus, X E L C n-1. A similar (but simpler) a r g u m e n t shows t h a t X C C n-1. Therefore, X ~ #n. T h e proof is finished. [2] r

In order to o b t a i n other i m p o r t a n t properties of the m a p fn we need some preliminary statements. T h e proof of the following one is, in fact, contained in

150

4. MENGER MANIFOLDS

the proof of L e m m a 4.2.6. LEMMA 4.2.16. For each e > O, there is a 5 > 0 such that for any map ~o: S n-1 --~ F R , , with diam(im(~o)) < 5, and any 5-homotopy H : S n-1 x[0, 1]---, B n+l, with Ho = pn~o, there exists a h o m o t o p y G: S n-1 x [0, 1] ~ FR, such that Go = ~o, pnG = H and d i a m G ( S n-1 x [0, 1]) < e. LEMMA 4.2.17. Let Y be an L C n - l - c o m p a c t u m , K be a finite simplicial complex and ~o: Y --~ Ig(n+l)l be a map. Then the fibered product (pullback) X of the spaces FRK(,~) = 1-'n+l and Y with respect to the maps Pn+l -- PRK(n ) : Pn+l --+

IK(r'+l)l and ~o: Y --, Ig(n+l)l is an L C n - l - c o m p a c t u m . i i c~ l:~ n PROOF. Let c~ (~,~=1~.i ) be the one-point compactification of a discrete collection of n-dimensional disks B.n and let x be the compactifying point. Let S~'-1 denote the boundary of the disk B~'. Assume the contrary. Then there exists a map r (U~IS~ '-1) ---, X such that for each i and for any extension of r -1 (to the disk B~) the diameter of its image is more than some positive number a. Let a be an (n + 1)-dimensional simplex in K (n+l) containing the point y = pn+l~O~r Here ~o~:X --+ Fr,+l denotes the canonical projection (parallel to ~o) of the fibered product X. The map q: X ~ Y has similar meaning (see the following diagram):

a (u~,S~ '-I)

r

, X

,- Fn+l

Pn+l

y

r

,._

IK(n+l) I

We identify the simplex a with the unit (n + 1)-dimensional ball B n+l. Also, we identify the restriction of the many-valued retraction R g ( n ) : l K ( n + l ) l --~ Ig(n)l to lal with the many-valued retraction R n : B n+l ~ OB n+l (see the beginning of this section). We consider two cases. C a s e 1. Assume first that y f~ OBg +1, and take a neighborhood V of y in IK(n+l)l disjoint from OB~ +1. The restriction of Pn+l to this neighborhood is n-soft by construction. But then the restriction of q to q-l(~o-l(Y)) is also nsoft. It only remains to observe that the latter fact, coupled with the condition Y E L C n - l , contradicts the assumption made in the beginning. C a s e 2. Now assume that y E OB~ +1. Since Y c L C n-1 we can conclude that there is an extension r (U~__lBn) ~ Y of the composition qr Take

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

151

an e > 0 such that for each set F C X, the inequalities diam((t'(F)) < e and diam(q(F)) < e imply the inequality d i a m F < a. Next, choose 5 as in Lemma 4.2.16. Let b~ be the center of the ball B~. Connect the points 4(b~) and qr by paths ~i" [0, 1] ~ Y so that limi--.oo diam~i([O, 1]) = 0. Then there is a number k such that

max{diam(rlk([O, 1]) U r

diam((trlk([O , 1]) U (t(~(B~))} < 5.

Let Ht" S~ -1 x [0,1] --~ Y be a homotopy which c o n n e c t s r -1 with the constant map to the point 4(bk) inside of the image 4(B~), and then sends this constant map along the path ~k to another constant map to the point qr By L e m m a 4.2.16, the homotopy (tilt can be lifted to a homotopy Gt . S kn - 1 x [0, 11 --. Fn+l so that Go = (t'r G I ( S ~ -1) C_ Pn~I(Y) and d i a m ( G t ( S ~ -1 x [0, 1]) < e. Since Rn+I(Y) -1 "~ S n and the convex hall of G I ( S ~ -1) in the sphere S n has diameter less than e, we may conclude that, also shrinking c o n v ( G l ( S ~ -1) to a point, the homotopy Gt can be extended to an e-homotopy Lt" S~ -1 x [0, 1] --* Fn+l. Extending the homotopy Ht by adding the identity map, we obtain a homotopy Nt" S~ -1 • [0, 1] ~ Y such that (tiNt -- Pn+lLt. But then (recall that X is the fibered product in the above indicated diagram) we get a well-defined a-homotopy Mr" S~ -1 x [0,1] ~ X such that M0 = r -1 and M I ( S ~ -1) = r This contradiction finishes the proof. V1 LEMMA 4.2.18. Let K be an m-dimensional finite simplicial complex and f" X --+ [K[ be a map, constructed in Lemma 4.2.12. Then for each L C n - l - c o m p a c t u m Y , m > n, and for each map (t" Y ~ [K[, the fibered product Z of X and Y with respect to f and (t is an L C n - l - c o m p a c t u m . PROOF. Denote by X1 the inverse limit of the following diagram Fm

Fm-1

IKI

IK(m-1) [

Fn+2

...

[K(n+2) [

IK(n+l)[

and assume that p~" X1 ~ IK(i)l and q~" X1 ~ F~ denote the corresponding projections, n-softness of the projection p~ implies t h a t the fibered product y I of X1 and Y with respect to Pm I and (t is an L C n - l - c o m p a c t u m . Let (t~ Y~ ~ X1 be the projection of this fibered product, parallel to (t. It is easy to see that X is also the fibered product of spaces X1 and Fn+l with respect to maps ! . X l '"+ IK ( n + l ) [ and pn+l . Fn+l ---+ [ K ( n + l ) [. Then Z itself is the fibered Pn+l product of spaces Fn+l and Y~ with respect to Pn+l: Fn+l --~ IK(n+I)I and

152

Pn+l

4. MENGER MANIFOLDS

~ . y~

"-+ Ig

lemma is proved.

(n+l)

I" By L e m m a 4.2.17, Z is an L C n - l - c o m p a c t u m .

The

F'I

The following statement expresses one of the most important properties of the map fn: I~n --+ I~ constructed above. THEOREM 4.2.19. For each LCn-l-compactum Y contained in the Hilbert cube I ~, the inverse image f ~ l ( y ) is a #n-manifold. Additionally, i f Y E C n-l, then f ~ - l ( y ) is homeomorphic to #n. PROOF. We use the same notations as in the proof of Theorem 4.2.15. Recall t h a t the map f n : # n --~ I • was constructed as the limit projection of the inverse sequence S = {Mi, c~ +1} consisting of I ~ -manifolds (M1 -- I W) and n-invertible, polyhedrally n-soft and ( n - 1)-soft short projections. If Y is an LCn-l-compactum contained in I 0~ , then the inverse image fnZ(Y) is the limit space of the induced inverse sequence S ' - {Yi, r~+l}, where Y1 = Y, Yi+l = (c~+l)-l(Yi) and r~+1 = ~+l/Yi+l. Since the restriction of an ninvertible, polyhedrally n-soft and ( n - 1)-soft map onto the inverse image still has all these properties, we see that all short projections of the spectrum ,~' = {Yi, r~ +1} are n-invertible, polyhedrally n-soft and (n - 1)-soft. While proving Theorem 4.2.15 we have already seen t h a t the inverse limit of an inverse sequence consisting of L C n - l - c o m p a c t a and polyhedrally n-soft short projections is an LCn-l-compactum. Therefore it suffices to show that Yi E LC n-1 for each i. We prove this fact by induction. By assumption, Y1 E LC n-1. Assume t h a t Yi E LC n-1. The projection a~ +1 can be represented as the composition ")'i+1~i+1, where ~i+1 is a trivial bundle with fiber the Hilbert cube and "yi+l is homeomorphic to the map f~+l x i d i ~ . Here f~+l: X~+I ~ Ki is the map from Lemma 4.2.12 (see the diagram in the proof of Lemma 4.2.12). Consequently, by Lemma 4.2.18, ~i+l(Yi+l) is an LCn-l-compactum. But then Yi+l ~ ~i+l(Yi+l) x I ~ is also an LCn-l-compactum. Thus f ~ l ( y ) c LC n-1. Polyhedral n-softness of the short projections guarantees that if, in addition, Y E C n - l , then f n l ( Y ) e C n-1. Obviously, dim f ~ l ( y ) < dim #n = n. Theorem 4.2.15 shows that dim f n l ( Y ) ~n. Theorem 4.2.15 also guarantees that any map of an at most n-dimensional c o m p a c t u m into f~-I ( y ) can be arbitrarily closely approximated by embeddings. Therefore, by Theorem 4.1.19, f ~ Z ( y ) is a #n-manifold (and is homeomorphic to #n i f Y E Cn-1). [:]

Remark 4.2.20. In fact a stronger result can be proved: for any LC n-1c o m p a c t u m Y and for any map ~: Y ~ I W (not only for embeddings, as in Theorem 4.2.19) the fibered product of Y and #n with respect to ~ and fn is an L C n - 1_co m pact um. Summarizing the results proved above, we have.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

153

THEOREM 4.2.21. There are maps fn: #n ~ i W and gn: ttn ---* #n satisfying the following properties: (i) The maps fn and gn are n-invertible, ( n - 1 ) - s o f t and polyhedraUy n-soft. (ii) All the fibers of the maps fn and gn are homeomorphic to #n. (iii) The inverse images of L C n - l - c o m p a c t a under the maps fn and gn are #n-manifolds. (iv) The maps fn and gn both satisfy the parametric version of D D n P , that is, any two maps ~, ~ : I n ~ #n can be arbitrarily closely approximated by maps a I, ~l: I n ~ #n such that f n a l = f n ~ , f n ~ I = f n ~ , g n d = gnu, g n f l ' - - gnZ and i m ( a ' ) A im(Z') = O. Using the map fn as a guide, additional considerations allow us to obtain the following result. THEOREM 4.2.22. Any metrizable A ( N ) E ( n + 1)-compactum is an U V n - I image of: (i) An (n + 1)-dimensional A ( N ) E - c o m p a c t u m . (ii) The Hilbert cube I ~ (an I W -manifold, respectively). (iii) The (2n + 1)-dimensional cell ( a (2n + 1)-dimensional topological manifold, respectively). A simple comparison of the major ingredients of the Hilbert cube manifold theory (see Chapter 2) with the corresponding results of Menger manifold theory presented so far, shows that from a certain point of view the n-dimensional analog of the Hilbert cube Q should be considered to be, not the usual ndimensional cube I n, but the n-dimensional universal Menger compactum #n (moreover, the Hilbert cube itself may be viewed as the "infinite-dimensional Menger compactum"). In addition, one can observe a fairly deep analogy between the theories of #n-manifolds and Q-manifolds themselves. On the other hand, at first glance it is not clear what is the analog of the operation of "taking the product by Q" in #n-manifold theory - the operation which is involved in the formulations of triangulation (Theorem 2.3.28) and stability (Theorem 2.3.10) theorems for Q-manifolds. A decisive step in finding a "full" analog of this operation in #n-manifold theory is based on Theorems 4.2.15 and 4.2.19. First observe that taking the product X • Q of a space X and the Hilbert cube Q may be interpreted as taking the inverse image lr~-l(X) of a space X c Q, where r l " Q • Q ~ Q denotes the natural projection onto the first coordinate. It turns out that the map gn : f n / . f n l ( # n ) " #n ___+ttn in Theorem 4.2.15 may be thought of as the analog of the projection ~1 in the theory of #'~-manifolds. If this is agreed, everything then falls in place. The following statement is a triangulation theorem in #n-manifold theory. THEOREM 4.2.23. For any #n-manifold M , there is an n-dimensional polyhedron P such that for any embedding of P into #n the inverse image g~-l(p) is homeomorphic to M .

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PROOF. We consider the compact case. The locally compact case can be proved similarly. Take an n-dimensional finite polyhedron P and a map ~" P --~ M which induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i _< n - 1. E m b e d P into #n. It is easy to see that the composition

~gn/g~l(P) 9gnl(P)~

P ~ M

also induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n 1. Observe also t h a t the inverse image g ~ l ( p ) is a #n-manifold (we use Theorem 4.2.19). Therefore, by Theorem 4.1.21, M and g ~ l ( p ) are homeomorphic. [7 Here is the promised stability theorem for #"-manifolds. THEOREM 4.2.24. For any #n-manifold M in tt n, the inverse image g~-l(M) is homeomorphic to M . PROOF. If M is compact, then, by Theorem 4.2.19, the inverse image g n l ( M ) is a tin-manifold. Since the restriction g n / g n l ( M ) " g n l ( M ) ~ M is polyhedrally n-soft, we see t h a t it induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n - 1. By Theorem 4.1.21, g ~ l ( M ) is homeomorphic to M. The proof of the non-compact case is similar. Kl We conclude this section with the following statement which shows, in particular, that the map fn cannot be made n-soft. THEOREM 4.2.25. There is no n-soft map of an n-dimensional compactum onto a higher dimensional cube (the Hilbert cube in particular). PROOF. If n -- 0, the validity of our statement is obvious: the open image of a zero-dimensional c o m p a c t u m is zero-dimensional. Thus we may assume that n _> 1. If such a map g" Z --* I m existed, then Z must be connected (notice that 1-soft maps are monotone). Recall that cell-like maps cannot raise cohomological dimension (see Remark 3.1.11) and that the cohomological dimension of any cube coincides with its Lebesgue dimension: Therefore, g cannot be cell-like. It is clear now t h a t it suffices to prove the following claim. C l a i m . A n y non-constant n-soft map of a connected n-dimensional compactum is cell-like. Proof of Claim. Let f" X ---, Y be an n-soft map, where X is connected, d i m X = n and I Y I> 1. Since the fiber f - l ( y ) 6 L C n - I N C n - 1 for each y e Y we see t h a t g k ( f - l ( y ) ) = [-Ik(fkl(y)) = 0 for all k < n. Here g k denotes the singular homology with respect to the group of integers a n d / ~ k the (;'ech-homology. Then it follows that I ~ k ( f - l ( y ) ) = 0 for all k < n. Let us show that [-In(f-~(y)) = 0. If so, using the standard criterion that a finite-dimensional Peano continuum with trivial cohomology has trivial shape, we obtain the desired conclusion. Assume the contrary. Namely, t h a t / ~ n ( f - l ( y ) ) __/=0 for some y E Y. Take a map ~" f - l ( y ) ~ g ( Z , n ) (the Eilenberg-Maclane complex) which is not

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

155

homotopic to a constant map. Consider a point x E Y different from y. W i t h o u t loss of generality we may assume t h a t I f - l ( x ) I 1 (otherwise shrink the fiber f - l ( x ) into a point). Since dimX - n, there is an extension ~: Z ~ K(Z,n). Next dengte by Y0 the set of all those points z E Y for which the restriction ~ / f - l ( z ) is homotopic to a constant map. Obviously, Yo is not empty. Indeed, x E Yo. Let us show t h a t Y0 is an open set in Y. Take a point z E Yo, and let h: f - l ( z ) x [0,1] ~ g ( Z , n ) be a h o m o t o p y connecting ~ / f - l ( z ) with a constant map to some point c E K ( Z , n), i.e.

h / ( f - l ( z ) x {0}) = Cp/f-l(z) and h / ( f - l ( z ) x {1}) = c. Consider the closed subset A -- (X x {0, 1} U ( f - l ( z ) x [0, 1]) of the product X x [0, 1] and define the m a p hi" A ~ K ( Z , n) by letting

hl(X

X

{ 1 } ) = c, h l / ( X x { 0 } ) = ~ a n d

hl/(f-l(z)

x

[0, 1 ] ) = h.

Since the CW-complex K (Z, n) is an absolute extensor with respect to the class of compact spaces, we can extend hi to a map h" U ---, K(Z, n), where U is a neighborhood of A in the p r o d u c t X x [0, 1]. T h e n U contains an open set of the form G x [0, 1], where G is a neighborhood of the point z in Y. Clearly G is contained in Y0 (see the definition of Y0). Since z was an arbitrarily chosen point of Y0, we conclude t h a t Y0 is open in Y. Next observe t h a t the complement Y - Y0 is also a n o n - e m p t y set, since it contains y. Let us now show t h a t this complement is also open in Y. Assume the contrary. T h e n there exists a sequence {zk} of points of Y0 such t h a t z = limzk E Y - ]I0. We construct a sequence {gk" f - l ( z ) --+ f--l(zk)} of maps, which converges to the identity map idy-~(z) in the space C ( f - l ( z ) , X ) . Since the space F -- f - l ( z ) x ({zk" k e N } U { z } ) is at most n-dimensional, n-softness of the m a p f guarantees the existence of a map r F --. X such t h a t

r

x

{z}) = idy-l(z)

and r

x

{zk}) C f - l ( z k )

for each k E g .

Let gk -- e l ( f - l ( z ) X {Zk}). T h u s we have a sequence {~gk}of maps each of which is homotopic to a constant map and which converges to the m a p ~ / f - 1 (z). This contradicts the fact t h a t sufficiently close maps into A N R-space are homotopic. Consequently, Y - Y0 is open in Y. Connectedness of Y shows t h a t this is impossible F-1

Historical and bibliographical notes 4.2. T h e o r e m 4.2.1 and Proposition 4.2.2 (a weaker version of it - w i t h o u t stating an n-invertibility) were proved in [33]. T h e construction of n-invertible, ( n - 1)-soft maps presented in this Section, as well as T h e o r e m s 4.2.23 and 4.2.24 in the compact case, are taken from [127](see [90] for the non-compact case). This result completes a circle of works of various

156

4. MENGER MANIFOLDS

authors concerning the existence of dimension raising maps. Historically, the first example of an open map (of the one-dimensional compactum onto the twodimensional "Pontryagin surface") of this sort was constructed by Kolmogorov in [196] in 1937. Further examples, with some additional properties, have been constructed in [11], [12], [15], [319], [320], [189], [190], [191], [200], [211], [245], [248], [304] etc. Theorems 4.2.22 and 4.2.25 are taken from [129] and [125]. Theorem 4.2.22 extends an earlier result from [71].

4.3. n-soft m a p p i n g s of P o l i s h spaces~ r a i s i n g d i m e n s i o n It has already been remarked in Section 4.2 that the maps fn and gn, constructed in that section, cannot be made n-soft. Also, both of them fail to satisfy the property of preservation of Z-sets in the inverse direction. In this section we construct a map with the last property which is "almost" n-soft. We begin with the following technical statement. LEMMA 4.3.1. If f" X ---. Y is an n-soft map, then the inverse image of each Zn-set in Y is a Zn-set in X . For each simplex a, denote the first and second barycentric subdivisions of a by fla and ~2a respectively. Ma denotes the closed star of the barycenter va of a in the triangulation ~2a" Ma = St(Va,~2a). We put No = Int(Ma). Finally let ra "(I a I - { v a } ) ~ ] Oa ]denote the canonical deformation retraction. LEMMA 4.3.2. For each n > 0 and each countable locally-finite simplicial complex K , there exist a countable locally finite simplicial complex B ~ and proper simplicial maps f~" IBm( I--*1 g I and g~" IBm( I---*1 g ( n + l ) I satisfying the following conditions: (i) f ~ is an (n + 1)-invertible u y n - m a p . (ii) If a is a simplex of K, then g ~ ( ( f ~ : ) - l ( ] a ])) C_I a (n+l) ]. (iii) f ~ c / ( f ~ ) - l ( [ K (n+l) I ) = g ~ / ( f ~ : ) - l ( ] K(n+l)I). (iv) There exists a subspace Ang of ] B ~ l such that the restriction f ~ / A ~ " A ~ --*1K[ is an (n + 1)-soft map and the complement l B ~ l - A ~ is a a Z-set in ] B ~ ]. (v) If Z is a Zn+l-set in I g I, then ( f ~ c ) - l ( z ) is a Zn+l-set in ] B ~ ]. PROOF. Let us consider an arbitrary simplex a and define two compact-valued retractions ~ , r l a I---*1 a(n+l) I. Definitions are given by induction on the (n + 1 +/)-dimensional skeleta of a. If x el a (n+l) I, then we put ~ ( x ) = x = r If T is an (n + 2)-dimensional face of a, then

n(x) __ { 107"1, ~~

rr(x),

if x e Nr, i f x e ITI--NT

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

157

and

[OT[, r

--

r~-(x),

i f x 9 M~, if x 9 t~-1- M~

If T is an (n + 3)-dimensional face of a, then

~(~)

=

f I(0~)(-+1) I, t ~2(~(~)),

if x 9 Nr, if x 9 [ v [ - - N r

{ I(0T)(~+I)[, Can(rr(x)),

if x 9 Mr, if x 9 I~1- Mr.

and Can(x) =

Continuing this process, we obtain the desired retractions of [a] onto [a(n+l)[. Finally, if K is an arbitrary locally-finite simplicial complex, then the retractions ~ : , ~b~-IK} ~ ]K(n+l) 1 are defined as the unions of ~ and ~pa n, a 9 g . Now denote by B ~ the standard triangulation (induced by K) of the polyhedron {(x, y) 9 [g[ x ]g(n+l)[ 9 r ~ y}. We put also f~: = 7rl/[B~;[ and g~: = ~2/[B~I, where lr1" IKI • [K(n+l)l--* IKI and 7r2" IKI • IK("+~)I--' IK(n+l)[ denote the natural projections. Straightforward verification shows that: (a) the compact-valued retraction ~ : " [K[ ~ [K("+I)[ is lower semi-continuous. (b) the compact-valued retraction r IKI ---, IK(n+I)I is upper semi-continuous. (c) ~ - ( x ) C_ r for each point x 9 IKI. (d) the collection {r : x 9 IKI} is connected and uniformly locally connected in all dimensions less than n + 1 (we consider the standard metric

on

IKI).

Therefore, properties (i)-(iii) of the lemma are satisfied. Let us verify condition (iv). First of all, consider the subspace A~r = {(x,y) e ] K I • tK(n+l)l" y 9 qo~:(x)}

of IB~I and note that the complement I B ~ I - A~: is an Fa-subset of IB~I. By conditions (a), (c) and Theorem 2.1.15, the restriction f ~ / A ~ is an (n + 1)-soft map. Let us show that the complement IB~:I- A~. is a aZ-set in IB~I. Clearly it is sufficient to show that the last fact is true for the (n + 2)-dimensional simplex a. It follows from the construction that IB~:I- A~c = OMo • Io(-+a)l- T, where Z = {(~, y) e Iol • Io(~+1)1" ~ ( ~ ) = y}. Since OMa is a Z-set in Ma, we can conclude that OMo x [a(n+l)[ is a Z-set in M a x [a(n+l)[. Consequently, [B'~[-A'~ is a aZ-set in M a x [a(n+l)[. Consider now an open subspace U = [ B ~ [ - T of the polyhedron [B~[ Evidently, [ B ~ [ - A n C U C Mo x [a(n+l)[ Then the complement tBant- Aan is a aZ-set in U. Finally, for the same reason, we can conclude that [ B ~ ] - A~ is a aZ-set in [Ban[.

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4. MENGER MANIFOLDS

The last condition is an easy consequence of condition (iv). Indeed, let Z be a Zn+~-set in [g[. Then, by (iv) and Lemma 4.3.1, the set

C 1 -=

{h e C ( I n + I , A ~ ) " ira(h) n ((S~7)-l(z) n A~:) -- 0}

is a dense G~-subset of C ( I n + I , A ~ ) . At the same time, by (iv), the set

6 2 - - {h e C ( I nq-1, [ B~z [)" ira(h) C Ang}

is also a dense G6-subset of C ( I n+l, IB~[). A Baire category argument finishes the proof, fl

THEOREM 4.3.3. Let n >_ 0 and let X be the limit space of an inverse sequence ~qx ---- {[ Xi [, p~+l} all spaces of which are locally finite polyhedra and all bonding maps of which are proper, simplicial and (n-b 1)-soft. Then there exists an (n-b 1)invertible proper UVn-surjection f ~ " M x ---* X of some #n+l-manifold M x onto X satisfying the following conditions: (i) For each (n -b 1)-dimensional locally compact space Z, closed subset Zo of Z, open cover Lt of M x , and proper map h" Z ~ M x , such that h/Zo is a fibered Z-embedding (with respect to f ~ ) , there exists a fibered Z-embedding (with respect to f ~ ) g" Z ---. M x which is U-close to h and such that f ~ g - f ~ h and g / Z o - - h/Zo. (ii) There exists a subspace A x of M x such that the restriction f ~ / A z " A x ---* X is an ( n + 1)-soft map and the complement M x - A x is a a Z-set in M x . (iii) If Z is a Zn+l-set in X , then ( f ~ ) - l ( Z ) is a Z-set in M x .

PROOF. For simplicity we consider only the compact case. The general case can be handled similarly. A standard Baire category argument reduces the proof to the case when Z0 - 0. After making these assumptions, we proceed as follows. We construct another polyhedral inverse sequences SM -- {[Ki[ , qi _i+1~~, all bonding maps of which are simplicial UVn-surjections. Further, we shall construct a family of ( n + 1)-invertible simplicial UYn-surjections fi: [Ki[ ---* [Xi[ which forms a strictly commutative (n+l)-invertible UYn-morphism {f~: [K~] [Xi[}: SM ~ 8 x . This means that if we fix an index i and consider the naturally arising diagram

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

159

IKi+ll

i+1

r~

iKil

p~+l

........

.fi

~ [Xi[

~ , i + l A .e then the characteristic map li = ~i ~ J i + l of the diagram is an ( n + 1)-invertible UVn-surjection (here ITil denotes the fibered product of IKi{ and IXi+ll with _i+1 respect to maps fi and ~'i ; si and ri denote the corresponding projections of this fibered product). Then we shall obtain the space M x as the limit of t.~M and the map f ~ as the limit map of the morphism {f i}. Let K1 = Bnxl and f l = f ~ l " IKll --~ IXll (see L e m m a 4.3.2). We can suppose, of course, that mesh(Kx) < 1 (if not, then we consider a sufficiently fine subdivision of K1 and denote it again by K1). Suppose now that we have already 1 and simplicial constructed finite simplicial complexes Km with m e s h ( K m ) < ~-~ n UV -surjections fm" [Kml ~ IXml and qm-lm . IKml __~ IKm_ll, m _< i, in such a way that the characteristic maps of all the naturally arising rectangular diagrams are (n + 1)-invertible UVn-surjections. In order to perform the inductive step, consider the fibered product ITil of IKil and IX~+ll with respect to the maps f~ and p~+l and denote by ri" ITil--* Igil and si" ITi[--~ IXi+ll the corresponding projections. Now consider the map f ~ , ' l B L,I n ~ ILil where Li denotes the natural triangulation of the polyhedron ITil x i2n+3 (we again use L e m m a 4.3.2). 1 Let K i + l -- BnL~, assuming at the same time t h a t mesh(K{+l) < 2~--Tf" Also _i+l define f~+l = sili and qi = ril +i, where li = ~lf~, and ~1" ITil x i2n+3 __~ iTil denotes the projection onto the first coordinate. One can easily verify that all our requirements are satisfied and consequently the inductive step is complete. As already remarked, we let M x = limSM and f ~ = lim{fi}. Since all mentioned rectangular diagrams strictly commute (because, by the construction, their characteristic m a p s - li's - are surjective) and all fi's are UVn-surjections, we can conclude that their limit map f } " M x --~ X is also a VVn-surjection. The standard argument (see C h a p t e r 6) shows that (n + 1)invertibility of all fi's and a l l / i ' s implies (n + 1)-invertibility of f ~ . Let us now verify condition (i). Fix an index m and an open cover L/m of IKml such that qml(Llm) refines b/(recall that qi: M x --~ Igil and pi: X ~ IX~I denote the limit projections of the spectra SM and S x respectively). We are going to

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4. MENGER MANIFOLDS

construct maps gi" Z ~ [Ki[ in such a way t h a t the following conditions are satisfied: (a) If i > m, then gi is an embedding. (b) q i~+1g i + l - - g i .

(c) f~g~ = p ~ / ] h . (d) If i > m and c~i" i n + l __~ IK~I is an arbitrary map, then there exists a m a p 13i . I n+l ~ IKil such t h a t f i ~ __. f i a i , q i - l ~ i ~ q ii_ l a i and im(Z~) n i~(g~) = O. Let gi = qih for each i _< m. We now indicate how the map gi+l can be constructed. First of all fix an e m b e d d i n g u" Z ~ I2n+3 (we use the inequality d i m Z _< n + 1). Now consider the diagonal p r o d u c t g m A p m + l f ~ h , which maps Z into ITm] 9 T h e n the diagonal product (gin • P m + l f xnh ) A u will be an embedding of Z into the product ILml = ITml • I2n+3. By L e m m a 4.3.2, the map f~,," [Kin+l[ ~ [Lml is (n + 1)-invertible. Consequently, there exists an embedding gm+l" Z ---* I g m + l l such t h a t f~,.,,gm+l = ( g m A P m + l f ~ h ) A u . A simple verification shows t h a t conditions (b) and (c) are satisfied. Now consider any m a p a m + l " i n + l ~ IKm+ll" Since u is an e m b e d d i n g and dim Z _ n + 1, there is a point a E i2n+3 such t h a t a qf ira(u). Let the same letter a denote the constant map which sends the whole cube I n+l to the point a. Consider the m a p (lmo~m+l/ka) . I n+l ----* ILml and observe t h a t the image of I n+l under this m a p does not intersect the image of Z under the composition f L m g m + l . Consider now any lifting ~m+l" I n+l ~ IKm+l[ of the product (lmoLm+lAa). Again, straightforward verification shows t h a t condition (d) is also satisfied. Therefore, continuing this process we obtain the maps g~ for each i. By (b), the diagonal p r o d u c t g of all g~'s maps Z into M x . By (a), g is an embedding. By (c), the desired equality f ~ g - f ~ h also holds. T h e choice of an index m and the equalities qmg = gm "-- qmh (which are true by our construction) show t h a t g and h are/g-close. Finally, let us show t h a t ira(g) is a fibered Z-set with respect to the map f ~ . Fix an open cover l; of M x and any map c~" I n+l --+ M x . Clearly we can assume t h a t there exist an index j _> m and an open cover l)j of ]Kj[ such t h a t q~-l(1)j) refines "g. As above we shall inductively construct maps ~i" I n+l ---* [Ki[ in such a way t h a t the following conditions are satisfied" - i + l ~ i + 1 = ~i. (e) qi (f) f ij3i = pif~ca.

(g) im ( ~ + ~) n im (g~+ ~) = r (h) Zj = q ~ . We let ~i ---- qia for each i < j (consequently, the last condition is automatically satisfied). Let us construct the m a p ~3j+1. By (d) (assuming t h a t i = j + 1), we obtain a map/3j+1" I n+l -~[ g y + l [ such t h a t i m ( Z j + l ) ~ i m ( g j + l ) = 0 (i.e. j+l condition (g) is satisfied), qj j3j+l = ~j and fj+lJ3j+l = p j + l f ~ a . For i > j + 2 , we can construct maps ~i" I n+l ~ [ Ki [ in a similar way to the construction of /~y+l. It only remains to r e m a r k t h a t if ~ is the diagonal p r o d u c t of all ~i's, then one can easily verify t h a t ira(Z) ~ ira(g) = O, f ~ Z = f ~ a and t h a t fl and a are

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

161

])-close. This finishes the verification of condition (i). Let us now show that M x is an #n+l-manifold. Since mesh(Ki) ~ 0 we can conclude (conditions (ii) and (iii) of Lemma 4.3.2) that M x admits small maps into (n + 1)-dimensional polyhedra. Consequently, d i m M x _< n + 1. The inverse inequality is obvious, because, by the above verified condition (i), M x contains a copy of every (n + 1)-dimensional compactum. Since M x is, by the construction, the limit space of the polyhedral inverse sequence SM all bonding maps of which are UVn-surjections we conclude that M x is an LCn-compactum. Again, by (i), it follows immediately that M x satisfies the disjoint (n + 1)-disk property and hence, by Theorem 4.1.19, M x is an #n+l-manifold. Next, we verify condition (ii). By Lemma 4.3.2, there exists a subspace A1 of [KI[ such that the restriction f l / n l : A1 ~ [XI[ is an (n + 1)-soft map and the complement [ K I [ - A1 is a aZn+l-set of ]KI[. It follows from our construction that the fibered product R1 of A1 and [ X2 [ w i t h respect to the maps f l / A 1 and p2 is a subspace of [TI[. Clearly the natural projections of R1 onto A1 and IX2] coincide with the restrictions rl/R1 and Sl/R1 respectively. Since f l / A 1 is (n + 1)-soft, we conclude (see Lemma 6.2.5) that sl/R1 is also (n + 1)-soft. Consider the set R1 x i2n+3 and denote by A2 its inverse image under the (n + 1)-soft map f~l (see Lemma 4.3.2). Then the restriction f 2 / A 2 : A 2 ~ IX2[ is (n + 1)-soft. Since the bonding map p2 is (n + 1)-soft, using again Lemma 6.2.5, we see that the map rl is also ( n + l ) - s o f t . Consequently, by Lemma 4.3.1, the set ]TI[-R1 is a aZn+l-set in ]T1]. In this situation one can easily observe that, again by Lemma 4.3.1, the complement [ K 2 [ - A2 is a a Z n + l - s e t in [K2[. Continuing i + l / A i+ 1 ) and an in such a manner we obtain an inverse sequence SA = {Ai, ~i (n + 1)-soft morphism (in the sense of Chapter 6) morphism { f i / A i } of Sn to S x . Clearly, the limit space A x of the spectrum $A is a subspace of M x and the restriction f ~ / A x " A x ---* X , which coincides with the limit map of the morphism {f~/Ai}, is an (n + 1)-soft map (here we use L e m m a 6.2.7). It only remains to remark that the complement M x - A x is a aZ-set in M x . Condition (iii) is a direct consequence of condition (ii) and Lemma 4.3.1 (compare with the proof pf L e m m a 4.3.2). The proof is complete. [::] We also need the following statement. PROPOSITION 4.3.4. Each #n+lmanifold M can be represented as the limit space of an inverse sequence SM -- {Mi ,Pi i+1 } consisting of locally compact poly-

h~d~a a~d p~op~ ~imptic~al (n + 1)-~~t~bt~, n-~oft a~d polyh~d~aUy (,~ + 1)-~oft bonding maps. PROOF. By Proposition 4.1.10, there exists a proper n-homotopy equivalence a : ] K ]---~ M, where g is an at most (n + 1)-dimensional countable locally finite simplex. Let K0 = K. Suppose that the countable locally finite simplicial simplexes K~ and proper simplicial maps Pi-1 ~ 9 ]K~[ ~ [K~-I[, satisfying conditions

162

4. MENGER MANIFOLDS

(i) - (iii) from L e m m a 4.3.2, have already been c o n s t r u c t e d for each i _< m. We m a y also assume t h a t the m a p Pi-1 i 9 IKil - . IKi_ll has the following property: 9 For any m a p a" I n+l --, IKi_ll there exist two maps ill, f~2" I n+l --~ IKil such t h a t P~-lflj i ---- a, j -- 1,2, and im(f~l) N im(fl2) -- 0. In order to construct these objects for the next step, we apply L e m m a 4.3.2 to a sufficiently fine triangulation of the p o l y h e d r o n IKml x [0, 1]. In this way we get the next simplicial complex g m + l ---- B gmx[0,1] n m+l -- 7rlfg,,~ n x [0,1], 9 We let pm where ~rl"lgml x [0, 1] ~ Igml is the projection. As in the proof of Theorem 4.3.3, we can verify t h a t the limit space M ~ of the inverse sequence 8M -{Mi, Pi i + 1 } is a #n+l-manifold. Clearly, the limit projection p0" M ' --, IKI, being a p r o p e r U V n - m a p between at most (n + 1)-dimensional locally finite L C nspaces, is a proper n - h o m o t o p y equivalence. Therefore M and M ~ are properly n - h o m o t o p y equivalent. T h e o r e m 4.1.21 finishes the proof. V-1 Applying T h e o r e m 4.3.3 and P r o p o s i t i o n 4.3.4 we get the following s t a t e m e n t . THEOREM 4.3.5. Let n > 0 and X E ( # k . k >_ n + 2} U {I ~ }. Then there exists an (n + 1)-invertible UYn-surjection f ~ " #n+l __~ X , satisfying conditions ( i ) - (iii) of Theorem 4.3.3. We conclude this section with the following s t a t e m e n t . THEOREM 4.3.6. Let n ~_ O. For each locally compact polyhedron K , there exists a proper (n + 1)-invertible UYn-surjection h~" M ~ +1 ---. g of some #n+l_ manifold M ~ +1 onto K satisfying the following conditions: (i) xf L i~ ~ ~lo~d ~bpoly~d~o~ of K , t ~ it~ i ~ ~ im~g~ (h~)-I(L) is a #n+l-manifold. (ii) If L is a closed subpolyhedron of K and Z is a Z-set in L, then the

i~,~

im~g~ (h~)-l(z)

i~ ~ z - ~ t

i~

(h~)-~(L).

PROOF. R e p e a t the proof of T h e o r e m 4.3.3, first observing that if L is a s u b c o m p l e x of a countable locally finite simplicial complex K, then the simplicial complex B~ from L e m m a 4.3.2 is a s u b c o m p l e x of the complex B ~ and the map f ~ coincides with the restriction f ~ / I B ~ ]. []

Historical and bibliographical notes 4.3. L e m m a 4.3.1 was proved in [279]. All o t h e r results of this Section were o b t a i n e d by the a u t h o r [79], [84], [88], [97], [98]. The non-separable case was considered in [108].

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

163

4.4. Further properties of M e n g e r manifolds Using the existence of the dimension raising maps constructed in Sections 4.2 and 4.3, we now discuss other major ingredients of Menger manifold theory. 4.4.1. n - h o m o t o p y kernel and Open E m b e d d i n g T h e o r e m . The open embedding theorem for I ~ -manifolds states (see C h a p t e r 2) that for each I ~ manifold X, the product X x [0, 1) can be embedded into I ~ as an open subspace. Observe that identifying X with X x [0, 1] (stability of I"~-manifolds), the product X x [0, 1) may be viewed as the complement of the image of an appropriately chosen Z-embedding of X into itself. Using this remark as a guide, we introduce the following notion. Consider a #n+l-manifold M and two Z-embeddings f , g : M --~ M each of which is properly n-homotopic to the identity map idM. Then the homeomorphism g f - 1 . f ( M ) --. g(M) is properly n-homotopic to idf(M) and, consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending g f - 1 . Clearly the restriction h / ( M - f ( M ) ) is a homeomorphism between the complements M - f ( M ) and M - g(M). This shows that the following definition does not depend on the choice of a Z-embedding.

DEFINITION 4.4.1. The n-homotopy kernel K e r n ( M ) of a #n+l-manifold M is defined to be the complement M - f ( M ) , where f : M ~ M is an arbitrary Z-embedding properly n-homotopic to idM.

PROPOSITION 4.4.2. Let M and N be #n+l-manifolds. Then the following conditions are equivalent: (i) N admits a proper UVn-surjection onto the product M x [0, 1). (ii) N is homeomorphic to K e r n ( M ) . PROOF. It suffices to show that K e r n ( M ) also admits a proper UVn-surjecti on onto the product M x [0, 1). Take a proper v v n - s u r j e c t i o n f : M1 : M x [0, 1], where M1 is also a #n+l-manifold (see Theorem 4.2.1). Consider the quotient space M2 of M1 with respect to the partition whose nontrivial elements are fibers f - l ( m , 1) over the Z-set M x {1} of the product M • [0, 1]. Clearly, by Theorem 4.1.19, M2 is a #n+l-manifold. Moreover, if we consider the naturally induced proper UVn-surjection of M1 onto M2, then we conclude, by Theorem 4.1.21, that M1 and M2 are even homeomorphic. Next, it is easy to see that the set g - l ( M • {1}) is a Z-set in M2 and the restriction g / g - l ( M • {1}) is a homeomorphism. By Theorem 4.1.20, the composition 7rMg, where ~rM: M x [0, 1] ~ M is the natural projection, can be arbitrarily closely approximated by homeomorphisms. In particular, there exists a homeomorphism h: M2 ---+ M that is properly n-homotopic to 7rMg. T h e n the map r -- h g - l i , where

164

4. MENGER MANIFOLDS

i: M --* M x {1} is a n a t u r a l h o m e o m o r p h i s m , is a Z - e m b e d d i n g properly nh o m o t o p i c to idM. Indeed, r - h g - l i ~n --p 7rMgg-1 i -- zrMi -- idM. Thus, by Definition 4.4.1, we conclude t h a t the c o m p l e m e n t M - r ( M ) is h o m e o m o r p h i c to K e r n ( M ) . Consequently, the space M 2 - g - l ( M • { 1 } ) - h - l ( M - r ( M ) ) i s also h o m e o m o r p h i c to K e r n ( M ) . It only remains to note t h a t the space M 2 g - l ( M x {1}) a d m i t s a p r o p e r UVn-surjection onto the p r o d u c t M x [0, 1). V1

Now we are ready to prove the open e m b e d d i n g t h e o r e m for Itn+l-manifolds.

THEOREM 4.4.3. The n-homotopy kernel of each #n+l-manifold admits an open embedding into Itn+l.

PROOF. First of all let us show t h a t every Itn+l-manifold M admits a p r o p e r U V n - m a p onto a certain I w -manifold X. For this we take a proper UVn-map ~" M ~ P , where P is a locally c o m p a c t polyhedron (see T h e o r e m 4.2.23). Clearly the p r o d u c t P • I w --- X is a IW-manifold (see C h a p t e r 2). By T h e o r e m 4.2.1, there is a p r o p e r UVn-surjection r M ' ~ M • I w of some #n+l-manifold onto the locally c o m p a c t L C n - s p a c e M x I w . Since the composition r M r M ' M is also a proper U V n - m a p , we conclude, by T h e o r e m 4.1.20, that M ' and M are homeomorphic. It only remains to observe t h a t the required proper UV nsurjection h" M --* X is given by the composition (~o • idlw )r Now we proceed with the direct proof of our theorem. Take a proper UV nsurjection h" M ---+ X, where X is a I w -manifold. By the open e m b e d d i n g t h e o r e m f o r / W - m a n i f o l d s (see T h e o r e m 2.3.25), one m a y suppose that the product Z • [0, 1) lies in I w as an open subspace. Consider a p r o p e r (n + 1)-invertible UVn-map g" M1 ---* M x [0, 1], where M1 is a # n + l - m a n i f o l d (we use T h e o r e m 4.2.1). Since the m a p g is (n q- 1)-invertible and dim M -- n q- 1, we may assume, w i t h o u t loss of generality, t h a t the restriction g / g - l ( M x {1}) is a homeomorphism and A = g - l ( M • {1}) is a Z-set in M1. As above, we conclude, using T h e o r e m 4.1.20, t h a t the # n + l - m a n i f o l d s M and M1 are homeomorphic. Therefore, it only remains to show t h a t the c o m p l e m e n t M1 - A (which is obviously h o m e o m o r p h i c to K e r n ( M ) ) admits an open e m b e d d i n g into #n+l. Consider the m a p f" # n + l ___, i w of T h e o r e m 4.3.5 and let M2 denote the # n + l - m a n i f o l d f - l ( X x [0, 1)), which is open in # n + l . All t h a t remains to be shown is t h a t the # n + l - m a n i f o l d s M2 and M1 - A are homeomorphic. The last fact can be observed in the following way: b o t h # n + l - m a n i f o l d s MI - A and M2 admit proper U Y n - m a p s onto X • [0, 1) (consider the maps (h • id)g and f ) . Therefore, by T h e o r e m 4.1.21, they are homeomorphic. T h e following d i a g r a m helps to reconstruct the complete argument.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS M2

f

M1-A

M1

g

~ M x [0, 1)

g

~M

x

[0,1]

~t n + l

f

hxid

•

h

x [0,1]

•

165

,~ I w

71"

M This finishes the proof.

~X

K]

4.4.2. n - h o m o t o p y C l a s s i f i c a t i o n T h e o r e m . Theorem 4.1.21 completely describes the proper n-homotopy classification of #n+l-manifolds. In particular, n-homotopy equivalent compact #n+l-manifolds are homeomorphic. Obviously, the last fact is incorrect in the non-compact case: compare #n+l and # n + l - { p t } . The main result of this subsection solves the n-homotopy classification problem of arbitrary ttn+lmanifolds (compare with Theorem 2.3.26). PROPOSITION 4.4.4. For each #n+l-manifold M, the spaces K e r n ( M ) and Kern(Kern(M)) are homeomorphic. PROOF. As above, take a proper UVn-surjection g: M --~ X of M onto a I w -manifold X. By Proposition 4.4.2, there exists a proper UVn-surjection f : Kern(M) ~ M x [0, 1). For the same reason, there exists a proper UV nsurjection h: K e r n ( K e r n ( M ) ) --~ Kern(M) x [0, 1). Consequently, we have two proper UVn-surjections:

p = (g x id)f: Kern(M) ~ M x [0, 1) and

q = (p x id)h: K e r n ( K e r n ( M ) ) ~ K e r n ( M ) x [0, 1),

166

4. MENGER MANIFOLDS

where id denotes the identity map of [0, 1). Since X is a Hilbert cube manifold, the product X x [0, 1] is homeomorphic to X (see Chapter 2). Remarking that the spaces [0, 1) x [0, 1) and [0, 1] x [0, 1) are homeomorphic, we have X x [0,1) x [ 0 , 1 ) ~ X x [0,1] x [0,1) ..~ X x [0, 1). Consequently, the #n+l-manifolds K e r n ( M ) and K e r n ( K e r n ( M ) ) admit proper UVn-surjections onto the same I ~ -manifold X x [0, 1). Therefore they are properly n-homotopy equivalent. Theorem 4.1.21 finishes the proof. V1 PROPOSITION 4.4.5. Let M be a #n+l-manifold. I r A is a Z-set in K e r n ( M ) , then the spaces K e r n ( M ) and K e r n ( K e r n ( M ) - A) are homeomorphic. PROOF. As in the proof of Proposition 4.4.4 consider three proper UV nsurjections:

g: M ~ X, f : K e r n ( M ) - - * M x[0, 1 ) a n d p =

(gxid)f: Kern(M)~

Xx[0,1),

where X is a I ~ -manifold and id denotes the identity map of [0, 1). Now we redefine the map p in such a way that the set p(A) will be a Z-set in X x [0, 1). For this, consider any Z-embedding r: A ~ X x [0, 1) properly n-homotopic to the restriction p/A. By Proposition 4.2.2, there exists a proper UVn-surjection q: K e r n ( M ) ---* X x [0,1) such that q/A = r. Moreover, as in the proof of Proposition 4.4.2, we can additionally suppose that A = q-lq(A). Consequently, the restriction

q / ( K e r n ( M ) - A): K e r n ( M ) -

A ---. X x [0, 1 ) - q(A)

is a proper UVn-surjection. As above (compare with the proof of Proposition 4.4.4) this implies that K e r n ( K e r n ( M ) - A) admits a proper UYn-surjection onto the product (X x [0, 1 ) - q(A)) x [0, 1). By the same argument, there exists a proper VYn-surjection of K e r n ( K e r n ( M ) ) onto the product X x [0, 1) x [0, 1). By Proposition 4.4.4, K e r n ( M ) and K e r n ( K e r n ( M ) ) are homeomorphic. Therefore, by Theorem 4.1.21, it suffices to show that the spaces (X • [ 0 , 1 ) - q ( A ) )

x [0,1) and X x [ 0 , 1 ) x [0,1)

are homeomorphic. Indeed, since q(A) is a Z-set in X x [0, 1), we conclude that the I~-manifolds X x [0, 1) and X x [0, 1 ) - q ( A ) are homotopy equivalent. Then, by the homotopy classification theorem for I W-manifolds (Theorem 2.3.26), the products X x [0, 1) x [0, 1) and (X x [0, 1 ) - q(A)) x [0, 1) are homeomorphic. This finishes the proof. E] PROPOSITION 4.4.6. Let a #n+l-manifold M be a Z-set of a #n+l-manifold N , and suppose the inclusion i: M --+ N is an n-homotopy equivalence. Then there exists a Z-set A in N such that the complement N - A is homeomorphic to K e r n ( M ) .

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

167

PROOF9 By Theorem 4.2.23, there exist (n + 1)-invertible proper UVn-sur jections ~: M ---. L and r N ---. K1, where L and K1 are (n + 1)-dimensional locally compact polyhedra. The (n + 1)-invertibility of ~ implies the existence of a proper map s: L ~ M with ~s - idL. Then the composition r L --. K1 is properly homotopic to some proper piecewise linear (PL) map p: L ~ K1. Form the mapping cylinder M(p) -- K of the map p. Recall that this is the space obtained from the disjoint union (L • [0, 1])@ K1, by identifying (/, 1) with p(l) for each l E L. At the same time we identify L with L x {0}. Clearly L x [0, 1) can be considered as an open subspace of K. Since p is a proper PLmap, L • {0} and K1 are subpolyhedra of the polyhedron K. Let c: K ~ K1 be the collapse to the base, i.e. the natural retraction defined by sending (/, t) to p(l). Clearly, c is a proper CE-surjection that is a homotopy equivalence. Now we consider an (n + 1)-invertible proper UYn-surjection f~(" M~(+1 --. K , satisfying the conditions of Theorem 4.3.6. Clearly, the composition cf~" M~(+1 ---* K1 is a proper UVn-surjection, and hence, by Theorem 4.1.21, the #n+l-manifolds M ~ +1 and N are homeomorphic. By Theorem 4.3.6, the inverse image ( f ~ ) - l ( L x {0}) is a #n+l-manifold that, again by Theorem 4.1.21, is homeomorphic to M. One can easily verify, using the assumption and the specifics of the above construction, that the natural inclusion of ( f ~ ) - l ( L x {0}) into M ~ +1 is an n-homotopy equivalence. Moreover, by Theorem 4.3.6, the above inverse image is a Z-set in M ~ +1 (since L • {0} is a Z-set in K). Now redefining the above objects for simplicity, we have the following situation. A proper UVn-surjection f : N ~ K, satisfying the conditions of Theorem 4.3.6, is given, M -- f - l ( L • {0}) is a Z-set in N and the inclusion M ~ N is an n-homotopy equivalence. Clearly, K - K1 -- L x [0, 1), and hence the inverse i m a g e / - I ( L • [0, 1)) admits a proper UYn-surjection onto L • [0, 1). On the other hand, K e r n ( M ) admits a proper vVn-surjection onto M • [0, 1), and hence onto the product L • [0, 1) as well. Thus, by Theorem 4.1.21, the inverse image f - l ( L • [0, 1)) and K e r n ( M ) are homeomorphic. Consequently, to finish the proof it remains to construct an open embedding h: f - l ( L x [0, 1)) ---. N such t h a t the complement A -- g - h ( f - l ( L • [0, 1)) is a Z-set in N. Since N is a #n+l-manifold, by Theorem 4.1.19, there is a countable dense subset {~k: k -- 1 , 2 , . . . } of c ( I n + l , g ) consisting of Z-embeddings. As observed above, f - l ( L • {0}) is a Z-set in N and the inclusion f - l ( L x {0}) ~ N is an n-homotopy equivalence. It easily follows from Proposition 4.2.2 that in this case there exists a retraction rl: N --. f - l ( L x {0}) that is n-homotopic to idN. Consider the restriction 81 = r l / 9

9 f-l(n

x {0}-)U~I(/n+l)

---~ f - l ( n

1

x [0, ~)).

Clearly 81 is a proper map. By Theorem 4.1.19, Sl is properly n-homotopic to a Z-embedding

gl 9 I - I ( L x { 0 } . ) U ~ I

(/n+l

)-"> .f --1 (L x [0, 1 )) Z

168

4. MENGER MANIFOLDS

that coincides with the identity map on f - l ( L x {0}). By Theorem 4.1.15, there exists a homeomorphism GI" N --~ N extending gl. Put hi = G~-1. Then hi is a homeomorphism such t h a t (a) h l / f - l ( L x { 0 } ) - id. (b) (ill(/n+l) C_ h l f - l ( L x [0, 1)). 1 Consider now the polyhedron K - (L x [0, ~)). Since the set L x { 89 is a Z-set in K (L x [0, ~)), 1 we can conclude, by Theorem 4.3.6, that the set h l f - l ( L x { 1}) is a Z-set in a #n+l-manifold N - h l f - l ( L • [0, 89 Moreover, since the inclusion -

1 L x {~}~K-L

1 x [0,~))

is a homotopy equivalence, we conclude that the inclusion

hlf-l(L x { })~

N-hl.f-l(L

x [0,~))

is an n-homotopy equivalence. Again, using the above construction, we see that there is a homeomorphism

h ,2. N - h l f

-1

1

(L x

))---+ N

which is the identity on h l f - l ( L x {89 ~o2(I n+l) C'I ( N -

hlf-l(L

x [0,

-

h 1 f - l ( L x [0, 1

and for which ))) C_ h ~ h l f - l ( L x [~,2-

)).

Extend h~ to a homeomorphism h2 defined on N by defining h2 = id on

h l f - l ( L x [0, 89 Then we have 1

~o1(In+l) U ~o2(I~+1) C_ h 2 h l f - l ( L x [0,2 9 g)). Inductively continuing this process, we construct homeomorphisms hk" N ~ N in such a way that hk+l = id on h k f - l ( L x [O,k. k--~]) and 1

V~l(In+l)IJ...iJ(Pk(In+l ) C_ h k h k _ l . . . h l f - l ( L x [O,k. k + 1) ). Define an open embedding h" f - l ( L x [0, 1)) ~ N by h(x) = limk--.oo h k ' " hi(x) for each x E f - l ( L x [0, 1)). Clearly U{~k ( i n + l )" k = 1 , 2 , . . . } _C h f - l ( L x [0, 1)) and consequently, by the choice of the family {~k" k = 1, 2 , . . . }, the complement [-1

N - h f - l ( L x [0, 1)) is a Z-set in N.

Now we are ready to prove the n-homotopy classification theorem for #n+l_ manifolds. THEOREM 4.4.7. #n+l-manifolds are n-homotopy equivalent if and only if their n-homotopy kernels are homeomorphic.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

169

PROOF. Let M and N be n - h o m o t o p y equivalent tin+l-manifolds. Take maps n n a" M ~ N and/3" N ~ M such t h a t ~ a " ~ i d M and a ~ " ~ i d N . As above, we can find (n + 1)-invertible proper UYn-surjections f " K e r n ( M ) ~ M • [0, 1) and g" K e r n ( N ) ~ N • [0, 1). Let ~" M x [0, 1) ~ N x [0, 1) and r g x [0, 1) --, M • [0, 1) be a proper maps such t h a t ~ is homotopic to a x id and r is homotopic to fl x id, where id denotes the identity map of [0, 1) (compare with [69, L e m m a 21.1]) . Since g is (n + 1)-invertible and dim K e r n ( M ) = n + 1, there exists a proper m a p r" K e r n ( M ) ~ K e r n ( N ) such t h a t gr = ~o.f. Similarly we have a proper map s" K e r n ( N ) ~ K e r n ( M ) such t h a t f s = Cg. In this situation one n

can verify directly t h a t sr "~ idKer,~(g). Moreover, by T h e o r e m 4.1.19, we can additionally suppose t h a t r and s are Z-embeddings. Now consider the Z-set r ( g e r n ( M ) ) in a # n + l - m a n i f o l d K e r n ( N ) . It follows immediately from the above construction t h a t the inclusion map

r(Kern(M)) ~ Kern(N) is an n - h o m o t o p y equivalence. By Proposition 4.4.6, there is a Z-set A in K e r n ( N ) such t h a t K e r n ( N ) - A is h o m e o m o r p h i c to K e r n ( r ( g e r n ( M ) ) ) -K e r n ( K e r n ( M ) ) (recall t h a t r is an embedding). Then, by Propositions 4.4.4 and 4.4.5,

K e r n ( M ) .~ K e r n ( K e r n ( K e r n ( M ) ) ) ..~ K e r n ( K e r n ( N ) as desired. T h e second part of the t h e o r e m is trivial.

A) ..~ K e r n ( N )

V1

4.4.3. n - s h a p e a n d t h e C o m p l e m e n t T h e o r e m . T h e famous Complement T h e o r e m for I • -manifolds [69] states t h a t if X and Y are Z-sets in I ~ , then their complements I ~ - X and I ~ - Y are h o m e o m o r p h i c if and only if the shapes of X and Y coincide, i.e. S h ( Z ) = S h ( Y ) . T h e obvious form of T h e C o m p l e m e n t T h e o r e m fails for #n+l. T h e equality of shapes of two Z-sets X and Y in # n + l is sufficient, but far from necessary, for the c o m p l e m e n t s # n + l - X and # n + l _ y to be homeomorphic. Indeed, it can be easily seen t h a t if the (n + 1)dimensional sphere S n+l is e m b e d d e d into # n + l as a Z-set, t h e n # n + l _ s n + l is h o m e o m o r p h i c to tt n+l - {pt}. At the same time S h ( S n+l) ~ Sh(pt). T h e problem was solved in [87] (see also [90], [92]) where the notion of n - s h a p e was introduced. T h e relation between n - S H A P E and n - H O M O T O P Y categories is of the same n a t u r e as t h a t between the categories of S H A P E and H O M O T O P Y . Roughly, n - S H A P E is a "spectral completion" of n - H O M O T O P Y . T h e main result in this direction is the following. THEOREM 4.4.8. Let X and Y be Z-sets in #n+l. The complements tt n+l - X and # n + l _ y are homeomorphic if and only if n - S h ( X ) -- n - S h ( Y ) .

170

4. MENGER

MANIFOLDS

We would like to mention some corollaries of this theorem and the definition of n-shape itself. COROLLARY 4.4.9. I f S h ( X ) -- S h ( Y ) , then n -

S h ( X ) -- n - S h ( Y ) .

COROLLARY 4.4.10. I f X and Y are at most n-dimensional, then S h ( X ) S h ( Y ) if and only if n - S h ( X ) -- n - S h ( Y ) .

=

COROLLARY 4.4.11. I f Z - s e t s X and Y in #n+l are #n+l-manifolds, then the complements #n+l _ X and #n+l _ y are homeomorphic if and only if the compacta X and Y are homeomorphic. Let us emphasize that the notion of n-equivalence, introduced by Ferry [150] as a generalization of Whitehead's notion of n-type, coincides in several important cases with the notion of n-shape. Relations between these two concepts have been studied in [97].We conclude this section by noting that Theorem 4.4.8 was extended [280] to a larger class of subspaces than Z-sets. These are the so-called weak Z-sets.

4.4.4. M e n d e r m a n i f o l d s w i t h b o u n d a r i e s . The problem of putting a boundary on various types of manifolds were considered in [57] (PL manifolds), [282] (smooth manifolds) and [70] (I ~ - manifolds). It was proved in [70] that if an I W-manifold M satisfies certain minimal necessary homotopy-theoretical conditions (finite type and tameness at oo), then there are two obstructions a ~ ( M ) and Tc~(M) to M having a boundary. The first one is an element of the group li+___m{/C0rl(M - A ) " A C M A is compact}, where ]C0~rl is the projective class group functor. If c o o ( M ) -- O, then the second obstruction can be defined as an element of the first derived limit of the inverse system li.___m{YVhrl( M - A ) " A C M A is compact}, where ~Vh~rl is the Whitehead group functor. It was shown in [70] t h a t the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms o f t h e group l i m { Y ~ h ~ l ( M - A ) " A C M A is compact}. It should be emphasized that the above mentioned obstructions essentially involve the Wall's finiteness obstruction [303]. The natural analog of Wall's obstruction vanishes in the n-homotopy category. This is exactly what was stated in 4.1.12. We will see that this observation significantly simplifies the situation for #n+l-manifolds. First of all we need the following corollary of Proposition 4.1.12. PROPOSITION 4.4.12. I f a # n + l - m a n i f o l d M is n - h o m o t o p y dominated by an at m o s t (n + 1)-dimensional L C n - c o m p a c t u m , then M is n - h o m o t o p y equivalent to a compact #n+l-manifold.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

171

We say t h a t a #'~+l-manifold M admits a boundary if there exists a compact # n + l manifold N such t h a t M = N - Z, where Z is a Z-set in N. In this case we shall say t h a t N is a compactification of M corresponding to the b o u n d a r y Z, and conversely, Z is a b o u n d a r y of M corresponding to the compactification N. We also need the following definition [100]. DEFINITION 4.4.13. A space X is said to be n - t a m e at c~ if for each compactum A C X there exists a larger compactum B C X such that the inclusion X - B ~-~ X - A factors up to n-homotopy through an at most (n + 1)-dimensional finite polyhedron. PROPOSITION 4.4.14. I f a #n+l-manifold M is n-tame at c~, then M is nhomotopy equivalent to a compact #n+l-manifold. PROOF. Take a proper U V n - r e t r a c t i o n r: M ~ P of the given # n + l - m a n i f o l d M onto some (n + 1)-dimensional locally compact polyhedron P. It follows from elementary properties of proper U V n - m a p s t h a t P is n - t a m e at c~ as well. Using Proposition 4.1.8, one can easily see t h a t P is n - h o m o t o p y d o m i n a t e d by an at most (n + 1)-dimensional compact polyhedron. Proposition 4.4.12 finishes the proof. [::] Let us recall t h a t an I ~ -manifold M lying in a larger I ~ -manifold N is said to be clean if M is closed in N and the topological frontier of M in N is collared b o t h in M and N - I n t M . For obvious dimensional reasons we cannot directly define the corresponding notion for #n+l-manifolds. Nevertheless, the following notion is sufficient for us. DEFINITION 4.4.15. A #n+l-manifold M lying in a #n+l-manifold N is said to be n-clean in N provided that M is closed in N and there exists a closed subspace 5 ( M ) of M such that the following conditions are satisfied:

(i) 6(M) /~ ~ (ii) (iii) (iv) (v)

,~+~-m~ifold.

( N - M ) t2 5 ( M ) is a #n+l-manifold. 5 ( M ) is a Z - s e t in M . 5(M) is a Z - s e t in ( N - M ) U S(M). M - 5 ( M ) is open in g .

Sometimes we say t h a t M is n-clean with respect to 5 ( M ) . Let us indicate the s t a n d a r d situation in which n-clean submanifolds arise naturally. Suppose t h a t L is submanifold of a combinatorial PL-manifold P . Consider a proper UVn-surjection f : N ~ P of a # n + l - m a n i f o l d N from Theorem 4.3.6. Using the properties of f , it is easy to see t h a t M = f - l ( L ) is an n-clean submanifold of N with 5 ( M ) = f - l ( O L ) . Generally speaking, .f is not an open map and consequently 5 ( M ) does not necessarily coincide with the topological frontier of M in N.

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MANIFOLDS

LEMMA 4.4.16. Let N be a #n+l-manifold which is n - t a m e at oo. Suppose that M is a compact and n-clean submanifold of N . Then the #n+l-manifold (N - M ) U 5 ( M ) is n-homotopy equivalent to a compact #n+l-manifold. PROOF. By Proposition 4.4.14, it suffices to show t h a t the #n+l-manifold ( N - M ) U S ( M ) is n - t a m e at 00. Let A be a compact subspace of ( N - M ) U 6 ( M ) . Clearly, K1 = A U M is compact. Since N is n - t a m e at 00, there exists a c o m p a c t u m K2 such t h a t Kz C_ K2 C_ N and the inclusion N - K 2 ~ NK1 factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Let B = ((N - M ) U 6 ( M ) ) gl K2. Clearly B is a c o m p a c t u m and A c B. Note t h a t D

((N-M)U6(M))-B=N-K2

and N - K I C _

((N-M)U6(M))-A.

Consequently, the inclusion ((N-M)U6(M))-B

~

((N-M)U6(M))-A

factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Hence ( g - M ) U 6 ( M ) is n - t a m e at c~. FI LEMMA 4.4.17. A n y #n+l-manifold M can be written as a union M = Ui=IM~ such that all M i ' s are compact and n-clean and M~ C M i + l - 6 ( M i + 1 ) for each i= 1,2,.... oo

PROOF. It suffices to show t h a t for each c o m p a c t u m K C_ M, there exists a compact and n-clean M1 C_ M such t h a t K C_ M1 - 6 ( M 1 ) . As before, take a proper UVn-surjection g" M ~ X, where X is a I ~ -manifold. There is a compact and clean Y _c X such t h a t g ( K ) C_ I n t ( x ( Y ) (see [70]). By the relative triangulation theorem for I~-manifolds (see T h e o r e m 2.3.31), there exists a polyhedron P which can be w r i t t e n as a union of two subpolyhedra P1 and /)2 such t h a t X = P x I ~ , Y = P1 x I ~ , X - I n t x ( Y ) = P2 x I W and B d x ( Y ) = (P1 CI P2) x I W . Note also t h a t the subpolyhedron P1 F1P2 is a Z-set b o t h in P1 and P2. Consider now a proper UVn-surjection f" N ~ P of a #n+l-manifold N onto the polyhedron P satisfying the conditions of T h e o r e m 4.3.6. Consequently, we have two proper u v n - s u r j e c t i o n s f" N ~ P and 7rpg" M --+ P (here ~p" P x I ~ --, P denotes the n a t u r a l projection) of two #n+l-manifolds onto the polyhedron P. Consider an open cover b / - {P - 7 r g g ( K ) , I n t p ( P 1 ) } of P. By T h e o r e m 4.1.20, there exists a h o m e o m o r p h i s m h" M --+ N such t h a t the compositions 7rgg and f h are b/-close. Let M1 -- h - l f - l ( p 1 ) and 6(M1) = h - l f - l ( p 1 rl P2). By the properties of the map f, M1 is compact and n-clean. It only remains to note t h a t K C_ M1 - 6 ( M 1 ) . This finishes the proof. F1 We also need the following s t a t e m e n t , which is a direct consequence of the characterization T h e o r e m 4.1.19.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

173

PROPOSITION 4.4.18. Let a space M be the union of two closed subspaces M1 and M2. If M1, M2 and Mo = M1 A M2 are #n+l-manifolds and Mo is a Z-set both in M1 and M2, then M is a #n+l-manifold. PROOF. It suffices to show t h a t for any m a p f : X ~ M of an at most (n + 1)dimensional c o m p a c t u m X into M , and any open cover U E cov(M), t h e r e exists an e m b e d d i n g g: X ~ M , U-close to f . Let us consider the case when f ( X ) N Mi 7~ 0 for each i = 0, 1, 2. All o t h e r cases are trivial. By P r o p o s i t i o n 4.1.7, t h e r e exists an open cover 12 E coy(M) refining b / s u c h t h a t the following condition is satisfied: (.)~ for any at most (n + 1)-dimensional c o m p a c t u m B, closed s u b s p a c e A of B, and any two P-close m a p s c~1,c~2: A --. M such t h a t c~1 has an extension r B --. M , it follows t h a t C~2 also a d m i t s an extension r B ~ M which is/,/-close to ~1. Let Xi = f - l ( M i ) , i = 0, 1,2. Since M0 is a # n + l - m a n i f o l d , there is a Ze m b e d d i n g go: X0 --~ M0 such t h a t go and f / X o are ]2-close. By (*)n, t h e r e is an extension h: X --. M of G - ) such t h a t h and f are V-close. Since Mo is a Z-set in b o t h M1 and M2 we conclude t h a t go(Xo) is a Z - s e t b o t h in M1 and M2. Consequently, by T h e o r e m 4.1.19, for each i = 1, 2 there is a Z - e m b e d d i n g gi: Xi ~ Mi such t h a t gi/Xo = go and gi is U-close to h / X i . At the same time, w i t h o u t loss of generality we can assume t h a t one of these maps, say gl, has the following p r o p e r t y : g l ( X 1 - X0) N M0 = 0 (we once again use the fact t h a t M0 is a Z-set in M1). T h e n the map g, coinciding with gi on Xi (i = 1,2), is an embedding. It only remains to note t h a t g and .f are N-close. [] LEMMA 4.4.19. If a #n+l-manifold M is n-tame at oo, then we can write M = U~__IMi such that all Mi's are compact and n-clean, Mi C M i + l - ~ ( M i + l ) and the inclusion 5(Mi) r ( M i + l - Mi) U 5(Mi) is n-homotopy equivalence for each i = 1 , 2 , . . . . PROOF. Choose any c o m p a c t and n-clean submanifold A of M . By L e m m a 4.4.17, it suffices to find a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B 5(B) and the inclusion 8(B) ~-~ ( M - B ) U 5(B) is an nh o m o t o p y equivalence. By L e m m a 4.4.16, the # n + l - m a n i f o l d ( M - A)8(A) is n - h o m o t o p y equivalent to some c o m p a c t # n + l - m a n i f o l d X. Fix the corresponding n - h o m o t o p y equivalence r ( M - a ) U S ( A ) + X and its n - h o m o t o p y inverse ~o1" X ~ (M - A ) U 5(A). Obviously there is a m a p ~92" (M - A ) 5 ( A ) ~ X such t h a t r 5(A) --~ X is a Z - e m b e d d i n g a n d r is as close to r as we wish. Similarly, there is a Z - e m b e d d i n g ~o2" X ~ ( M - A ) U 5(A) which is as close to ~Ol as we wish. In particular, we can assume t h a t r and ~o2 are n - h o m o t o p y equivalences. If r and ~o2 were chosen sufficiently close to r and ~ol respectively, then, by T h e o r e m 4.1.18, t h e r e exists a h o m e o m o r phism h" ( M - A ) U 5(A) ~ ( M - A ) U 5(A) which e x t e n d s the h o m e o m o r phism qo2r

5(A) ~

qo2r

and which is sufficiently close to the

174

4. MENGER MANIFOLDS

identity map of (M - A ) U 5(A). In particular, we can assume that h is nhomotopic to id(M_A)oh(A ). T h e n the n - h o m o t o p y equivalence ~p - h -1~2" X --~ (iA ) U 6(A) is a Z - e m b e d d i n g and 6(A) C ~ ( X ) -- Y . Since Y is a compact #n+l-manifold, there exists a U V n - r e t r a c t i o n s" Y ~ K onto a finite (n + 1)-dimensional polyhedron K (see T h e o r e m 4.2.23). Similarly, take a proper V Y n - r e t r a c t i o n r" ( M - A) U 6 ( A ) --, T, where T is a polyhedron. Let i" Y "--. ( M - A ) U 6 ( A ) denote the inclusion map and j" K ~ Y be a section of s (i.e. s j - i d g ) . Note t h a t i is an n - h o m o t o p y equivalence. Let p" K --, T be a P L - m a p homotopic to the composition r i j . Form the mapping cylinder M ( p ) -- P of the map p. For the reader's convenience, we again recall t h a t P is the space obtained from the disjoint union ( g x [0, 1]) @ T, by identifying (k, 1) with p ( k ) , k E g . At the same time we identify g with g x {0}. Since p is a P L - m a p , K x {0} and T are s u b p o l y h e d r a of the polyhedron P. Let c" P --. T be the collapse to the base, i.e. the natural retraction defined by sending (k, t) to p ( k ) for each (k, t) E K • [0, 1]. Obviously, c is a proper cell-like map t h a t is a proper h o m o t o p y equivalence. Now consider a proper U V n - s u r j e c t i o n f" N --. P of some # n + l - m a n i f o l d N onto P, satisfying the conditions of Theorem 4.3.6. T h e compact #n+l-manifolds Y and N1 = f - l ( K • {0}) admit U V n - s u r j e c t i o n s s" Y ~ K x {0} and f / N l " N1 ---* K • {0} onto the same polyhedron. Consequently, by T h e o r e m 4.1.21, there exists a homeomorphism n gl" Y ---* N1 such t h a t f g l ~ - s . Similarly, we have two proper U V n - s u r j e c t i o n s r" ( M - A ) U 5 ( A ) --. T and c f " N --~ T. As above, there is a homeomorphism g2" ( M - A ) U 5 ( A ) --. N such t h a t c f g2 ~ p r. By the construction and the corresponding properties of proper U V n - s u r j e c t i o n s , we have n

n

n

c f gl ~-- cs -- ps ~ r i j s ~_ ri ~ c f g2i.

Since c f is a proper n - h o m o t o p y equivalence, we conclude t h a t gl" Y ~ N and g 2 / Y " Y ~ N are n-homotopic. Consider the h o m e o m o r p h i s m

= ~g2--1 /g2(Y)" g 2 ( Y ) ~ N~. Clearly n

--1

~-- g2g2 / g 2 ( Y ) -- idg2(y ).

By the properties of the map f, N1 is a Z-set in N. Note also that, by our construction, g2(Y) is a Z-set in N as well. By T h e o r e m 4.1.15, we can find a h o m e o m o r p h i s m G: N ~ N extending a. Let H -- G g2. Note that H ( Y ) -G g 2 ( Y ) = a g 2 ( Y ) = g l . Finally, let B -- A U H - I ( f - I ( K

• [0, ~])) and 5(B) -- H - I ( f - I ( K

• { })).

It follows from the properties of the map f and Proposition 4.4.18 t h a t B is a compact and n-clean submanifold of M , A C B - 5(B), and the inclusion 5(B) ~ ( M - B ) U S ( B ) is an n - h o m o t o p y equivalence. To see this, observe t h a t

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS 1 the m a p p, and consequently the inclusion K x {~} ~ n - h o m o t o p y equivalence. [-1

175

P - ( K x [0, ~]), 1 is an

LEMMA 4.4.20. Let a #n+l-manifold M be a Z - s e t in a compact # n + l - m a n i fold N . I f the inclusion i" M --~ N is an n-homotopy equivalence, then there exists a UVn-retraction of N onto M . PROOF. Let j" N --~ M be an n - h o m o t o p y inverse of i. By T h e o r e m 4.1.21, ?1 there is a h o m e o m o r p h i s m h" N ~ M such t h a t h n J. T h e n hi rn~ j i ~_ idM. Consequently, by P r o p o s i t i o n 4.2.2, there is a UV'~-surjection r" N ~ M such t h a t ri = idM. U] T h e following result gives us a c h a r a c t e r i z a t i o n of t i n + l - m a n i f o l d s with b o u n d aries. THEOREM 4.4.21. A #n+l-manifold admits a boundary if and only if it is n-tame at oo. PROOF. Let M be a # n + l - m a n i f o l d which is n - t a m e at cx~. By L e m m a 4.4.19, we can represent M as a union M -- U ~ I M i such t h a t all t h e M i ' s are c o m p a c t a n d n-clean, Mi C M i + l - 5 ( M i + I ) and the inclusion 5(Mi) ~ ( M i + l - Mi) U 5(Mi) is an n - h o m o t o p y equivalence for each i = 1 , 2 , . . . . By L e m m a 4.4.20, for each i there exists a u y n - r e t r a c t i o n fi" ( M i + l - Mi) U 5(Mi) ~ 3(Mi). Let the u v n - r e t r a c t i o n ri" M i + l --* Mi coincide with fi on M i + l - Mi and with t h e identity on Mi. T h e n we have an inverse sequence S = {Mi, ri} consisting of c o m p a c t # n + l - m a n i f o l d s and U V n -retractions. By T h e o r e m 4.1.20, ri is a nearh o m e o m o r p h i s m for each i. By [58], each limit p r o j e c t i o n of the s p e c t r u m S is a n e a r - h o m e o m o r p h i s m as well. Consequently, N = lim S, being h o m e o m o r p h i c to M1, is a c o m p a c t # n + l - m a n i f o l d . Since 5(Mi) is a Z-set in Mi for each i, we conclude t h a t the subset Z = l i m { 5 ( M i + l ) , r i / 5 ( M i + l ) } is a Z - s e t in N. It only remains to note t h a t N - Z is n a t u r a l l y h o m e o m o r p h i c to M . Conversely, suppose t h a t the # n + l - m a n i f o l d M a d m i t s a b o u n d a r y . This m e a n s t h a t there are a c o m p a c t # n + l - m a n i f o l d N and a Z - s e t Z in N such that M = N-Z. Let us show t h a t M is n - t a m e at cx~. Let A be a c o m p a c t subspace of M . As in the proof of L e m m a 4.4.17, t h e r e exists a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B - 5 ( B ) . It suffices to show t h a t ( M - B ) U 5(B) is n - h o m o t o p y equivalent to an at m o s t (n + 1)-dimensional finite p o l y h e d r o n . Indeed, it is easy to see t h a t ( M - B ) U 5 ( B ) is n - h o m o t o p y equivalent to a c o m p a c t # n + l - m a n i f o l d ( N - B ) U S ( B ) . It only remains to a p p l y P r o p o s i t i o n 4.1.10. El Not all # n + l - m a n i f o l d s a d m i t b o u n d a r i e s in the above sense. To see this, consider the 3-dimensional (topological) manifold W ( c o n s t r u c t e d by W h i t e h e a d ) which is defined as the c o m p l e m e n t in S 3 of a c o n t i n u u m W h which, in t u r n , is the intersection of a nested sequence of tori in S 3. T h e manifold W has an

176

4. MENGER MANIFOLDS

infinitely generated f u n d a m e n t a l group at c~. Let n >_ 1, and consider a #n+l_ manifold M and a proper U V n - s u r j e c t i o n f " M ---. W . Since n + 1 >_ 2, we see t h a t f induces an isomorphism of f u n d a m e n t a l groups of ends. T h e n it is easy to see t h a t M is not 1-tame at c~ and, therefore, cannot have a boundary. On the other hand, it can shown t h a t the Freudenthal compactification of any connected #l-manifold contains its end as a Z-set. Consequently, the Freudenthal compactification of any connected #1manifold is homeomorphic to #1. In other words, any connected # l - m a n i f o l d has a boundary. We also mention two related results. PROPOSITION 4.4.22. I f the compactum X is a boundary for a #n+l-manifold M , then the compactum Y is also a boundary for M if and only if dim Y < n + 1 andn-Sh(Y)=n-Sh(X). Two compactifications N and T of the same space M are said to be equivalent if for every c o m p a c t u m A C M there is a h o m e o m o r p h i s m of N onto T fixing A point wise. Of course, if the #n+l-manifolds N and T are compactifications of a #n+l_ manifold M , then the inclusions M ~ N and M --~ T are n-homotopy equivalences (because, N - M and T - M are Z-sets in N and T respectively). Consequently, N and T are h o m e o m o r p h i c as n - h o m o t o p y equivalent compact #n+l-manifolds (Theorem 4.1.21). A stronger result can be obtained. PROPOSITION 4.4.23. Every two # n + l - m a n i f o l d compactifications of a given # n + l - m a n i f o l d are equivalent in the above sense. T h e problem of w h e t h e r a # n + l - m a n i f o l d has a b o u n d a r y which is itself a # n + l - m a n i f o l d was also considered in [104]. DEFINITION 4.4.24. A proper map f " Y ---, X between at most (n + 1)-dimensional locally compact spaces is an n - d o m i n a t i o n near c~ provided that there exists a cofinite subspace X1 of X ( i.e. X1 is closed and X - X1 has compact closure) and a proper map g" X1 ---* Y such that f g is properly n-homotopic to the inclusion map X1 ~-* X . If, in addition, for some cofinite subspace Y1 of Y the composition g f /Y1 is properly n - h o m o t o p i c to the inclusion map ]I1 ~ Y , then we say that f is an equivalence near c<). DEFINITION 4.4.25. A space X is finitely n - d o m i n a t e d near 0o if there exists a finite polyhedron P and an n - d o m i n a t i o n near c~, f" (P • [0, 1)) (n+l) ---, X . If, in addition, f is an equivalence near oc, then we say that X has finite n-type near c<). T h e following s t a t e m e n t corresponds to Proposition 4.1.12 in proper n-homotopy category. PROPOSITION 4.4.26. Each finitely n - d o m i n a t e d near oc, at most (n + 1)dimensional, locally compact L C n - s p a c e has finite n-type near c~.

4.5. HOMEOMORPHISM GROUPS

177

Using Proposition 4.4.26, it is possible to prove [104] the following theorem. THEOREM 4.4.27. A ttn+l-manifold M has a boundary which itself is manifold if and only if M is finitely n - d o m i n a t e d near c<).

a

#n+l_

It follows from Proposition 4.4.22 and Theorem 4.1.21 that if such a boundary exists, then it is uniquely determined.

Historical and bibliographical notes 4.4. Theorem 4.4.27 (as well as Proposition 4.4.26) was proved in [104]. All other results of this Section are due to the author. More precisely, the concept of the n-homotopy kernel of a #n+l-manifold was introduced in [98]. Theorem 4.4.3 and 4.4.7 also appear in [98]. A weaker version of Theorem 4.4.3 was proved earlier in [90]. The complement theorem for Z-sets in Mender compacta (Theorem 4.4.8 and its corollaries) were proved in [87] (also, see the related papers [90] and [92]). The results of Subsection 4.4.4 appear in [100].

4.5. Homeomorphism Groups Let M be a Mender manifold and A u t h ( M ) its group of autohomeomorphisms. The study of A u t h ( M ) is well-developed. It is in their spaces of autohomeomorphisms that we see one of the major differences between I ~ -manifolds on the one hand, and Mender manifolds on the other. Homeomorphism groups of I ~ manifolds are /2-manifolds (see Chapter 2), while homeomorphism groups of #n-manifolds M, as it will be shown below, are totally disconnected. If n -- 0, then A u t h ( M ) is 0-dimensional, and if n >_ 1, then A u t h ( M ) is 1-dimensional. Hence, the only compact Lie groups to act effectively on a #n-manifold are finite groups. On the other hand, all compact 0-dimensional metric groups act effectively on all Mender manifolds. The algebraic properties of homeomorphism groups of Mender manifolds are rather similar to the properties of homeomorphism groups of n- and I "~-manifolds. For an autohomeomorphism H" X ~ X , s u p p H ---- cl{x E X" H(x) -~ x}. I f s u p p H is contained in a s u b s e t A, we say that H is supported on A. 4.5.1. D i m e n s i o n of A u t h ( # n ) . We begin with the following Definition. DEFINITION 4.5.1. A c o n t i n u u m X is locally setwise homogeneous if there exists a basis 34 of connected open subsets of X and a dense subset B C_ X such that for each E E Lt and a, b E B M E there exists h E A u t h ( X ) supported on E such that h(a) = b. Clearly, a locally setwise homogeneous continuum is locally connected. The Sierpinski curve M 2 is locally setwise homogeneous but not homogeneous. The solenoids are homogeneous but not locally setwise homogeneous.

178

4. MENGER MANIFOLDS

Suppose t h a t X is a locally setwise h o m o g e n e o u s c o n t i n u u m and e > 0 is a sufficiently small number. Let U be a n e i g h b o r h o o d of i d x in A u t h ( X ) with d i a m U < e. Let x , y E X with d(x, y) -- e, and let A be a small d i a m e t e r arc from x to y in X. One can define, using local setwise homogeneity, a convergent sequence {h~} in A u t h ( X ) "sliding x towards y along A" and such t h a t h -limh~ E A u t h ( X ) and h E Bd(U). Thus, each e - n e i g h b o r h o o d of i d x has none m p t y b o u n d a r y , and so dim A u t h ( X ) >_ 1. Therefore we have the following statement. THEOREM 4.5.2. Let X

be a locally setwise homogeneous continuum. A u t h ( X ) is at least 1-dimensional.

Then

S t r o n g local h o m o g e n e i t y of # " - m a n i f o l d s (Corollary 4.1.17), implies their local setwise homogeneity. Consequently, we have the following. COROLLARY 4.5.3. If M is a compact #"-manifold, then dim A u t h ( M ) >_ 1. Recall t h a t a space X is almost O-dimensional if it has a basis B of open sets such t h a t for each B E B, X - c l B -- U(Ui 9 i E N } where each Ui is b o t h open and closed. Clearly, each 0-dimensional space is almost 0-dimensional. Also, it is easy to see t h a t every almost 0-dimensional space is totally disconnected. T h e c o m p l e t e Erdhs space ~ = (x E 12" xi is irrational for each i} is a 1-dimensional space which is almost 0-dimensional. PROPOSITION 4.5.4. Each almost O-dimensional space is at most 1-dimensi-

onal. SKETCH OF PROOF. Let X be an almost 0-dimensional space and let B be a c o u n t a b l e basis witnessing this fact. Let ~ = ( f i " i E N } be a collection of continuous functions f i " X ~ (0, 1} such t h a t if B , B ' E 13 with cl(B) Mcl(B') -0, there is a f i E ~ ' w i t h f~(B) = 0 and f i ( B ' ) = 1. Let p be the metric o n X given by p(x, y) = ~-~i 2 - i I f i ( x ) - f i ( y ) ]. Let d be a t o t a l l y bounded metric on X. It suffices to show t h a t the metric dimension # d i m ( X , d ) < 1. Now, d ~ -- d + p is also a t o t a l l y b o u n d e d metric on X. Let Y be the completion of X with respect to d ~. It suffices to show t h a t for each t > 0, there is an open set U of Y containing X such t h a t each c o n t i n u u m in U has d i a m e t e r less t h a n t. Let 34 - {U open in Y 9 diam U < t / 3 and U MX E B}. Let C be any continu u m in W - U/d. I f B , B ' E B with B M C ~ 0 ~ B ' M C , then cl(B) N c l ( B ' ) =fi 0, for otherwise there is fi E 9r with f~(B) = 0 and f i ( B ' ) = 1. But f~(X) = {0, 1} and so, by the definition of d', Y = c l y ( f ~ - l ( O ) ) U c l y ( f ~ - l ( 1 ) ) , where cly(f/--l(0)) and c l y ( f ~ - l ( 1 ) ) are disjoint closed sets. Hence C, being a continuum, c a n n o t meet b o t h c l y ( f / - l ( 0 ) ) and c l y ( f ~ - l ( 1 ) ) which is a contradiction. It follows t h a t d i a m C < t. V1 PROPOSITION 4.5.5. If M is an Mkn-manifold with 0 ~_ n < k < oe, then A u t h ( M ) is almost O-dimensional.

4.5. HOMEOMORPHISM GROUPS

179

SKETCH OF PROOF. Let g E A u t h ( M ) and ~ > 0. Let h E A u t h ( M ) with d(g,h) > e. We shall show that there is an open and closed set U containing h such that d ( g , j ) > e for each j E U. Now, d(g, h) = c + 45 for some 5 > 0, and there is x E M so that d(g, h) = d(g(x), h(x)). Choose an n-sphere S in M such that g(S) c_ N ( g ( x ) , 5 ) and h(S) C_ N ( h ( x ) , 5). Since dim M = n, there is a retraction r : M ~ h(S) such that r ( M - N ( h ( x ) , 2 5 ) ) is constant. Let U -- {f E A u t h ( M ) : r f / S ~ .}. Then U is both open and closed in A u t h ( M ) because close maps into S are homotopic. Also h E U. Let f E N(g, e). T h e n f ( S ) C N ( g ( x ) , c + 5). Hence, r . f ( S ) is a point and f ~ U. D COROLLARY 4.5.6. Let 1 < n < cx~. If M is a compact #n-manifold, then d i m A u t h ( M ) - - 1. PROOF. Apply Corollary 4.5.3 and Propositions 4.5.4 and 4.5.5.

[:]

4.5.2. S i m p l i c i t y . Anderson [16], [17] originated a technique for identifying minimal, non-trivial normal subgroups of A u t h ( X ) for spaces with certain dilation and homogeneity properties. DEFINITION 4.5.7. Let X be a space. A subset A of X is deformable if for every non-empty open set U in X , there is h E A u t h ( X ) with h(A) C_ U. Let V be an open set. A collection ({Bi : i E N } , h) is called a dilation system in U if {B~} is a sequence of disjoint non-empty open sets in U with lim Bi -- {p} for some p E U and h E A u t h ( X ) supported on U such that h(B~+l) ---- B~ for each i. PROPOSITION 4.5.8. Let X be a metrizable space in which each non-empty open set contains a dilation system. Let G be a subgroup of A u t h ( X ) generated by all homeomorphisms which are supported by deformable subsets of X . I f G :fi {e}, then G is the smallest non-trivial normal subgroup of A u t h ( X ) . If X is a finite-dimensional manifold without boundary, then A u t h 0 ( X ) , the subgroup of homeomorphisms isotopic to the identity, is simple (see [151] and [141]). It is also known (see [220] and [323]) that Auth(/2) and A u t h ( I ~ ) are simple. DEFINITION 4.5.9. Let M be a #n+l-manifold. A pair (W, 5 ( W ) ) is an nclean pair if W is n-clean with respect to 5 ( W ) in the sense of Definition 4.4.15 and if, in addition, both W and 5 ( W ) are homeomorphic to #n+l. By Theorem 4.1.15 and the existence of n-clean pairs in #n+l, (compare with [184]), it follows that every open set in #n+l has a dilation system. Also, every proper closed set in #n+l is deformable. Since every element of A u t h ( # n+l) is stable (see Theorem 4.5.13 below) we have the following. THEOREM 4.5.10. A u t h ( # n+l) is simple.

180

4. MENGER MANIFOLDS

4.5.3. Stability of homeomorphisms. An a u t o h o m e o m o r p h i s m of a space X is said to be stable [32] if it can be expressed as the composition of finitely m a n y a u t o h o m e o m o r p h i s m s each of which is the identity on some n o n - e m p t y open subspace of X. It is well-known t h a t all a u t o h o m e o m o r p h i s m s of the Hilbert cube I ~ and the Hilbert space 12 are stable (see, for example, [32]). Every orientation-preserving h o m e o m o r p h i s m of R n is stable [194]. LEMMA 4.5.11. For each #n+l-manifold M , there is a Z-embedding a" M ---+ M which is properly n-homotopic to idM and which satisfies the following condition: (.) If F e A u t h ( M ) and F / a ( M ) = id, then F can be expressed as the composition of two autohomeomorphisms of M each of which is the identity on some open subspace of M . PROOF. Take a proper UVnsurjection g" M ---+ K , where K is an at most (n + 1)-dimensional locally compact p o l y h e d r o n ( T h e o r e m 4.2.23). Consider also a p r o p e r UVn-surjection f" MI" K x [-1, 2], satisfying conditions of T h e o r e m 4.3.6. It can easily be checked t h a t the inverse image M -- f - l ( g x [0,2]) is also a copy of the # n + l - m a n i f o l d M. Moreover, since the composition 7 r l f / M --* g x {0}, where ~rl" g x [0, 2] --. g x {0} denotes the projection, is a proper U V nsurjection, we conclude t h a t there exists a h o m e o m o r p h i s m a" M --, f - l ( K x {0}) such t h a t f a ..~n 7 r l f / M o T h e n a is properly n - h o m o t o p i c to idM. By the mp properties of f , we see t h a t a" M --~ M is a Z-embedding. Consequently, it only remains to show t h a t if F E A u t h ( M ) and F / f - l ( g x {0}) -- id, then F can be expressed as the composition of two a u t o h o m e o m o r p h i s m s of M each of which is the identity on some open subspace of M . Let U - F - l ( f - l ( g x (1,2])). Since F / a ( M ) - id we conclude t h a t U n f - l ( K x {0}) = 0. Consequently, there exists a sufficiently small tl > 0 such t h a t f - l ( K x [O, t l ] ) N ( U U F ( U ) ) = O. Let V : f - l ( K x [0, t0)), where to = l t l l + t1l . Consider the h o m e o m o r p h i s m ~" [0, tl] --* [ - 1 , t l ] defined as v~(x) = t-LL--x--1 1+tl Consider also an open cover ZX=

{ [ K x

--1,--

,K x

--~,tl

]}

of the p r o d u c t K x [-1, tl]. Again, by the properties of f , the inverse images N1 = f - l ( K x [0, tl]) and N2 = f - l ( K x [-1, tl]) are #n+l-manifolds. Clearly these a d m i t proper u v n - s u r j e c t i o n s f l = (id x t p ) f / N l " N1 ---+ K x [-1, tl] and f2 = f / N 2 " K x [-1, tl] onto the polyhedron K x [-1, tl]. T h e n we see t h a t the inverse image f - l ( K x {tl}) is a Z-set b o t h in N1 and N2. Moreover, since, v~(tl) = tl, we conclude t h a t the identity h o m e o m o r p h i s m h o f . f - l ( K x {tl}) satisfies the equality f2h = fl/f-l(K x {tl}). Consequently, there exists a h o m e o m o r p h i s m T" N1 ---+ N2 such t h a t T / f - I ( K x {tl}) = id and the composition f 2 T is U-close to f l . Now

4.5. HOMEOMORPHISM GROUPS

181

consider the homeomorphism H" M ~ M1 such that H / f - I ( K x [0, tl]) = T and H / f - I ( K x [tl,2]) = id. It is easy to verify that H ( V ) N M = 0 and H/(UUF(U))=id. Let f

a(x) = ~F(x),

[ x,

ifx eM ifxEM1-M.

Since F / f - I ( K x {0}) = id, we conclude t h a t G E Auth(M1). Now let F2 -H - 1 G H e A u t h ( M ) . If x e U, then H ( x ) - x. Hence, G H ( x ) - F ( x ) e F ( V ) . Consequently, H - 1 G H ( x ) -- F ( x ) . In other words, F2/U -- F l U . Similarly, if x e V, then H ( x ) e M1 - M . Hence G H ( x ) = H ( x ) and H - 1 G H ( x ) = H - 1 H ( x ) = x. This means that F 2 / V = id. Obviously, F -- F2F1, where F1 = F 2 1 F . It only remains to note t h a t F 1 / U -- id and F 2 / V -- id. 0 LEMMA 4.5.12. Let M be an #n+l-manifold and ~" M ~ M be a Z-embedding properly n-homotopic to idM. I f G E A u t h ( M ) and G / ~ ( M ) -- id, then G can be expressed as the composition of two autohomeomorphisms each of which is the identity on some open subspace of M . PROOF. Take a Z-embedding a satisfying condition (.) of L e m m a 4.5.11. Then the homeomorphism / 3 a - l : a ( M ) ~ ~ ( M ) is properly n-homotopic to the inclusion a ( M ) ~ M. Consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending ~ a -1. Consider the homeomorphism h - l G h E A u t h ( M ) . Evidently, h - l G h / a ( M ) -- id. By the choice of a, there exist F1, F2 E A u t h ( M ) and open subspaces 1/1 and 1/2 of M such that F i / V i = id, i -- 1, 2, and h - l G h -- F2F1. Then G -- G2G1, where Gi -- hFih -1, i -- 1, 2. It only remains to note t h a t G i / h ( V i ) -- id, i -- 1, 2. V] THEOREM 4.5.13. Let M be a #n+l-manifold and F E A u t h ( M ) . properly n-homotopic to idM, then F is stable.

I f F is

PROOF. Let ~" M --~ M be a Z-embedding properly n-homotopic to idM. Let a" M ---+ M be another Z-embedding properly n-homotopic to idM and such that a ( M ) N ( ~ ( M ) U F ( ~ ( M ) ) ) = 0. Consider the homeomorphism S" a ( M ) U f l ( M ) ~ a ( M ) U F ( ~ ( M ) )

which coincides with the identity on a ( M ) and with F on /~(M). Clearly, f is properly n-homotopic to the inclusion a ( M ) U ~ ( M ) ~ M . Therefore there exists an extension F2 E A u t h ( M ) of f . Let F1 = F 2 1 F . By L e m m a 4.5.12, F = F2F1 can be expressed as the composition of four a u t o h o m e o m o r p h i s m s each of which is the identity on some open subspace of M. V1 Since each map of #n+l into itself is (properly) n-homotopic to the identity, we obtain the following statement.

COROLLARY 4.5.14. Every autohomeomorphism of#n+1 is stable.

182

4. MENGER MANIFOLDS

4 . 5 . 4 . G r o u p a c t i o n s o n M e n d e r m a n i f o l d s . T h e well-known HilbertSmith conjecture asks w h e t h e r every c o m p a c t group acting effectively on a manifold is a Lie group. This is equivalent to asking w h e t h e r the group A n of p-adic integers acts effectively on a manifold. This long s t a n d i n g problem is still open. T h e situation is r a t h e r different for Mender manifolds. For instance, it is known [13] t h a t any c o m p a c t metrizable zero-dimensional topological group G acts freely on #1 so t h a t the orbit space # I / G is h o m e o m o r p h i c to tt 1. There are several constructions of group actions on #n-manifolds. THEOREM 4.5.15. Let M be a # " - m a n i f o l d . T h e n : (i) E v e r y c o m p a c t z e r o - d i m e n s i o n a l metrizable group G acts on M so that the orbit space M / G

is h o m e o m o r p h i c to M

(ii) Ap acts freely on M so that d i m M / A p -- n + 1 (iii) Ap acts on M so that dim M / A p = n + 2 T h e r e are u n e x p e c t e d ties between group actions on Mender c o m p a c t a and the H i l b e r t - S m i t h problem mentioned above. Namely, a positive solution to the following conjecture would prove t h a t there is no free Ap-action on a connected (topological) manifold M with dim M / A p < 0r

CONJECTURE 4.5.16. Let m and n be positive integers and G be a zero-dim e n s i o n a l c o m p a c t m e t r i c group. I f #mWn and #n are free G-spaces, then there is no equivariant map #mTn ~ #n.

Historical and bibliographical notes 4.5. T h e o r e m 4.5.2 was proved in [54]. Corollary 4.5.6 a p p e a r s in [244]. Corollary 4.5.14 is due to the author [94]. It was shown later t h a t all a u t o h o m e o m o r p h i s m s of all connected Mender manifolds are stable [266]. Item (i) of T h e o r e m 4.5.15 was proved in [131] (see also [265]). Items (ii) and (iii) a p p e a r in [219] (the last s t a t e m e n t is based on the work [257] ). Conjecture 4.5.16 appears in [1].

4.6. w-soft map

of a onto

E

In this Section, using T h e o r e m 4.2.21, we show t h a t there exists an "almost soft" m a p from a onto E. DEFINITION 4.6.1. A m a p f : X c o u n t a b l e - d i m e n s i o n a l I space B , g: A ---, X

~

Y

is called w - s o f t if f o r each strongly

closed subspace A

and h: B ---, Y with f g -- h / A ,

of B

and any two m a p s

there exists a m a p k ~

B ---, X

such that k / A -- g and f k -- h, i.e. if the following diagram

1Recall that a space is strongly countable-dimensional if it can be represented as the countable union of finite-dimensional closed subspaces.

4.6. w-SOFT MAP OF a ONTO E

X

A

183

~Y

~

,.-B

commutes.

Obviously, e v e r y w-soft m a p is n-soft for each n E w. In p a r t i c u l a r , e v e r y wsoft m a p is s u r j e c t i v e a n d o p e n ( c o m p a r e w i t h t h e p r o o f of i m p l i c a t i o n (ii) ~

(i)

in P r o p o s i t i o n 2.1.34).

LEMMA 4.6.2. Let

T

,-X

i

,~ X I

,~Y

be a c o m m u t a t i v e diagram, consisting of compact spaces, where i is an embedding and d i m T < n. T h e n there exists a c o m m u t a t i v e n - s o f t diagram 2

2This means that the diagonal product k~Ah ~, considered as the map of T ~ onto its image, is n-soft (see page 159).

184

4. M E N G E R M A N I F O L D S

TI

\

h!

~

T

,

~

XI

h

f

Z

g

~Y

where j is an embedding and d i m T ~ <_ n + 1.

PROOF. L e t K be t h e fibered p r o d u c t of Z a n d X I w i t h r e s p e c t to t h e m a p s g" Z --~ Y a n d f~" X ~ ---, Z.

Let ~" K

---, Z a n d r

K

--, X ~ d e n o t e t h e

r e s t r i c t i o n s of t h e n a t u r a l p r o j e c t i o n s Z x X ~ --, Z a n d Z x X ~ ~

X ~ onto K

r e s p e c t i v e l y . B y T h e o r e m 4.2.21, t h e r e exists an n - s o f t m a p m" T ~ ~ K , w h e r e d i m T ~ < n + 1. Since d i m T < n we c a n find a n e m b e d d i n g j " T ~ T I such t h a t

m j = k A i h . It o n l y r e m a i n s to o b s e r v e t h a t t h e d e s i r e d m a p s are h ~ = C m a n d k l = ~m. [:]

PROPOSITION 4.6.3. There exists an w-soft map f" X ~ I ~ , where X is a a-compact strongly countable-dimensional AR-space.

PROOF. W e c o n s i d e r t h e H i l b e r t c u b e I w to be t h e limit s p a c e of an inverse _i+1 }, w h e r e Yi = i i a n d qii+1 " i i + l ~ i i d e n o t e s t h e projecs e q u e n c e , ~ = {Yi, ~i t i o n o n t o t h e c o r r e s p o n d i n g s u b p r o d u c t . W e n o w c o n s t r u c t a n inverse s e q u e n c e ,~ -- { X i , p_i+1 i }, a c o l l e c t i o n of c o m p a c t s u b s p a c e s Zi,z C_ Zi,2 C_ . . . C Zi,i -- X i for e a c h i a n d a m o r p h i s m

{ a i " X i ~ Yi}" 8 ~ 8 '

such t h a t t h e following c o n d i t i o n s are satisfied: (i) d i m Zi,k <_ k -t- 1, w h e n e v e r k < i.

4.6. w-SOFT MAP OF a ONTO IE

185

(ii) T h e d i a g r a m

a~+~/z~+~.k Zi+l,k

~)i,k =

i+l i

k i pi+l/Zi+l,

a~/zi,k

Z~,k is k-soft. (iii) T h e d i a g r a m

o~i+1

Xi+l

~)i "--

~ Yi+l

i+l i

i+l i

Xi

,.

'-- Yi

is soft.

Let X1 = I x I and a l = r l " I x I ~ I. Assume t h a t for each i _< j, the m a p s a i and the spaces Zi,k, k = 1, 2 , . . . , i, have already been defined. Let Zj+I, 1 be ' the fibered p r o d u c t of the spaces Zj,1 and Yj+I with respect to the maps aj/Zj,1 and ~j A + I , and ~o" Zj+I, , 1 --, Zj,1 and r Zj+I, 1 ~ Yj+I be the corresponding projections of the fibered product. By T h e o r e m 4.2.21, there exists a i-soft m a p m " Z j + I , 1 --. Z j'+ I , I ~ where dim Z j + l , 1 = 2. We define the maps ~,j ~.j+l and a j + l by means of their restrictions to the subspaces Zj+l,i.

Let pj j+ l / Z j + I , 1 - - ~ o m

and o z j + l / Z j + l , 1 = Cm. Using L e m m a 4.6.2, we get the sequence of spaces Zj+I,1 c_ Zj+I,2 c _ . . . c Z j + l d with dim Zj+l,k _< k + 1, and maps _j+l -

rk--pj

/Zj+l,k" Zj+l,k ~ Zj,k, ~k = aj+l/Zj+l,k" Zj+l,k ~ Yj+I

such t h a t r k / Z k - 1 = rk-1, ~k/Zk-1 = r

a n d the diagrams T~j,k are k-soft.

186

4. MENGER MANIFOLDS

If k = j , let l be an e m b e d d i n g of t h e (j + 1 ) - d i m e n s i o n a l space Z j + I , j into t h e c u b e i2j+3. L e t

X j + l -- X j x Yj+I • i 2 j + 3

Zj+I,j is e m b e d d e d into Xj+I by t h e m a p l' = ~ X j be t h e r e s t r i c t i o n of t h e p r o j e c t i o n onto t h e first c o o r d i n a t e a n d a j + l " X j + I ~ Yj+I be t h e r e s t r i c t i o n of t h e p r o j e c t i o n onto t h e second c o o r d i n a t e . It is easy to see t h a t c o n d i t i o n s (i)-(iii) are satisfied. For each n a t u r a l n u m b e r k E N , let a n d a s s u m e t h a t t h e space

r j / k ~ j / k l 9 L e t t,j ,.j+l " X j + I

Zk = lim{Zi,k,pi + l i / Z i + l , k , i},

X = U~=IZk

and

f = lim{cri}/X" X ~ lim,9' = I ~ . Obviously, t h e space X is a s t r o n g l y c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t space. N e x t we show t h a t t h e m a p f is w-soft. C o n s i d e r t h e c o m m u t a t i v e d i a g r a m

X

~I

A t

.B

w

w h e r e A is a closed s u b s e t of a s t r o n g l y c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t space B. L e t r -- { r a n d r - {r where r A -+ X i a n d r B -+ Yi are t h e coord i n a t e m a p s . W e also let B -- Uj= l c ~ B j , w h e r e t h e B j are at m o s t j - d i m e n s i o n a l s u b s e t s of B a n d B1 C_ B2 C_ . . . . Let A j -- A A B j . T h e n A - U~=IA j. N o t e t h a t , w i t h o u t loss of generality, we can a s s u m e t h a t r C_ Zj. Now we are going to c o n s t r u c t a m a p (I)" B -+ X as (I) -- {q)i}, w h e r e (I)i" B --+ Xi are c o o r d i n a t e m a p s . We c o n s t r u c t s t h e (Ih's by i n d u c t i o n . L e t (I)l" B - , X I be a m a p such t h a t ~1(I)1 -- r a n d ~ 1 / A 1 -- ~1. Now a s s u m e t h a t for every i < j m a p q)i has a l r e a d y b e e n c o n s t r u c t e d so t h a t r C_ Zi,k, k < i. Using t h e 1-softness of t h e s q u a r e d i a g r a m T)j,1 we choose a m a p (I)j(1) + l " B1 --+ Zj+I,1 such t h a t t h e following c o n d i t i o n s are satisfied:

= Cj/B 9 Pjj+1~(1) =j+l _

m(1) 9 ~j+l~j+l 9 r

j+l/r

=

1.

Cj+I/B1.

(A N Zj+l,1) = Cj+l/r

(A N Zj+l,1).

4.6. w-SOFT MAP OF a ONTO E Using 2-softness of the square d i a g r a m T)j , 2 we choose a m a p (I)(2) j+l" such t h a t 9 p- Jj + l d ~~(j2+)l = (~j/B2.

187 B2 ~

Zj+l,2

d~(2) 9 ~j+l~j+l

-- Cj+I/B2. ~(2) /.,.-1 9 "~j+I/Wj+I (A A Zj+I,2) = C j + l / r -1 (A N Zj+l,2) 9 ~(2) = j + l / B 1 = ~(1) =j+l" C o n t i n u i n g this process we obtain a m a p ~ j + l " B j --, Z j + I j conditions" 9 Pjj + l m~ j(+j )l = ~ j / B j 9 aj+l~j+l -- C j + I / B j . ~(j) -1 9 xj+l/~)j+l(A A ZjTl,j) = Cj+I/~);_~I(A

satisfying t h e

N Zj+l,j).

9 ff~" 1 / B j -1 - - ~ j + l " Finally, using softness of t h e d i a g r a m 7)j we can choose a m a p (I)j+l : B ~ X j + I such t h a t _j+l 9 ~j

(I)j+l = (I)j.

9 aj+l(I)j+l

= r

9 ( ~ j + I / B j - - d~(J) =j+l" 9 Cj+I/A = r It follows easily from the c o n s t r u c t i o n t h a t the m a p (I) = ((I)j} is well defined, r = r and f(I) = r It only remains to observe t h a t t h e w-softness of f g u a r a n t e e s t h a t X is an AR-space. O THEOREM 4.6.4. There exists an w-soft map r

a --+ E.

PROOF. Let f" X ~ I ~ be the m a p from P r o p o s i t i o n 4.6.3 a n d let X ~ = f - l ( E ) . Clearly, X ~ is a strongly c o u n t a b l e - d i m e n s i o n a l a - c o m p a c t AR-space. B y P r o p o s i t i o n 2.5.5, the p r o d u c t X ~ x a is h o m e o m o r p h i c to a. T h e desired m a p is r - fir- a --, E, where ~" X ~ x a ~ X ~ denotes the p r o j e c t i o n onto t h e first coordinate. [l

Historical and bibliographical notes 4.6. T h e o r e m 4.6.4 was proved in [325]. A s o m e w h a t weaker version of it was o b t a i n e d earlier in [229].

CHAPTER 6

G e n e r a l T h e o r y of A b s o l u t e E x t e n s o r s Dimension n and n-soft Mappings

in

6.1. AN E (n )-spaces a n d n-soft m a p p i n g s 6.1.1. D i s c u s s i o n o f t h e c o n c e p t o f A b s o l u t e R e t r a c t . It is well known (and easy to see) t h a t a retract of a Hausdorff space is closed in the space under consideration. It is for this reason that an absolute retract in some class 7) of Hausdorff spaces is defined to be a space X E :P such t h a t for any closed embedding of X into an arbitrary space Y E P, there exists a retraction of Y onto X. This definition of the concept of an absolute retract has an essential deficiency of a logical nature. The fact of the m a t t e r is t h a t retracts are not only closed, but are also C-embedded in the ambient spaces. But, by the Brouwer-Tietze-Urysohn extension theorem, the property of closed subspaces being C-embedded characterizes normal spaces, and therefore the classical definition of the concept of an absolute retract is satisfactory only for various subclasses of the class of normal spaces. However, the indicated deficiency already manifests itself fully in the class of Tychonov spaces. This is confirmed by a result of Hanner [162] to the effect that each absolute retract in the class of all Tychonov spaces is compact. Consequently, under this definition neither the Euclidean space R n, nor the Hilbert space 12, is an absolute retract in the class of Tychonov spaces (although both are such in the class of Polish spaces) and this makes it impossible to use the theory of retracts in studying non-metrizable topological linear spaces and the associated problems of infinite-dimensional topology and functional analysis. Because of this, it becomes necessary to add the requirement of being Cembedded to the classical definition of the concept of an absolute retract. Moreover, it turns out t h a t in a number of cases we can even omit the requirement of being closed and confine ourselves to the requirement of being C-embedded. This is possible, for instance, in the case of spaces for which any C-embedding is automatically closed. W h a t are the spaces characterized by the last property? These are, according to Proposition 1.1.16, precisely the realcompact spaces, 227

228

6. ABSOLUTE EXTENSORS

i.e. the spaces homeomorphic to closed subspaces of arbitrary powers of the real line R (the class of realcompact spaces is sufficiently broad: it contains all the compact Hausdorff spaces and all the Polish spaces; at the same time, it is sufficiently geometric, and in the study of it such classical objects of topology as infinite powers R r and N ~ of a Hilbert space and of a Baire space are brought to the forefront). But what happens outside the class of realcompact spaces? It may seem that a logical inconsistency analogous to that considered above emerges when the requirement of closeness is waived. But this is not so, since each absolute retract (under the altered definition) in the class of Tychonov spaces turns out to be realcompact. A more precise interpretation can be given to the above discussion. Consider the category T Y C H of Tychonov spaces and their continuous maps on the one hand, and the category V E C T of vector spaces their linear maps on the other. The contravariant functor g: T Y C H ~ V E C T assigning to each space X the vector space C ( X ) of all continuous real-valued functions defined on X, and to each continuous map f : X ---, Y the induced linear map C ( f ) : C ( Y ) ~ C ( X ) , enables us to single out the class of morphisms of T Y C H corresponding to the epimorphisms of V E C T . It can be shown that C ( f ) is epimorphic if and only if f is a C-embedding (i.e., an embedding with C-embedded image). In other words, C-embeddings turn out to be "monomorphisms" in T Y C H (we use the contravariance of the functor g). But since the concept of an absolute retract is categorical, we obtain the following definition. DEFINITION 6.1.1. A Tychonov space X is said to be an absolute retract (notation: X E A R ) if for any C-embedding of X into an arbitrary Tychonov space Y , there exists a retraction of Y onto X . DEFINITION 6.1.2. A hood retract (notation: trary Tychonov space Y , Y and a retraction of U

Tychonov space X is said to be an absolute neighborX E A N R ) if for any C-embedding of X into an arbithere exist a functionally open neighborhood U of X in onto X .

6.1.2. D e f i n i t i o n o f A N E ( n ) - s p a c e . The notion of A ( N ) R - s p a c e admits another approach as well. Before we give the corresponding definitions, let us call a functionally open neighborhood U of a subset A in a space X stable if there is a functionally closed subset Z of X such that A C_ Z C U. Observe that if a subset A is C-embedded in X, then every functionally open neighborhood of A is stable (compare with [159]). DEFINITION 6.1.3. A space X is called an absolute (neighborhood) extensor in dimension n (shortly, A ( N ) E ( n ) - s p a c e ) , n - O, 1 , . . . , c~, if]or each at most n-dimensional space Z and each subspace Zo of Z, any map f : Zo ---* X , such that C ( f ) ( C ( X ) ) C C ( Z ) / Z o , can be extended to (some stable functionally open neighborhood of Zo) Z. An A ( g ) E ( o c ) - s p a c e is called an absolute (neighborhood) extensor, or shortly, an A ( N ) E - s p a c e .

6.1. A N E ( n ) - S P A C E S AND n-SOFT MAPPINGS PROPOSITION 6.1.4.

The class o f A ( N ) R - s p a c e s

229

c o i n c i d e s with the class o f

A(N)E-spaces.

PROOF. We prove only the p a r e n t h e t i c a l case. T h e proof of t h e a b s o l u t e version is c o m p l e t e l y analogous. First of all, we note t h a t each A N E - s p a c e is an A N R - s p a c e . This i m m e d i a t e l y follows from t h e c o r r e s p o n d i n g definitions. Sup-

pose now t h a t X is an A N R - s p a c e . We can a s s u m e t h a t X is C - e m b e d d e d in R r for some cardinal r (see Subsection 1.1.2). Take a f u n c t i o n a l l y o p e n neigh-

b o r h o o d V of X in R r a n d a r e t r a c t i o n r" V --~ X . C o n s i d e r also a space Z, its s u b s p a c e Z0, and a m a p f " Z0 --* X such t h a t C ( f ) ( C ( X ) ) C_ C ( Z ) / Z o . Since, as is easily seen, t h e space R r is an A E - s p a c e , we see t h a t t h e r e exists a m a p g" Z --~ R r such t h a t g / Z o = f . Since X is C - e m b e d d e d in R r , the functionally o p e n n e i g h b o r h o o d U = g - l ( V ) of Z0 in Z is stable. It only r e m a i n s to note t h a t t h e m a p r g / U " U --* X is t h e desired extension of f . [-1 PROPOSITION 6.1.5. E a c h A N E ( O ) - s p a c e is an A E ( O ) - s p a c e . PROOF. Let X be an A N E ( O ) - s p a c e a n d f - Z0 --* X be a m a p , defined on a s u b s p a c e Z0 of a zero-dimensional space Z, such t h a t C ( f ) ( C ( X ) ) C_ C ( Z ) / Z o . T h e n f has an e x t e n s i o n g" U --, X , where U is a stable f u n c t i o n a l l y open neighb o r h o o d of Z0 in Z. T h e stability of U g u a r a n t e e s t h a t t h e r e is a functionally closed subset F in Z such t h a t Z0 c_ F C_ U. Since Z is zero-dimensional, t h e r e exists an open and closed subset G in Z such t h a t F C_ G C_ U . O b v i o u s l y t h e m a p g / G " G - , X has an extension h" Z --~ X onto Z. It only r e m a i n s to note t h a t h e x t e n d s t h e m a p f . [::] COROLLARY 6.1.6. E a c h A N E ( n ) - s p a c e

is an A E ( O ) - s p a c e .

PROPOSITION 6.1.7. E a c h A E ( O ) - s p a c e is realcompact. PROOF. E m b e d X into R r as a C - e m b e d d e d s u b s p a c e for some cardinal T. Take a p r o p e r m a p g" N r -+ R r , a n d consider its r e s t r i c t i o n f - g / g - l ( X ) onto t h e inverse image of X . Since X is C - e m b e d d e d in R r , we can conclude that C(f)(C(X)) C_ C ( N r ) / g - l ( X ) . B u t d i m N r = 0. T h e r e f o r e t h e r e exists an e x t e n s i o n h" N r --~ X of t h e m a p f . By t h e p r o p e r n e s s of g, t h e m a n y - v a l u e d map r = hg -1" R r --~ X

is c o m p a c t - v a l u e d and u p p e r semi-continuous. Since r is t h e i d e n t i t y on X , we see t h a t X is closed in R r . P r o p o s i t i o n 1.1.13 finishes t h e proof. ["1 PROPOSITION 6.1.8. E a c h A E ( O ) - s p a c e is p e r f e c t l y a - n o r m a l and has a countable S u s l i n n u m b e r .

PROOF. As in t h e proof of t h e preceding s t a t e m e n t , e m b e d an A E ( 0 ) - s p a c e X into R r as a C - e m b e d d e d subspace. Take a p r o p e r m a p g" N r --, R r and consider a m a p h" N r --, X which e x t e n d s the m a p g / g - l ( X ) .

T h e surjectivness

230

6. ABSOLUTE EXTENSORS

of g implies t h e surjectivness of h. Therefore, since the Suslin n u m b e r of N r is c o u n t a b l e , we conclude t h a t X also has a c o u n t a b l e Suslin n u m b e r . Let us show now t h a t X is perfectly n - n o r m a l . Take an a r b i t r a r y open subset V of X a n d let U - X - c l x V. T h e p r o p e r n e s s of the m a p g g u a r a n t e e s t h a t t h e sets

V' = R r - gh-l(x

- Y ) and U' = R ~ - g h - l ( X

- U)

are o p e n in R r . Observe also t h a t

v I n x = V, uI n x = u and V I N U I - - 0 . As was r e m a r k e d just before P r o p o s i t i o n 1.1.21, t h e space R r is perfectly nn o r m a l . Consequently, t h e set Z - ClRr V ~ is functionally closed in R r . Also, Z N U ~ -- 0. It only r e m a i n s to note t h a t c l x V -

Z n X.

Vi

PROPOSITION 6.1.9. A space X is an A ( N ) E ( n ) - s p a c e if and only iS X is realcompact and the following condition is satisfied: 9 For each at most n-dimensional realcompact space Z, each closed subspace Zo of Z, any map f" Zo --~ X , such that C ( f ) ( C ( X ) ) C C ( Z ) / Z o , admits an extension onto (a stable functionally open neighborhood of Zo

i ~ z ) z. PROOF. By P r o p o s i t i o n 6.1.7, each A N E ( n ) - s p a c e is r e a l c o m p a c t and obviously satisfies t h e above condition. Let us prove t h e converse. Let X be a r e a l c o m p a c t space s a t i s f y i n g t h e above f o r m u l a t e d condition. C o n s i d e r an a r b i t r a r y space Z of dimension at most n, an a r b i t r a r y subspace Zo of Z and a m a p f " Zo --* X such t h a t C ( $ ) ( C ( X ) ) c_ C ( Z ) / Z o . D e n o t e by Ao t h e closure of the set Z0 in the H e w i t t realcompactification v Z of Z. Since Z is C - e m b e d d e d in v Z , we have C ( $ ) ( C ( X ) ) C_ C ( v Z ) / Z o . By P r o p o s i t i o n 1.3.12, t h e m a p f can be e x t e n d e d to a m a p g" A0 --* X so t h a t C ( g ) ( C ( X ) ) C_ C ( v Z ) / A o . Since d i m v Z - d i m Z _< n, our a s s u m p t i o n g u a r a n t e e s t h e existence of a stable functionally open n e i g h b o r h o o d V of the set A0 in v Z a n d a m a p h" V - , X with h/Ao -- g. It only r e m a i n s to observe t h a t U -- V n Z is a stable f u n c t i o n a l l y open n e i g h b o r h o o d of Z0 in Z and the m a p

h / U e x t e n d s t h e originally given m a p f .

D

PROPOSITION 6.1.10. Each Polish space is an AE(O)-space. If n >_ 1, then the class of Polish A N E ( n ) - s p a c e s (respectively, A E ( n ) - s p a c e s ) coincides with the class of Polish L C n - l - s p a c e s (respectively, L C n - I N Cn-l-spaces). PROOF. Let X be a Polish space and Z0 be a closed subset of a zero-dimensional r e a l c o m p a c t space Z. Consider a m a p f : Z0 --~ X satisfying the inclusion C ( f ) ( C ( X ) ) c_ C ( Z ) / Z o . B y T h e o r e m 1.3.10, t h e space Z can be represented as t h e limit space of a factorizing w - s p e c t r u m S = { Z a , p ~ , A } consisting of zerod i m e n s i o n a l Polish spaces. Since Z0 is closed in Z, t h e limit space of the induced

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

231

spectrum

A}

So=

coincides with Z0 (Corollary 1.2.6). By Proposition 1.3.13, there exist an index a EA andamap

f~: clz~ p,~(Xo)

---, x

such that f = f a p a / Z 0 . Since d i m Z a = 0, the map f a has an extension ga onto Za (see Proposition 2.1.13). Then the composition g = gap~ is the desired extension of f onto Z. Consequently, by Proposition 6.1.9, X is an AE(0)-space. The remaining part of our s t a t e m e n t can be proved in a completely similar way (compare with Theorem 2.1.12). E] PROPOSITION 6.1.11. Let X be a compact space. Then X is an A ( N ) E ( n ) space if and only if the following condition is satisfied: 9 For each at most n-dimensional compactum Z and for each closed subspace Zo of Z, any map f : Zo ---* X has an extension to (a neighborhood of Zo in Z ) Z. PROOF. Obviously, if X is an A ( N ) E ( n ) - c o m p a c t u m , then it satisfies the above formulated condition. Let us show t h a t the converse is also true. Suppose t h a t a c o m p a c t u m X satisfies the above condition. We need to show t h a t X is an A ( g ) E ( n ) - s p a c e in the sense of Definition 6.1.3. Consider an arbitrary space Z of dimension at most n, its arbitrary subspace Z0, and a map f : Z0 ---* X such t h a t C ( f ) ( C ( X ) ) C C ( Z ) / Z o . Denote by A0 the closure of Z0 in the Stone-Cech compactification flZ of the space Z. Following the proof of Proposition 1.3.12, we can conclude t h a t the map f can be extended to a map g: A0 ---* X. By our assumption and the equality d i m f l Z - d i m Z , the map g can be extended to a map h: V ~ X, where V is a functionally open neighborhood of the set A0 in flZ. Let U -- V NZ. Clearly, U is a stable functionally open neighborhood of Z0 in Z and the map h / U : U --, X extends the originally given map f. V1 As follows from the last two statements, the above defined notion of A ( g ) E ( n ) space is completely compatible with the known definition of this notion in the classes of Polish and compact spaces. Of course, within Polish spaces (and, in general, in the class of metrizable spaces) these concepts have been exploited for decades, and an extensive literature is devoted to their study. On the other hand, until the middle of the nineteen seventies almost nothing, except the definition itself, was known about non-metrizable A ( N ) E ( n ) - c o m p a c t a . The first serious investigation in this direction was the work of Haydon [164] in which the spectral characterization of non-metrizable A E ( 0 ) - c o m p a c t spaces was obtained. Moreover, it was shown in [164] t h a t the class of A E ( 0 ) - c o m p a c t a coincides with the class of Dugundji compacta (the latter class, in different terms, was defined in [249]). In Section 6.3 we consider this problem more formally.

232

6. ABSOLUTE EXTENSORS

Concluding this Subsection, we remark that the compactness of X is absolutely essential in Proposition 6.1.11. More formally: 9 There exists a Polish space, satisfying the condition of Proposition 6.1.11 (for n = c~), which is not an absolute retract. This fact follows from Corollary 3.2.16. As already mentioned in Subsection 2.1.2, this was observed by van Mill (see [22]).

6.1.3. D e f i n i t i o n of n - s o f t m a p p i n g s . The following definition shows that the notion of an n-soft map is a categorical counterpart of the notion of A E ( n ) space. D E F I N I T I O N 6 . 1 . 1 2 . A m a p f : X --+ Y is s a i d to be n - s o f t , n -- 0, 1 , . . . , c~, if for each at most n-dimensional space Z, its two subspaces Zo and Z1 with Zo C Z1 and any two maps g: Zo ~ X and h: Z1 ~ Y such that C ( g ) ( C ( X ) ) C C ( Z ) / Z o , C ( h ) ( C ( Y ) ) C C ( Z ) / Z 1 and f g = h/Zo, there exists a map k: ZI: X such that f k = h, g = k/Zo and C ( k ) ( C ( X ) ) C_ C ( Z ) / Z 1 . Maps that are c~-soft are called soft The following diagram helps to understand the situation described in the above Definition.

X

,~Y

go t

,~Z1 t

~Z

We are now going to establish some elementary properties of n-soft maps. We begin with the following statement, the proof of which is trivial and hence omitted. LEMMA 6.1.13. Each n-soft map is surjective. LEMMA 6.1.14. A finite composition of n-soft maps is n-soft. PROOF. It suffices to consider the case of two maps. Let f l : X1 ---+ X2 and f 2 : X 2 ---* X3 be n-soft maps. Consider an at most n-dimensional space Z, its subspaces Z0 and Z1 with Z0 c Z1, and two maps g: Z0 ---* X1 and h: Z1 --~ X3 satisfying the conditions

C ( g ) ( C ( X l ) ) C C ( Z ) / Z o , C ( h ) ( C ( X 3 ) ) C C ( Z ) / Z 1 and f 2 f l g = h/gO.

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

233

Straightforward verification shows that C ( f l g ) ( C ( X 2 ) ) c_ C ( Z ) / Z o . Consequently, by the n-softness of the map f2, there exists a map k~: Z1 ---, X2 such that f 2 k ' = h, f i g -

k ' / Z o and C ( k ' ) ( C ( X 2 ) ) C

C(Z)/Zl.

Further, the n-softness of the map f l guarantees the existence of a map k: Z1 X1 satisfying the following conditions: f l k = k' k / Z o = g and C ( k ) ( C ( X l ) ) C C ( Z ) / Z 1 . It only remains to observe that f 2 f l k -- f2 k t = h.

D

LEMMA 6.1.15. Let f = f 2 f l and f l and .f be n-soft maps. an n-soft map.

Then .f2 is also

PROOF. Consider two maps f l : X1 ---* X 2 and f 2 : X 2 ~ X 3 and assume t h a t f l and f - f 2 f l are n-soft. Consider also an at most n-dimensional space Z, its subspaces Z0 and Z1 with Z0 C Z1, and two maps g: Z0 ~ X2 and h: Z1 ~ X3 satisfying the following conditions: C(g)(C(X2)) C C(Z)/Zo,

C(h)(C(X3)) C C(Z)/Z1

and f2g = h/Zo.

Consider an arbitrary point z0 E Z0. By L e m m a 6.1.13, there is a point xl E X1 such that .fl(Xl) = g(zo). By the n-softness of the map f l , there exists a map g~: Z0 ~ X1 satisfying, in particular, the following conditions: C(gt)(c(x1)) C C(Z)/Zo,

and f l g ' = g.

The n-softness of the map f guarantees the existence of a map k~: Z I : X1 such that C(k')(C(Xl)) C C(Z)/Z1,

k ' / Z o = g' and f k ' =

h.

Finally observe t h a t the map k = f l U : Z1 ~ X2 has all the required properties from Definition 6.1.12 with respect to the map f2. [--1 PROPOSITION 6.1.16. A map f : X ~ Y of a realcompact space X onto an A N E ( n ) - s p a c e Y is n-soft if and only if the following condition is satisfied: 9 For each at most n-dimensional realcompact space Z, for its closed subspace Zo, and for any two maps g: Zo -+ X and h: Z ~ Y such that f g = h/Zo and C ( g ) ( C ( X ) ) C C ( Z ) / Z o , there exists a map k: Z ~ X such that k / Z o = g and f k = h. PROOF. Obviously every n-soft map satisfies the above stated condition. Suppose now that f is the map given in our statement. Let us show its n-softness. Let Z, Z0, Z1, g and h satisfy the conditions of Definition 6.1.12 with respect to the map f . Consider the Hewitt realcompactification u Z of the space Z, and the closures A0 and A1 of Z0 and Z1, respectively, in uZ. Since every space

234

6. ABSOLUTE EXTENSORS

is C-embedded in its Hewitt realcompactification, we have the following two inclusions:

C ( g ) ( C ( X ) ) C C ( v Z ) / Z o and C ( h ) ( C ( Y ) ) C C ( v Z ) / Z 1 . By Propositions 6.1.7 and 1.3.12, the maps g and h have extensions ~" A0 ~ X and h" A1 ---* Y such that

C ( ~ ) ( C ( X ) ) C_ C ( v Z ) / A o and C ( h ) ( C ( Y ) ) C_ C ( v Z ) / A 1 . Since f ~ / Z o = h/Zo, and since Z0 is dense in A0, we conclude that f[7 = h/Ao. Since Y is an ANE(n)-space, and since dim vZ - dim Z < n, there exists a map h I" U ---, Y, defined on a stable functionally open neighborhood U of A1 in vZ, extending h. Corollary 1.1.15, Lemma 1.1.18 and Proposition 1.3.18 imply that U is an at most n-dimensional realcompact space. Observe also that A0 is closed in U. Consequently, according to our assumption, there exists a map k t" U --* X such that f k ~ = h ~ and k~/Ao = ~.. The restriction k~/Z1 will be denoted by k. It follows from the construction that f k -- h and k/Zo -- g. All that remains to be verified is the inclusion C ( k ) ( C ( X ) ) C C(Z)/Z1. Let E C ( X ) . By the construction, U is a stable functionally open neighborhood of the set A1, and hence of the set Z1, in vZ. Take a functionally closed subset F o f v Z such that Z1 C F C U. The f u n c t i o n ~ k E C(Z1) can be extended to a function ~k ~ E C(U). In this situation, Proposition 1.1.23 guarantees that the function ~k~/F E C(F) has an extension onto vZ. Denote this extension by r Obviously, the restriction r is an extension of ~k onto Z. This completes the proof. E1 COROLLARY 6.1.17. Let f" X ---, Y be an n-soft map between realcompact spaces and Iio be an ANE(n)-subspace of Y. Then the restriction

/ / / - l ( y 0 ) ' / - l ( y 0 ) - ~ Yo is also n-soft. PROOF. It is easy to see that the inverse image f - l ( Y 0 ) is realcompact. Therefore we can use Proposition 6.1.16. Let Z0 be a closed subspace of an at most n-dimensional realcompact space Z. Consider two maps g" Zo ~ f - l ( y o ) and h" Z ~ II0, satisfying the conditions

f g = h and C ( g ) ( C ( f - l ( y o ) ) ) C_ C ( Z ) / Z o . One can easily verify that

c(g)(c(x)) c c(z)/zo. Consequently, by the n-softness of the map f, there exists a map k" Z ~ X such that k/Zo = g and f k = h. The last equality guarantees that k(Z) C

/-~(v0). D

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

235

Combining the proofs of Proposition 6.1.11 and 6.1.16, we obtain the following statement. PROPOSITION 6.1.18. A map f : X ~ Y between compact spaces is n - s o f t if and only if the following condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l compactum Z , its closed subspace Zo, and any two maps g: Zo ~ X and h: Z ~ Y with f g = h / Z o , there exists a map k ~ Z ~ X such that k / Z o = g and f k = h.

Combining the proofs of Proposition 6.1.10 and 6.1.16 we get PROPOSITION 6.1.19. A map between Polish spaces following condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l Polish space and any two maps g: Zo ~ X and h: Z --+ exists a map k ---, Z ~ X such that k / Z o = g

in n - s o f t if and only if the Z , its closed subspace Zo, Y with f g = h / Z o , there and f k = h.

R e m a r k 6.1.20. If the compact spaces X and Y in Proposition 6.1.18 are metrizable, then we can assume that the compacta Z and Zo are also metrizable. Therefore the last two Propositions, coupled with the fact just mentioned, show that in the class of Polish spaces the definition n-soft map (Definition 6.1.12) coincides with the definition of the same notion given in Chapter 2 (see Definition 2.1.33). It is also useful to observe the following simple fact: 9 in Corollary 6.1.17 the assumption that Y0 is an A N E (n )-space can be removed if (a) X, Y and Yo are compact spaces, or (b) X, Y and ]Io are Polish spaces.

PROPOSITION 6.1.21. Let f " X ~ Y be an n - s o f t map. A ( g ) E ( n ) - s p a c e if and only if Y is an A ( g ) E ( n ) - s p a c e .

Then X

is an

PROOF. First assume that X is an A ( N ) E ( n ) - s p a c e . Consider an arbitrary at most n-dimensional space Z, its subspace Z0, and a map g" Zo ~ Y such t h a t C ( g ) ( C ( Y ) ) c C ( Z ) / Z o . Take a point z0 e Z0 and, using L e m m a 6.1.13, consider a point x0 E X such that f ( x o ) -- g(zo). Let go(zo) -- xo. The nsoftness of the map f guarantees the existence of a map gl. Zo ~ X such that fg'g and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . By our assumption, there exist a stable functionally open neighborhood U of the set Z0 in Z and a map h/" U ~ X, extending g~. Finally, observe that the composition h -- f h t" U --. Y is an extension of the originally given map g. Therefore Y is an A ( N ) E ( n ) - s p a c e . Conversely, assume that Y is an A ( N ) E ( n ) - s p a c e . Consider an arbitrary at most n-dimensional space Z, its subspace Z0, and a map g" Z0 --* X such t h a t C ( g ) ( C ( X ) ) c_ C ( Z ) / Z o . Then, by our assumption, there exist a stable functionally open neighborhood U of the set Z0 in Z and a map h ~" U --* Y such that h~/Zo = f g . By Lemma 1.1.18 and Proposition 1.3.18, we have dim U _ n.

236

6. ABSOLUTE EXTENSORS

Consequently, the n-softness of the map f guarantees the existence of a map h" U --~ X with h / Z o = g. This shows t h a t X is an A ( N ) E ( n ) - s p a c e . [:] Next we introduce some additional concepts needed in the sequel. DEFINITION 6.1.22. A map f " X ~ Y is called n-invertible if for any map h" Z ---, Y of any at most n - d i m e n s i o n a l space Z into Y , there is a map g" Z ---, X such that f g = h. Obviously, every n-soft map is n-invertible. In turn, every n-invertible map is surjective. It is known [171] t h a t each 0-invertible map between compact spaces has a so-called regular averaging operator (see Definition 6.1.24) and t h a t for maps between metrizable c o m p a c t a the converse is also true. Every metrizable c o m p a c t u m is a 0-invertible image of the Cantor cube D • (see [233], [171]). In C h a p t e r 4 we have already seen t h a t a more general s t a t e m e n t is true: each metrizable c o m p a c t u m is an n-invertible image of an n-dimensional c o m p a c t u m (the universal Menger c o m p a c t u m #n for example; see T h e o r e m 4.2.21). In the meantime, the map constructed in T h e o r e m 4.2.21 has several additional properties (for instance, t h a t m a p is open for each n _> 1). If we do not require the presence of these additional properties, then the above fact can be obtained as an immediate consequence of T h e o r e m 1.3.10. Indeed, take an arbitrary metrizable c o m p a c t u m X and consider all possible maps ga" Za ~ X of at most n-dimensional metrizable c o m p a c t a Za into X. Let Z denote the discrete sum of these c o m p a c t a Za. T h e maps ga naturally induce the map g" Z --~ X. Consider the Stone-(~ech extension g~" f~Z ---, X of g. Since dim f~Z = n we can, applying T h e o r e m 1.3.10, find an n-dimensional metrizable c o m p a c t u m Y and two m a p s p " f~Z --~ Y and f" Y ---, X such t h a t g~ = f p . It only remains to observe t h a t the map f is n-invertible. DEFINITION 6.1.23. A map f : X --~ Y is said to be functionally open (respectively, functionally closed) if the image of any f u n c t i o n a l l y open (respectively, f u n c t i o n a l l y closed) subset of X is functionally open (respectively, functionally closed) in Y DEFINITION 6.1.24. We say that a surjection f " X - , Y admits a regular averaging o p e r a t o r if there exists a linear continuous I map v" C ( X ) --+ C ( Y ) satisfying the following conditions" (a) v ( 1 x ) = 1y. (b) /f ~ >_ 0, then v(~ >_ O. (c) v . C ( f ) = i d c ( y ) . T h e following s t a t e m e n t will be exploited below. 1Here C(X) and C(Y) are endowed with the compact-open topology.

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

237

PROPOSITION 6.1.25. For each infinite cardinal n u m b e r T, there exists a functionally closed, proper and O-invertible m a p g" N r ---. R r admitting a regular averaging operator. PROOF. For each a 6 r , t a k e a 0-invertible m a p ha" D ~a ~

[0, 1]a of t h e

C a n t o r cube D ~ o n t o t h e closed interval [0, 1]a (see t h e a b o v e discussion). Let L a = h~-l((0, 1)a) a n d ga = h a / L a , ~ 6 T. D e n o t e by g t h e p r o d u c t of ga's, i.e. g = •

~ e ~}.

Since L a is a z e r o - d i m e n s i o n a l , n o n - c o m p a c t , locally c o m p a c t space w i t h a c o u n t a b l e basis, we c o n c l u d e (recall t h a t T is infinite c a r d i n a l ) t h a t t h e p r o d u c t l-I{n~" ~ 9 r } is h o m e o m o r p h i c to N r (use T h e o r e m 1.1.5). In t u r n , t h e p r o d u c t YI{(o, 1)a. a 6 T} is obviously h o m e o m o r p h i c to R r . T h e 0-invertibility of t h e m a p g follows from t h e 0-invertibility of t h e ga's, a 6 T. It is easy to see t h a t t h e m a p g is p r o p e r a n d f u n c t i o n a l l y closed. T h u s , it only r e m a i n s to show t h a t g admits a regular averaging operator. It easily follows from t h e a b o v e c o n s t r u c t i o n t h a t we are in t h e following s i t u a t i o n . A 0-invertible m a p ( n a m e l y t h e p r o d u c t of h a ' s , a 6 7-) h" D r ~ I r is given, R r C I r , h - I ( R r ) is h o m e o m o r p h i c to N r a n d t h e m a p g" N r ~ R r coincides w i t h t h e r e s t r i c t i o n h / h - I ( R

r ).

Take a r e g u l a r a v e r a g i n g o p e r a t o r

u for t h e m a p h. It follows from e l e m e n t a r y p r o p e r t i e s of r e g u l a r a v e r a g i n g o p e r a t o r s [249] t h a t for each c o m p a c t u m K in R r , t h e m a p h g = h / h - I ( K ) also a d m i t s a r e g u l a r a v e r a g i n g o p e r a t o r UK. T h e s e o p e r a t o r s are c o m p a t i b l e in t h e following sense: 9 If K1 a n d K2 are c o m p a c t a in R r a n d K1 C K2, t h e n for each f u n c t i o n 6 C ( h - l ( g 2 ) we have

UK2(~p)/gl -- UKl ( C f l / h - l ( g l ) ) . U s i n g these p r o p e r t i e s of o p e r a t o r s u g , a n d also t h e fact t h a t b o t h spaces N r a n d R r are f g - s p a c e s 2, it is not h a r d to see t h a t by l e t t i n g

v(~)(y) = u{u}(~/g-l(y)),

y 6 Rr

a n d ~p e C ( N r )

we define a r e g u l a r a v e r a g i n g o p e r a t o r for t h e m a p g.

[-1

PROPOSITION 6.1.26. Every O-soft m a p between A E ( O ) - s p a c e s is f u n c t i o n a l l y open. PROOF. L e t f" X ~ Y be a 0-soft m a p a n d Y be an A E ( 0 ) - s p a c e . W i t h o u t loss of g e n e r a l i t y we can a s s u m e t h a t for s o m e infinite c a r d i n a l n u m b e r T t h e s p a c e Y is C - e m b e d d e d into R r , the space X is C - e m b e d d e d into t h e p r o d u c t R r x R r , a n d t h e m a p f coincides w i t h t h e r e s t r i c t i o n of t h e p r o j e c t i o n ~1" R r • R r ~ R r onto X . Take a m a p gl" N r --~ R r w i t h t h e p r o p e r t i e s of P r o p o s i t i o n 2X is an fK-spa~e if any real-valued function, defined on X and having continuous restrictions on every compact subspace of X, is continuous on X; see [193]

238

6. ABSOLUTE EXTENSORS

6.1.25. It follows from t h e c o n s t r u c t i o n s p r e s e n t e d in the proof of P r o p o s i t i o n 6.1.25 t h a t t h e p r o d u c t g2 -- gl

•

gl : N~

x N T

-~

R T

x R ~

is also p r o p e r a n d functionally closed (moreover, g2 is 0-invertible and a d m i t s a regular averaging o p e r a t o r ) . Observe also t h a t 7rig2 - glA1, where A I : N r x N r --, N r d e n o t e s the p r o j e c t i o n onto t h e first coordinate. Since Y is a Ce m b e d d e d A E ( 0 ) - s u b s p a c e of R r , there exists a m a p h i : N r --~ Y coinciding w i t h gl on t h e set g ~ l ( y ) . Similarly, since X is a C - e m b e d d e d A E ( 0 ) - s u b s p a c e of t h e p r o d u c t R r x R r , and since t h e m a p f is 0-soft, we conclude t h a t there is a m a p h2: N r x N r --, X coinciding with g2 on the set E g 2 1 ( X ) and such t h a t f h 2 -- hlA1.

Now consider c o m p a c t - v a l u e d and u p p e r semi-continuous

retractions

rx--h2g21:

R r x R r ---,X and r y =

h l g l l : R r ---,Y.

It follows from t h e c o n s t r u c t i o n s t h a t f r x - ryTrl. In particular, since r y is t h e identity on Y, we see t h a t for each point b E r ~ - l ( Y ) we have the equality f ( r x ( b ) -- ~rl(b). T h e functional closedness of t h e m a p g2 g u a r a n t e e s t h a t the set ~(v)

= (b e R ~ • R ~ : ~ x ( b ) c U }

is functionally open in the p r o d u c t R r • R r for each functionally open subset U of X . It is also easy to see t h a t the projection r l is functionally open. Consequently, t h e set v = ~(~x~(V)) n Y is functionally o p e n in Y. T h u s , in order to c o m p l e t e the proof, it only remains to show t h a t V = f (U). Since U c_ r x l ( U ) N l r ~ l ( Y ) , we see t h a t

f ( u ) = ~ ( u ) c_ ~ (~x~(U) n ~ ( y ) )

= ~(~x~(U))n

If y E V, t h e n there is a point b E r x l ( U ) N

Y = v.

zr~-l(Y) such t h a t 7rl(b) -- y.

As r e m a r k e d above, for t h e point b we have f r x ( b )

-- 7rl(b). B u t r x ( b ) C_ U.

Consequently,

y = ~ ( b ) = f ~ x ( b ) c_ f ( u ) . This implies the desired equality V = f ( U ) .

D

It is useful to c o m p a r e the following i m m e d i a t e consequence of P r o p o s i t i o n 6.1.26 w i t h P r o p o s i t i o n 2.1.34. COROLLARY 6.1.27. A surjection between Polish spaces is O-soft if and only

if it is open.

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

239

Historical and bibliographical notes 6.1. Definitions 6.1.22 and 6.1.24 appeared in [171] and [39] respectively. The other notions and results of this Section are due to the author (see, for example, [79], [84], [80]).

6.2. M o r p h i s m s of s p e c t r a and square d i a g r a m s Every morphism between spectra induces the limit map between a limit spaces of given spectra (see Subsection 1.2.2). On the other hand, for "good" spectra the converse is also true: every map between the limit spaces of these spectra is induced by some morphism (see Section 1.3). Consequently, every topological property of these limit maps can be described in terms of morphisms of these spectra. From this point of view, the study of the structure of morphisms of spectra themselves becomes very important. There are several possible approaches to this problem. One of them deals with the description of properties of morphisms using the language of square diagrams, consisting of elements of morphisms and short projections of the corresponding spectra. We begin with the following definition. DEFINITION 6.2.1. A characteristic map of a commutative square diagram

/2 X~

Xl

"-Y2

fl

~-Yl

is the diagonal product p A f2 , considered as a map from the space X2 into the fibered product (i.e. pullback) of the spaces X1 and Y2 with respect to the maps

f l : X1 ---* Y1 and q: Y2 ---+ Y1. Square diagrams with surjective characteristic maps are called bicommutative (see, for example, [202], [245], [278]). A much finer way of distinguishing square diagrams is contained in the following definition. DEFINITION 6.2.2. A commutative square diagram is called n-soft, n - 0, 1, .. , co, if its characteristic map is n-soft. The co-soft diagrams are called soft. DEFINITION 6.2.3. A commutative square diagram is said to be a Cartesian square if its characteristic map is a homeomorphism.

240

6. ABSOLUTE EXTENSORS

LEMMA 6.2.4. Suppose that the spaces Y1 and ]I2 in the above square diagram, consisting of realcompact spaces, are A N E ( n ) - s p a c e s . I f the map f l is n-soft and the diagram is a Cartesian square, then f2 is also n-soft. PROOF. By P r o p o s i t i o n 6.1.16, it suffices to consider an a r b i t r a r y at most ndimensional r e a l c o m p a c t space Z, its closed subspace Zo, and two maps g2 : Zo ---+ X2 and h 2 : Y 2 satisfying the following conditions

f2g2-

h2/Zo and C ( g 2 ) ( C ( X 2 ) ) C_ C ( Z ) / Z o ,

and to prove the existence of a m a p k2: Z ---. X2 such t h a t f2k2 -- h2 and

g2 = k2/Zo. Let gl - p g 2 and hi - - q h 2 . Clearly fig1-

h l / Z o and C ( g l ) ( C ( X 1 ) ) C C ( Z ) / Z o .

T h e n-softness of the m a p f l g u a r a n t e e s the existence of a m a p k l : Z --+ X1 satisfying the following two conditions: f l k l -- hi and gl -- k l / Z o . Since our d i a g r a m is a C a r t e s i a n square, the diagonal p r o d u c t k -- k l A h 2 of the maps kl and h2 m a p s the space Z into the space X2. It only remains to observe t h a t

f2k2 -- h2 and g2 - k2/Zo.

E]

LEMMA 6.2.5. Let an n-soft square diagram, consisting of realcompact spaces, be given. I f the spaces Y1 and Y2 are A N E(n)-spaces and the map f l is n-soft, then f2 is also n-soft. Moreover, for any at most n-dimensional realcompact space Z , its closed subspace Zo, and any three maps g2: Zo ---* X2, h2: Z ---, ]I2 and kl : Z ---, X1 satisfying the conditions

f l k l = qh2, f2g2 -- h2/Zo, Pg2 -- k l / Z o and C ( g 2 ) ( C ( X 2 ) ) C C ( Z ) / Z o , there exists a map k2: Z ---, X2 such that g2 - k2/Zo, h2 -- f 2k2 and pk2 - kl. PROOF. Let X denote the fibered p r o d u c t of the spaces X l and ]I2 with respect to the m a p s f l : X 1 ~ Y1 and q: Y2 ~ ]I1. Let ~ : X --. X1 and r X ~ Y2 d e n o t e the n a t u r a l projections of the fibered product. Since the diagram

X1

fl

~Y1

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

241

is a C a r t e s i a n square, we conclude, by L e m m a 6.2.4, t h a t the m a p r is n-soft. T h e characteristic m a p h = p/k f2 of the given d i a g r a m is n-soft by our assumption. Consequently, by L e m m a 6.1.14, the m a p f2 (being the c o m p o s i t i o n of the maps h and r is also n-soft. T h e validity of the second part of our s t a t e m e n t can be shown as follows. First, observe t h a t the diagonal p r o d u c t k l / k h 2 : Z ---+ X satisfies the following equalities: ~o(kl/kh2) = kl,

h2 and hg2 = ( k l / k h 2 ) / Z o .

r

T h e n-softness of the m a p h g u a r a n t e e s the existence of a m a p k2: Z --+ X2 such t h a t g2 = k2/Zo and hk2 = ( k l A h 2 ) . It only r e m a i n s to note t h a t

f2k2 = Chk2 = r

= h2

and

pk2 = Chk2 = ~ ( k l A h 2 ) = kl. This completes the proof.

I-1

LEMMA 6.2.6. Let S x -- { X ~ , p ~ , T} be a well ordered continuous spectrum, consisting of realcompact spaces X a and n-soft short projections paa + l . I f Xo is

an A N E ( n ) - s p a c e , then the limit projection p0: l i m S x ~ Xo is n-soft. PROOF. T h e limit space of the s p e c t r u m X = S x is r e a l c o m p a c t (see Subsection 1.1.2). Therefore we can use P r o p o s i t i o n 6.1.16. Take an a r b i t r a r y at most n - d i m e n s i o n a l r e a l c o m p a c t space Z, a closed s u b s p a c e Z0 of Z, and any two m a p s g: Z0 --~ X and h: Z ~ X0 such t h a t

Pog = h / Z o and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . Let k0 = h. Suppose t h a t for each a < /~ we have a l r e a d y c o n s t r u c t e d m a p s k~: Z ~ X a such t h a t

pag = k a / Z o whenever a < and

ka=p~k.y

whenever a < 7

We now c o n s t r u c t a m a p kz: Z ---, XZ. If/3 is a limit ordinal, then, by t h e continuity of the s p e c t r u m S x , the diagonal p r o d u c t

obviously satisfies the required properties. If/~ = a + 1, t h e n the existence of the m a p kz follows i m m e d i a t e l y from the n-softness of the short projection p~. Thus, the maps ka: Z --. X a are c o n s t r u c t e d for each a < T and satisfy the conditions

pay -- k a / Z o and ka -- p~ak~ w h e n e v e r a < ~ < T. It only remains to r e m a r k t h a t the diagonal p r o d u c t k = /k{ka : a < T} satisfies the required equalities g - k / Z o and pok -- h. IN

242

6. ABSOLUTE EXTENSORS

LEMMA 6.2.7. Let S x = { X a , p ~ , T} and S y = {Ya, q~, T} be two well ordered continuous spectra consisting of realcompact spaces. Let

{fa: X a -+ Ya; oz E T} be a morphism between these spectra such that all arising adjacent square diagrams are n-soft. If Yo is an A N E ( n ) - s p a c e and the maps fo and qg+l, ~ < T are n-soft, then the limit map of the morphism { f a } is also n-soft.

PROOF. Let X = l i m S x , Y = l i m S y and f = l i m { f a } . By P r o p o s i t i o n 6.1.21, X is a r e a l c o m p a c t space and, by L e m m a 6.2.6, Y is an A N E ( n ) - s p a c e . Consider an a r b i t r a r y at most n - d i m e n s i o n a l r e a l c o m p a c t space Z, a closed subspace Z0 of Z, and any two maps g: Z0 ~ X and h: Z ~ Y satisfying the conditions

f g = h/Zo and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . By P r o p o s i t i o n 6.1.16, in order to establish the n-softness of the map f, it suffices to c o n s t r u c t a m a p k: Z---, X such t h a t g - k/Zo and f k - - h. Let g a - P a g and ha = qah for each c~ < T. Observe t h a t

faga - ha/Zo and C ( g a ) ( C ( X a ) ) C_ C ( Z ) / Z o whenever c~ < T. By our a s s u m p t i o n , the m a p f0 is n-soft and, consequently, there exists a map k0: Z ~ X0 such t h a t go - ko/Zo and foko - ho. Suppose t h a t for each c~ < f~, where f~ < T, we have already c o n s t r u c t e d maps ka: Z ~ X a so t h a t the following conditions are satisfied: (a) f a k a -- ha. (b) ga = ka/Zo. (c) k a - - p ~ k , r whenever c~ < 3' < ~. Let us c o n s t r u c t the m a p ko: Z ~ X O with the required properties. If ~ is a limit ordinal, t h e n let

Obviously, by continuity of the s p e c t r a S x and S y , we have

f~k/3 = hi3, g/3 = k/3/Zo a n d ka = p~k/3 whenever c~ < f~. If f~ = a + 1, then, by the a s s u m p t i o n , the d i a g r a m

6.2. MOI:tPHISMS OF SPECTRA AND SQUARE DIAGRAMS

Xt~

~Y

X o,

.- Y~

243

is n-soft. Therefore, by L e m m a 6.2.5, there exists a m a p k~: Z --+ Xt~ such t h a t

g[3- k~/Zo,

f [ 3 k ~ - h E and ka --p~k~.

Thus, the maps ka are now constructed for each c~ < r. We need only to observe t h a t the map k = A { k ~ : ~ < T} satisfies all the desired properties.

[:]

Remark 6.2.8. T h e proofs of Lemmas 6.2.4 - - 6.2.7, coupled with Proposition 6.1.18, allow us to conclude t h a t if all participating (in these Lemmas) spaces are compact, t h e n we can drop the requirement t h a t the spaces X0, Y0, Y1 and I:2 are A N E(n)-spaces. We have the same situation when all spaces are Polish and the length of the spectra, i.e. the cardinal n u m b e r T, does not exceed Wl. We call a m o r p h i s m between spectra bicommutative if all naturally arising square diagrams are b i c o m m u t a t i v e in the above sense. PROPOSITION 6.2.9. Let S x

-- {Xa, p~a, A} and S y = {Ya, q~, A} be two factorizing w-spectra with surjective limit projections and f : X ~ Y be a proper map between their limit spaces. Then the following conditions are equivalent: (i) f is functionally closed. (ii) f is induced by a bicommutative cofinal and w-closed morphism, consisting of proper maps. PROOF. First assume t h a t f is the limit m a p of some b i c o m m u t a t i v e morphism {f~: X a ~ Yc,, A'} : ,Sx/A'--+ S y / A ' , where A ~ is a cofinal and w-closed subset of the indexing set A and all maps fa, c~ E A ~, are proper. We wish to show t h a t f is a functionally closed map. Consider a functionally closed subset Z of X. Since ,Sx is a factorizing spectrum, there exist an index c~ E A ~ and a closed subset Za of the space X a such t h a t Z = p ~ l ( z a ) . T h e b i c o m m u t a t i v i t y of the m o r p h i s m {f~} implies the equality

f ( Z ) = f ( p - ~ l ( z ) ) = q~l(fo,(Zo~)).

244

6. ABSOLUTE EXTENSORS

Using the continuity of the projection qa and the properness of the map fa, we see from the above equality t h a t the set f (Z) is indeed a functionally closed set. Suppose now t h a t f is functionally closed. We are going to find cofinal and w-closed subspectra of the spectra 8 x and S y , and we shall construct a bicommutative morphism between these subspectra. W i t h o u t loss of generality, applying Theorem 1.3.4, we can assume from the very beginning that a morphism

{fa" Xa --+ Ya, A}" S x --~ S y with f = lim{fa} is already given. We are now going to perform a spectral search (see Subsection 1.3.2) with respect to the following relation L C_ A 2. L : {(a,/3) E L ' a __ a, and closed subsets Fn3, n E w, of the space Y3 such t h a t

fp-l(z~)

-- q~l(Fn3),

for each n E w.

By Proposition 1.1.29, the set A I of all L-reflexive elements is cofinal and w-closed in A. Let us show t h a t the morphism {fa" a E A I} is bicommutative. Obviously it suffices to show that for each a E A t and for any two points x~ E Xa and y E Y such that f,~(x,~) = q,~(y), we necessarily have the equality p ~ l ( x a ) A f - l ( y ) # 0. Assume the contrary. Then for some index a E A ~, and for some points xa E Xa ~nd y e V with f ~ ( ~ ) = q~(y), we h~v~

p'~l(xa) n f - l ( y ) __ 0. Represent the point xa as the intersection r){Zn~, i E w}, where {Zn~" i E w} is a decreasing sequence of elements of the basis {Zn~" n E w}. Then, obviously, p~( ~~) -

= n{p~(z

~,) 9 i e

~}.

The properness of the map f guarantees that

f(M{p-~l(zn~) 9 i E w}) = M{f(p-~l(Zn~)) " i e

w}.

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

245

Therefore y f[ M { f ( p ~ l ( Z n ~ ) ) " i E co}. By the L-reflexivity of the index a, for each i E co we have the equality f (p~l (Zn~) _ qgl (Fn~). Consequently, qa(y) r M{Fna~ " i E co}. On the other hand, q.(u) = f.(~.)

e n ( ] . ( ~ . )Z a

. i e ~ } = n F ~.a . i e

~}.

This contradiction shows the bicommutativity of the morphism {fa}. The properness of the map f and the bicommutativity of the morphism {fa} imply the compactness of the fibers of all maps .fa with a E A ~. Finally, let us show that each such f a is a closed map. Take a closed subset Z in the space Xa, and represent it as the intersection M{Zn~ } of a decreasing sequence of elements of the basis {Zn~" n E co}. The compactness of the fibers of f a guarantees that 12g

f a ( Z ) -- f a ( M { Z n ,

.

O~

i e co)) -- N { f a ( Z n , )" i G co}.

Consequently, f a ( Z ) is closed in Ya.

ff]

A similar argument proves the following statement. PROPOSITION 6.2.10. Under assumptions of Proposition 6.2.9, the map f is open if and only if it is induced by a bicommutative, cofinal and co-closed morphism, consisting of open (and proper) maps. The next result extends Theorem 4.2.21 to non-metrizable compact spaces. THEOREM 6.2.11. Let n >_ 1. Every compactum of weight T >_ co is an ninvertible, (n - 1)-soft image of an n-dimensional compactum of weight T. PROOF. We proceed by induction on r. For metrizable compacta (i.e. in the case ~- - co) our statement is contained in Theorem 4.2.21. Suppose that the theorem has already been proved for all compacta with weight < T, and consider an arbitrary c o m p a c t u m X with w ( X ) = T. Represent X as the limit space of a continuous well ordered spectrum S x = {X~,p~, ~}, consisting of compacta of weight < T (and surjective projections). Next we construct, by induction, a continuous well ordered inverse spectrum S~- = {Ya, q~, A}, "parallel" to the above indicated spectrum S x and consisting of n-dimensional compacta, and a morphism {fa}" S x --* S y consisting of ninvertible and ( n - 1)-soft maps. By the inductive hypothesis, there exist an n-dimensional compactum ]I0 of weight w ( Y o ) = w ( X o ) and an n-invertible and ( n - 1)-soft map f0" ]I0 ~ X0. Suppose now that for each c~ < -y (where -y < ,k) we have already constructed (a) an n-dimensional compactum Ya of weight w ( Y ~ ) = w ( X ~ ) , (b) an n-invertible and ( n - 1)-soft map fa" Ya ~ Xa, and (c) maps q~'Y/3--* Ya (here c~
246

6. ABSOLUTE EXTENSORS (ii) all square diagrams of the form

Ya--1

~Xc~+l

pg+l

q~+l

Y~

~X~

are n-invertible and ( n - 1)-soft (whenever a + 1 < 7). Let us now construct the compactum Y~ and maps

f~" Y~ -..--+X~ and q~" Y.~ ~ Yo,, for (a < 7). If 7 is a limit ordinal, then Y~ = lim{Ya, q~, 7} and f7 = lim{f~" a < 7}. Obviously, dim Y7 - n and w(YT) - w(XT). By L e m m a 6.2.7 and Remark 6.2.8, the map f7 is ( n - 1)-soft. Essentially the same argument (in fact, an even simpler version of it) shows that f7 is n-invertible as well. The maps q~ are naturally defined as the limit projections of the above spectrum, the limit space of which was Yr" Now suppose t h a t 7 = a + 1. Consider the fibered product Z of the compacta Ya and X7 with respect to the maps f~ and p~. Denote by ~" Z ~ Ya and r Z - . X~ the canonical projections of this fibered product. By Lemma 6.2.4, the map r is ( n - 1)-soft. It is not hard to see the n-invertibility of r as well. Since the compactum Z is contained in the product of the spaces Ya and XT, we see that w(Z) < T. Consequently, by the inductive hypothesis, there exist an n-dimensional compactum Y7 ofweight w(YT) = w ( Z ) and an n-invertible and ( n - 1)-soft map h: Y7 ~ Z. Let f~ -- Ch and q~ = ~oh. Obviously, the map f~ is n-invertible and ( n - 1)-soft (Lemma 6.1.14). The newly formed square diagram, the characteristic map of which coincides with h, is, by definition, n-invertible and ( n - 1)-soft. This completes the construction of the spectrum S y and of the morphism {f~}, satisfying all the required properties. Finally, let Y = l i m S y and f = lim{f~}. Clearly, Y is an n-dimensional c o m p a c t u m of weight T. It can be shown as above t h a t f : Y --~ X is n-invertible and (n - 1)-soft. [--1 Denote by Dnr an n-dimensional compactum of weight T, admitting an ninvertible and ( n - 1)-soft map f : Onr ~ I r onto the Tychonov cube I v (apply

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

247

Theorem 6.2.11). The (n - 1)-softness of this map guarantees (Proposition 6.1.21) t h a t Onr is an A E ( n - 1)-compactum. If X is an at most n-dimensional c o m p a c t u m of weight w ( X ) < n, then there obviously exists an embedding i" X ~ I r . Since dim X < n and f is n-invertible, we conclude t h a t there is a map j" X ~ D nr such that f j = i Clearly, in this situation j is an embedding. Consequently, the c o m p a c t u m D~ contains a topological copy of every at most n-dimensional c o m p a c t u m of weight < T. Below, factorizing w-spectra consisting of Polish spaces and surjective projections will be called Polish. A space is spectrally complete if it can be represented as the limit space of at least one Polish spectrum. An embedding of R A into I A, where A is an arbitrary set, is said to be standard, if t B ( R A) = R B and t B / R A = 7rB for each subset B C A (here tB" I A ---* I B and ~rB" R A ---* R B denote the projections onto the corresponding subproducts). LEMMA 6.2.12. Let f" T ~ I A be a O-soft map of a compactum T of weight ]A I > w. Suppose that R A is standardly embedded in I A. Then the space K = f - I ( R A ) is spectraUy complete. Moreover, K can be represented as the limit space of some factorizing Polish spectrum, the indexing set of which coincides with expojA. PROOF. By Proposition 6.1.21, T is an A E ( 0 ) - c o m p a c t u m . Consequently, by Proposition 6.1.8, T is perfectly x-normal. Represent T as the limit space of an w-spectrum ST

--

{TB, p~, expwA },

consisting of metrizable compacta and surjective projections. In turn, let

S I = {I B, t~, expojA} and SR = { R B, lr~, e x p ~ A } be the standard factorizing w-spectra associated with I A and R A respectively. By Proposition 6.1.26, the map f is open. Consequently, by Proposition 6.2.10, we may assume without loss of generality t h a t the map f is the limit map of a bicommutative morphism

{fB" TB ~ IB; expwA}" S T ---* SI, consisting of open maps lB. Let K B -~ p B ( K ) for each B E e x p ~ A and qB = p B / K B whenever C C B and C, B E expojA. Let us show t h a t the spectrum

S K -- { K B , q~, expojA} is a factorizing Polish spectrum, the limit space of which coincides with K. First we show t h a t lira S K - - K. In order to establish this fact it suffices to show that K -- A { p B I ( K B ) 9 B E e x p ~ A } (see Proposition 1.2.4). Since R A is

248

6. ABSOLUTE EXTENSORS

standardly embedded into I A, and since the morphism {fB} is bicommutative, we conclude that K B -- p B ( K ) -- p B f - I ( R

A) -- f B l t B ( R A) -- f B I ( R B)

for each B E e x p ~ A . Therefore, p B I ( K B ) - pBl f - I B ( R

B) -- f - - l t B I ( R B ), B e expwA.

But in this situation we have N{pBI(KB)

9B e expwA}-

f--I(N{tBI(RB

) " B E exp~A}-

f-I(RA)-

K.

Thus, K -- lira 8 K. A completely similar argument shows the continuity of the spectrum S g . Obviously, all the spaces KB, by the equalities, K s -- f ~ I ( R B ) , B E e x p ~ A , are Polish. The construction immediately implies that the projections of the spectrum ~ g are surjective. Therefore, ~ g is a Polish spectrum. Let us show now that SK is a factorizing spectrum. Since f is open, K is dense in T. Therefore, by Proposition 1.1.21, K is z-embedded in T. By Proposition 1.1.22, for each function ~o E C ( K ) there exist a set L, which can be represented as the countable intersection of functionally open subsets of T, and a function r E C ( L ) such that K C L and r = e l K . Clearly, L is a LindelSf space. Observe also that since the spectrum 3T is factorizing, there exists a G~-subset -1 (LBo). Then LBo of the space TBo (for some Bo E e x p ~ A ) such that L = PBo L -- lim 3L, where SL = { L B = P B ( L ) , PC, B B D Bo, B E e x p ~ A } .

Obviously, the spectrum ~L is Polish (with proper projections). Its limit space (i.e. the space L), as was already mentioned, is LindelSf. Consequently, by Corollary 1.3.2, ~L is a factorizing spectrum. This means that there exist an element B E e x p ~ A , B D_ Bo, and a function CB E C ( L B ) , such that r = r Since L B D_ K B , we have ~ s = r E C ( K B ) . It only remains to observe that ~ = r D PROPOSITION 6.2.13. Let T >_ w and n = O, 1 , . . . , o0. Then there exist spectraUy complete realcompact spaces K1 and K2 and maps ~" K2 ---+ K1, gl" K1 --~ R r and g2" K2 ~ R r x R r , satisfying the following conditions: (a) The maps gi, i = 1, 2, are proper, functionally closed and n-invertible. (b) For each f u n c t i o n ~ e C ( K 1 ) there is a f u n c t i o n r E C ( R r ) such that if ~o is constant on the fiber g~-l(y), Y e R r , then r = ~(g~l(y)). (c) For each f u n c t i o n ~ E C(K2), there is a f u n c t i o n r e C ( R r x R r ) such that if ~ is constant on the fiber g21(y), y e R r x R r , then r -- ~(g21(y)). (d) d i m K i -- n for each i - 1,2. (e) R - w ( g i ) = w ( g i ) = T for each i = 1,2.

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

249

(f) T h e square d i a g r a m g2 K2

*-R ~" x R r

7rl

K1

gl

~- R r

is c o m m u t a t i v e .

PROOF. If n -- c<), t h e n the validity of our s t a t e m e n t is obvious. Indeed, it suffices to let K1 -- R r , K2 -- R r x R r , gl -- i d R r , g2 ---- i d R r x R r and - ~rl. If n -- 0, the s t a t e m e n t is a consequence of P r o p o s i t i o n 6.1.25. T h u s we only have to consider the case when n is a n a t u r a l number. Take an n - d i m e n s i o n a l c o m p a c t u m T1 of weight r and an n-invertible and ( n - 1)-soft m a p f l : T1 ---+ I r from T h e o r e m 6.2.11. We m a y a s s u m e t h a t R r is s t a n d a r d l y e m b e d d e d in I r . D e n o t e by K1 the inverse image f ~ - l ( R r ), a n d by gl the restriction of f l on K1. By L e m m a 6.2.12, the space K1 is a s p e c t r a l l y c o m p l e t e r e a l c o m p a c t space. T h e m a p gl is open, p r o p e r and, consequently, functionally closed. For each function ~ e C ( K 1 ) , the function r e C ( R r ), defined by letting

r

egll(y)},

y e R ~',

is continuous (here we use t h e fact t h a t the m a p gl is open and closed) and obviously satisfies condition (b). As r e m a r k e d in t h e proof of L e m m a 6.2.12, t h e space K1 is z - e m b e d d e d in T1. Consequently, by P r o p o s i t i o n 1.3.18, dim K1 <_ n. T h e reverse inequality follows from the n - i n v e r t i b i l i t y of t h e m a p gl and t h e universality of t h e space R r ~ Observe also t h a t since the space K1 is c o n t a i n e d in a c o m p a c t u m of weight T and also a d m i t s an o p e n m a p onto a space of weight T (namely, onto R r ) , we conclude t h a t the weight of t h e space K1 equals r. Since the weight of any space c a n n o t exceed its R-weight we see, a p p l y i n g L e m m a s 1.3.15 and 6.2.12, t h a t R - w ( K 1 ) = r. Consider the following d i a g r a m

250

6. ABSOLUTE EXTENSORS

h

T2

TI x I r,

fl • id

,- I r x I r

7rl

K2

h/K2

x id

.K1

7rl

~R ~

T1 ......f l

K 1 .....

gl

,_ I r

R r

Here h: T2 -* T1 x I r denotes an n-invertible and (n - 1)-soft map of an ndimensional c o m p a c t u m 7'2 of weight r onto the product T1 x I r (Theorem 6.2.11). Denote by K2 the inverse image of the product K1 x R r under the map h. Finally, A denotes the composition of the map h / K 2 and the projection Trl: K 1 x R r --~ K 1 and g2 denotes the composition of the restriction h / K 2 and the product gl x i d . Since T1 is an A E ( O ) - c o m p a c t u m , the product T1 x I r is also an AE(0)-space. Further, by Proposition 6.1.21, the space T2 is an AE(0)-space and, hence, by Proposition 6.1.8, is perfectly a-normal. An argument, completely similar to that presented above, shows that all the constructed objects satisfy the desired conditions. Finally, we remark that the characteristic map of the diagram in condition (f) coincides with the map h / K 2 and, therefore, is surjective by the construction. [-1

H i s t o r i c a l a n d b i b l i o g r a p h i c a l n o t e s 6.2. The concept of an n-soft diagram was introduced by the author [78], [79]. The main result of this Section - - Theorem 6.2.11 was proved in [125]. In the case n = 1, weaker results have been obtained in [148] and [78] (see also [245]). Proposition 6.2.13 was proved in [84]. Propositions 6.2.9 and 6.2.10, for the class of compact spaces, was obtained in [278] It has already been mentioned in the main text that the compacta D r are universal with respect to the class of all at most n-dimensional compact spaces of weight < r. The first results in this direction were simultaneously and independently obtained in [246], [247] and [324]. 9

?%

6.3. SPECTRAL CHARACTERIZATIONS OF n-SOFT MAPPINGS

251

6.3. Spectral characterizations of n-soft m a p p i n g s We say t h a t a m a p f : X --~ Y has a Polish kernel if there exists a Polish space P such t h a t X is C - e m b e d d e d in the p r o d u c t Y x P so t h a t f coincides with the restriction r y / X of the projection r y : Y x P --, Y onto X . Observe t h a t any m a p between Polish spaces has a Polish kernel. T h e following t h e o r e m is one of the main results of this C h a p t e r .

THEOREM 6.3.1. Let n = 0, 1 , . . . , oo. A map f : X ---, Y between A ( N ) E ( n ) spaces is n-soft if and only if there exist factorizing well ordered continuous spectra S x - { X a , p ~ , T } , S y - - {Ya, q~, T} and a morphism {fa}" S x ---* B y , satisfying the following conditions: (i) X - - l i m , ~ x , Y - - l i m , ~ y and f = l i m { f a } . (ii) The spaces Xo and Yo are Polish A ( g ) E ( n ) - s p a c e s and the map f 0 : X 0 Yo is an n-soft map between them. (iii) The spaces X a and Ya are A ( g ) E ( n ) - s p a c e s and the map fa: Z a -~ Ya is an n-soft map between them, a < T. (iv) All short projections in the spectra ,~x and S y are n-soft and have Polish kernels. (v) All adjacent square diagrams (generated by the short projections of the spectra S x and S y and the corresponding elements of the given morphism) are n-soft and their characteristic maps have Polish kernels. (vi) If the map f itself has a Polish kernel, then all square diagrams indicated in (v) are Cartesian squares.

PROOF. One p a r t of our t h e o r e m follows from L e m m a s 6.2.6 and 6.2.7. Let us prove the converse, i.e. show t h a t n-softness of t h e m a p f implies the existence of the i n d i c a t e d s p e c t r a and m o r p h i s m w i t h the c o r r e s p o n d i n g properties. We consider only the absolute case. T h e r e m a i n i n g p a r t can be proved in a c o m p l e t e l y similar way. Let T = m a x { R - w ( X ) , R w(Y)}. T h e non-trivial case is the case of u n c o u n t a b l e T. Take any set A of cardinality IAI -- T. W i t h o u t loss of generality, we may assume t h a t the space Y is C - e m b e d d e d in Tl A, the space X is Ce m b e d d e d in t h e p r o d u c t R A x R A and the m a p f : X --* Y coincides w i t h t h e restriction 7rl/X of the projection r l : R A x R A ---+ R A onto X. Next consider spectrally c o m p l e t e r e a l c o m p a c t spaces K1, K2 a n d m a p s gl, g2 a n d A from P r o p o s i t i o n 6.2.13. Since Y is C - e m b e d d e d in R A a n d dim K1 -- n, there exists a m a p h i : g l -* Y such t h a t h l / g ~ l ( Y ) -- g l / g ~ l ( Y ) . Similarly, the fact t h a t X is C - e m b e d d e d in TIA x R A, the e q u a l i t y d i m K 2 - n, the n-softness of the m a p f , a n d the obvious equality f g 2 / g 2 1 ( X ) = h l A / g 2 1 ( X ) g u a r a n t e e the existence of a m a p h2" K2 -* X such t h a t h 2 / g 2 1 ( X ) -- g 2 / g 2 1 ( X ) and f h 2 -- hlA. Consequently, the d i a g r a m

252

6. ABSOLUTE EXTENSORS

g2

K2

, RA • R A

h2

-~(x)

7[1

g2

K1

gl1(y)

gl

RA

~y.

is commutative. It follows from these constructions that the compact-valued and upper semicontinuous retractions ri - hid:( -1 i -- 1 2, satisfy the equality f r 2 - r1~1 Let us now introduce some notations. If C C B C A, then 7r~" R B ---, R C and r B" R A ---* R B

denote the projections onto the corresponding subproducts. By YB (respectively, by X B ) we denote the space 7rB(Y) (respectively, the space (~rB x ~ B ) ( X ) ) , and by Y ( B ) (respectively, by X ( B ) ) we denote the inverse image (TrBI(YB) (respectively, the inverse image (~rB • 7 r B ) - I ( X s ) ) . The restriction of the projection lr~ onto YB (respectively, the restriction of 7r~ x 7r~ onto XB) is denoted by q~ (respectively, by p~). A similar meaning is assigned to the symbols qB and PB. Finally, by f B we denote the restriction of the projection 7rB" R s • R B ---+ R B onto XB. A subset B C A is called admissible if for each point x 6 X (B) we have the equality

Straightforward verification shows that the union of an arbitrary collection of admissible subsets is admissible. Let us investigate other properties of admissible sets. First of all, observe that for each admissible set B and for each point y E Y (B) we have the following equality: q B r l ( Y ) -- 7rB(y). Indeed, let y 6 Y ( B ) . Take a point x 6 X ( B ) such that 7rl(x) = y. Then q B r l ( y ) -- q B h l g 1 1 ( y ) -- q B h l g l l ( T r l ( x ) ) .

6.3. SPECTRAL CHARACTERIZATIONS OF n-SOFT MAPPINGS

253

By condition (f) of P r o p o s i t i o n 6.2.13, we have --1 1 gl (Trl(X))--- )~g2 (x).

Consequently, qBrl(y) = qBhlAg21(x) = qBfh2g21(x) = qBfr2(x) =

Let us now show t h a t for each admissible subset B, the subspace YB is closed a n d C - e m b e d d e d in R B. Take a section iB: R B ~ R A of the projection ~B and let r lB = qBrZiB. Since r ~ is a c o m p a c t - v a l u e d u p p e r semi-continuous r e t r a c t i o n of R B onto YB, we conclude t h a t YB is closed in R B and, consequently, is realc o m p a c t . Let ~ E C ( Y B ) . T h e n ~ q B h l E C ( K 1 ) . Take a function r E C ( R A ) , satisfying condition (b) of P r o p o s i t i o n 6.2.13 with respect to the c o m p o s i t i o n ~ q B h l . T h e function @, defined by letting

~(y) = r

y ~ R B,

e x t e n d s ~ to R B. This shows t h a t YB is C - e m b e d d e d in R B. A similar a r g u m e n t proves t h a t X B is C - e m b e d d e d into R B • R B for the admissible set B. Next we show t h a t for each admissible set B the space YB is an A E ( n ) space. T h e r e a l c o m p a c t n e s s of YB allows us to use P r o p o s i t i o n 6.1.9. Consider an a r b i t r a r y at m o s t n - d i m e n s i o n a l r e a l c o m p a c t space Z, its closed s u b s p a c e Z0, a n d a m a p so: Zo ~ YB satisfying the inclusion C ( s o ) ( C ( Y B ) ) C_ C ( Z ) / Z o . Since R B is an A E - s p a c e , t h e r e is an extension s2" Z ~ R B of the m a p so, considered as a m a p of Z0 into R B. By condition (a) of P r o p o s i t i o n 6.2.13, gl is an n-soft map. Therefore t h e r e exists a m a p s l : Z ~ K1 such t h a t g l s l = iBs2. Let s = q B h l S l . Using the above indicated p r o p e r t i e s of admissible sets, we see t h a t the map s: Z ~ YB is an extension of so. A c o m p l e t e l y similar a r g u m e n t shows t h a t X B is also an A E (n )-space for each admissible set B. Now we are going to prove t h a t for each admissible set B the m a p qB is nsoft. Since the space Y, by our a s s u m p t i o n , and the space YB, by the above proof, are b o t h A E ( n ) - s p a c e s , it suffices, in order to show the n-softness of qB, to a p p l y P r o p o s i t i o n 6.1.16. Thus, let Z be an a r b i t r a r y at most n - d i m e n s i o n a l r e a l c o m p a c t space, Z0 be its closed subspace, and so: Z0 ~ Y and s: Z ~ YB be a r b i t r a r y m a p s satisfying the conditions qBso = s / Z o and C ( s o ) ( C ( Y ) ) C_ C ( Z ) / Z o .

Since the projection 7rB is a soft map, t h e r e is a m a p k2" Z ---+ R A such t h a t 7rBk 2 :

s

and

k2/Zo

:

so.

T h e n-invertibility of the m a p gl, coupled w i t h the inequality dim Z __ n, guarantees the existence of a m a p kl" Z ~ K1 such t h a t g l k l -- k2. Let k -- h l k l . S t r a i g h t f o r w a r d verification shows t h a t k is the required lifting of the m a p s,

254

6. ABSOLUTE EXTENSORS

extending the map so. This proves the n-softness of the map qB. A similar argument shows the n-softness of the map PB for each admissible set B. The n-softness of the maps f , PB and qs imply, by Lemmas 6.1.14 and 6.1.15, the n-softness of the maps f B , P~ and q~ for admissible sets C and B with CCB. Suppose now that C and B both are admissible subsets of A and C C_ B. Denote by T ( C , B ) the following commutative diagram:

fB XB

XC

* YB

fc

,- Y

Obviously this diagram is the subdiagram of the following commutative diagram, denoted by R ( C , B ) "

RB • RB

~

RB

•

RC x Rc

lrC

, RC

Our first goal is to establish the n-softness of the diagram T ( C , B ) . First, let us show that the diagram T ( C , B ) is bicommutative. Let x c 6 X c , YB 6 YB and f c ( x c ) -- q~(YB). Since the diagram R(C, B) is obviously bicommutative (it is not hard to see that this diagram is even soft), there is a point zo 6 R B x R B such that (lr~ x 7 r ~ ) ( z o ) - x c

and IrlB(z0).

Clearly there is a point z 6 X (C) such that zo - (~rB x 7rB)(Z ). Consider a point x of the c o m p a c t u m r2(z) and let x B : pB(x). It is easy to see that B

P c ( X B ) -- x c and f B(XB) -- YB.

This shows that T ( C , B ) is indeed a bicommutative diagram.

6.3. SPECTRAL CHARACTERIZATIONS OF n-SOFT MAPPINGS

255

DenGte by X0: X B ~ To the characteristic map of the diagram T ( C , B ) (here To stands for the corresponding fibered product) and by X: R B • R B "--* T the characteristic map of the diagram R ( C , B ) (recall that, as remarked above, X is a soft map; here T is the corresponding fibered product, which as can easily be seen, itself is a power of the real line and, hence, is an absolute extensor). Since T ( C , B ) is a subdiagram of the diagram R ( C , B ) , one can see that To is a subspace of T and that the map X0 coincides with the restriction of X to X B . Summarizing the above data, we arrive at the following diagram: RB x RB

pg

x

Xc

nc

7rB

f

,~ R B

~

C

nc

* YC

By Proposition 6.1.21 and Lemma 6.2.4, To is an A E (n )-space and, consequently, in order to show the n-softness of the map X0 we can use Proposition 6.1.16. Let Z be an arbitrary at most n-dimensional realcompact space, Z0 be a closed subspace of Z, and so: Z0 --* XB and s: Z --. To be maps satisfying the conditions Xoso = s / Z o and C ( s o ) ( C ( X B ) ) C C ( Z ) / Z o . T h e softness of the map X implies the existence of a map k2: Z ~ R B • R B such that so - k 2 / Z o and s - xk2. Consider a section iB : R B --~ R A of the projection r s . Since the map g2 is n-invertible, there is a map kl: Z ~ K2 such that g2kl -- (iB X iB)k2.

Let k -- p B h 2 k l . Straightforward verification shows that k is the desired lifting of the map s, which extends so. Therefore T ( C , B ) is n-soft diagram. If the map f has a Polish kernel, then a similar argument shows t h a t the characteristic map of the diagram T ( C , B ) is a homeomorphism. Indeed, in this case instead of diagram R ( C , B ) we deal with a diagram of the type

256

6. ABSOLUTE EXTENSORS

7r I

~R s

RSxp

~rB • i d p

Rc • p

~rl

,R c

where P is a Polish space witnessing the fact that f has a Polish kernel. It is easy to see that the above diagram is the Cartesian square, i.e. the corresponding characteristic map is a homeomorphism. This fact, coupled with the bicommutativity of the diagram T ( C , B ) , obviously implies that T ( C , B ) is also the Cartesian square. We return to the general situation. Represent the product R A • R A as the limit space of the standard factorizing Polish spectrum

s = (R ~ • R ~, ~3 • ~ , ~ p ~ A } Then the space X is the limit space of the induced factorizing w-spectrum

Since the r-weight of a spectrally complete realcompact space K2 equals IAI (see condition (d) from Proposition 6.2.13), we conclude that K2 -- lim Sg2, where

s ~ = {k~, tg, r

}

is a factorizing Polish spectrum. Since g2 is proper and functionally closed (condition (a) from Proposition 6.2.13), by Proposition 6.2.9, we may assume that g2 - lim g2B, where

is a bicommutative morphism, consisting of proper maps. By Spectral Theorem 1.3.6, the map h2" K2 ~ X can also represented as the limit map of a morphism

Therefore, the collection of those element B E exp,.,A for which the diagram

6.3. SPECTRAL CHARACTERIZATIONS OF n-SOFT MAPPINGS

g2

K2

257

, RA X RA

h2 7rB X 7rB

1

x ,~ R B

•

RB

B

XB

is commutative is w-closed and cofinal in e x p ~ A (apply Proposition 1.1.27). The bicommutativity of the morphism {gS} implies the equality

hf/(~')-~(z~) =

~/(g~')-~(x~).

We claim that each of the indicated countable subsets B C A is admissible. Indeed, let x E X ( B ) . Then pB~2(~) = p . h 2 g ; ~ ( x ) = h:B t . g 2- 1 (~).

Consequently, p.~(~)

= h f ( g f ) - ~ ( ~ . • ~ . ) ( ~ ) = g~(g~)-~(~. • ~ . ) ( ~ ) = (~. • ~.)(~).

This shows that exp~,A contains an w-closed and cofinal subcollection, consisting of admissible subsets of A. Since IA] = T, we can write A -- { a a : a < T}. Since the collection of countable admissible subsets of A is cofinal in exp,,,A, each point aa E A is contained in a countable admissible subset B a C_ A. Let Aa = U{Bf~: f~ _ a} and X a -- XAo, , Paa+l --PA,,, A,,,+I , Y a - -

A'~+~andfaYA,~, qa+l a -- qA,,,

fA,, a < T.

It follows, from the properties of admissible subsets proved above, t h a t the spectra ,Sx = { X a , p ~ , T}

,Sy = { Y a , q ~ , T}

satisfy all of the required properties. 6.3.1. El

and the morphism {fa}" S x ---+ S y This completes the proof of Theorem

While proving the above theorem, we in fact have obtained the another spectral characterization of n-soft maps. Sometimes this version is more convenient.

258

6. ABSOLUTE EXTENSORS

THEOREM 6.3.2. Let n = 0, 1 , . . . , co. A map f : X ---, Y between A ( N ) E ( n ) spaces is n-soft if and only if there exist factorizing Polish spectra 8 x = {Xa,p#a,A}, S y = {Ya, q~,A} , consisting of Polish A(g)E(n)-spaces, and a morphism { f a } : S x ---+ S y , consisting of n-soft maps, such that the following conditions are satisfied: (i) X = l i m S x , Y = l i m S y and f = lim{f~}. (ii) All limit projections of the spectra S x and S y are n-soft. (iii) All limit square diagrams, generated by limit projections of spectra S x and S y , by elements of the morphism { f a } and by the map f , are n-soft. (iv) If the map f has a Polish kernel, then all square diagrams indicated in (iii) are the Cartesian squares. As an i m m e d i a t e corollary of T h e o r e m 6.3.2 we have the following statement. PROPOSITION 6.3.3. Let n = O, 1 , . . . , oo and ~" be an infinite cardinal number. If f : X ---, Y is an n-soft map between A(Y)E(n)-spaces, then there exist factorizing T-spectra SX = {Xa,p~, A} , 8 y = {Ya, q~, A} , consisting of A(Y)E(n)-spaces, and a morphism { / a } : S x ---, S y , consisting of n-soft maps, such that the following conditions are satisfied: (i) X = l i m S x , Y = l i m S y and f = l i m { / a } . (ii) All limit projections of the spectra S x and S y are n-soft. (iii) All limit square diagrams, generated by limit projections of spectra S x and 8 y , by elements of the morphism { f a } and by the map f , are n-soft. PROOF. T h e desired spectra can be obtained by taking T-completions of the w-spectra from T h e o r e m 6.3.2. T h e desired m o r p h i s m is then generated by the m o r p h i s m from t h a t Theorem. 1-1 Let us now consider a particular case of the above s t a t e m e n t . Namely, assume t h a t X = Y and t h a t f is the identity map. T h e n we obtain the following two results. PROPOSITION 6.3.4. A space X is an A(N)E(n)-space if and only if it is the limit space of a factorizing well ordered continuous spectrum 8 x = {Xa,p~, T} , all short projections of which are n-soft and have Polish kernels, and the first element Zo of which is a Polish A ( g ) E ( n ) - s p a c e . PROPOSITION 6.3.5. A space X is an A ( g ) E ( n ) - s p a c e if and only if it is the limit space of a factorizing T-spectrum consisting of A ( g ) E ( n ) - s p a c e s and n-soft limit projections.

Remark 6.3.6. As a useful (and manageable) exercise we suggest t h a t the reader verifies t h a t if the spaces X or Y in the above s t a t e m e n t s are (~ech complete and LindelSf, then all projections in the corresponding spectra are proper.

6.3. SPECTRAL CHARACTER/ZATIONS OF n-SOFT MAPPINGS

259

Remark 6.3.7. In the case when all spaces are compact and n - 0, Proposition 6.3.4 was obtained by Haydon [164]. He first investigated spectra with open projections having metrizable kernel (observe t h a t in the case of compact spaces, the latter class of maps coincides with those with Polish kernels in our sense). Such spectra are called Haydon spectra. Thus, A E ( 0 ) - c o m p a c t a are precisely the limit spaces of Haydon spectra having a point (or any other metrizable compactum) as their first element. In general, the investigation of A E ( 0 ) - c o m p a c t a by means of analysis of the corresponding Haydon spectra is similar to Pontrjagin's method of investigation of (locally) compact topological groups by means of their spectral representations in terms of Lie series (we will see in Chapter 8 that the mentioned analogy between Haydon spectra and Lie series leads us in fact to the proof of the fact that each compact topological group is an A E ( 0 ) - c o m p a c t u m ) . Consider some of corollaries of the above results.

COROLLARY 6.3.8. The weight and the R-weight of a non-discrete A E ( O ) space coincide. PROOF. Let X be a non-discrete AE(0)-space of weight T. Since X is nondiscrete, T > w. By Proposition 6.3.5, the space X can be represented as the limit space of a factorizing r - s p e c t r u m 8 x - { Z a , p ~ , A } , consisting of AE(0)-spaces of R-weight v and 0-soft limit projections. By Theorem 1.3.6, there exist an index a E A and a map is" X a ~ X such t h a t ic~pa = i d x . Being a 0-soft map, f is surjective (Lemma 6.1.13) and open (Proposition 6.1.26). Consequently, pa is a homeomorphism. But then R - w ( X ) = R - w(X,~) = T. E] COROLLARY 6.3.9. I f a space X admits an n-soft map with a Polish kernel onto an uncountable product yI{Ya: a E A } of non-trivial 3 Polish AE.(n)-spaces, then X is homeomorphic to the product x0 • II{Y..

,,

A - A0},

where Ao is a countable subset of the indexing set A and Xo is a Polish space, admitting an n-soft map onto the product 1-I{Ya : a E A0}.

PROOF. The product l"I{Ya : a E A} is an A E (n)-space. Consequently, by Proposition 6.1.21, X is also an A E (n )-space. Represent Y as the limit space of the standard factorizing Polish spectrum S = {YB, ~rB, e x p ~ A } , where YB -I-I{Ya" a E B} and ~r~" YB --* YC are the natural projections (C C_ B). Consider also factorizing Polish spectra S x and S y from Theorem 6.3.2. Since the Rweight of the space Y coincides with IAI and since f has a Polish kernel, one can easily conclude that the R-weight of the space X is also T. Therefore, without loss of generality, we may assume t h a t the indexing sets of the spectra S x and ,~y coincide with exp~A. Further, by Theorem 1.3.6, again without lost of generality, we may assume that the spectra ,~ and ,~y are identical. Consider now the morphism { f s } from Theorem 6.3.2. Then the square diagram 3i.e. containing at least two points.

260

6. ABSOLUTE EXTENSORS

X

*~Y

7rB

PB

XB

fB

~ YB

is a Cartesian square. Denote X B by X0 and let A0 -- B. Since the projection ~rB : Y ~ Y B is a trivial bundle with fiber l"I{Ya : a E A - A0}, we see that (since the above diagram is the Cartesian square) the projection PAo of the spectrum S x is also a trivial bundle with the same fiber. [-]

COROLLARY 6.3.10. I f a c o m p a c t u m a d m i t s an n - s o f t with a Polish k e r n e l m a p onto the T y c h o n o v cube I r , T > w, t h e n it is h o m e o m o r p h i c to the p r o d u c t o f the cube I r a n d a m e t r i z a b l e c o m p a c t u m which a d m i t s an n - s o f t m a p onto the Hilbert cube I ~ .

It should be observed that the analog of Corollary 6.3.10 (as well as of Corollary 6.3.9) is not true in the metrizable case. Indeed, we saw in Chapter 4 that the Hilbert cube is an n-soft image of the universal (n + 1)-dimensional Mender compactum #n+l.

H i s t o r i c a l a n d bibliographical n o t e s 6.3. The results of this Section form the foundation of the general theory of non-metrizable A N E ( n ) - s p a c e s . In particular, they solve the so-called adequacy problem (of A N E ( n ) - s p a c e s and n-soft maps) posed in [278]. Without fear of overstatement one might say that the entire circle of problems has its roots in the work of Haydon [164], which in fact contains the proof of the indicated adequacy within the class of compacta for n - 0 (i.e. Proposition 6.3.4 for n - 0). The case n - cr in Proposition 6.3.4 was considered in [272] and the case n - 1 in [148] and [242]. The remaining cases of Proposition 6.3.4 (as well as Proposition 6.3.5 in the class of compacta) have been obtained in [125]. The remaining results in this direction, covering the non compact case and even the case of maps (Theorems 6.3.1 and 6.3.2, Propositions 6.3.3 - - 6.3.5), belong to the author [79], [84]. Corollaries 6.3.8 6.3.10 were also obtained by the author [79], [80].

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

6.4. F u r t h e r

properties

261

of AE(0)-spaces

In this Section, b a s e d on t h e results o b t a i n e d in Section 6.3, we e s t a b l i s h several i m p o r t a n t p r o p e r t i e s of A E ( 0 ) - s p a c e s . T h e s e p r o p e r t i e s will be l a t e r used in t h e d e v e l o p m e n t of I v - a n d R v - m a n i f o l d theories.

6.4.1. General

properties

of AE(0)-spaces.

W e begin w i t h t h e following

and gl : N r ~

be O-soft m a p s , a n d s u p p o s e

statement. LEMMA 6.4.1. L e t f : X ~

Y

Y

T h e n t h e r e e x i s t s a O-soft m a p g2: N v • N W --~ X

that f has a Polish kernel.

s u c h t h a t f g2 -- glTrl, w h e r e lr1: N v • N W ~

N ~

is the p r o j e c t i o n .

Moreover,

the r e s u l t i n g s q u a r e d i a g r a m g2 N r X N ~

~X

7rl

N r

91

, y

is O-so/t. PROOF. F i r s t we consider t h e case T -- w. In this case, t h e spaces X a n d Y are b o t h Polish, a n d t h e m a p s f a n d gl are o p e n a n d surjective. Let Z b e t h e fibered p r o d u c t of spaces N Wa n d X w i t h r e s p e c t to m a p s gl" N ~ ~ Y a n d f'X

~Y.

Denote bye"

Z ~

N W ands"

Z ~

X the canonical projections.

O u r goal will be a t t a i n e d if we c o n s t r u c t an o p e n s u r j e c t i o n h" N W x N ~ ~

Z

satisfying t h e e q u a l i t y r l - Ch. It is clear t h a t t h e m a p s r a n d ~o are o p e n a n d Z is a Polish space. T a k e a n y c o m p l e t e m e t r i c d on Z a n d c o n s t r u c t a s e q u e n c e (b/k} of c o u n t a b l e o p e n covers of Z w i t h t h e following p r o p e r t i e s : (i) U k - - {Vii ..... i k ' i j E N } , j E N , k E N . (ii) c l z U ~ l ..... ~k,ik+l C_ U~ ..... ~k, i k + l E N . (iii) U~ ..... ~, - - U { U ~ ..... ~k,ik+~'ik+l E N } .

(iv) d i a m d ( V ~

..... ~

n r

< ~, y e N ~ .

(v) r ..... i x ) - NW, i j E N , k e N . To prove t h a t such a s e q u e n c e of o p e n covers exists it suffices to show t h a t if U is an o p e n s u b s e t of Z such t h a t r

= N ~ , t h e n for a n y e > 0 a n d a n y

p o i n t x0 E U t h e r e is an o p e n s u b s e t V of Z such t h a t :

(a) ~o ~ v . (b) c l z V C _ U .

262

6. ABSOLUTE EXTENSORS

(c) r ~. (d) diamd(V M r < e for any point y 9 N ~ . Fix a m a p s: N w ---. U such t h a t Cs -- i d N w and xo 9 A -- s ( N ~ ). T h e set A is closed in Z and, consequently, t h e r e exists an open n e i g h b o r h o o d OA of A in Z whose closure (in Z) is c o n t a i n e d in U. For any point a 9 A we take an open n e i g h b o r h o o d Oa of a of d i a m e t e r less t h a n e which is contained in O A . Choose a c o u n t a b l e subfamily { O a i : i 9 w} of the collection {Oa: a 9 A} t h a t covers A. In a c o m b i n a t o r i a l m a n n e r , we refine the open cover { r i 9 w} of N W by a c o u n t a b l e disjoint o p e n cover {Gi: i 9 w} w i t h the condition t h a t r 9 Go. T h e n we let

V = U{Oai s

i 9 w}.

T h e reader can easily verify t h a t V is the desired set. Let us now define the m a p h : N " x N '~ ---, Z as follows. (a,b) = ( a , ( i l , . . . , i k . . . . )) 9 N " x g ~ let

h(a,b) = r

For any point

M N { U i , ..... ik" k E N } .

It follows from t h e completeness of Z and from properties (ii) and (iv) t h a t h(a, b) is n o n - e m p t y and consists of precisely one point. T h e easy verifications of the continuity, surjectivity and o p e n n e s s of h are left to the reader. Next we consider the case of u n c o u n t a b l e T. Clearly X and Y are A E ( O ) spaces ( P r o p o s i t i o n 6.1.21). By T h e o r e m 6.3.2, applied to b o t h of the m a p s f and to gl, there exist Polish spaces X0 and Yo, open surjections f0" X0 --* Yo and go: N ~ --~ Y0, and 0-soft m a p s p: X ~ X0 and q: Y ---, Y0 such t h a t the following conditions hold: (e) T h e square d i a g r a m

X

Xo

is a C a r t e s i a n square. (f) T h e square d i a g r a m

~Y

/0

'~ Yo

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

263

gl N r

Nw

~ Y

go

~ YO

where 7r: N r --+ N ~ , is 0-soft. Let Z d e n o t e the fibered p r o d u c t of X a n d N T with respect ro f and gl, a n d let ~ : Z ~ X a n d r Z ~ N r be the canonical projections. T h e symbols Z0, ~o0 a n d r have analogous meaning. T h e following d i a g r a m helps us to s u m m a r i z e the situation:

N r

x N W

h

~r x i d

gl

N'r

NO., X N W

ho

// ~o

,~Z

,_Y

~o0

,~Zo

N /W

go

,~X

,-X0

-Yo

Here A denotes the restriction of the p r o d u c t p x ~r onto Z. N o t e t h a t A is surjective. Observe also t h a t t h e c o n s t r u c t i o n a n d p r o p e r t y (a) imply t h a t : (g) T h e square d i a g r a m

264

6. ABSOLUTE EXTENSORS

Z

, - N ~"

~)o

Zo

~ N w

is a Cartesian square (only general categorical arguments are needed while making this conclusion [212]). By the case already considered above, there exists an open surjection

ho" N w

x N w --~ Z o

such that 71"1 - - - - r Obviously, (h) The square diagram 7r 1 N r

x N w

~N r

~r x i d

N W x N ~

7rl

,~ N w

is a Cartesian square. Since all the diagrams constructed up to now are commutative, one can easily see that the map

h-

lrl/kho(Tr

x idNw

)" N "

x N W ---+ Z

makes the whole diagram commutative. By properties ( g ) a n d (h), the square diagram

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

N r x N w

265

'-- Z

lr • id

N o.' x N w

ho

~ Zc

is a C a r t e s i a n square. Consequently, by L e m m a 6.2.4, h is a 0-soft map. It only remains to note t h a t the c o m p o s i t i o n g2 = ~oh satisfies all the required properties. T h i s c o m p l e t e s the proof of t h e lemma. [-] THEOREM 6.4.2. The class o f A E ( O ) - s p a c e s o f w e i g h t <_ T c o i n c i d e s w i t h the class o f O-soft i m a g e s o f N ~ .

PROOF. One p a r t of this s t a t e m e n t is a direct consequence of P r o p o s i t i o n 6.1.21 (coupled w i t h the fact t h a t N ~ is an A E ( 0 ) - s p a c e ) . Let us prove the r e m a i n i n g part, i.e. t h a t each A E ( 0 ) - s p a c e of weight < T is a 0-soft image of N r . If T = w, as a l r e a d y observed in Subsection 1.1.1 (see T h e o r e m 1.1.7), the t h e o r e m is true. Suppose t h a t T > w and consider an A E ( 0 ) - s p a c e X of weight T. By Corollary 6.3.8, R - w ( X ) = T. Therefore, by P r o p o s i t i o n 6.3.4, the space X is h o m e o m o r p h i c to t h e limit space of a factorizing well ordered c o n t i n u o u s s p e c t r u m 8 x = { X a , p~, T } , all short p r o j e c t i o n s paa+l , c~ < T, of which are 0-soft and have Polish kernels, and such t h a t X0 is a Polish space. Let f0" N ~ -* X0 be an open surjection, the existence of which is g u a r a n t e e d by T h e o r e m 1.1.7. F u r t h e r , suppose t h a t for each a, with a < / 3 < T, we have a l r e a d y c o n s t r u c t e d 0-soft m a p s

s." (N~)" -~ x . such t h a t the square d i a g r a m

f~ (N')'~

,-X.y

p~

(NW) ~

~Xa

266

6. ABSOLUTE EXTENSORS

is 0-soft whenever a, 7 < fl (here 7r~ denotes the corresponding natural projection). We are now going to construct a map ff~: (N~) fl --. X#. If fl is a limit ordinal, then the desired map can be obtained by using L e m m a 6.2.7. Observe, in the meantime, t h a t all newly arising square diagrams are also 0-soft. If fl = a + 1, t h e n the existence of a map f#, satisfying the required properties, follows from L e m m a 6.4.1. Thus, the 0-soft map f a : ( N ~ ) a --. X a is constructed for each c~ < T. Consider the corresponding 0-soft (by construction) morphism

{f~}" S

--+ 8 x ,

where lim S = N r and let

f = lim{f~}" N r ---, X. It only remains to note that, by L e m m a 6.2.7, f is a 0-soft map.

El

COrtOLLArtY 6.4.3. Let X be an AE(O)-space of weight w ( X ) > r > w and = { F t ' t E T } be an arbitrary f a m i l y of subsets of X each of which can be written as the intersection of not more than r f u n c t i o n a l l y open subsets of X . Then there exists a subfamily ~ C_ ~ of cardinality < r such that clx U~" = clx U~ "~. Moreover, the set clx U~" can also be represented as the intersection of not more than r f u n c t i o n a l l y open subsets of X .

PROOF. By T h e o r e m 6.4.2, there is a 0-soft map f " N w(g) ---, X . The family f - l ( ~ - ) = { f - l ( F t ) . t e T } consists of sets t h a t can be represented as the intersection of not more than r functionally open subsets of N ~~ By the result of [251], there is a subfamily ~'~ C_ ~" of cardinality < r such t h a t ClN~(X > Uf -1 (~') -- ClN~(X) Uf -1 (9~"). Since f is an open map (Proposition 6.1.26), we have clx U~" = clx U Y . Let us now prove the second part of our statement.

Represent X as the l i m i t

space of a factorizing r - s p e c t r u m S x = {Xa, p~, A } , consisting of AE(0)-spaces and 0-soft limit projections (Proposition 6.3.5). It is not hard to see t h a t there exists an index c~ E A such t h a t Ft = p~l(p,~(Ft)) for each Ft E .T". Since w ( X a ) < v, we easily see t h a t the set F = clx.

U{pa(Ft)" Ft E .T"}

can be represented as the intersection of not more t h a n T functionally open subsets of the space X a . But then its inverse image p ~ l ( F ) also can be represented as the intersection of not more t h a n T functionally open subsets of X. T h e openness of the limit projection pa guarantees t h a t clx U.T" -- p ~ l ( F ) . F1

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

267

COROLLARY 6.4.4. Let X be an AE(O)-space of uncountable weight. Then an arbitrary family of functionally closed subsets of X contains a countable subfamily with the same closure. Moreover, this closure itself is functionally closed. COROLLARY 6.4.5. An AE(O)-space with a dense subset, consisting of points of countable pseudocharacter, is Polish. COROLLARY 6.4.6. The closure of any Gs-subset of an arbitrary AE(O)-space is functionally closed. COROLLARY 6.4.7. Each Gc-subset of an arbitrary AE(O)-space is z-embedded. COROLLARY 6.4.8. Each functionally closed subset of an arbitrary AE(O)space is C-embedded. PROPOSITION 6.4.9. The property of being an AE(O)-space is hereditary with respect to subspaces that can be represented as countable intersections of functionally open subsets. PROOF. Let X be an AE(0)-space and Y = M{Gn : n C w}, where each of the sets Gn is functionally open in X. Represent X as the limit space of a factorizing Polish spectrum S x = {Xa, p~, A} with 0-soft limit projection. Then there is an index a E A such t h a t the space X a contains a G~-subset Ya with the property y _ p~l(y~). It only remains to note t h a t the restriction p a / Y : Y --~ Ya is 0-soft (Corollary 6.1.17), and then to apply Propositions 6.1.10 and 6.1.21. [3 Since the space N r has the Baire property (i.e. the intersection of countable many open dense subsets is dense), Theorem 6.4.2 implies the following result. COROLLARY 6.4.10. Each AE(O)-space has the Baire property. COROLLARY 6.4.11. An open realcompact subspace of an arbitrary AE(O)space is functionally open. PROOF. Let U be an open realcompact subspace of an AE(0)-space X. It follows from the results of Subsection 1.1.2 t h a t the complement X - U can be represented as the union of functionally closed subsets of X. By Corollary 6.4.4, X - U is functionally closed in X. [3 A bit finer argument proves the following. COROLLARY 6.4.12. A realcompact Gs-subspace of an arbitrary AE(O)-space can be represented as the intersection of a countable collection of functionally open subsets.

268

6. ABSOLUTE EXTENSORS

6 . 4 . 2 . D i m e n s i o n a l p r o p e r t i e s o f A E ( 0 ) - s p a c e s . It was shown in Section 6.2 t h a t for each n _> 0 and for each T >_ w, t h e r e exists an (n 4- 1)-dimensional c o m p a c t u m Dnr+l of weight T a d m i t t i n g an n-soft m a p onto the T y c h o n o v cube I r . Obviously, this c o m p a c t u m serves as an e x a m p l e of an (n 4- 1)-dimensional AE(n)-compactum of weight T. It can be shown [125] t h a t we cannot lower the dimension of such c o m p a c t a . More precisely: 9 I f n >_ 1, t h e n each n - d i m e n s i o n a l AE(n)-compactum is metrizable (if n -- 0 this s t a t e m e n t is not true take the C a n t o r cube D r of any weight T). T h e reason for this lies in the m e t r i z a b l e case. N a m e l y in T h e o r e m 4.2.25, which states t h a t t h e r e is no n-soft m a p of an n - d i m e n s i o n a l metrizable comp a c t u m onto the Hilbert cube I W . On the o t h e r hand, as shown in T h e o r e m 5.1.10, every Polish space is an n-soft image of an n - d i m e n s i o n a l Polish space. Analysis of this s i t u a t i o n leads us to the hypothesis t h a t perhaps, in contrast to t h e c o m p a c t case, for each n > 0 and for each T _> w there exist n-dimensional AE(n)-spaces of weight T. In order to establish this principal fact, we need two p r e l i m i n a r y results. LEMMA 6.4.13. Let s -- (Xk, "Pk k+l ,w} be an inverse sequence, consisting of n-dimensional AYE(n)-spaces and n-soft projections. Then d i m ( l i m 8 ) < n. PROOF. In our s i t u a t i o n each of the spaces X k, k E w, can be C - e m b e d d e d in X - l i m 8 . Consequently, R - w(Xk) < R - w ( X ) for each k E w. Having in m i n d this observation, we can, based on P r o p o s i t i o n 6.3.5, represent the spaces Xk, k E w, as well as the space X , as the limit spaces of factorizing w-spectra

S I c - (X~,q~a,k,A} and S x - - (X~,p~a,A}, consisting of A N E(n)-spaces and n-soft limit projections, and with the same indexing set A. F u r t h e r , by T h e o r e m s 1.3.6 and 1.3.10, we can w i t h o u t loss of generality assume t h a t all spaces in the s p e c t r a 8k are at most n-dimensional. A p p l y i n g T h e o r e m 6.3.2 to each of the short projections of the sequence 8, one can see t h a t for each k E w there is a b i c o m m u t a t i v e (even n-soft) m o r p h i s m M k+l

k

e

k4-1,a

= lPk

}" S k + ~ - ~ S k

the limit m a p of which coincides with the projection pk+l In this situation, for k 9 each index a E A we get an inverse sequence k+l,a

consisting of n - d i m e n s i o n a l Polish spaces (and n-soft projections). T h e bicomm u t a t i v i t y of all n a t u r a l l y occuring square d i a g r a m s g u a r a n t e e s t h a t X a = lim 8 a for each a E A. Consequently, each of the spaces Xa, a E A, as the limit of the sequence 8a, consisting of n - d i m e n s i o n a l Polish spaces, is n-dimensional. A p p l y i n g L e m m a 1.3.7, we conclude t h a t dim X < n. T h e reverse inequality

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

269

follows from the remark made at the beginning of the proof (that X contains a C-embedded copy of each Xk). [-q LEMMA 6.4.14. Let S x = {Xa,pBa,~"} be a continuous well ordered spectrum, consisting of n-dimensional ANE(n)-spaces and n-soft projections. Then dim(lim S x ) = n. PROOF. We consider two cases. First, if the spectrum S x contains a countable cofinal subspectrum, then our s t a t e m e n t follows from L e m m a 6.4.13. Suppose now than there is no countable cofinal subspectrum in S x . Then this spectrum is w-complete. By L e m m a 6.2.6 and Proposition 6.1.21, the limit space lim S x is an A N E (n )-space. Consequently, by Proposition 6.1.8, its Suslin number is countable. Therefore, based on Proposition 1.3.3, we can conclude t h a t S x is a factorizing spectrum. Now the inequality dim(lira S x ) <_ n follows from L e m m a 1.3.7. The reverse inequality is obvious. I--1 THEOREM 6.4.15. Let n = O, 1,... and T >_w. Each ANE(n)-space of weight T is an n-soft image of n-dimensional space of weight n. PROOF. We proceed by induction on T. For ANE(n)-spaces of countable weight, i.e. for Polish A N E (n )-spaces (see Proposition 6.1.10), our s t a t e m e n t is contained in Theorem 5.1.10. Suppose t h a t the theorem has already been proved for AgE(n)-spaces of weight < ~- and consider an arbitrary AgE(n)-space Z of weight T. By Proposition 6.3.4, X can be represented as the limit space of a continuous well ordered spectrum

Sx = {x~,pg +~, ~} consisting of ANE(n)-spaces of weight < T and n-soft projections. By induction, let us now construct a continuous well ordered spectrum

s v = {y~, q~+~, ~ } "parallel" to the given one and consisting of n-dimensional ANE(n)-spaces. We also construct a morphism {f~}: S y -:-+ SX, consisting of n-soft maps. By the inductive hypothesis, there exist an n-dimensional A N E (n )-space ]Io of weight w(Yo) = w(Xo) and an n-soft map fo: Yo ~ Xo. Suppose t h a t for each a < 3', where 3' < A, we have already constructed: (a) an n-dimensional ANE(n)-space Y~ of weight w ( Y a ) = w(X~). (b) an n-soft map f a : Ya ~ X~. (c) n-soft maps q ~ ' Y 3 ~ Y~ (a < 3 < 7, satisfying the following two conditions: _ q~=q~qt~ for e a c h a 5 , 3 w i t h a < 5 < 3 < - y

270

6. ABSOLUTE EXTENSORS All square diagrams of the type

Y.+~

* Xa+l

p~,+~

qg+l

Y~

~X~

are n-soft (a + 1 < "r). We now construct the space X r and maps f r and qa7 with a < 7. If "r is a limit ordinal, then X r --lim{Y~,q~,'r}

and f r -

lim{f~" a < "r}.

By L e m m a s 6.2.6 and 6.4.14, X r is an n-dimensional ANE(n)-space of weight w(Yr) -- w(Xr). L e m m a 6.2.7 guarantees the n-softness of the map f r : Yr X r. We define the maps qa7 as the limit projections of the spectrum the limit space of which coincides with Yr" These maps are n-soft by L e m m a 6.2.6. It is easy to see t h a t all naturally occuring square diagrams are n-soft. Next we consider the case "r - a + 1. Consider the fibered p r o d u c t X of spaces Y~ and X r with respect to the maps fa and p7a. Denote by ~o" Z ~ Ya and r Z ~ X r the canonical projections of this fibered product. By L e m m a 6.2.4, ~o and r are n-soft maps. Consequently, the space Z is an A N E (n )-space (Proposition 6.1.21). Since the space X is contained in the p r o d u c t Y~ x X r , its weight is less t h a n r. Consequently, by inductive hypothesis, there exist an n-dimensional ANE(n)-space Yr of weight w(Z) and an n-soft map h ' Y r ~ Z. Let f r - Ch and q~ - ~oh. Obviously, these maps are n-soft ( L e m m a 6.1.14) and the newly arising square diagram is also n-soft (because its characteristic m a p coincides, by construction, with nsoft map h). Thus the s p e c t r u m 8 y and the morphism {f~}, satisfying all the required properties, is constructed. Finally, let Y - lim Sy and f -- lim{fa }. Applying L e m m a s 6.2.6, 6.2.7 and 6.4.14, we conclude t h a t Y is an n-dimensional V1

ANE(n)-space of weight w(X) and f : Y --. X is an n-soft map.

Consider now an n-dimensional space R~ of weight T which admits an nsoft map onto R r (we have just proved the existence of such spaces in T h e o r e m 6.4.15). If n -- 0 we can, by T h e o r e m 6.4.2, assume t h a t R~ = N r . These

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

271

spaces have specific universality properties. We will investigate these in the next Section. Thus, in contrast with the c o m p a c t case, for each n _ 0 there are n-dimensional A E (n )-spaces of any weight. Is it possible to lower the dimension of such spaces? In order to answer this question we need the following lemma. LEMMA 6.4.16. Let F" X --~ Y be an n-soft map between Polish spaces and d i m Y -- k < n. I f [ f - l ( y ) [ >_ 2 for each point y 6 Y , then d i m X > k. PROOF. Since dim Y -- k, there is a point y0 6 Y such t h a t every neighborhood of y0 in Y is exactly k-dimensional. By our assumption, the fiber f - l ( y o ) contains at least two points, say x0 and Xl. By the n-softness of the m a p f , we can find two maps i0, i1" Y ~ X satisfying the following conditions: (a) fire = i d y for each m = 0, 1. (b) im(YO) -- Xm for each m -- 0, 1. Let U - {y 6 Y" io(y) 7~ il(y)}. Obviously, U is an open n e i g h b o r h o o d of the point y0 in Y. By the choice of y0, we have d i m U - - k. Let V - - f - l ( u ) and g - f / V . T h e n g" V ~ U is an n-soft m a p with two sections (namely: j m - ira~U, m = O, 1) with disjoint images. Since dim U = k, a t h e o r e m on partitions (see, for example, [146, page 67]) shows t h a t there exists a collection T -- ((Ap, Bp)" 1 < p < k} of closed subsets of V such t h a t Ap N Bp -- q) for each p - 1, 2 , . . . , k and for any choice of partitions Cp (in U) between Ap and Bp their intersection N{Cp" 1 < p <_ k} is non-empty. It is not hard to verify (we leave the corresponding details to the reader) t h a t the collection T ' - - { ( A p x [0,1],BpX [0,1])"

l<_p<_k}U{U

x {0},V x

{I})

of closed subsets of U x [0, 1] also has the m e n t i o n e d property. Namely, for each choice of partitions (in U x [0, 1]) between elements of pairs of the collection T', their intersection is non-empty. Next consider in the space V the following collection of pairs of disjoint closed subsets:

{g-l(Ap),g-l(Bp))"

1 <_ p <_ k} LJ ( j o ( U ) , j l ( U ) ) .

Suppose t h a t Cp is a partition in V between the sets g - l ( A p ) and g - l ( B p ) , p -- 1 , 2 , . . . , k , and C is a partition in Y between the sets j o ( V ) a n d j l ( V ) . Consider the m a p ~" U x {0, 1} ~ V, defined by letting

~o(y,t) = jr(y) for each ( y , t ) 6 V x {0, 1}. Since k < n, we have dim(V x [0, 1]) < n and, consequently, by the n-softness of the m a p g, there is a m a p h" U x [0, 1] ~ V such t h a t

h/(V x {0,1})=~

and g h = r v ,

where ~ru" V x [0, 1] ~ V is the n a t u r a l projection. T h e sets h - l ( C p ) , p -1, 2 , . . . , k and h - l ( c ) are partitions in U x [0, 1] between the elements of the

272

6. ABSOLUTE EXTENSORS

corresponding pair of the collection T'. Consequently, as noted above, h-l(c)

N M{h-l(Cp) 9 1 < p <

k} ~ O.

T h e n the intersection C ;'1N{cp. 1 _ p _< k} is also non-empty. By the above mentioned theorem on partitions, we have dim X >_ dim V > k. [:] PROPOSITION 6.4.17. If X is an non-metrizable ANE(n)-space, then d i m X

>n. PROOF. Obviously, our s t a t e m e n t is true for n - 0. It is also clear that it suffices to consider only the case of spaces of weight wl. Thus, let X be an ANE(n)-space of weight wl, n > 1 and suppose the contrary, i.e. assume t h a t dim X - k < n. W i t h o u t loss of generality, according to Corollaries 6.4.4 and 6.4.5, we may assume t h a t there is no point of countable pseudo-character in X. Represent X as the limit space of a factorizing Polish spectrum S x = { X ~ , p ~ , A } , consisting of Polish ANE(n)-spaces and n-soft limit projections (Proposition 6.3.5). By T h e o r e m s 1.3.6 and T:3.2.23, we may additionally assume t h a t dim X a - k for each c~ E A. Take an arbitrary index c~ E A. If x is a point in X a , then the fiber p-gl(x) is functionally closed in X. Consequently, the absence of points of countable pseudo-character in X guarantees t h a t this fiber contains at least two points. But then there is an index/~x E A with/3x > c~ such t h a t the fiber (pflax)-l(x)is also non-trivial. The openness of the projection p~x guarantees the existence of an open neighborhood Vx of the point x in X a such t h a t I ( p ~ ) - l ( y ) ! >_ 2 for each point y E Vx. Take a countable subcover {Vx,~ " m E w} of the open cover {Vx" x E Z a } of the Polish space Xa and let /~ - sup/~x.~. Next observe t h a t for each point x E X a we have ](p~)-l(x)] > 2. By L e m m a 6.4.16, dim X~ > k. This contradiction completes the proof. D

COROLLARY 6.4.18. Every finite-dimensional ANR-space is metrizable.

Historical and bibliographical notes 6.4. T h e results of this Section are due to the a u t h o r [80], [84], [85]. A particular case of Corollary 6.4.3 can be found in [251]. In the compact case Corollaries 6.4.3, 6.4.5, and 6.4.18, as well as L e m m a 6.4.16 were obtained in [275], [276]. For the same class of spaces, Proposition 6.4.9 appears in [171]. T h e metrizability of 1-dimensional A N R-compacta was established earlier by Isbell [179].Characterizations of AE(0)-spaces in terms of many-valued maps are given in [109].

6.5. STRONGLY UNIVERSAL SPACES

273

6.5. S t r o n g l y u n i v e r s a l s p a c e s

6.5.1. S p a c e s o f m a p s ( c o n t i n u a t i o n ) . Let X and Y be arbitrary Tychonov spaces and T be an arbitrary infinite cardinal number. We are going to introduce a topology, depending on T, on the set C(Y, X) of all continuous maps from Y into X. The space thus obtained will be denoted by Cr(Y, X ) . We recall that coy(X) denotes the collection of all countable functionally open covers of the space X. For each map f : Y --~ X the sets of the form

B(f,{blt: t 9 T } ) = {g 9 C r ( Y , X ) : g is L/t - close to f for e a c h t 9 T}, where ITI < T and L/t 9 coy(X) for each t 9 T, are declared to be open basic neighborhoods of the point f in C~(Y, X). The maps, contained in the neighborhood B ( f , {L/t: t 9 T}) are called {L/t: t 9 T}-close to f. If a space X has a countable basis, then, obviously, the space C~o(Y,X) (for any space Y) coincides with the space C ( Y , X ) , endowed with the topology introduced in Subsection 2.1.1. We will make use of the following simple lemma. LEMMA 6.5.1. Let X be an AE(o)-space of infinite weight 7. Then there exists a subcollection {L/t: T} of the collection coy(X) such that for any space Y and for any two maps f, g: Y - . X the following conditions are equivalent: (i)

f

= g.

(ii) The maps f and g are {L/t: t 9 T}-close. PROOF. If X is a Polish space, then take a metric d generating the topology of X, and denote b y / i n an arbitrary countable open cover of X, the diameters of whose elements are less than 1_. Obviously, {b/n} is the desired collection. n Next consider the case when X is a non-metrizable AE(0)-space. Represent X as the limit space of a factorizing Polish spectrum S x = { X t , P tt ~, T}. Here the indexing set T of the spectrum S x satisfies the equality IT] = w ( X ) . In each of the Polish spaces Xt, take a countable collection {L/~" n 9 N } as considered above. Then the collection {ptl(Unt)" n e N , t 9 T} has all the required properties.

El

The above topology on the set of continuous maps allows us to give a precise meaning to the notion of strong universality in the general case (compare with Definition 2.3.12). Consider an arbitrary subclass P of the class of all Tychonov spaces. If r _ w and n = 0, 1 , . . . , c o , we let

"Pr,n= { X C'I9" R - w ( X )

<_r and d i m X _ n } .

274

6. ABSOLUTE EXTENSORS

A space X E Pr,n is said to be strongly T~r,n-universal if for each space Y E 7")r,n the set of all C-embeddings is dense in the space C r ( Y , X ) . We consider only two classes: the class ,4 of all Tychonov spaces and the class B of all compact spaces. It should be observed that if w < T < T', then the T'-topo1ogy on the set C ( Y , X ) is stronger than the T-topology for any spaces X and Y. In other words, the identity map id" C r , ( Y , X )

---+ C r ( Y , X )

is continuous. Moreover, the fundamental meaning of the hierarchy of the concept of strong universality generated above is, in fact, an increased refinement of the concept of "closeness". For instance, as shown in Subsection 2.3.2 (Proposition 2.3.13), the Hilbert cube I~is strongly B~,~-universal, i.e. every map of any metrizable compactum into I ~ can be arbitrarily closely (in the space C~(-, I "~ )) approximated by embeddings (recall t h a t every compactum is C embedded in any ambient space). One can easily verify that this fact remains valid for the Tychonov cube I r as well. This means that the set of all embedding of any c o m p a c t u m Y of weight _ T into I r is dense in the space C ~ ( Y , I r ). It turns out t h a t if T > w, the Tychonov cube I r satisfies a much stronger version of the strong universality property. Namely: the cube I r is strongly Br,~-universal. Let us see why. Take a map f " Y ---. I r , where Y is a compactum of weight < T. Let O f be an arbitrary neighborhood of the point f in the space C r ( Y , I r ). Our goal is to show that the neighborhood O f contains at least one embedding. W i t h o u t loss of generality we may assume that O f : B ( f , {/4t" t E T}), where ITI < r and bit E c o v ( I r ) for each t E T. Let A be an arbitrary set of cardinality T. We identify the cube I r with I A. Since ITI < T, there exist a subset B C_ A of cardinality A = max{w, ITI} < T and functionally open covers/4tB E c o v ( I B) such that Ut = 7rBI(UB) for each t E T (here ~rB" I A --* I B denotes the natural projection onto the corresponding subproduct). Since A < r, obviously, I A - B ~ I r and, consequently, there is an embedding j" Y --+ I A - B (recall that w ( Y ) <_ T). Next, consider the embedding i - l r B f / k j " Y --+ I B x I A - B = IAo

Since r B i = r B f we see, by the choice of B, that i E O f . proved the following statement. LEMMA 6.5.2. Let T >_ w.

Therefore we have

T h e n the T y c h o n o v cube I r is strongly Br,oo-

universal.

In a similar way, one can prove the 0-dimensional counterpart of the above result. LEMMA 6.5.3. Let T >_ w. T h e n the C a n t o r cube D r is strongly Br,o-universal. Next we consider the non-compact case.

6.5. STRONGLY UNIVERSAL SPACES

275

LEMMA 6.5.4. Let T > w. T h e n the space N r is strongly Jtr,o-universal. P R O O F . F i r s t c o n s i d e r t h e c a s e T -- w.

L e t f " Y -+ N w b e a m a p of a n

a r b i t r a r y z e r o - d i m e n s i o n a l P o l i s h s p a c e Y. L e t / 4 -

{Un" n E w} be a c o u n t a b l e

o p e n cover of N ~ . O u r g o a l is to c o n s t r u c t a closed e m b e d d i n g g" Y --. N W , /4-close to f .

Without

loss of g e n e r a l i t y , we m a y a s s u m e t h a t /4 c o n s i s t s of

p a i r w i s e d i s j o i n t sets. L e t Yn = f - l ( U n ) , s i o n a l P o l i s h s p a c e , n E w.

n E w. T h e s p a c e Yn is a z e r o - d i m e n -

T h e s p a c e Un (as a n o p e n s u b s p a c e of N ~ ) d o e s

not contain an open compact subspace and, consequently, by Theorem is h o m e o m o r p h i c

to N ~ , n E w.

By Proposition

1.1.5,

1.1.6, t h e r e e x i s t s a c l o s e d

e m b e d d i n g gn" Yn --'+ Un for e a c h n E w. I t o n l y r e m a i n s to n o t e t h a t t h e m a p g" Y --~ N ~ , c o i n c i d i n g w i t h gn on Yn, n E w, is a c l o s e d e m b e d d i n g / g - c l o s e

to

S. L e t us now c o n s i d e r t h e case T > w. T a k e a n a r b i t r a r y set A of c a r d i n a l i t y r a n d let Y be a z e r o - d i m e n s i o n a l s p a c e o f R - w e i g h t < T. C o n s i d e r also a m a p

f" Y ~

N A a n d its n e i g h b o r h o o d O f in t h e s p a c e C r ( Y , N A ) . W i t h o u t

loss of

g e n e r a l i t y we m a y a s s u m e t h a t

o f = {g ~ C~(Y, N A ) 9 ~Bg = ~ . f } w h e r e B is a s u b s e t of A of c a r d i n a l i t y < T a n d r B " N A ---+ N B s t a n d s for t h e natural projection onto the corresponding subproduct. space Y admits a C-embedding

B y T h e o r e m 6.4.2, t h e

i n t o N ~ 4. Since IA - B I = T, it follows t h a t

t h e r e is a C - e m b e d d i n g j" Y --~ N A - B .

Then the desired C-embedding

of Y

into N A can be defined by letting

i - ~vBf/kj. This completes the proof.

[-1

LEMMA 6.5.5. Let r >_ w. T h e n the space R r is strongly .Ar,oo-universal. P R O O F . If

r

--

w, t h e n t h e c o n c l u s i o n follows f r o m P r o p o s i t i o n 2.3.14.

An

a r g u m e n t s i m i l a r t o t h a t u s e d in L e m m a 6.5.4 verifies t h e r e m a i n i n g c a s e s (i.e.

r>w).

r-1

T h u s , if n = 0 or n = oo, s t r o n g l y A r , n - u n i v e r s a l s p a c e s e x i s t for e a c h r _> to. I n t h e m e a n t i m e , as s h o w n in T h e o r e m

5.1.12, s t r o n g l y A w , n - u n i v e r s a l s p a c e s

e x i s t for e a c h n E to. D o we h a v e a s i m i l a r s i t u a t i o n in t h e n o n - m e t r i z a b l e c a s e ? The following statement answers this question. PROPOSITION 6.5.6. Let T > w a n d n -- 0, 1 , . . . , c~.

Each n-dimensional

space o f weight r , a d m i t t i n g an n - s o f t m a p onto R T , is strongly ~4r,n-universal. 4Indeed, consider a 0-soft map ~a" N r --~ R r . Since R - w(Y) ~_ r, we may assume that Y is C-embedded in R r . Since dim Y -- 0, we conclude, by the 0-softness of ~, that there is a map r Y --~ N r such that ~0r = idy. It only remains to note that r is a C-embedding.

276

6. ABSOLUTE EXTENSORS

PROOF. Let f " X ~ R r be an n-soft map, where X is an n-dimensional space of weight T. Consider an arbitrary at most n-dimensional space Y of R-weight _ r. Consider also a map g" Y ~ X and its neighborhood B ( g , {lgt" t e T}), [T I < r, /At E c o y ( X ) ,

t E T

in the space C r ( Y , X ) . Let A = max{w, [T[}. Since r is uncountable, we have A < T. Represent the space R ~ as the limit space of the s t a n d a r d (factorizing) A-spectrum S = {Za, lrO, A} where each of the spaces Za is homeomorphic to R x and each of the limit projections 7ra" R r --. Z a is the trivial bundle with fiber R r . Since X, as an n-soft preimage of R r , is an A E ( n ) - s p a c e , we conclude, applying Proposition 6.3.3, t h a t there exist a factorizing A-spectrum S x = { X a , p ~ , A } , consisting of A E ( n ) - s p a c e s of weight A, and a morphism {f~}" S x -+ S, consisting of n-soft maps, such t h a t (i) Z - - l i m S x and f = lim{fa}. (ii) All the naturally occuring 5 limit square diagrams are n-soft. Since S x is a factorizing ,k-spectrum and [T I < ,X, there exist an index a E A and functionally open countable covers/A~ E c o v ( X a ) , t E T , such that /At = pffl(L/~) for each t E T. Consider the corresponding limit square diagram:

Y X

Pc~

X0

,R r

Z

for

lr~

* Za

Here Z denotes the fibered p r o d u c t of the spaces X~ and R r with respect to the maps fa and 7ra. As usual r Z -~ R r and ~o: Z ~ X a denote the canonical projections of this product. T h e map h: X ~ Z is the characteristic map of the indicated diagram, which is n-soft by condition (ii). T h e m a p 7ra is the trivial bundle with fiber R r . Consequently, by the inequality R - w ( Y ) < T, there is a C - e m b e d d i n g i: Y ~ R r such t h a t 7 r a i - 7raf g. T h e n f a P a g = 7raf g = 7rai.

5Generated by the limit projections of the spectra Sx, S, elements of the morphism { f a } and by the map f.

6.5. STRONGLY UNIVERSAL SPACES

277

B u t in this s i t u a t i o n the diagonal p r o d u c t j = p a g / k i maps the space Y into Z. Since i is a C - e m b e d d i n g , we see t h a t j is also a C - e m b e d d i n g . O b s e r v e also that Cj--i and ~ j - p a g . Since d i m Y < n, the n-softness of the characteristic m a p h g u a r a n t e e s the existence of a m a p g~" Y -+ X such t h a t hg ~ -- j . Clearly, g~ is a C - e m b e d d i n g . It only remains to show t h a t !

g ~_ B ( g , { b l t ' t e T}).

This fact follows i m m e d i a t e l y from the following equalities I

P a g = ~ h g I = ~ J = Pag

and from the choice of t h e index a E A. T h e proof is completed.

[-]

A similar a r g u m e n t proves t h e following s t a t e m e n t . PROPOSITION 6.5.7. L e t r > w.

Each compactum

of weight r, admitting a

soft m a p o n t o I r , is s t r o n g l y B r , c ~ - u n i v e r s a l .

THEOREM 6.5.8. L e t r >_ w a n d n = O, 1 , . . . , c~. T h e n there e x i s t s a s t r o n g l y .Ar,n-universal AE (n)-space R~.

PROOF. If n = 0 or n = oc, t h e n the result follows from L e m m a s 6.5.4 and 6.5.5. If T = w and n is a positive integer, it suffices to a p p l y T h e o r e m 5.1.12. Finally, if T > w, apply T h e o r e m 6.4.15 and P r o p o s i t i o n 6.5.6. El

H i s t o r i c a l a n d bibliographical n o t e s 6.5. All t h e results of this Section are due

to the a u t h o r [84], [83].

CHAPTER

7

T o p o l o g y of N o n - M e t r i z a b l e M a n i f o l d s

7.1. N o n - m e t r i z a b l e

manifolds

7.1.1. D e f i n i t i o n o f m a n i f o l d s . We are now ready to define the f u n d a m e n tal concepts of R r -manifolds and I r -manifolds for r > w. First let us ask: is it possible to extend the usual definition of R ~ - (or I W-) manifold to the nonmetrizable case? In other words, is the following definition, stating t h a t R r - ( I r - ) manifolds are precisely those spaces locally h o m e o m o r p h i c to R r (respectively, I r ), satisfactory? Let us discuss this question more formally. Consider the simplest case. Let T > w and x be an a r b i t r a r y point in R r . Let U = R r - x . Obviously, U is locally h o m e o m o r p h i c to R r . In the meantime, U is not an A N R - s p a c e and, moreover, is not realcompact 1. But u n d e r a "good" definition, each R r -manifold must be realcompact (since the model space R r is an absolute retract; compare with Corollary 6.1.6 and Proposition 6.1.7). Therefore the s t a n d a r d definition has to be a d a p t e d to the non-metrizable case. In order to u n d e r s t a n d exactly w h a t modifications have to be performed, we continue our analysis and ask the next question: which open subspaces of R r are realcompact? T h e complete answer to this question is provided by Corollary 6.4.11 these are precisely the functionally open subspaces of R r . In turn, each functionally open subspace U of R r can be represented as the p r o d u c t V x R r , where V is an open subspace of R ~ , i.e. an R ~ -manifold. Obviously, V has a countable functionally open cover each element of which is h o m e o m o r p h i c to R ~ . Therefore, the space U also admits a countable functionally open cover each element of which is h o m e o m o r p h i c to R ~ x R r ~ R ~ . W i t h these observations in mind, we arrive at the following definition. DEFINITION 7.1.1. A s p a c e is c a l l e d a n R r - m a n i f o l d countable

functionally

open

cover

each

element

of which

R r .

1The last fact immediately follows from Proposition 1.1.24. 279

(T >_ w ) ,

if it has

is h o m e o m o r p h i c

a to

280

7. NON-METRIZABLE MANIFOLDS

Of course, in t h e case ~- -

w t h e a b o v e definition coincides with t h e u s u a l

definition of R ~ -manifolds. In a d d i t i o n , it is obvious t h a t each functionally o p e n s u b s p a c e of R r is an R r -manifold. A similar discussion leads to t h e following definition. DEFINITION 7.1.2. A space is called an I r -manifold (T > w), if it has a countable functionally open cover each element of which is homeomorphic to a f u n c t i o n a l l y open subspace of I r . Obviously, each I r - m a n i f o l d is locally c o m p a c t a n d LindelSf. O u r n e x t goal is to show t h a t R r - a n d I r -manifolds are A N R-spaces. This allows us to a p p l y the s p e c t r a l t e c h n i q u e d e v e l o p e d in C h a p t e r 6. T h e p r o o f of t h e following s t a t e m e n t is trivial and is left to the reader. LEMMA 7.1.3. A functionally open subspace of an A N R-space is an A N R space. PROPOSITION 7.1.4. I f a space X is represented as a countable union of its f u n c t i o n a l l y open subspaces, each of which is an A N R-space, then X is also an A N R-space. PROOF. If X is m e t r i z a b l e , t h e n t h e s t a t e m e n t follows from the well-known result of [161]. C o n s i d e r t h e case of n o n - m e t r i z a b l e X . It follows from L e m m a 1.1.18 a n d P r o p o s i t i o n 1.1.26 t h a t the space X is r e a l c o m p a c t . Therefore, we can a s s u m e t h a t X is a closed a n d C - e m b e d d e d s u b s p a c e of t h e space R r for s o m e u n c o u n t a b l e T. Let X ---- U{Xi" i E N } , w h e r e each Xi is a functionally o p e n A N R - s u b s p a c e of X . T h e p r o o f of t h e s t a t e m e n t is di vi ded into t h r e e parts. Case

1. First consider t h e case w h e n t h e r e are only finitely m a n y m e m b e r s

in t h e a b o v e r e p r e s e n t a t i o n of X .

In this case, w i t h o u t loss of generality, we

can a s s u m e t h a t X - X1 U X2. Let ~" X ~ [1,2] be a c o n t i n u o u s function such t h a t ~ - 1 ( i ) - X - X i , i - 1,2. Since X is C - e m b e d d e d in R r , t h e r e is a f u n c t i o n ~" R ~ --* [1,2] such t h a t ~ / X - ~. T h e sets ~ - 1 ( i ) , i - 1,2, are disjoint f u n c t i o n a l l y closed s u b s e t s of R r . C o n s e q u e n t l y , t h e r e exist functionally o p e n s u b s e t s O1 a n d 0 2 in R ~ such t h a t ~ - 1 ( i ) _c Oi a n d O1 M 0 2 - ~, i = 1, 2. Let Zi - R r - Oi, i -- 1, 2. E v i d e n t l y t h e sets Z1 a n d Z2 are functionally closed in R r a n d satisfy t h e following conditions:

Z1 U Z2 -

R r a n d Z0 M X C_ X0,

w h e r e Zo - Z1 M Z2 a n d Xo - X1 M X2. Let us show t h a t X0 is C - e m b e d d e d in Xo U Z0. In o r d e r to do this we have to verify t h e following two properties: (i) X0 is z - e m b e d d e d in X0 U Z0. (ii) If Z is f u n c t i o n a l l y closed in X0 U Z0 a n d Z M X0 - 0, t h e n Z and X0 can be f u n c t i o n a l l y s e p a r a t e d in Xo U Zo. Let Z be a f u n c t i o n a l l y closed s u b s e t of Xo. Since X0 is z - e m b e d d e d in X , t h e r e is a f u n c t i o n a l l y closed s u b s e t Z ~ in X such t h a t Z = Z ~ M X0. Since X is even C - e m b e d d e d in R ~ , t h e r e is a f u n c t i o n a l l y closed s u b s e t Z ~ in R r such that Z ~MX

-

Z ~. Let F -- Z ~ M ( X O U Z 0 ) .

It can easily be seen t h a t F is

7.1. NON-METRIZABLE MANIFOLDS

281

functionally closed in Xo U Zo and F M Xo = Z. This finishes the verification of property (i). Let us now verify property (ii). If Z is a functionally closed subset of Xo U Zo and Z M X0 = 13, then Z C Z0 - X. Since, by Corollary 6.4.8, every functionally closed subset of R r is C - e m b e d d e d in R r , we conclude t h a t Z is functionally closed in R r . But X is C - e m b e d d e d in R r . Consequently, X and Z are functionally separated in R r . This shows t h a t Xo and Z are functionally s e p a r a t e d in X0 U Z0 and finishes the verification of p r o p e r t y (ii). Thus, X0 is C - e m b e d d e d in X0 U Zo. By L e m m a 7.1.3, X0 is an A N R - s p a c e . Therefore there exists a retraction r0" V ~ X0, where V - is a functionally open neighborhood of Xo in X0 U Z0. Obviously the set (X0 U Z0) - V is functionally closed i n X o U Z o and ( X o U Z o ) - V c Z0-X. Therefore, ( X o U Z o ) - V is functionally closed in R r . Once again using C - e m b e d d e d n e s s of X in R r, we can conclude t h a t there exists a functionally open subset U of R r such t h a t X C U and U M (X0 U Zo) c_ V. It can now be easily seen t h a t by setting

ri(a) = { a,~'~

ifif aa Ee xi(X~176

CI Zi M U

we obtain a m a p ri" ( X M Zo) U (Zo M U) ~ X i such t h a t C ( r i ) ( C ( X i ) ) C C ( U M Z i ) / ( ( X N Zi) U (Z0 M U)), i = 1,2. Since X i is an A N R - s p a c e , there exists an extension gi" Gi ~ X i of the m a p ri, where Gi is a functionally open neighborhood of the set ( ( X M Z ~ ) U ( Z o M U ) ) in UMZ~, i = 1,2. A straightforward a r g u m e n t shows t h a t the set G = G1 U G2 is a functionally open n e i g h b o r h o o d of X in R r . It only remains to observe t h a t the map r" G --. X , defined by setting r(a) = ri(a) for a e Gi (i = 1,2), is a retraction. C a s e 2. Now we consider the case when X = U { X i " i E N } and Xi M X j = 13 for i ~ j , i , j E N . Consider a function ~o" X --+ R defined by setting ~o(Xi) = i, i E N. Since X is C - e m b e d d e d in R r , there is a function r R r --. R e x t e n d i n g ~o. T h e pairwise disjoint sets Ui = r 89 i + 1)) are functionally open in t:l r , and U i M X = X i for each i E N. Obviously, Xi is C - e m b e d d e d in Ui, i E N. Consequently, since each Xi is an A N R-space, there is a retraction ri" Vi ~ X i , where 1I/ is a functionally open in Ui (and, hence, in R r ) n e i g h b o r h o o d of Xi. Let V = U{Vi" i E N } and r(a) = ri(a) for each a E 1I/ and i E N . Clearly, r ~ V --. X is a retraction and V is a functionally open n e i g h b o r h o o d of X in Rr . C a s e 3. We are now in position to consider the general case. Let Gi = X1 U 9 .. U X i , i E N . T h e sets Gi are functionally open in X and U{Gi" i E N } = X . By (a), Gi is an A N R - s p a c e , i E N . T h e C - e m b e d d e d n e s s of X in R r g u a r a n t e e s the existence of a countable subset T C T and open subsets Oi in ~rT(X) such t h a t ~rTl(Oi)gl X = Gi (~rT" R r --* R T denotes the n a t u r a l projection onto the corresponding s u b p r o d u c t ) , i E N. Since T is countable, l r T ( X ) has a countable base and, consequently, we can fix a metric d which induces the topology of

282

7. NON-METRIZABLE MANIFOLDS

lrT(X).

Let

f~ = (~ 9 -r(X)" d ( ~ , ~ r ( X ) - O~)> 1}. It is easy to see t h a t cl(F~) C F i + l and ~ r T ( X ) -- U{Fi" i E N}. Let { F~, K i --

ifi=l,2 if i = 3, 4 , - . .

Fi - c l ( F i _ 2 ) ,

Let Li -- ~rTI(Ki) MXi, i E N. T h e n X - U{Li" i E N}, and L j M L i + 2 = 0 for each i E N and j _ i. Obviously, L~ is functionally open in G~ and consequently, by L e m m a 7.1.3, is an A N R - s p a c e , i E N . By Case 2, the spaces A ---- U{L2i-I: i E N } and B ---- U{L2~: i E N} are A N R - s p a c e s . Observe now t h a t A and B are functionally open subspaces of X and X - A U B. Therefore, by Case 1, X is an A N R - s p a c e . [-1 COROLLARY 7.1.5. E v e r y R r - m a n i f o l d is an A N R - s p a c e . COROLLARY 7.1.6. E v e r y I r - m a n i f o l d is an A N R - s p a c e . LEMMA 7.1.7. L e t Y E { R r, [-1, 1]r}, T >_ w. T h e n there exists a h o m e o m o r p h i s m f " y N • I ---, Y N

• I s u c h t h a t the f o l l o w i n g c o n d i t i o n s are satisfied:

(i) ~ r l f ( y , t ) -- t f o r each ( y , t ) e y g

• I , w h e r e 7ri" y g

• I --~ I d e n o t e s

the p r o j e c t i o n o n t o the s e c o n d f a c t o r .

(ii) f ( ( 0 , . . - , 0 , . . - ) , t ) = ( ( 0 , . - . , 0 , . . - ) , t ) f o r each t e I . (ii) f (y, O ) = (y, O ) / o r each y e y g . (iv) /((Yl,Y2, Y3,"" , Y n , ' " ) , 1) = ((Y2, Y3,"" , Y n , ' " ) , 1) / o r each { Y n } e yN.

PROOF. Let g" Y x Y x I ~ Y x Y be a reflection isotopy from L e m m a 2.3.4. T h e desired h o m e o m o r p h i s m / " Y N x I --~ Y N X I can be defined as follows: f((Yl,Y2, Y3,''" , Y n , ' ' ' ) , t )

-- ((g(Yl,Y2, t ) , y 3 , ' ' ' , Y n , ' ' ' ) , t ) .

[3 T h e following L e m m a is an i m m e d i a t e consequence of the above statement. LEMMA 7.1.8. L e t Y ~ . y g x y N x I --+ Y N

(i)

7ri~o(x , y,

E { R r, [-1, 1]r}, T > w.

T h e n there exists a m a p

x I s u c h t h a t the f o l l o w i n g c o n d i t i o n s are satisfied:

t ) - - t f o r each ( y , x , t) E y g x y N x I .

(ii) T h e r e s t r i c t i o n ~ / Y g x Y g

X (0,1)'yNxyN

x (0,1) ~ y N x

(0,1)

is a h o m e o m o r p h i s m .

(ii) ~ ( y , x , 0 ) = (y, 0) f o r each (y,x) e y Y x y g . (iv) ~ ( y , x , 1 ) = (x, 1) f o r each ( y , x ) E y N x y g . (v) ~o((0, . . . , 0, . . . ), (O, . . . , O, . . . ), t) -- ((O, . . . , O, . . . ), t) f o r each t e I .

7.1. NON-METRIZABLE MANIFOLDS

283

(vi) For each n >_ 1, there exists e E (0, 1) such that if t E [0, e] and 1 <_ i < n, then . . . , y~, . . . , y,~, . . . ), ~, t)

~((y~,

=

(w,

. . . , w)

f o r each ( { y n } , x , t ) E y N X y N X I (lri" y N x I ~ y i denotes the projection onto the product of the first i factors). (vii) For each n > 1, there exists e e (0, 1) such that if t C [ 1 - e, 1] and l <_i < n, then ~(y,

(~,...

, ~,...

, ~,...

), t)

=

(~1,...

, ~)

f o r each (y, { X n } , t ) E y N X y N X I.

COROLLARY 7.1.9. Let Y E { R r, [-1, 1]r}, T > w. Suppose that f o r a Polish space X and a map a" X ~ I, there is a h o m e o m o r p h i s m f " X x y N ~ X x y N such that ~ r x f ( x , y ) -- x f o r each ( x , y ) E X x y N . Then there exists a h o m e o m o r p h i s m f a " X x Y N __. X x Y l q satisfying the following conditions: (i) 7rxf(x(x, y) - x f o r each (x, y) e X • y N . (ii) / f x E a - l ( 0 ) , then f a ( x , y ) = f ( x , y ) f o r each y E y N . (ii) I f x E a - l ( 1 ) , then f a ( x , y ) = ( x , y ) f o r each y E y N . PROOF. The formula i ( x , y ) = ( x , y , a ( x ) ) , ( x , y ) E X x y N defines the map i" X x YN ~ X x YN x I. The desired homeomorphism f a can be now defined as the composition of the five maps indicated in the diagram X xyN

i--~X x y N x I

idx •

-1

~

X xY

N

xY

N

xI

----+X x y N X y N X I ~d-5--5~~X x y N X I ~

f •

---,

•

X x yN,

where ~ denotes the homeomorphism from L e m m a 7.1.8.

1"1

THEOREM 7.1.10. Let p" E ~ B be a locally trivial bundle with fiber Y E ( R r , I r } , T >_ W. I f the base space B is Polish, then p is trivial. PROOF. Take a countable, locally finite open cover (Un" n E w} of the base space B such t h a t for each n E w there is a homeomorphism hn" Un x y N __~ p - l ( u n ) satisfying the equation p h n -- ~rv~. We wish to construct a homeomorphism h" B x y N __. E such that ph -- lrs. Choose a closed cover { C n ' n E w} of B which refines { U n ' n E w} combinatorially. Consider the set ,4 consisting of all possible pairs ( A , h ( A ) ) , where A C w and h ( A ) " U {Cn" n e A } • y N __, p - l ( u { c

n.

n

E A})

284

7. NON-METRIZABLE MANIFOLDS

is a h o m e o m o r p h i s m satisfying the e q u a t i o n ph(A) = ~ru{Cn : nEA}. Let us introduce a p a r t i a l order on the set A. We say t h a t ( A I , h ( A 1 ) ) '< (A2, h(A2)) if and only if A1 C A2 a n d the h o m e o m o r p h i s m s h(A~) a n d h(A2) coincide on the set

(U{Cn" n 9 A 1 } - U { U n " n 9 A 2 - A1}) x y N . Let us now consider an a r b i t r a r y chain B in .4 and show t h a t it has an u p p e r b o u n d in .4. D e n o t e by T t h e collection {A: ( A , h ( A ) ) 9 B}, a n d let A* = U{A: A 9 T} a n d C(A) = U{Cn: n 9 A}, where A 9 T U { A * } . For each point b 9 C(A*) choose an e l e m e n t Ab 9 T such t h a t b 9 Ub C C(Ab), where Ub is open in C(A*) and Ub n u n - 0 for each n 9 A* - Ab. Obviously the relations

h(A*)/Ub x y N _ h(Ab)/Ub x y g , b 9 C(A*) define a h o m e o m o r p h i s m h(A*)" C(A*) x Y g ---. p - l ( C ( A * ) ) such t h a t ph(A*) -re(A*). It is not hard to see t h a t (A*,h(A*)) is an u p p e r b o u n d of the chain B in A. Since the chain B was chosen arbitrarily, we conclude, by Zorn's Lemma, t h a t the p a r t i a l l y o r d e r e d set .A has a m a x i m a l element. Denote it by ( A , h ( A ) ) . Obviously it only remains to show t h a t A -- w. Assume, to the contrary, t h a t this is not the case. Therefore there is an element m C w - A. Let C =

U{C,.,'n 9 A} a n d h = ( h ( A ) ) - l h r n / ( C n Urn) x y g . Consider a function fl" C --~ I such t h a t C n Crn C fl-l(O) and g -- cl(fl-l([O, 1]) C C N Urn. Let a = fl/C N Urn. By Corollary 7.1.9 (replacing in it X by C n UM and f by h), we have a h o m e o m o r p h i s m ha" (C NUrn) x y g ~ (C nurn) x y g satisfying the following two conditions h a / ( C n Crn) x y N = h / ( C n Crn) x y N and

h~i((C n urn)- K)

x y N - id.

T h e n h(A)ha" (C N Urn) x y N ~ p - l ( C n Urn) is a h o m e o m o r p h i s m such t h a t

h ( A ) h a / ( C N Cm) x y N = hrn/(C n Crn) x y N and,

h ( A ) h a / ( C n Um) - K) • y N - h ( A ) / ( C n U r n ) - K) x y N Consequently, t h e formula

g(b,y) = {

h(A)(b,y), h(A)ha(b,y), h~(b,y),

if (b,y) e ( C - K ) x y g if (b,y) e (C n Urn) x y N if (b, y) E Crn x y Y

7.1. NON-METRIZABLE MANIFOLDS

285

defines a h o m e o m o r p h i s m g: (C U Cm) • y N ___+p - l ( c U Gin). S t r a i g h t f o r w a r d verification shows t h a t (A U { m ) , g ) E .4 and ( A , h ( A ) ) -< (A U { m ) , g ) . Since the last relation c o n t r a d i c t s the m a x i m a l i t y of the element (A, h(A)), the proof is complete. 1-1

7.1.2. R ~ -manifolds.

THEOREM 7.1.11. The following conditions are equivalent for any space X : (i) X is an R ~"-manifold. (ii) There is an R ~ -manifold (unique up to a homeomorphism) P such that X is homeomorphic to the product P • R r. PROOF. For ~- -- w, the s t a t e m e n t follows from the results of C h a p t e r 1. C o n s e q u e n t l y we assume below t h a t T > w. First note t h a t it i m m e d i a t e l y follows from Definition 7.1.1 t h a t the p r o d u c t P x R ~ is an R r - m a n i f o l d for each R ~ -manifold P . Let us prove the implication (i) ~ (ii). By Corollary 7.1.5 a n d P r o p o s i t i o n 6.3.5, X is h o m e o m o r p h i c to the limit space of some factorizing w - s p e c t r u m S z { X a , p~, A ) consisting of Polish A N R - s p a c e s and soft limit projections. Let X = U { X i : i E N ) , where each Xi is functionally open in X and is h o m e o m o r p h i c to R ~ . T h e factorizability and w-completeness of the s p e c t r u m S x implies the -1 (X~~ where X ~ ~ -- P~o (Xi), existence of an index s 0 E A such t h a t Xi - Pao i E N . For each i E N consider the s p e c t r u m

8i-

{ p a ( X i ) , p ~ , ~ E A, ~ >_ s o ) .

Obviously the s p e c t r a Si are w-complete. All projections of these s p e c t r a are open, since the projections of the s p e c t r u m S z are open (see P r o p o s i t i o n 6.1.26). P r o p o s i t i o n 1.3.3 allows us to conclude t h a t the s p e c t r a Si, i E N , are factorizing. Note also t h a t X i = l i m S / , i E N. By our a s s u m p t i o n , X i ~ R r for each i E N. Consequently, r e p r e s e n t i n g R r as the limit space of the s t a n d a r d factorizing w-spectrum, consisting of c o u n t a b l e s u b p r o d u c t s of R r a n d the corr e s p o n d i n g s u b p r o d u c t s , and a p p l y i n g T h e o r e m 1.3.6 ( c o u n t a b l y m a n y times), we conclude t h a t for some index a E A, a _ s0, the spaces p a ( X i ) , i E N , are all h o m e o m o r p h i c to R W , and the restrictions p a / X i : X i --+ p a ( X i ) , i E N are all trivial bundles with fiber R r . Therefore, the space X a is an R ~ -manifold (because it is r e p r e s e n t e d as the union of open subspaces p a ( X i ) , i E N , each of which is h o m e o m o r p h i c to R W ) and the limit p r o j e c t i o n p c : X ----> X a is a locally trivial bundle with fiber R r (because, for each i E N, the r e s t r i c t i o n of t h e p r o j e c t i o n pa onto the s u b s p a c e X i is a trivial b u n d l e with fiber R r ). By T h e o r e m 7.1.10, the p r o j e c t i o n p c : X ---+ X a is a trivial b u n d l e with fiber R r . Therefore, the space X is h o m e o m o r p h i c to the p r o d u c t X a x R ~. S u p p o s e now t h a t P x R r ~ L x R r, where P a n d L are R ~ -manifolds. R e p r e s e n t R r as the limit space of the s t a n d a r d factorizing w - s p e c t r u m S -

286

7. NON-METRIZABLE MANIFOLDS

((R~)a,r~,exp~T}.

T h e n t h e p r o d u c t s P x R ~ a n d L x R r are the limit spaces

of t h e factorizing w - s p e c t r a

Sp-

( P • (RW)a, i d p • ~r~, expwT} a n d S L -

( L • ( R W ) a , i d L x ~r~,expwT}

respectively. A p p l y i n g T h e o r e m 1.3.6 to t h e s e s p e c t r a , we concl ude t h a t t h e p r o d u c t s P x R w a n d L x R ~ are also h o m e o m o r p h i c . T h e stability of R ~ m a n i f o l d s g u a r a n t e e s t h a t , in this case, P ~ L. T h i s shows t h e uni queness (up to a h o m e o m o r p h i s m ) of o u r r e p r e s e n t a t i o n . 13 COROLLARY 7.1.12. I f X is an R r -manifold, then X ~ X • R r. PROOF. B y T h e o r e m 7.1.11, X Consequently, X •

r~P•

~

P • R r, w h e r e P is an R ~ -manifold.

r xR r~P

•

r~X.

D

COROLLARY 7.1.13. Every R r -manifold is triangulable, i.e. can be represented as the product K • R r, where K is a locally compact polyhedron. PROOF. T h e case r -- w was c o n s i d e r e d in C h a p t e r 1. If T > w, t h e n T h e o r e m 7.1.11 allows us to r e p r e s e n t a given R v - m a n i f o l d X as t h e p r o d u c t P • R r, w h e r e P is an R ~ -manifold.

B y t h e t r i a n g u l a t i o n t h e o r e m for R ~ -manifolds,

t h e r e exists a locally c o m p a c t p o l y h e d r o n K such t h a t P ~ K • R ~. Obviously, X ~P

xR v~K

xR ~ xR r~K

COROLLARY 7.1.14.

x R r.

D

The following conditions are equivalent for any space X :

(i) X is an R r -manifold. (ii) X is homeomorphic to a functionally open subspace of R r . PROOF. T h e case T = w was c o n s i d e r e d in C h a p t e r 1. C o n s i d e r t h e case T > w. As i n d i c a t e d above, each f u n c t i o n a l l y o p e n s u b s p a c e of R r is an R r manifold. Conversely, s u p p o s e t h a t X is an R r -manifold. By T h e o r e m 7.1.11, X P • R r, w h e r e P is an R ~ -manifold. Obviously, P is h o m e o m o r p h i c to an o p e n s u b s p a c e of R ~ . C o n s e q u e n t l y , X is h o m e o m o r p h i c to a functionally o p e n s u b s p a c e of t h e p r o d u c t R ~ x R r ..~ R v. [--l COROLLARY 7.1.15. H o m o t o p y equivalent R ~" -manifolds are homeomorphic. PROOF. T h e case T -- w was c o n s i d e r e d in C h a p t e r 1. ~" > w.

C o n s i d e r t h e case

Let X a n d Y be h o m o t o p y e q u i v a l e n t R r -manifolds.

By T h e o r e m

7.1.11, X ~ P • R r a n d Y ..~ L • R r, w h e r e P a n d L are R ~ -manifolds. Since h o m o t o p y e q u i v a l e n t R ~ - m a n i f o l d s are h o m e o m o r p h i c , it suffices to show t h a t P a n d L are also h o m o t o p y equivalent. R e p r e s e n t X as t h e limit space of factorizing w - s p e c t r u m S x = ( X a , p~, A } , all spaces X a of which are h o m e o m o r p h i c to t h e p r o d u c t P x R ~ a n d all limit p r o j e c t i o n s pa of w h i c h are soft (even trivial b u n d l e s w i t h fiber R r ; c o m p a r e w i t h t h e p r o o f of T h e o r e m 7.1.11). Similarly, let Y -- S y , where Sy

= (Ya, q ~ , A } is a factorizing w - s p e c t r u m c o n s i s t i n g of spaces Y~,

7.1. NON-METRIZABLE MANIFOLDS

287

h o m e o m o r p h i c to the p r o d u c t L x R ~, a n d soft limit projections. C o n s i d e r two a d d i t i o n a l factorizing w-spectra: 81 - ( X , x I , p ~ , A } a n d 82 -- (Ya x I , q ~ , A } . It can easily be seen t h a t lim 31 - X x I a n d lim 82 = Y x I. B y our a s s u m p t i o n , t h e r e are m a p s f : X --, Y, g: Y --, X , H : X x I - , X and F : Y x I --, Y satisfying the following conditions: (i) H (x, 0) -- x for each x E X . (ii) U ( x , 1) -- g f ( x ) for each x E X . (ii) F ( y , O ) - y for each y E Y. (iv) F ( y , 1 ) - f g ( y ) for each y E Y. A p p l y i n g T h e o r e m 1.3.6 to t h e m a p s f, g, H a n d F , we can a s s u m e w i t h o u t loss of generality t h a t for some index c~ E A we have m a p s f a : X a --* Ya, ga : Ya --* X a , H a : X a x I --, X a and F a : Ya • I --~ Ya satisfying the following conditions: (v) H a ( x a, O) -- x a for each x a E X a . (vi) H a ( x a , 1) -- g a f a ( x a ) for each x a E X•. (vii) F a ( y a , O) -- Ya for each ya E Ya. (viii) F a ( y a , 1 ) - f a g , ( Y a ) for each ya E Y~. Therefore g a l a ~-- i d x . and f a g a ~- i d y . , i.e. the p r o d u c t s P • R w a n d L • R w are h o m o t o p y equivalent. B u t P ~ P • R w a n d L ~ L • R w. Consequently, P and L are h o m o t o p y equivalent as desired. [-] COROLLARY 7.1.16. A contractible R r - m a n i f o l d is h o m e o m o r p h i c to R r . COROLLARY 7.1.17. Let r > w and Y be a G~-subset of an R r - m a n i f o l d . T h e n the following conditions are equivalent: (i) Y is an R r -manifold. (ii) Y is an A N R-space. PROOF. T h e implication (i) ~ (ii) follows from Corollary 7.1.5. Let us show t h e validity of t h e implication (ii) ~ (i). S u p p o s e t h a t Y is a G~subset of an R r -manifold X . B y (ii) a n d P r o p o s i t i o n 6.1.7, Y is r e a l c o m p a c t . Consequently, by Corollary 6.4.12, Y can be r e p r e s e n t e d as an intersection of c o u n t a b l y m a n y functionally o p e n subsets of X . We can also a s s u m e t h a t X -P x R r, where P is an R ~ -manifold ( T h e o r e m 7.1.11). In this s i t u a t i o n , Y -7r-l~r(Y), where 7r" ( P x R ~) x R r - , P x R ~ denotes t h e projection. T h e softness of the p r o j e c t i o n 7r, coupled with P r o p o s i t i o n 6.1.21, g u a r a n t e e s t h a t ~r(Y) is a Polish A N R - s p a c e . Therefore, Y ~ 7r(Y) x R r ~ ~r(Y) x R ~ x R r. By T h e o r e m 2.3.9, t h e p r o d u c t lr(Y) x R ~ is an R ~ - m a n i f o l d . Consequently, by T h e o r e m 7.1.11, Y is an R r - m a n i f o l d . !1 COROLLARY 7.1.18. E a c h R r - m a n i f o l d is strongly ,4r, oo-universal. PROOF. A p p l y T h e o r e m 7.1.11 a n d L e m m a 6.5.5.

C1

288

7. NON-METRIZABLE MANIFOLDS

7.1.3. I r - m a n i f o l d s . corollaries.

T h e following s t a t e m e n t also has several i m p o r t a n t

THEOREM 7.1.19. T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t f o r any space X : (i) X is an I r - m a n i f o l d . (ii) There is an I ~ - m a n i f o l d ( u n i q u e up to a h o m e o m o r p h i s m ) P such that X is h o m e o m o r p h i c to the p r o d u c t P x I r.

T h e proof of this theorem is similar to the P r o o f of T h e o r e m 7.1.11, and is therefore omitted. COROLLARY 7.1.20. I f X is an I r - m a n i f o l d , t h e n X .~ X x I r.

COROLLARY 7.1.21. E v e r y I r - m a n i f o l d is triangulable, i.e. can be r e p r e s e n t e d as the p r o d u c t K x I r, where K is a locally c o m p a c t polyhedron.

COROLLARY 7.1.22. I f X is an I r - m a n i f o l d , t h e n the p r o d u c t X x [0, 1) is h o m e o m o r p h i c to a f u n c t i o n a l l y open subspace o f I r .

PROOF. Represent X as the p r o d u c t P x T r, where P is an I ~ -manifold ( T h e o r e m 7.1.19). By T h e o r e m 2.3.25, the product P x [0, 1) is homeomorphic to an open subspace of the Hilbert cube I W . It only remains to note t h a t the p r o d u c t X x [0, 1) ..~ P x [0, 1) x I r is functionally open in I W x I r ~ I r. [-7 T h e converse to Corollary 7.1.22 will be proved in the next section. COROLLARY 7.1.23. T h e I r - m a n i f o l d s X a n d Y are h o m o t o p y equivalent if a n d only if the p r o d u c t s X x [0, 1) a n d Y x [0, 1) are h o m e o m o r p h i c .

PROOF. Obviously the projections ~rx : X x [0, 1) ~ X and r y : Y x [0, 1) -~ Y are h o m o t o p y equivalences, and consequently, if the products X x [0, 1) and Y x [0, 1) are h o m e o m o r p h i c we can conclude t h a t the spaces X and Y are h o m o t o p y equivalent. Suppose now t h a t t h e / r - m a n i f o l d s X and Y are h o m o t o p y equivalent. Represent, by T h e o r e m 7.1.19, the spaces X and Y as the products P x I r and L x I r respectively, where P and L are I ~ -manifolds. As in the proof of Corollary 7.1.15 it can be seen t h a t the I ~ -manifolds P and L are homotopy equivalent. Consequently, by T h e o r e m 2.3.26, the products P x [0, 1) and L x [0, 1) are homeomorphic. If so, it only remains to note t h a t X x [0,1) ~ P x [0,1) x I r and Y x [ 0 , 1 ) ~ L x [ 0 , 1 ) x I r. [:]

COROLLARY 7.1.24. E v e r y contractible c o m p a c t I r - m a n i f o l d is h o m e o m o r p h i c to I r .

PROOF. Represent a given contractible compact I r - m a n i f o l d X as the product P x I r, where P is a compact I~ -manifold. Obviously, P is contractible and consequently, P ~ I W. T h e n X .~ P x I r ..~ I "~ x I r ,~ I r. [-7

7.2. I~ -MANIFOLDS

289

We conclude this section with the following t w o / r - c o u n t e r p a r t s of Corollaries 7.1.17 and 7.1.18. COROLLARY 7.1.25. Let T > w and Y be a Gs-subset of an I T - m a n i f o l d . the following conditions are equivalent: (i) Y is an I T -manifold. (ii) Y is an A N R-space.

Then

COROLLARY 7.1.26. Each I T -manifold is strongly Br,oo-universal.

Historical and bibliographical notes 7.1. The definitions of R r - and I r -manifolds presented here were given in [84], although the initial definition of I r manifolds was given by SSepin in [277]. Theorem 7.1.10 is due to C h a p m a n [67]. The results of subsection 7.1.2 were obtained in [84]. The results of subsection 7.1.3, except Corollaries 7.1.23 and 7.1.26, are due to SSepin [277]. Corollaries 7.1.23 and 7.1.26 were proved in [83].

7.2. T o p o l o g i c a l c h a r a c t e r i z a t i o n o f I v - m a n i f o l d s We begin this Section with some auxiliary considerations. DEFINITION 7.2.1. A space X is said to be homogeneous with respect to pseudo-character if any two points X l and x2 in X have the same pseudo character, i.e. r -- r X). Since the weight of any non-discrete AE(0)-space coincides with its pseudocharacter (this easily follows from the results of Section 6.4), we get the following simple observation. LEMMA 7.2.2. I f X is a non-discrete, homogeneous with respect to pseudocharacter, AE(O)-space of weight T > w, then r = T for each point x E X . LEMMA 7.2.3. Let T >_ w and S x = { X a , p ~ , A } be a factorizing T-spectrum, consisting of A N R - s p a c e s and soft projections. Suppose that the limit space X of the spectrum ,~x is homogeneous with respect to pseudo-character and w ( X ) > T. Then for each index ~ E A there exist an index ~' E A, ~' > ~, a n d t w o sections s "0 za ,.o i 1 . X a --* X a , of the projection p~ , such that z a ( X , ) f'l i l ( x a ) = 0. PROOF. Let a0 -- a. Take an arbitrary point z E X a o and consider its inverse --1 image Pao (z). Since the weight of the space X a o does not exceed T, the pseudocharacter of any point in Xa o is < T. Consequently, the pseudo-character of the set Pao -1 (z) in X is also <_ T. By our assumption, X is homogeneous with respect to pseudo-character. Therefore, by L e m m a 7.2.2, we may conclude that the fiber pa-lo(Z ) contains at least two points. Then, obviously, there exists an index o~1 e A such that c~1 > c~0 and I ( p ~ ) - ! ( z ) l > 1. The softness of the

290

7. NON-METRIZABLE MANIFOLDS

projection Paoal guarantees t h a t it has two sections i ~ i~- Xao -~ X a l such that ~ ill(Z). Let

i~

=

9

Clearly, Vx is a n o n - e m p t y 2 open subset of Xao. Let 7 < v+ (by v + we denote the minimal cardinal n u m b e r greater than T). Suppose t h a t for each f~, 1 < f~ < 7, we have already constructed an index a 0 e A, an open subset Vf~ in Xao and two sections i~, i~" Xao ~ Xf~ of the projection p~" X~ ~ Xao, satisfying the following conditions: (i) a~ < a~, whenever 1 _ 6 < / 3 < 7. (ii) a ~ - sup{a6" 6 < f~}, whenever f~ is a limit ordinal. (iii) V6 c Vf~, whenever 1 _ ~ < f~ < 7. (iv) V0 -- U{V~" 6 < f~}, whenever/3 is a limit ordinal. (v) = p,~, z~, whenever 1 s 6 < f~ < 7 and k = 0, 1. (vi) i~ = A{i~: ~ < f~}, whenever f~ is a limit ordinal and k = 0, 1. (vii) Va = {x ~ X,~ o " i~ ~ i~(x)}. We shall now construct an index a. r ~ A, an open subset V7 of X~ o and two sections iTk" X~ o ~ XT, k = 0, 1, of the projection Pao ~ , satisfying all the required properties. First assume t h a t 7 is a limit ordinal. It follows from conditions (i) and (ii) t h a t the collection { a a - f~ < 7} forms a chain in the indexing set A. Since 7 < r + , the cardinality of this chain does not exceed r. By the r-completeness of A there exists an index a7 = sup{aft" f~ < 7} E A. By the continuity of the s p e c t r u m 3 x , the space X 7 is naturally homeomorphic to the limit space of the s p e c t r u m {Xaa , pa~,/3, as 6 < 7 }. Therefore, by conditions (v) ~ .k __ i ~ and (vi), the maps z9k" Xao --* Xa~, k -- 0, 1, satisfying the equalities paaz~ for each k -- 0, 1 and each/3, with 1 < f~ < 7, are well defined. In particular, the 9k k 0, 1 are sections of the projection P(~0 a~ m a p s ~7, Let V.~- {x e Xao" i~ ~ i~(x)}. We shall verify conditions (iii) and (iv). Let/3 < 7. Consider an arbitrary point x e Vf~. By condition (vii), i~ ~ i~(x). But, by construction, pan.~Ta':k ~ = i~, k -- 0, 1. Consequently, Pa~Z7

91 (x) . This proves the inclusion V~ C V~ (condition (iii)). and obviously i~ ~ z.y Consequently, 117 D U{V~" fl < 7}. Let us prove the converse inclusion (and 9 # finish the verification of condition (iv)). Let x E V.r. By definition, z~ It is not hard to see t h a t there exists an index f~ < 7 such t h a t i~ ~ i~(x). Consequently, by condition (vii), x E V~ _c U{V~" f~ < 7}. Thus the above constructed objects satisfy all the required properties. 2Since z E 171.

7.2. Ir -MANIFOLDS Consider now the case ~ - f l + l . z E X(, o - V~, and let

291

Suppose t h a t V~ ~ X a o. Take a point

9 ~ = ~(~) ~ ~(X,o) c x,~. Consider the fiber p-l(x/~). Since the weight of the space Xao does not exceed af~ T, and the space X is homogeneous with respect to pseudo-character and has -1(~) weight strictly more than T, we conclude, by L e m m a 7.2.2, t h a t the fiber Pa~ contains at least two different points. Then there is an index a~ E A such t h a t a~ > a~ and

I(p~,)-~(~)l > 1 f~ Softness of the projection Pa~ a~ guarantees the existence of sections

9o ,.;1 . X a ~ ._~ Xa.r of the projection such that j~(xf~) ~ j ~ ( x z ) . The sections ,~, o~,,t Pao can now be defined by letting 9k

.k .k

~,.y = 3 / ~ / ~ ,

k

=

0,

1.

As above, if we let

0 v~ = {~ e X.o" ~(~) # ~~(~)}, then Vf~ _C VT. Observe also that, by construction, z E V ~ - V#. Thus the construction of the above objects, satisfying conditions (i) (vii), can be continued and carried out for each/~ < r +. Then we obtain an increasing (by conditions (iii) and (iv)) collection {V#:/~ < T +} of open subsets of Z o o . Since w ( X a o ) <_ T, this collection must stabilize, i.e. there must exist an index fl0 < r + such t h a t V#o = V/~ whenever/~0 _ ~ T +. It follows from the above construction t h a t this is possible only if V#o = X~ o. Let a ' = rio. It only remains to note that the sections za, .k , k = 0, 1, of the projection Pao a' have disjoint images in X a , . IN LEMMA 7.2.4. Let f : X --~ Y be a soft map between A N R-spaces. I f the space ~ d ~ ( X ) > ~ ( g ) >_ ~ , t h ~ f has two sections with disjoint images.

x i~ ho.~og~eo~ ~ith ~ p ~ a to p ~ d o - ~ h a ~ a ~

PROOF. Let T = w ( Y ) and represent X as the limit space of a factorizing T-spectrum S x -- { X a , p ~ , A } , consisting of A N R - s p a c e s and soft limit projections (Proposition 6.3.5). By T h e o r e m 1.3.4, there exist an index a E A and a map f a "* Y such t h a t f -- f ~ p a . It follows from L e m m a 7.2.3 t h a t the projection pc" X ---. Xo, of the spectrum S x has two sections i0 and il with disjoint images. Softness of f implies softness of the map f a (Lemma 6.1.15). Therefore f a has a section i: Y ~ X a . T h e n the desired sections of the m a p f can be defined as the compositions j0 = ioi and j l = ill. r]

292

7. NON-METRIZABLE MANIFOLDS

LEMMA 7.2.5. Let a square diagram

X

~Y

qo

Xo

Po

'~Yo

be a Cartesian square. I f q is a retraction and the map p has two sections with disjoint images, then the map po also has two sections with disjoint images. PROOF. Let r : Yo ---* Y be a section of the r e t r a c t i o n q and il, i2: Y ~ X be sections of the m a p p with disjoint images. Let j k -- qoikr, k -- 1, 2. Obviously t h e m a p s jk, k -- 1, 2 are sections of the m a p p0. Suppose t h a t j l (Y0) CI j2(Yo) 7~ 0, a n d take a point x0 from this intersection. T h e n there is a point y0 E Y0 such t h a t Jl(yo) -- x0 - J2(y0). Since the d i a g r a m is a C a r t e s i a n square, there is a unique point x E X with the p r o p e r t i e s qo(x) -- xo and p(x) - r(yo). At the same time, it is easy to check t h a t the two different points i l ( r ( y o ) ) and i2(r(yo)) of X also have the p r o p e r t i e s qo(ik(r(yo))) -- Jk(YO) -- xo and p ( i k ( r ( y o ) ) ) -- r(yo), k -- 1, 2. This c o n t r a d i c t i o n s shows t h a t t h e sections j l and j2 have disjoint images.

[-1

LEMMA 7.2.6. Let S -- { X n , p ~ + l , w } be an inverse sequence, consisting of locally compact and LindelSf A N R-spaces and soft proper short projections with Polish kernels. I f each of these projections has two sections with disjoint images, then the limit projection P0: l i m b -~ X0 is the trivial bundle with fiber the Hilbert cube I ~ . PROOF. If w ( X o ) ----w, the s t a t e m e n t has already been proved in Subsection 2.4.4 ( P r o p o s i t i o n 2.4.38). Therefore, we may assume t h a t w ( X o ) - T > w. Each of the spaces X n , n E w can be r e p r e s e n t e d as the limit space of a factorizing w - s p e c t r u m Sn - { X ~ , q ~ ' n , A } consisting of locally c o m p a c t A N R-spaces and soft limit projections ( P r o p o s i t i o n 6.3.5 and R e m a r k 6.3.6). In the m e a n t i m e , observe t h a t these s p e c t r a have the same indexing set A. This follows from the

7.2. /r-MANIFOLDS

293

equalities w ( X n ) = r , n E w. Since all short projections pnn+l of t h e s p e c t r u m ,9 are proper, soft and have Polish kernels, we m a y assume w i t h o u t loss of generality (by T h e o r e m 6.3.2) t h a t for each n E w the p r o j e c t i o n pr~+1 is t h e limit m a p of a Cartesian morphism

Mnn+l= {pn-bl,o. xn+l _.+ X n a , A } .

,gn+l --+ ,gn

consisting of p r o p e r and soft m a p s between locally c o m p a c t A N R - s p a c e s with c o u n t a b l e bases. Take an a r b i t r a r y index a E A and consider the c o u n t a b l e inverse s p e c t r u m , n + l , o , w } . Let X o = l i m S o and X = lim,9. T h e limit m a p of the s.= m o r p h i s m {qan} " ,9 --* ,5,, is d e n o t e d by qo. T h u s we have the following infinite commutative diagram X

~

~ Xn+l 9

"

*- X n

;-

9

"

"

~-XO

qn+l

qo

Xo

pn+l

"

qn

O

~

~ X~+l

9 ..

O

~n+l,o

~n

~._ X

?1' O

~

" " "

I'-Xo

0

S t r a i g h t f o r w a r d verification shows t h a t all t h e square s d b d i a g r a m s of t h e above d i a g r a m are C a r t e s i a n squares. In particular, the d i a g r a m P0 X -"

*-X0

qO

qo

x. is also a C a r t e s i a n square.

.x ~

By L e m m a 7.2.5, each of t h e m a p s Pn ..n+l 'o , n E W , has two sections with disjoint images. In this situation, the limit p r o j e c t i o n p~" X o -+ X ~ of the s p e c t r u m So is the trivial b u n d l e with fiber t h e Hilbert cube I "~ ( P r o p o s i t i o n 2.4.38). Since t h e last d i a g r a m is a C a r t e s i a n square, we easily conclude t h a t t h e limit p r o j e c t i o n p0: X -+ X0 of t h e s p e c t r u m ,9 is also t h e trivial bundle with fiber I w . [--I

294

7. NON-METI:tIZABLE MANIFOLDS

We need one more auxiliary lemma. LEMMA 7.2.7. Let X be a locally compact and L i n d e l 6 f space of weight r > w. T h e n X a d m i t s a closed embedding into a p r o d u c t P x I r , where P is an I ~ manifold. PROOF. Consider the o n e - p o i n t compactification a X of t h e space X. Identify a X with a (closed) subspace of the cube I r . Obviously X is functionally open in X . Therefore t h e r e is a functionally open subset V of I r such t h a t a X A V - X. T h e n X is closed in V. It only remains to note t h a t every functionally open s u b s p a c e of I r a d m i t s a r e p r e s e n t a t i o n as a p r o d u c t P x I r , where P is a I wmanifold. El We are now r e a d y to prove the following c h a r a c t e r i z a t i o n theorem. THEOREM 7.2.8. Let r >_ w. T h e n the following conditions are equivalent f o r every locally c o m p a c t and L i n d e l h f A N R - s p a c e X o f weight r" (i) X is an I r - m a n i f o l d .

(ii) X is strongly Br,oo-universal. (iii) F o r each c o m p a c t u m Y o f weight < r, the set of embeddings is dense in the space C r ( Y , X ). (iv) T h e set of embeddings is dense in the space C r ( D , X ).3 (v) X is h o m o g e n e o u s with respect to the p s e u d o - c h a r a c t e r . PROOF. T h e validity of t h e implications (i) ~ (ii) has been established in Corollary 7.1.26. I m p l i c a t i o n s (ii) ==~ (iii), (iii) ==~ (iv) and (iv) = ~ (v) are obvious. Let us prove t h a t (v) ~ (i). Recalling L e m m a 7.2.7, we may assume t h a t X is a closed subspace of t h e p r o d u c t P x I A, where P is an I ~ -manifold and A is an a r b i t r a r y set of c a r d i n a l i t y r. Since every closed subset of any n o r m a l space in C - e m b e d d e d , we see t h a t there is a r e t r a c t i o n r" V ---, X, where V is a functionally open n e i g h b o r h o o d of X in the p r o d u c t P x I A. T h e space V can be r e p r e s e n t e d in the same m a n n e r as the p r o d u c t L x I A, where L is a I ~ -manifolds. T h u s we have the following situation: T h e space X is closed in the p r o d u c t L x I A (where L is a I w -manifold) and t h e r e is a r e t r a c t i o n r" L x I A ~ X . We need some a d d i t i o n a l notations. If C c B C A, then A~" L x I B ~ L x I C and AB" L x I A ---. L x I B

d e n o t e the m a p s idL • r ~ and idL • 7rB respectively. Here r B" I B ---, I v and r B " I A ---+ I B d e n o t e t h e n a t u r a l projections onto the c o r r e s p o n d i n g subprod-

ucts. T h e subspace AB(X) of the p r o d u c t L x I B shall be denoted by X B . T h e properness of the m a p AB g u a r a n t e e s t h a t X B is closed in L x I B. Let 3Recall that D stands for the two-point discrete space.

7.2. I r - M A N I F O L D S

295

X ( B ) -- ABI(XB). The restriction of the map A~ onto X B is denoted by p~. Similarly, pB -- AB / X. A subset B C A is said to be admissible if for each point x E X (B) we have

pB~(~) = A~(x). As in the proof of Theorem 6.3.1 one can see t h a t admissible subsets of A have the following two properties: (a) the union of an arbitrary collection of admissible subsets is admissible. (b) each countable subset of A is contained in a countable admissible subset of A. (c) if C C_ B C A, and C and B are admissible subsets of A, then the maps PB" X --+ X B and p~" X B ---+X c are proper and soft. Next consider an arbitrary admissible subset B of A of cardinality [B[ < -i-. We are now going to show t h a t there is a countable admissible subset B' of A such t h a t the map p B u B ' . X B U B , '+ X B is the trivial bundle with fiber the Hilbert cube I • . Since the map PB" X ---+X B is soft, and since the space X is homogeneous with respect to pseudo-character, we conclude by L e m m a 7.2.4, t h a t there are two sections i0, i l" XB ' + X o f the map P B such t h a t

io(XB)ni~(x~)=O. Obviously, the sets io(XB) and i l ( X B ) are closed in the product L x I A. Consequently, there is a countable subset B1 of A such t h a t

~,B~ (~o(XB)) n ~B~ ( ~ ( x ~ ) ) = O. By property (b) of admissible sets, we may assume t h a t B1 is an admissible subset of A. In this situation, the map PBuBoBUBz" X BUB1 -'+ X BUBo

has two sections (namely, PBuBlio and PBuBIil) with disjoint images (here we assume that B0 -- 0). Continuing this process by induction, we construct countable admissible subsets Bn of A such t h a t the map B U BoU...U B,,U B,,+ z PBUBoU...UB, " X BUBoU...UB,+~ --* X BUBoU...UB,,

has two sections with disjoint images, n E w. The countability of each of the sets Bn ensures t h a t each of the indicated maps P BUBoU...UB,~+I BUBoU...UB,~ has Polish kernel and, by properties (a) and (c), are proper and soft. Therefore we obtain the inverse sequence BUBoU-..UBB+I {XBuBoU'"UB,,PBUBoU...UB,

,n E u)}

consisting of locally compact and Lindelhf A N R-spaces and proper and soft short projections with Polish kernels admitting t w o s e c t i o n s with disjoint images.

296

7. NON-METRIZABLE MANIFOLDS

Let B ~ -- U { B n : n E w}. T h e limit space of t h e above inverse sequence is obviously h o m e o m o r p h i c to the space X B u B ' , and the limit projection 4 of t h a t sequence coincides w i t h the m a p pgOB'.

X BUB ' ~

X B.

L e m m a 7.2.6 allows us to c o n c l u d e t h a t the m a p pBBUB' is the trivial bundle with fiber I W. Observe also t h a t the set B U B ~ is, by p r o p e r t y (a), also an admissible subset of A. F u r t h e r , by transfinite induction ( r e p e a t i n g the above described process of passing from B to B U B ~) we c o n s t r u c t a continuous well ordered s p e c t r u m ,.q = { X a , p aa + l , T} of length r, satisfying the following conditions: (i) X = l i m S . (ii) all spaces X a are locally c o m p a c t and Lindel6f ANR-spaces. (iii) all short projections p~+l are trivial bundles with fiber I W . (iv) the space X0 is a locally c o m p a c t A N R-space of c o u n t a b l e weight. In this situation, the space X is h o m e o m o r p h i c to the p r o d u c t X0 x I T . By condition (iv), the p r o d u c t X0 x I W is a n / W - m a n i f o l d ( T h e o r e m 2.3.21). It only remains to note t h a t , by T h e o r e m 7.1.19, the space X is an I T-manifold. V1 T h e following s t a t e m e n t , providing a topological c h a r a c t e r i z a t i o n of the Tychonov cube I T , follows directly from T h e o r e m 7.2.8. THEOREM 7.2.9. Let T > w. The following conditions are equivalent for each compact AR-space of weight 7": (i) X is homeomorphic to I T . (ii) X is strongly Bv,~-universal. (iii) for each compactum Y of weight < T, the set of embeddings is dense in the space Cv(Y, X ) . (iv) the set of embeddings is dense in the space C v ( D , X ) . (v) X is homogeneous with respect to pseudo-character. It follows from T h e o r e m 2.4.18 t h a t the equivalences (i) r (ii) in T h e o r e m s 7.2.8 and 7.2.9 are valid in the case ~- = w as well. T h e o t h e r conditions from T h e o r e m s 7.2.8 a n d 7.2.9 are not equivalent to condition (i). For instance, the closed unit interval is homogeneous with respect to p s e u d o - c h a r a c t e r a n d is an AR-compactum. T h e t h e o r e m s proved above have several corollaries. COROLLARY 7.2.10. Let T >_ w. The product X x I T is an I T-manifold if and only if X is a locally compact and LindelSf A N R - s p a c e of weight < T. PROOF. If T -- W, then the s t a t e m e n t is true ( T h e o r e m 2.3.21). Assume t h a t T > w. If the p r o d u c t X x I r is an I r - m a n i f o l d , t h e n the space X, as a r e t r a c t of an I r -manifold, is a locally c o m p a c t and Lindel/if A N R-space of weight __ T. 4which maps X B U B, onto the space Z B

7.2. I ~ -MANIFOLDS

297

Conversely, since the p r o d u c t X x I r is a locally c o m p a c t and Lindelbf A N R space of weight T, it suffices, by T h e o r e m 7.2.8, to verify the h o m o g e n e i t y with respect to pseudo-character of the p r o d u c t X x I ~ . The last fact is an i m m e d i a t e consequence of an easy observation t h a t the Tychonov cube I r is h o m o g e n e o u s with respect to pseudo-character. F-I Similarly we have the following.

COROLLARY 7.2.11. L e t T >_ w. T h e p r o d u c t X x I v is h o m e o m o r p h i c to the cube I v i f and only i f X

is a c o m p a c t A R - s p a c e o f w e i g h t <_ T.

The following two s t a t e m e n t s do not have metrizable c o u n t e r p a r t s . COROLLARY 7.2.12. L e t T > w. I f the p r o d u c t o f two spaces is an I V - m a n i f o l d (respectively, is h o m e o m o r p h i c to the cube I r ), t h e n at least one o f these spaces is an I r - m a n i f o l d (respectively, is h o m e o m o r p h i c to I ~" ).

PROOF. Suppose t h a t X1 x X2 is a n / r - m a n i f o l d . T h e n each of the spaces X I and X2 is a locally c o m p a c t and Lindelbf A N R - s p a c e of weight < T. Assume, contradicting our hypothesis, t h a t neither X1 nor X2 is an I r -manifold. Then, by T h e o r e m 7.2.8, there are points xz E X1 and x2 E X2 such t h a t r X1) < T and r X2) < T. It only remains to note t h a t in this situation the pseudo-character of the point (Xl, x2) in X1 x X2 is strictly less t h a n T. This contradiction completes the proof. [-! COROLLARY 7.2.13. I f the square o f a space X is h o m e o m o r p h i c to the Tyc h o n o v cube I r , T > w, t h e n X

is h o m e o m o r p h i c to I r .

We are now in position to prove the converse to Corollary 7.1.22. COROLLARY 7.2.14. I f the p r o d u c t X x [0, 1) can be e m b e d d e d in the T y c h o n o v cube I v , T > w, as a f u n c t i o n a l l y open subspace, t h e n X

is an I v - m a n i f o l d .

PROOF. Obviously the p r o d u c t X x [0, 1), as a functionally open subspace of I r , is an I r -manifold. By Corollary 7.2.12, X must be an I v -manifold. E] We conclude this Section by presenting the following s t a t e m e n t .

COROLLARY 7.2.15. L e t T > w.

E a c h c o m p a c t u m o f w e i g h t T, a d m i t t i n g a

soft m a p onto the T y c h o n o v cube I r , is h o m e o m o r p h i c to I r .

PROOF. It suffices to apply P r o p o s i t i o n 6.5.7 and T h e o r e m 7.2.9.

E]

H i s t o r i c a l a n d bibliographical n o t e s 7.2. The results of this Section are basically obtained in [277]. T h e equivalence of conditions (ii) and (iii) with conditions (i) and (iv) in T h e o r e m s 7.2.8 and 7.2.9 was observed in [83]. Corollaries 7.2.14 and 7.2.15 also are due to the author.

298

7. NON-METRIZABLE MANIFOLDS 7.3. T o p o l o g i c a l c h a r a c t e r i z a t i o n

of R r -manifolds

We begin with the following statement. LEMMA 7.3 91 9 L e t S = { X n , e"n+l n , w } be an inverse sequence, consisting of ANR-spaces

and soft short projections with Polish kernels.

Suppose that f o r

each n E w, the space X n + l contains a C - e m b e d d e d copy of the product X n x R ~ such that p n + l / ( x n

x R ~ ) = ~rx~, where r x , "

onto the first coordinate.

X n • R ~ -+ X n is the projection

T h e n the limit projection po: X = l i m S --~ X o is the

trivial bundle with fiber R ~ .

PROOF9 The case w ( X o ) = w was considered in Chapter 2 (see Proposition 29 Consider the case w ( X o ) = T > w. Since each sort projection of the spectrum S has Polish kernel, we conclude t h a t w ( X n ) = r for each n E w. Represent the space Xn, n E w, as the limit space of a factorizing w-spectrum S n - { X ~ , q~,n, A } , consisting of Polish A N R - s p a c e s and soft limit projections. Observe, in the meantime, that the indexing sets of all these spectra coincide with A (which has cardinality r). Since all short projections pn+l of the spectrum S are soft and have Polish kernels, we see, by T h e o r e m 6.3.2, that pr~+1, n E w, is the limit of some Cartesian morphism

Mnn + l = {pnn+l'~149 Xan+l --+ X n , A } " Sn+l --+ Sn, consisting of soft maps between Polish A N R - s p a c e s . For each a E A, consider the inverse sequence S,, - {X~,,pnn+l'a,w}, and let Xo = lim So. If/~ > a, then there is a Cartesian morphism M ~ = {q~,n. X ~ --+ x n , w } " S 8 --+ S o ,

consisting of soft maps q~,n, n E w. Denote by q~ the limit map of the morphism M~. Thus, the following infinite commutative diagram 5 arises" X

Dr "'"

pnn+l D- X n

~ Xn+l

,~ "'"

*-Xo

qn+l

qo

O

,...n+l,a

Xa

~

9 ..

,. x

n+l

Pn

5Compare with the proof of Lemma 7.2.6.

~

X

n o

o ~

...

7.3. Rr -MANIFOLDS

299

S t r a i g h t f o r w a r d verification shows t h a t the limit space of the s p e c t r u m S ' = { X a , q ~ , A } coincides with the space X , and t h a t all newly arising square diag r a m s are also C a r t e s i a n squares. B y a s s u m p t i o n , the space X n + I contains a C - e m b e d d e d copy of the p r o d u c t X n x R ~ so t h a t p n + l / ( x n x R ~ ) = 7rx,. Consider the factorizing w - s p e c t r u m S " = { X ~ x R ~ ,q~,n x idRw , A } ,

t h e limit space of which is t h e p r o d u c t X n x R "~ . Since X n x R "~ is C - e m b e d d e d in

Xn+l,

we

see t h a t the s p e c t r u m Sn+l/(Xn

x R w ) -- {clx2+1 q n + l ( x n x R w ), ua"B'n+l, A } ,

also having the p r o d u c t X n x R ~ as its limit, is factorizing. Consequently, by T h e o r e m 1.3.6, we m a y a s s u m e t h a t t h e s p e c t r a S " and 3 n + l / ( X n • R ~ ) are isomorphic. A n a l y z i n g the m a p s pnn+l and ~rx~ (using results of Section 6.3), we conclude t h a t the indexing set A contains a cofinal and w-closed subset A n , n E w, such t h a t for each c~ E A n t h e space X n + l contains a closed copy of the p r o d u c t X an x R ~ so t h a t p n T l , ~ jI ( x ~n

x

R w ) = 7rx:,,

w h e r e ~rx2" X ~ • R ~ -+ X an d e n o t e s the projection onto t h e first coordinate, n E w. Let A' -- n { A n : n E w}. B y P r o p o s i t i o n 1.1.27, t h e set A' is cofinal a n d w-closed in A. In particular, A' ~ 0. Take a E A'. T h e n t h e limit m a p p~" X a -+ X ~ of the s p e c t r u m S a is, by the case considered above (recall t h a t w ( X ~ - w ) , the trivial bundle w i t h fiber R "J . Using t h e fact t h a t the d i a g r a m po X

*X0

qO

q~

Xa

P~

~ X ao

is a C a r t e s i a n square, we see t h a t the limit p r o j e c t i o n p0 of t h e s p e c t r u m S is also the trivial b u n d l e with fiber R ~ .

KI

LEMMA 7.3.2. Let X be an A N R - s p a c e of weight T > w, and suppose that f o r each space Y of R - w e i g h t < T the set of C - e m b e d d i n g s is dense in the space C r ( Y , X ) . Suppose, in addition, that a soft m a p p" X -+ Z , where w ( Z ) < T, is also given. T h e n X contains a C - e m b e d d e d copy of the product Z • R ~ so that p / ( Z x R ~ ) -- ~rg, where ~rz" Z x R ~ -+ Z denotes the projection onto the first coordinate.

300

7. NON-METRIZABLE MANIFOLDS

PROOF. Since the map p is soft, Z is an A N R - s p a c e (Proposition 6.1.21). T h e n the p r o d u c t Z • R ~ is also an A N R - s p a c e and, consequently, by Corollary 6.3.8, the R-weight of the p r o d u c t Z • W is strictly less t h a n T. By L e m m a 6.5.1, there is a collection {/g~: t E T}, IT] -- w ( Z ) , of countable functionally open covers of the space Z, satisfying the conditions of t h a t Lemma. Let Ht - p-1 (L/~), t E T. By the softness of the m a p p, there is a section i: Z --~ X of p. T h e n the neighborhood B(i~rz, {/act: t E T})

of the point iTrz: Z x R '~ --. X , in the space C r ( Z x R '~ , X ) , contains at least one C-embedding. Denote it by g. By the choice of the above neighborhood, we have pg = lrz. Therefore, g is the desired C - e m b e d d i n g of the product Z x R ~~ into the space X. V! Now we are ready to prove the characterization theorem for R r -manifolds (and, in particular, of the space R r itself). THEOREM 7.3.3. Let r > w. The following conditions are equivalent for any A ( N ) R - s p a c e X of weight r:

(i) x i~ ho~omo~phi~ to n " ( ~ p ~ c t i ~ d y , x ~ ~n n " - m ~ i f o l d ) . (ii) X is strongly ~4r,~-universal. (iii) For each space Y of R - w e i g h t < r, the set of C-embeddings is dense in the space C r ( Y , X ) . PROOF. Implication (i) - - ~ (ii) was proved in Corollary 7.1.18. Implication (ii) ~ (iii)is trivial. Let us prove implication (iii) ==~ (i). We consider only the absolute case (the remaining case can be proved in a similar way). Since X is an AR-space of weight T, we may assume t h a t X is a retract of the space R A, where A is an a r b i t r a r y set of cardinality r. Let B be an arbitrary admissible subset of A (see the proof of T h e o r e m 6.3.1) of cardinality < T. T h e n the map 7 r s / X : X ~ i t s ( X ) , where ~rs: R A ~ R s is the projection onto the corresponding subproduct, is soft. Consequently, by L e m m a 7.3.2, the space X contains a C - e m b e d d e d copy of the product 7 r s ( X ) x R ~~ so t h a t 7rB/(TrB(X) • R w ) = 7r,rB(x), where 7r,rB(x) : 7rB(X) x R ~~ --. 7rB(X) is the projection onto the first coordinate. Since the R-weight of the product 7rB(X) x R ~ is less t h a n T, we see (using results of Section 6.3) t h a t there is a countable admissible subset B ~ of A, satisfying the following conditions: (a) T h e restriction of the projection 7rBUB, OIltO 7rB(X ) • R w (considered as a C - e m b e d d e d subspace of the space R A) is a homeomorphism. (b) T h e subspace 7rBuB,(TrB(X) X R ~~) is C - e m b e d d e d in 7rBuB,(X). In this situation the space 7 r B u s , ( X ) contains a C - e m b e d d e d copy of the p r o d u c t 7rS(X) X R ~~ so t h a t

~ "

/(~.(x)

• n~

)= ~.(~),

7.3. Rr -MANIFOLDS

301

where lrguB'. RBUB ' _._+R B is t h e p r o j e c t i o n onto the c o r r e s p o n d i n g s u b p r o d u c t . K e e p i n g in m i n d t h a t t h e union of an a r b i t r a r y collection of admissible subsets is admissible, by transfinite i n d u c t i o n ( r e p e a t i n g the c o r r e s p o n d i n g p a r t of the proof T h e o r e m 7.2.8) we c o n s t r u c t a continuous well ordered s p e c t r u m S x = { X ~ , p ~ , T} of length T SO t h a t the following conditions are satisfied: (1) X - lira S x . (2) All spaces X a are AR-spaces. (3) All short projections p ~ + l are soft and have Polish kernels (since the above c o n s t r u c t e d sets B ' are countable). (4) For each a < T the space X a + l contains a C - e m b e d d e d copy of the p r o d u c t X a x R ~ so t h a t p,~+l/(x,~ x R ~ ) = 7rx~, where ~rx~" X a x R w ~ X a is the projection onto the first coordinate. (5) X0 is a Polish AR-space. A p p l y i n g transfinite i n d u c t i o n one more time and using L e m m a 7.3.1, We see t h a t X is h o m e o m o r p h i c to the p r o d u c t X0 x R r ~ R r . T h e p r o o f is complete. I--1 This result has several consequences. COROLLARY 7.3.4. The product X x R r , T > w, is an R r -manifold if and only if X is an A N R-space of weight <_ T. PROOF. If T = W, t h e n the s t a t e m e n t has already been proved in C h a p t e r 2 (see T h e o r e m 2.3.22). Suppose t h a t T > w. If the p r o d u c t X x R r is an R r -manifold, t h e n the space X , as a r e t r a c t of R r -manifold, is an A N R-space of weight _ r. Let us prove the sufficiency of our condition. Thus, let X be an A N R - s p a c e of weight < ~-. Obviously, the p r o d u c t X x R r is an A N R - s p a c e of weight < T. Consequently, by T h e o r e m 7.3.3, it suffices to verify t h a t for each space Y of R-weight < r, the set of C - e m b e d d i n g s is dense in the space C r ( Y , X x R r ). Take an a r b i t r a r y set A of cardinality T, and consider a m a p f " Y ---+ X x R A and an a r b i t r a r y basic n e i g h b o r h o o d 0 - B ( f , {Lit" t E T}), ITI < T, of the point f in the space C r ( Y , X x R A ) . Let A = max{w, ITI}. T h e uncountability of T implies t h a t A < T. R e p r e s e n t X as the limit space of a factorizing A-spectrum S z = { X B , p ~ , e x p ~ A } , consisting of A N R - s p a c e s of weight A and soft limit projections. T h e space R A can also be r e p r e s e n t e d as the limit space of the s t a n d a r d factorizing A-spectrum S = { R B, 7rB, e x p ~ A }. T h e n the p r o d u c t X x R A is the limit space of the factorizing A-spectrum

• S = {X.

• R',p

•

In this situation, by L e m m a 6.5.1, there is an element B E e x p ~ A such t h a t

o'=

{g e c

(Y,X x RA) 9 (pB x

B)g = (;B x

B)f}

302

7. NON-METRIZABLE MANIFOLDS

is a neighborhood of f in the space C r ( Y , X • R A ) , contained in O. R - w ( Y ) <_ T, there is a C-embedding j " Y ~ R A - B . Let

Since

jB" X B • R B-+ X • R B

be a section of the map pB • i d R s .

Then the desired C-embedding of Y into

X x R A, contained in O ~ (and, consequently, in O), can be defined by letting i--jB(PB

This completes the proof.

X 7rB)f /k j " Y ~

X x R B x R A-B.

0

A similar argument proves the following statement. COROLLARY 7.3.5. T h e p r o d u c t X x R r is h o m e o m o r p h i c to R r , T >_ w, i f a n d only X

is an A R - s p a c e o f w e i g h t ~ T.

If r > w, then Corollary 7.3.5 can be proved using a different approach. Namely, we can apply the following statement to the projection ~rx" X x R r Rr .

COROLLARY 7.3.6. L e t r > w. E v e r y space o f w e i g h t r , a d m i t t i n g a soft m a p o n t o R r , is h o m e o m o r p h i c to R r .

PROOF. Apply Theorem 7.3.3 and Proposition 6.5.6.

E!

H i s t o r i c a l a n d bibliographical n o t e s 7.3. All results of this Section are due to the author [84]. As already shown (Corollary 7.1.14; compare with Corollary 7.1.22), the class of functionally open subspaces of R r coincides with the class of R r -manifolds. The problem of the satisfactory topological characterization of open (not necessarily functionally open) subspaces of R r is still open. Corollary 6.4.10 and Proposition 6.1.7 show that if an open subspace of R r is not functionally open, then it is not even an AE(0)-space and, consequently, we cannot apply the spectral theory developed here. A similar problem with respect to open subspaces of Tychonov cubes also remains open.

7.4. T r i v i a l b u n d l e s In this Section we extend Theorems 2.4.32 and 2.4.33 to the non-metrizable case.

7.4. TRIVIAL BUNDLES

303

7 . 4 . 1 . T r i v i a l b u n d l e s w i t h f i b e r R r . Let a cardinal n u m b e r T > w and a m a p f" X ~ Y be given. T h e n for each m a p g" Z ---, X , we denote by C ~ ( Z , X ) the subspace of the space C r ( Z , X ) , consisting of maps h" Z ~ X such t h a t fh=

fg.

THEOREM 7.4.1. Let T > w and f " X ~ Y be a soft map between A N R spaces with w ( X ) = 7-. Then the following conditions are equivalent: (i) f is the trivial bundle with fiber R r . (ii) For each space Z of R - w e i g h t <_ T and f o r any map g" Z ~ X , the set of C-embeddings is dense in the space C g r ( Z , X ) . PROOF. We begin with implication (i) ~ (ii). Consider the projection Try" Y • R A ~ Y , where A is an arbitrarily chosen set of cardinality T. Take a space Z of R-weight < T and any map g" Z ~ Y x R A. Let {b/t" t E T} C c o v ( Y x R A ) , ITI < T. Let A = max{wlTI} and observe t h a t A < r. From the results of Section 6.3 and L e m m a 6.5.1 it follows t h a t there exists a subset B of A of cardinality A such t h a t the set o =

{h e

c~(z, r

x

n ~ ) 9 ( i ~ . • ~ . ) h = (idy • ~B)g}

is a neighborhood of the point g in the space c g ( z , Y x R A ) , contained in the n e i g h b o r h o o d of the same point generated by the covers Ht, t E T. Also, consider a C - e m b e d d i n g ~" Z --+ R A - B . T h e n the desired C - e m b e d d i n g , c o n t a i n e d in the n e i g h b o r h o o d O, can be defined by letting h : ( i d y x l r B ) g A ~.

Let us now prove the implication (ii) ~ ordered continuous inverse spectra

(i).

Consider a factorizing well

SX = { X a , p ~ +1, 1 _< c~ < T}, S y = {Ya, qaa + l , 1 __< a < T} and a m o r p h i s m {fa" X a --* Za}" ,Sx --~Sy between them, satisfying conditions of T h e o r e m 6.3.1. We now need to introduce some additional notation: (a) Z~ +1 denotes the fibered p r o d u c t of the spaces X a a n d Ya+l with respect to the maps f a and q~+l; the canonical projections of this p r o d u c t are denoted by ~oa a+l" Z~ +1 --, X a and c a + l . Z ~ + I __~ Ya+l 1 < a < T (b) Za denotes the fibered p r o d u c t of the spaces X a and Y with respect to the maps f a and qa; the corresponding canonical projections are denoted by ~oa" Za --~ X a and Ca" Za --* Y, 1 _< a < T. (C) h~ +1" X a + l --+ Z~ +1 denotes the diagonal p r o d u c t of the maps p~+l a

'

--

"

and f a + l , 1 _< c~ < T. (d) ha" X ~ Za denotes the diagonal product of the maps pa and f , 1 _< C~
Z1 ~ Z0.

304

7. NON-METRIZABLE MANIFOLDS (f) taa + l " Z a + l

Z a denotes the diagonal p r o d u c t of the maps Paa + l ~ a + l and r 1 <_ a < T. (g) ra" Z a ~ Z~ +1 d e n o t e s the diagonal p r o d u c t of the maps ~ a and qa+lCa, 1 ~_< T. S t r a i g h t f o r w a r d verification shows t h a t the g e n e r a t e d inverse s p e c t r u m

s = { z . , tg +~, ~} has the space X as its limit, and the limit projection to" lira S --, Z0 coincides with the originally given m a p f . It follows from the above discussion t h a t the square d i a g r a m t~+ ~ Za--1

,~ Z a

ra

r

Xa+l

haa+ 1

,~Z~ +1

is a C a r t e s i a n square, 1 _< a < T. consequently, the maps tg +1, a E T, are soft and have Polish kernels (since, by T h e o r e m 6.3.1, the maps haa and fl have these properties). Let us show t h a t for each a, 1 _< a < T, there is an index f~(a) > a such t h a t the projection t~ (a)" Zf~(a) --* Za of the s p e c t r u m S has a Polish kernel and the space Z~(a) contains a C - e m b e d d e d copy of the p r o d u c t Z a x R • so t h a t tfla(a)/(Za x R " ) = ~rz~, where, as usual, 7rz." Za x R ~ ---, Z a is the projection onto the first coordinate. As follows from the proof of T h e o r e m 6.3.1 in order to show the last fact it suffices to prove the existence in X of a C - e m b e d d e d copy of the p r o d u c t Za x R w so t h a t h a / ( Z a x R ~ ) =Trz~. Since ha i s a s o f t map, there is a m a p i" Za x R ~ --, X such t h a t hai = 7rz.. By L e m m a 6.5.1, t h e r e is a subcollection {U~ e T e T} of cardinality [T[ = w ( X a ) < T of the collection c o v ( X a ) such t h a t the conditions of L e m m a 6.5.1 are satisfied. Let /.It = p~l(b/;), for each t c T. T h e n , by our a s s u m p t i o n , there is a C - e m b e d d i n g g" Z a x R ~ ---, X such t h a t the following conditions hold: (h) T h e maps g and i are { U t ' t e T}-close.

(i) /g = / i . We claim t h a t g is the desired C - e m b e d d i n g . that

Indeed, it suffices to observe

hag = ( p a A f )g = p a g A f g = p a i A f i = ( p a A f )i = hai = ~rz,:,.

7.4. TRIVIAL BUNDLES

305

Thus indexes f~(a), satisfying the above conditions, exist for each a. In this situation, applying L e m m a 7.3.1, by straightforward transfinite induction (compare with the proof of Theorem 7.3.3) we reach the final c o n c l u s i o n - f is the trivial bundle with fiber R r . D The following s t a t e m e n t is an immediate consequence of the above Theorem. COROLLARY 7.4.2. L e t T > w, .f" X spaces, a n d w ( X )

~_ r .

~

Y

be a s o f t m a p

T h e n the c o m p o s i t i o n f ~ z ,

between AN R-

w h e r e l r x " X x R r ---. X

is

the p r o j e c t i o n o n t o the f i r s t c o o r d i n a t e , is the trivial b u n d l e w i t h fiber R r .

COROLLARY 7.4.3. L e t T > w a n d f : X ---. Y be a s o f t m a p o f an R r - m a n i f o l d X

o n t o a space Y o f w e i g h t < r .

T h e n f is the trivial b u n d l e w i t h f i b e r R r .

PROOF. By Theorem 7.1.11, the R r -manifold X is homeomorphic to the product P • R r , where P is an R ~ -manifold. Represent the product P • R r as the limit space of the s t a n d a r d factorizing A-spectrum S-

{ P • (R;~) a, i d p • ~ ,

exp;~r}

where r~" (R)') f~ ~ (R)') a denotes the projection onto the corresponding subproduct and A = m a x { w , w ( Y ) } . Then there are an index a e e x p ~ r and a map f a : P x (R'X) a ~

Y

such t h a t f = f a ( i d p x 7ra), where ~a: R r ~ ( R X ) a is the projection. The softness of the maps f and i d p x ~a guarantees the softness of the map f~ (Lemma 6.1.15). Since the map i d p x ~a is the trivial bundle with fiber R r , Corollary 7.4.2 completes the proof. W1

7.4.2. T r i v i a l b u n d l e s w i t h f i b e r I r . Replacing Lamina 7.3.1 by Lamina 7.2.6 in the proof of T h e o r e m 7.4.1, we obtain the following result. THEOREM 7.4.4. L e t T > W a n d f " X spaces w i t h w ( X )

---+ Y

be a s o f t m a p

between ANR-

-- T. T h e n the f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

(i) f is the trivial b u n d l e w i t h fiber I r . (ii) F o r each c o m p a c t space Z o f w e i g h t ~_ r a n d f o r a n y m a p g" Z ---. X , the set o f e m b e d d i n g s is d e n s e in the space C g ( Z , X ) .

(iii) T h e set o f e m b e d d i n g s is d e n s e in the space C ~ ( X x D , X ), w h e r e ~r" X x D ~ X

is the p r o j e c t i o n .

COROLLARY 7.4.5. L e t T > w, f : X ---* Y be a p r o p e r a n d s o f t m a p b e t w e e n locally c o m p a c t a n d L i n d e l h f A N R - s p a c e s , f rx,

where r z: X x I r ~

trivial b u n d l e w i t h fiber I r .

X

and w(X)

~_ T. T h e n the c o m p o s i t i o n

is the p r o j e c t i o n o n t o the f i r s t c o o r d i n a t e , is the

306

7. NON-METI:tIZABLE MANIFOLDS

COROLLARY 7.4.6. L e t T > w a n d f " X --~ Y be a p r o p e r a n d soft m a p o f an I r -manifold X

o n t o a space Y o f w e i g h t < T. T h e n f is the trivial b u n d l e w i t h

fiber I r .

In connection with T h e o r e m 7.4.4, the following question arises naturally [277]: 9 Is the soft map between c o m p a c t A N R-spaces, whose fibers are all homeomorphic to I r and which satisfies condition (iii) for T - - w , the trivial bundle with fiber I T . T h e o r e m 7.4.4 allows us to answer the above question negatively. Indeed, consider an arbitrary soft map f : I ~1 ~ I wl , all fibers of which are homeomorphic to I ~1 and which does not a d m i t two sections with disjoint images (such maps exist: take an wl power of the soft map ~: I W ~ I ~ of the Hilbert cube onto itself, all fibers of which are the copies of the Hilbert cube and which, nevertheless, is not the trivial bundle with fiber I "~ [300]). Let r : I wl x I ~ ~ I wl be the projection onto the first coordinate. We claim the desired map is the composition f~r. Obviously, all fibers of the map fir are copies of the Tychonov cube I ~1. Since the projection lr satisfies condition (iii) of T h e o r e m 7.4.4, the composition f r also does. We are now going to outline why f ~ is not the trivial bundle. Assume the contrary, i.e. suppose t h a t f r is the trivial bundle with fiber I v . Take a section i: I ~1 ---. I ~1 of the map f and a section j : I "~1 ---, I "~ • I "~ of the projection ~r. Also take a countable collection {Lli: i E w} of open covers of I "~ satisfying the conditions of L e m m a 6.5.1. Let ))i - {I ~ x 3/1, i E w}. Then, by condition (iii) of T h e o r e m 7.4.4, there exist maps hk : I ~ ~ I ~1 • I ~ , k ---- 1,2, {~)i: i E w}close to the composition j i such t h a t h l ( I ~ ) M I ~1 = 0 and f~Thk -- f r j i = i d x ~ , k : 1, 2. Let gk : 7rhk, k -- 1, 2. Obviously, the maps gl and g2 are sections of the m a p f. It is also not hard to see t h a t g l ( I w ~ ) n g 2 ( I ~1) - 0 . This contradicts the choice of f. Therefore, the composition f r is not the trivial bundle. 7.4.3. A p p r o x i m a t i o n by h o m e o m o r p h i s m s . A map f : X ~ Y is said to be a T - n e a r - h o m e o m o r p h i s m , T > w, if f is in the closure of the set of homeom o r p h i s m s in the space C T ( X , Y ) . In Other words, f is a T-near-homeomorphism if for each collection {L/t: t E T} _C c o y ( Y ) with IT[ < r there is a homeomorphism h: X ~ Y {/it: t E T}-close to f . In Subsection 2.1.1 (see T h e o r e m 2.1.8) we have already discussed w-near-hom e o m o r p h i s m s (i.e. n e a r - h o m e o m o r p h i s m s in the usual sense). Near-homeomorphisms between R ~- and/W-manifolds also have been characterized (Propositions 2.4.20 and 2.4.21 respectively). In order to describe the general situation we need the following definition. DEFINITION 7.4.7. L e t T >_ w. A m a p f " X ~ Y b e t w e e n r e a l c o m p a c t spaces is called T-approximatively n-soft

(n

dimensional

r e a l c o m p a c t space Z ,

a r b i t r a r y closed s u b s e t Zo o f Z ,

{blt" t E T }

C coy(Y),

=

O, 1 , . . . , c r

i f f o r each at m o s t n -

w h e r e IT ] < T, a n d m a p s g" Zo ~

X

collection

and h" Z ~

Y

7.4. TRIVIAL BUNDLES

307

such that f g = h l Z o and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o , there is a map k" Z --+ X such that g = k / Z o and the composition f k is { b t t ' t E T } - c l o s e to h. The T-approxitively oo-soft m a p s are called T-approxitively soft. Obviously, every n-soft m a p is T - a p p r o x i m a t i v e l y n-soft for each 7 _> w. Observe also t h a t if, as in t h e case of n-soft m a p s (see P r o p o s i t i o n 6.1.18), the spaces X and Y in Definition 7.4.7 are also c o m p a c t , then the spaces Z and Z0 also can be a s s u m e d to be compact. In this ease the inclusion C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o is a u t o m a t i c a l l y satisfied. LEMMA 7.4.8. A map f " X ~ Y between Polish spaces is w-approximatively n-soft (n = O, 1 , . . . , oo) if and only if the foUowing condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l Polish space Z , for each closed subset Zo of Z , for each open cover lg E c o y ( Y ) and for any two maps g" Zo --+ X and h" Z --, Y with f g -- h / Z o , there is a map k" Z -+ X such that g -- k / Z o and the composition f k is U-close to h. PROOF. Obviously every w - a p p r o x i m a t i v e l y n-soft m a p b e t w e e n Polish spaces satisfies the above condition. Let us prove the converse. Thus, suppose t h a t a m a p f - X --+ Y has the above property. Take an a r b i t r a r y at m o s t n - d i m e n s i o n a l r e a l c o m p a c t space Z, closed subset Z0 of Z, and a collection {b/t" t c T} C c o y ( Y ) with IT] < w. Also take two maps g" Z0 -+ X and h" Z --+ Y such t h a t f g - h / Z o and C ( g ) ( C ( X ) ) C C ( Z ) / Z o . R e p r e s e n t the space Z as the limit space of factorizing w - s p e c t r u m

8-

{Za,p~,A}

consisting of at most n - d i m e n s i o n a l Polish spaces ( T h e o r e m 1.3.10). Since Z0 is closed in Z, the limit space of the induced s p e c t r u m 80 = {clz~, p~, A} coincides with Z0. By T h e o r e m 1.3.6, t h e r e exist an index a l E A and a m a p h a l " Z a l ~ Y such t h a t h -- halpa~. F u r t h e r , by P r o p o s i t i o n 1.3.13, t h e r e exist an index a2 E A and a m a p ga2" clz= 2 --+ X such t h a t g : ga2pa2/Zo. W i t h o u t loss of generality we m a y a s s u m e t h a t a l = c ~ 2 : C~. It is not h a r d to see t h a t the m a p s ha = hal and ga -- ga2 have the following properties"

h -- hapa,

g -- g a P a / Z o and f g a : h a l clz,~ p a ( Z o ) .

By our a s s u m p t i o n , there is a m a p ka" Z a ~ X such t h a t ga = k a / c l z , p a ( Z o ) and t h e c o m p o s i t i o n f k a is b/-close to ha, where b/ refines each of the covers b/t, t E T (recall t h a t T is finite). T h e n the desired m a p k can be defined as k -- kapa. [7 THEOREM 7.4.9. Let T >_ w. A map between R r -manifolds is a T-near-homeo m o r p h i s m if and only if it is ~--approximatively soft.

308

7. NON-METRIZABLE MANIFOLDS

PROOF. If ~- = w, t h e n the s t a t e m e n t follows from L e m m a 7.4.8 and Proposition 2.4.20. Consider the case T > w. Let f : X --* Y be a T-approximatively soft m a p between R r - m a n i f o l d s . Consider an a r b i t r a r y collection {b/t: t E T} C_ coy(Y) with IT] < T. We shall c o n s t r u c t a h o m e o m o r p h i s m h: X -~ Y which is {L/t E T}-close to f . Let ~ -- max{w, ITI}. Clearly, ~ < T. By T h e o r e m 7.1.11, the R r -manifolds X and Y are h o m e o m o r p h i c to the limit spaces of factorizing ,~-spectra 5'x = {Xa, p~, A } and 8 y = {Ya, q~, A } respectively. Moreover, these s p e c t r a consist of R~-manifolds and soft limit projections (even trivial bundles with fiber R r ). By T h e o r e m 1.3.6, the m a p f is the limit m a p of the m o r p h i s m { f a : X a ~ Ya, A } : ,Sx ~ S y . A s t a n d a r d a r g u m e n t shows t h a t there exist an index a E A and open covers L/~ E cov(Ya) such t h a t L/t = q~-l(L/~) for each t E T. Take a subcollection {V~: t E T t} of the collection cov(Ya) of cardinality IT'I = w(Ya) = ~ < T, satisfying the conditions of L e m m a 6.5.1. Let ~)t = q~-1(1)~), t E T ~. Let us show t h a t the m a p f a : X a -* Ya is soft. Take an a r b i t r a r y realcompact space Z, a closed subspace Z0 of Z, and two m a p s ga: Z0 --* X a and ha : Z --. Ya satisfying the conditions

faga = h a / Z o and C ( g a ) ( C ( X a ) ) C_ C ( Z ) / Z o . Take a section i: X a ---* X of the limit projection pa of the s p e c t r u m S x and let g -- iga. Obviously,

q a f g -- h a / Z o and C ( f g ) ( C ( Y ) ) C_ C ( Z ) / Z o . Consequently, by softness of the limit projection qa of the s p e c t r u m S y , there is a m a p h: Z --~ Y such t h a t f g = h/Zo and gah = ha. By our assumption, f is T - a p p r o x i m a t i v e l y soft. Therefore there is a m a p k: Z ~ X such t h a t g = k/Zo and the composition f k is {Yt: t E Tt}-close to h. Let ka = pak. Keeping in mind the choice of the collection { P t : t E Tt}, one can easily verify t h a t ga = k a / Z o and f a k a = ha. This shows the softness of the m a p fa. But then, by L e m m a 6.1.14, the composition faPa : X ---* Ya is also soft. By Corollary 7.4.3, this composition is the trivial b u n d l e with fiber R ~ . Similarly, the limit p r o j e c t i o n qa of the s p e c t r u m 8y is also the trivial bundle with fiber R ~ . T h e bases of these trivial bundles coincide with the space Ya. Therefore there is a h o m e o m o r p h i s m s: X ~ Y such t h a t qas = fapa. T h e last equality shows, by the choice of the index a, t h a t the h o m e o m o r p h i s m s is {L/t: t E T}-close to the m a p f . Thus, f is a T - n e a r - h o m e o m o r p h i s m . Let us prove the converse. Let f : X ~ Y be a T - n e a r - h o m e o m o r p h i s m between R r -manifolds. We are going to show t h a t f is 7 - a p p r o x i m a t i v e l y soft. Consider an a r b i t r a r y r e a l c o m p a c t space Z, a closed subspace Z0 of Z, a collection {L/t: t E T } C_ cov(Y) with IT[ < T, and two m a p s g: Z0 --~ X and h: Z --. Y such t h a t f g = h/Zo and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . As above, let A = max{w, IT[}. Obviously A < T. R e p r e s e n t the space Y as the limit space of

7.4. TRIVIAL BUNDLES

309

a factorizing A-spectrum ,gy -- {Ya, q~, A} consisting of RA-manifolds and soft limit projections. As above, we can find an index c~ E A and an open covers lg~ E c o v ( Y ~ ) such t h a t b/t - q~-l(b/~) for each t E T. Applying L e m m a 6.5.1, we get a collection { ~ : t E T ' } C c o v ( Y a ) , I T ' I - A < T with the properties of t h a t Lemma. Let ))t -- q~-l(V~), t E T ~. By our assumption, there is a homeomorphism s: X ~ Y {])t: t E T~}-close to f. By the choice of the collection {~)t: t E T~}, we have q a f -- qas. Therefore, the map q a f , as the composition of soft maps s and qa, is soft ( L e m m a 6.1.14). T h e n there is a m a p k: Z ~ X such t h a t g - k l Z o and q a f k -- qah. The last equality, and the choice of the index c~, guarantee t h a t the composition f k is (b/t: t E T}-close to h. This completes the proof. V1 A similar a r g u m e n t proves the following statement. THEOREM 7.4.10. Let T >_ w.

A proper m a p between I v - m a n i f o l d s is a T-

n e a r - h o m e o m o r p h i s m if and only if it is T - a p p r o x i m a t i v e l y soft.

COROLLARY 7.4.11. Let T >_ w and X be an R V - m a n i f o l d .

T h e n the projection

r x " X • R r --~ X is a T - n e a r - h o m e o m o r p h i s m .

COROLLARY 7.4.12. Let T >_ w and X be an I V - m a n i f o l d .

T h e n the projection

7rx" X • I v --~ X is a r - n e a r - h o m e m o r p h i s m .

Historical a n d bibliographical notes 7.4. T h e results of this Section were ob-

tained by the a u t h o r [83], [91].

Corollaries 7.4.5 and 7.4.6 can be found in

CHAPTER

8

Applications

8.1. U n c o u n t a b l e

In this D r for T > Theorems the proofs

powers of countable discrete spaces

Section we give topological characterizations of the spaces N r and w. Recall t h a t characterizations of the spaces N ~ and D ~ are given in 1.1.5 and 1.1.10, respectively. We also suggest t h a t the reader compare presented below to those of T h e o r e m s 7.3.3 and 7.2.8.

8.1.1. C h a r a c t e r i z a t i o n

o f N r . We begin with some auxiliary statements.

LEMMA 8.1.1. Let X be a z e r o - d i m e n s i o n a l Polish space and let G be open and closed in the product X x N ~ . T h e n the m a p ~r/G" G --. It(G) is the trivial bundle with fiber N ~ , where lr" X • N ~ ~ X denotes the projection onto the first factor.

PROOF. W i t h o u t loss of generality it can be assumed t h a t ~r(G) -- X. Fix complete b o u n d e d metrics pl and p2 on X and N ~ , respectively. On X • N ~ we consider the metric p defined by the formula p ( ( x l , a l ) , (x2, a2)) - 2 - 1 p l ( x l , x2) + 2 - 2 p 2 ( a l , a 2 )

where (xk, ak) E X • N w , k --- 1,2. Using a s t a n d a r d a r g u m e n t we construct a collection ( G n ' n E N ) of open and closed subsets in G such t h a t (a) G - - U { G n ' n e N } , (b) Gn N G m : 0 whenever n ~ m, (c) ~ r ( G n ) = X for each n e N, and (d) d i a m p ( l r - l ( x ) N Gn) < 2 -1 for any n e N and x e X . Let/~1 = {Gn" n e N}. R e p e a t i n g the procedure described above for each of the sets G n separately, and using induction, we construct a sequence {/gk" k E N } of c o u n t a b l y infinite open and closed disjoint covers of G with the following properties (compare with the proof of L e m m a 6.4.1)" (i) b / k - { a n , , . . . , n k " ni e g , i

e g},

k e g.

(ii) Gnl,...,nk --U{Gnl,...,nk,nk+~ " n k + l e g } . (iii) r ( G n ~ , . . . , n k ) - X for any k-tuple ( n l , - ' - , n k ) . 311

312

8. APPLICATIONS

(iv) d i a m p ( T r - l ( x ) N Gn~,...,nk) < 2 - k for any x 9 X and any k-tuple (nl,'"" ,nk). We define a m a p h: X x N W --+ G as follows: h(x,a)-- r-l(x)n

n(a.

k e

N}

for any point ( x , a ) - (x, ( n l , ' " , n k , ' " )) 9 X x N W . It follows from the completeness of p and properties (ii) and (iv) t h a t h(x, a) is n o n - e m p t y and consists of precisely one point. Direct verification shows t h a t h is a homeomorphism. It remains to observe that, by the construction, r h -- r . [-1 LEMMA 8.1.2. spaces is a trivial G of X admits a p ( G n ) - p(G) f o r

A n open surjection p: X ~ Y between zero-dimensional Polish bundle with fiber N W if and only if any open and closed subset countably infinite disjoint open cover {Gn : n 9 N } such that each n 9 g .

PROOF. T h e necessity follows from L e m m a 8.1.1. In order to prove the sufficiency we follow the a r g u m e n t used in the proof of L e m m a 6.4.1. As in t h a t proof (using, in addition, the zero-dimensionality of our spaces and the above condition) we can take a complete metric d on X and construct a sequence {b/k} of countably infinite disjoint open covers of X with the following properties: (i) U k - {Vii ..... ik" ij e g } , j E N , k e N . (ii) Uil ..... i~,ik+l C_ Uil ..... ik, ik+l E N . (iii) Uil ..... ik --U{Ui~ ..... ik,ik+~" ik+l E N } . (iv) diamd(Ui, ..... ik N p - l ( y ) ) < ~ , y 9 y . (v) p ( V i , ..... i k ) - Y , ij 9 N , k 9 g . T h e desired fiber preserving h o m e o m o r p h i s m h" Y x N • --+ X can be defined by letting for each point ( y , a ) -

(y, ( i l , . . . , i k , . . . ) )

9 Y x g ~ .

N

LEMMA 8.1.3. Let S -- { X n , p ~ + l , w } be an inverse sequence consisting of zero-dimensional AE(O)-spaces and O-soft short projections with Polish kernels. Suppose that f o r each n 9 w, the space X n + l contains a C-embedded copy of the product X n • ~ such t h a t p n n + l / ( X n x N ~ ) = 7rx,, w h e r e r x ~ " X n x N ~ --~ X n is the projection onto the first coordinate. Then the limit projection po" X -lim,.q --~ X o is the trivial bundle with fiber N W . PROOF. Apply the a r g u m e n t presented in the proof of L e m m a 7.3.1.

[:]

Combining L e m m a s 6.5.4, 8.1.3 and the proof of T h e o r e m 7.3.3 we obtain the following characterization of the space N r ,T > w. THEOREM 8.1.4. Let T > w. The following conditions are equivalent for any zero-dimensional AE(O)-space X of weight T: (i) X is homeomorphic to N r . (ii) X is strongly A t , o-universal.

8.1. UNCOUNTABLE POWERS OF COUNTABLE DISCRETE SPACES

313

(iii) For each z e r o - d i m e n s i o n a l space Y of R - w e i g h t < T the set of C-embeddings is dense in the space C r ( Y , X ) .

Applying Theorem 1.1.5 and the above characterization we get (compare with Corollary 7.3.4) the following statement. COROLLARY 8.1.5. The product X x N r , T >_ w, is h o m e o m o r p h i c to N r if and only if X is a z e r o - d i m e n s i o n a l A E ( O ) - s p a c e of weight ~_ r.

8.1.2. C h a r a c t e r i z a t i o n of D r . Recall that the Cantor cube D r is strongly Br,0-universal (Lemma 6.5.3). In this Subsection we are going to show that this property in fact characterizes the space D r . THEOREM 8.1.6. Let r > w. Then the following conditions are equivalent f o r every zero-dimensional compact (locally compact and i i n d e l S f ) A E ( O ) - s p a c e X of weight T: (i) X is h o m e o m o r p h i c to D r (to D r x N ) .

(ii) X is strongly Br,o-universal. (iii) For each z e r o - d i m e n s i o n a l compactum Y of weight < T the set of erabeddings is dense in the space C r ( Y , X ) .

(iv) The set of embeddings is dense in the space C r ( n , X ) . (v) X is homogeneous with respect to pseudo-character. The proof of this Theorem involves Lemma 6.5.3, and follows the proof of Theorem 7.2.8 with obvious changes of a technical nature (identical to those between the proofs of Theorems 7.3.3 and 8.1.4). In particular, the proof is based on the following statement (compare with Lemma 8.1.2). LEMMA 8.1.7. Let f : X - . Y be an open surjection between zero-dimensional metrizable compacta. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber n "~ . (ii) A l l fibers of f are h o m e o m o r p h i c to n ~ . Theorem 8.1.6 has several consequences. COROLLARY 8.1.8. The product X x D r , r >_ w, is h o m e o m o r p h i c to D r if and only if X is a z e r o - d i m e n s i o n a l A E ( O ) - c o m p a c t u m of weight ~_ r.

COROLLARY 8.1.9. The C a n t o r cube D ~1 is h o m e o m o r p h i c to its hyperspace e x p D W~.

PROOF. Observe that if f" X -~ Y is an open surjection between zerodimensional metrizable compacta, then the corresponding map e x p f " e x p X -+ e x p Y

314

8. APPLICATIONS

is also an open surjection. Next verify that e x p ~ is equivalent to ~, where r : D ~ x D ~ --+ D ~ denotes the projection onto the first coordinate. Finally apply Theorem 8.1.6. rl COROLLARY 8.1.10. T h e C a n t o r cube D ~2 is n o t h o m e o m o r p h i c to its hyperspace e x p D ~2 .

PROOF. Assuming the contrary and applying Theorems 8.1.6 and 1.3.4 we can easily conclude that the map e x p r , where ~r: D ~1 x D ~1 ~ D ~1 must be equivalent to ~r. Further observe t h a t not all fibers of the map e x p r are homeomorphic to the Cantor cube D ~ . This contradiction proves our statement. [::]

H i s t o r i c a l a n d bibliographical n o t e s 8.1. The results of Subsection 8.1.1 are due to the author. The equivalence of conditions (i) and (v) of Theorem 8.1.6 has been proved in [272]. The related result, Lemma 8.1.7, providing a characterization of trivial bundles with fiber D "~ was obtained in [286]. One can prove this statement by simply "parameterizing" the well known proof of Theorem 1.1.10. Another possible way involves Michael's Selection Theorem 2.1.14. Corollary 8.1.9 also appears in [286]. Corollary 8.1.10 was proved in [271].

8.2. Spectral representations of topological groups A classical result of Pontrjagin [253] asserts that each compact topological group can be expressed as the inverse limit of some transfinite inverse spectrum in which all short projections are continuous homomorphisms whose kernels are compact Lie groups (so-called Lie series). Here kernel (i.e. the inverse image of the unit element) has algebraic meaning. Recall also Proposition 6.3.5 (and Remark 6.3.7) which states, in particular, that the AE(0)-compacta are precisely those t h a t admit special spectral representations (the so-called Haydon series). A simple comparison of these two results immediately shows that Haydon's theorem is, in some sense, the analog of Pontrjagin's theorem in the category of compacta and continuous maps. This comparison can be extended to yield a proof of the fact t h a t each compact topological group is an AE(0)-compactum. Moreover, noticing that each locally compact topological group is LindelSf and applying R e m a r k 6.3.6, one can easily see that every locally compact topological group is an AE(0)-space. This allows us to study locally compact topological groups from the point of view of the general theory of AE(0)-spaces and 0-soft maps. For instance, it immediately follows from Theorem 8.1.6 that each uncountable, zero-dimensional compact (locally compact and LindelSf) group is topologically equivalent to D r (to the product N • D r ) [181], [204]. In this Section we extend these characterizations to the non-locally compact case. We begin with the following useful statement.

8.2. TOPOLOGICAL GROUPS

315

LEMMA 8.2.1. Let A >_ w. Each AE(O) topological group X of weight T > A is isomorphic to the limit space of a factorizing A-spectrum S x -- { X a , p ~ , A } all spaces of which are AE(O)-groups of weight A and all projections of which are O-soft homomorphisms. PROOF. By Proposition 6.3.5, the space X can be represented as the limit space of a factorizing A-spectrum 8 x - { X a , p ~ , A } consisting of AE(0)-spaces of weight A and 0-soft limit projections. Let us show that this spectrum contains a A-closed and cofinal subspectrum consisting of topological groups and limit projections that are (continuous) homomorphisms. Since X is a topological group, there is a continuous multiplication #: X x X ~ X. Clearly, X • X is the limit space of the spectrum

All projections of the spectrum S x x S x are 0-soft, and hence open (Proposition 6.1.26). The Suslin number of the product X x X is obviously countable. Consequently, by Proposition 1.3.3, the spectrum S x x S z is factorizing. Next we apply Theorem 1.3.4 to the spectra S x x S x , S x and to the map # between their limit spaces. Then we get a A-closed and cofinal subset B of A such that for each a E B we have a continuous map # a : X a x X a --* X a such that =

• p.).

Now we can define a continuous multiplication operation in Xa by letting xa . y a = # a ( x a , Ya) whenever (xa, y~) e Xa x Xa. It is easy to see that Xa, c~ E B, becomes a topological group with respect to this operation. Moreover, for all c~,/~ E B with a < f~, the projection p~" Xt~ --~ Xa becomes a homomorphism with respect to the above defined operations. [:] COROLLARY 8.2.2. I f (in the notations of L e m m a 8.2.1) X is n-dimensional, then all elements of the spectrum S x = { X a , p ~ , A } can also be assumed to be n-dimensional. PROOF. Apply Lemma 8.2.1 and Theorem 1.3.10.

[-1

We also need the following two simple statements. LEMMA 8.2.3. Let p: X ~ Y be a O-soft h o m o m o r p h i s m between zero-dimensional AE(O) topological groups. Then there exists a homeomorphism h: Y x kerp ~ X such that ph -- ~1, where r l : Y x kerp --~ Y denotes the natural projection onto the first factor. PROOF. Since p is 0-soft and dim Y - 0, there exists a continuous map i" Y X such that pi = i d y and i ( e y ) - e x ( e x and e y denote neutral elements of

316

8. APPLICATIONS

the groups X and Y respectively). Define a h o m e o m o r p h i s m h: Y • kerp ~ X by letting

h(y,a) = i(y).a,

(the dot r l . [::]

9

whenever (y,a) e Y • k e r p

denotes the m u l t i p l i c a t i o n o p e r a t i o n of the group X ) . Clearly ph

=

LEMMA 8.2.4. A n y zero-dimensional space of R-weight ~" admits a C-embedding into N r .

PROOF. 1 Since any zero-dimensional completely m e t r i z a b l e space can be considered as a closed subspace of the space N "~ ( P r o p o s i t i o n 1.1.6), the L e m m a is true for T -- w. Consider now a zero-dimensional space X of R-weight T > w. Obviously the Hewitt realcompactification v X of X is zero-dimensional and has the same R-weight as X . By T h e o r e m 1.3.10, v X can be represented as the limit space of some factorizing w - s p e c t r u m S = {Ya, q~, A} consisting of zerodimensional Polish spaces. Since S is factorizing, it is easy to see t h a t v X is C - e m b e d d e d (and closed) in the p r o d u c t 1-'[{Y~ : c~ e A}. This product, in turn, can be C - e m b e d d e d into N r (since each Ya can be e m b e d d e d into N • as a closed subspace and IAI = T). Consequently, X can be C - e m b e d d e d into N r . D

In the following T h e o r e m , k ( X ) denotes the minimal cardinal ~ such t h a t X contains a c o m p a c t s u b s p a c e K whose p s e u d o - c h a r a c t e r in X , r is T. Note t h a t if, for a topological group X , we have k ( X ) < w, t h e n X is locally

1One might formally apply Lemma 6.5.3.

8.2. TOPOLOGICAL GROUPS

317

compact. THEOREM 8.2.5. Let X be a non-locaUy compact zero-dimensional AE(O) topological group. Then X is homeomorphic to the product N a(X) x D w(X). PROOF. We consider two cases separately. Case 1" k ( X ) -- w ( X ) . We use transfinite induction. T h e s t a r t i n g point of the i n d u c t i o n (i.e. the case k ( X ) -- w ( X ) = w) is trivial, because topologically X is a zero-dimensional Polish space w i t h o u t open c o m p a c t subspaces. Consequently, by T h e o r e m 1.1.5, X is h o m e o m o r p h i c to the space of irrationals N ~ ..~ N W • D ~ . T h u s we can suppose t h a t in Case 1 our s t a t e m e n t is t r u e for all spaces of weight A, where w _< A < T, and consider a zero-dimensional AE(O) topological g r o u p X with k ( X ) = w ( X ) = T. To prove t h a t X is h o m e o m o r p h i c to the p r o d u c t N v x D r ~ N r ' we can use a topological c h a r a c t e r i z a t i o n of the l a t t e r space ( T h e o r e m 8.1.4). T r a n s l a t i n g condition (iii) of T h e o r e m 8.1.4 into s p e c t r a l language, we see t h a t it suffices to show the following: Consider a zero-dimensional space Y of R-weight A < ~- ( w i t h o u t loss of generality, we m a y a s s u m e t h a t A _ w), fix an a r b i t r a r y m a p f " Y --. X , fix a factorizing A-spectrum S x -- { X a , p ~ , A} , satisfying the conditions of L e m m a 8.2.1, and any index a E A. Our goal is to prove the existence of a C - e m b e d d i n g g" Y ---+ X such t h a t p a f = pad. First, let us t r y to find an index fl > a such t h a t the p r o j e c t i o n p~" X~ --~ X~ of the s p e c t r u m S x is (topologically) a trivial bundle with fiber N>'. Let a0 = a. Since the weight of the space X a o is strictly less t h a n the weight of the space X , we can conclude t h a t the fiber kerpao (i.e. the inverse image P~o -1 (e~o), where eao denotes the n e u t r a l element of the group Xao) is n o n - c o m p a c t . Indeed, otherwise we would have

k ( Z ) <_ r

) < r

A < ~-= w ( Z )

which contradicts our a s s u m p t i o n k ( X ) = w ( X ) . T h u s we can choose an index a l c A, with a l > a0, such t h a t kerpg~ is n o n - c o m p a c t . Suppose now t h a t for every ordinal 7 with 7 < 5, where 5 < A, we have a l r e a d y chosen indexes a 7 E A in such a way t h a t the following conditions are satisfied: (a) I f # < v < 6 , thena,
318

8. APPLICATIONS

and note that, as above, fl E A. Consider the well-ordered spectrum

s ' = {x~ p~+~ ~}. The A-continuity of the spectrum S x guarantees that the limit space of the spectrum S' is naturally isomorphic to the topological group X~ and the limit projection limS t~Xao=Xa is isomorphic to the projection p~" Xf~ --~ X a of the spectrum S z . By lemma 8.2.3, p~ is topologically a trivial bundle with fiber kerp~, i.e. there exists a homeomorphism hf~" Xa x kerp~ ---, X~ such that p~hf~ -- Irl, where r l " X a x k e r p ~ ~ X a denotes the natural projection onto the first factor. It follows from the properties of the spectrum S x that ker p~ is a zero-dimensional AE(O) topological group. By the construction of the index fl, we conclude immediately that k(kerp~) = w(kerp~) = s Consequently, by the inductive hypothesis, kerp~ is homeomorphic ( as a topological space) to N "x. Thus, by Lemma 8.2.4, there exists a C-embedding j" Y ~ kerp~. Let g/3 = h / 3 ( P a f A j ) . Obviously, gf~" Y ---+ X~ is a C-embedding such that p~g/3 = paY. Choose an arbitrary section i" X f~ ---, X of the 0-soft limit projection pz" X ---, X~ of the spectrum 8 x (this means that phi = idx~). Then the composition g=ig/3" Y ~ X

is the desired C-embedding of Y into X. This completes the verification of the condition mentioned above, and hence X is homeomorphic to N r . The proof of Case 1 is complete. Case 2: k ( X ) < w ( X ) . Since, by our assumption, X is non-locally compact we may suppose that w _< k ( X ) < w ( X ) = T. Take a factorizing A-spectrum (where A = k ( X ) ) 8 x = { X a , p ~ , A } satisfying the conditions of Lemma 8.2.1. Choose a compact subspace K C_ X such that r X) = A. Without loss of generality, we may assume that e C K, where e denotes the neutral element of the group X. Since S x is a factorizing A-spectrum and r X) = A, there exist an index c~ C A and a compact subspace Ka C_ Xa such that p-~l(Ka) = K . Consequently, kerpa is homeomorphic to the Cantor cube D r (as a compact topological group of weight T). By Lemma 8.2.3, the limit projection pa" X ---+Xa is topologically the natural projection 7r1" Xa x D r ---+Xa. Consequently, k ( X a ) = w(X,~) = A (otherwise we would have k ( X ) < A, a contradiction). But Xa is a zero-dimensional AE(O) topological group. As was established in Case 1, X~ is homeomorphic to N~. Therefore X is homeomorphic to the product N "x x D r . E]

8.3. LOCALLY CONVEX SPACES

319

Historical and bibliographical notes 8.2. The results of this section are taken from [29]. L e m m a 8.2.1 was first exploited in [93] (see Section 8.3).

8.3. L o c a l l y c o n v e x l i n e a r t o p o l o g i c a l

spaces

One of the fundamental results of infinite-dimensional topology and functional analysis - the well known theorem of Anderson-Kadets (see Proposition 2.4.30) - states that each separable infinite-dimensional Frgchet space is homeomorphic to the countable (infinite) power R • of the real line R. It is also well known [32] that each locally compact locally convex linear topological space is homeomorphic (even isomorphic) to some finite power R n of R (here we consider only Hausdorff linear topological spaces over the field of real numbers). The following reformulation of these two facts immediately follows from the corresponding definitions. THEOREM 8.3.1. The following conditions are equivalent for each locally convex linear topological space E: (i) E is homeomorphic to the countable (finite or infinite) power of the real line. (ii) E is a Polish space. (iii) E is an AE(O)-space of weight w. Our main goal in this Section is to extend the above result to spaces of arbitrary weight. We start with the following statement. PROPOSITION 8.3.2. Let A >__w. Each locally convex AE(O)-space is isomorphic to the limit space of some factorizing A-spectrum consisting of locally convex AE(O)-spaces (of weight A) and O-soft linear limit projections. In particular, each locally convex AE(O)-space is isomorphic to the limit space of some factorizing w-spectrum consisting of separable Frdchet space spaces and O-soft linear limit projections. PROOF. Let E be a locally convex AE(0)-space. Obviously E can be represented as the limit space of some factorizing A-spectrum S = { E a , p ~ , A } consisting of AE(0)-spaces (of weight A) and 0-soft limit projections. We are going to show t h a t this spectrum contains a A-closed and cofinal subspectrum consisting of locally convex spaces and linear limit projections. Obviously this will be sufficient for us. Let #: E • E --, E denote the continuous addition operation given in E as a linear topological space. Applying the argument of L e m m a 8.2.1, we get a A-closed and cofinal subset A1 of A and a morphism { # , : S o x Ea -* E , , A 1 } :

(S x S ) / A 1 -~ S / A 1

320

8. APPLICATIONS

whose limit map coincides with #. We then define a continuous addition operation + a in each Ea, a E A1, simply by letting xo, "t-a Ya = #a(xa, Ya) whenever (xa, Ya) E Ea x Ea. Similarly, consider a factorizing A-spectrum idR x S - - {R x Ea, idR x p~a,A} and the continuous operation v: R x E ---, E of multiplication by scalars. Applying Theorem 1.3.4 to the spectra idR x S and S (and to the map v between their limit spaces), we get a A-closed and cofinal subset A2 of A and a morphism {va: R x Ea ---* Eel, A2}: (idR • S ) / A 2 ~ S / A 2 with v - lim{va: a E A2}. As above, we now define the continuous operation of multiplication by scalars in Ea, a E A2, by letting t . a xa - va(t, xa) whenever (t, xa) E R x Za. The set A ~ -- A1 M A2 is A-closed and cofinal in A (Proposition 1.1.27). It only remains to note that for the spaces Ea, (~ E A ~ we have defined both operations from the definition of linear topological spaces. Both of these are continuous, and consequently all spaces of the spectrum S / A ~ are linear topological spaces. Moreover, all limit projections in the latter spectrum are linear with respect to the above defined operations. [-1 COROLLARY 8.3.3. Each locally convex AE(O)-space is isomorphic to a Cembedded (and closed) linear subspace of the product of separable Frgchet spaces. PROOF. Let E be a locally convex AE(0)-space. By Proposition 8.3.2, E is isomorphic to the limit space of some factorizing w-spectrum S - {Ea,p~, A} consisting of separable Frgchet spaces and 0-soft linear limit projections. Consider the product F = I-I{Ea: a E A}. Obviously, E is isomorphic to a linear (and closed) subspace of F (this subspace is the image of the space E under the diagonal product of all limit projections of the spectrum ,.q). Since S is factorizing, the mentioned subspace is C-embedded in F. Vl LEMMA 8.3.4. Let f : E ~ F be a O-soft linear map between locally convex AE(O)-spaces that has Polish kernel 5n the sense of Section 6.3). Suppose that f ( g ) -- T, where g and T are convex and C-embedded AE(O)subspaces of E and F respectively. Then the following conditions are equivalent: (i) The restriction f / K : K ~ T is a O-soft map. (ii) The restriction f / K : K ~ T is a soft map. PROOF. The non trivial part is the implication (i) ~ 8.3.2, there exist factorizing w-spectra SE= {Za,p~,A}

(ii). By Proposition

and SF---" {Fa, q ~ , A }

8.3. LOCALLY CONVEX SPACES

321

consisting of separable Frgchet spaces and 0-soft linear projections, the limit spaces of which are isomorphic to the spaces E and F respectively. By Theorem 6.3.2 we can suppose, without loss of generality, t h a t f is the limit map of some morphism { f a : Ea ~ Fa, A}: SE

--~

SF

consisting of 0-soft maps and such that all the corresponding limit square diagrams are Cartesian squares. Since K is a C-embedded and closed subspace of E we can conclude that the spectrum SK -- {Ka -- clE, ( p a ( K ) ) , p ~ / K ~ , A } , the limit space of which coincides with K, is also a factorizing w-spectrum. Since K is an AE(0)-space, we can apply Proposition 6.3.5 and Theorem 1.3.6 and suppose, without loss of generality, that for each a E A the map p a / K is 0-soft and p a ( K ) = g o . Similarly, we may assume that qa(T) = clF~(qa(T)) and the map q a / T is 0-soft whenever c~ E A. Finally, by Theorem 6.3.2, we can suppose (again without loss of generality) that each commutative diagram of the type

.f / K K

,~T

qa/T

pa/K

pa(K)

fa/pa(K)

,qo~(T)

is a Cartesian square and each map f a / p a ( K ) is 0-soft. In this situation, to complete the proof of the lemma it only remains to be proved t h a t the last map is soft (then we can use the fact that the above diagram is a Cartesian square to obtain, by L e m m a 6.2.4, the softness of the map f / K ) . Linearity of the limit projections of spectra SE and 8 F implies t h a t the spaces p a ( K ) and qa(T) are convex subspaces of the spaces E a and Fa, respectively. Moreover, it is easy to see t h a t each map f a is also linear. By Theorem 2.1.16, the map f a / p a ( K ) , as an open map between closed and convex subspaces of separable Frgchet spaces with closed convex fibers, is soft. The proof is complete. [-1 PROPOSITION 8.3.5. Let K be a C-embedded and convex subspace of a locally convex AE(O)-space. Then the following conditions are equivalent: (i) g is an AE(O)-space. (ii) K is an absolute retract.

322

8. APPLICATIONS

PROOF. It is sufficient to prove the implication (i) ~ (ii). By Corollary 8.3.3, we may suppose t h a t K is a convex and C - e m b e d d e d subspace of some p r o d u c t E -- 1-I{E~" a e A} of separable Frgchet spaces. If IAI _< w, then K is a convex and closed subspace of a separable Frgchet space and hence, by T h e o r e m 2.1.17, is an absolute retract. Now consider the case IAI - T > w. W i t h o u t loss of generality we may suppose t h a t each space Ea, a 9 A, contains at least two points, and consequently the space E is h o m e o m o r p h i c to R r (apply Proposition 2.4.23). Consider a functionally closed, proper and 0-invertible m a p f" N A ----+E having a regular averaging o p e r a t o r (see Proposition 6.1.25). Since dim N A = 0 and K is a C - e m b e d d e d A E ( 0 ) - s u b s p a c e of E, there exists a m a p h" N A ~ K such t h a t h/f-l(g) = f /f-l(g). We now introduce some notation. If C C_ B C_ A, then

~r~" H { E a " a 9 B}---+ H { E a " a 9 C} and

7rB'E-+H{Ea'aEB

}

denote the n a t u r a l projections onto the corresponding subproducts. Also

K B "- 1rB(K), K ( B ) - 7rBI(KB), PB -- 7rB/K and p~ -- 7r~/KB. We say t h a t a subset B of A is admissible if

lrB(x) - P B h f - l ( x )

for each point x 9 K ( B ) .

As in the proof of T h e o r e m 6.3.1, we can verify the following properties of admissible subsets: (a) T h e union of an a r b i t r a r y collection of admissible subsets is again an admissible subset. (b) For any admissible subset B, the set KB is closed and C - e m b e d d e d in the p r o d u c t YI{Ea" a E B}. (c) For any admissible subset B the space KB is an AE(0)-space. (d) For any two admissible subsets C and B with C C_ B, the maps PB and p ~ are 0-soft. (e) For any admissible subset B, the set K B is convex in the space

rI{Ea" a e B}. (f) T h e family of all countable admissible subsets of A is w-closed and cofinal in the w-complete set exp,,A. Since IAI = T, we may write A = {as" c~ < T}. By (f), each point a s of A is contained in some countable admissible subset B s . P u t A s = LJ{Bt~"/~ _< c~}, K s = KA,~ and p g + l = PA,~A'~+I (a < T).

8.3. LOCALLY CONVEX SPACES

323

It follows, from the properties of admissible subsets listed above, t h a t the limit space of the n a t u r a l l y induced well ordered continuous s p e c t r u m S - - { g a , p g +I,T} coincides with the given space K. T h e countability of the sets B a guarantees, by L e m m a 8.3.4, the softness of the short projections of this spectrum. Obviously, the space B0 is an absolute retract ( T h e o r e m 2.1.17). In this situation, Proposition 6.1.21 and L e m m a 6.2.6 conclude the proof. Vl We are now ready to prove the following s t a t e m e n t , which extends Proposition 2.4.30 to the non-metrizable case. THEOREM 8.3.6. Let T be an uncountable cardinal.

Then the following con-

ditions are equivalent f o r each locally convex linear topological space E of weight T"

(i) E is h o m e o m o r p h i c to R r (ii) E is an absolute retract. (iii) E is an AE(O)-space. PROOF. T h e implications (i) ==~ (ii) and (ii) ==~ (iii) are obvious. Let us prove the implication (iii) ~ (i). We use transfinite induction. If ~- - - w , then E is an infinite-dimensional separable Frgchet space and the conclusion follows from Proposition 2.4.30. Therefore we may suppose t h a t our s t a t e m e n t is true for all infinite-dimensional locally convex A E ( 0 ) - s p a c e s of weight A, w _ A < T, and consider a locally convex A E ( 0 ) - s p a c e E of weight r. By Proposition 8.3.5, E is an absolute retract. Consequently, by T h e o r e m 7.3.3 it suffices to show t h a t for any space Z of R-weight < T, the set of C - e m b e d d i n g s is dense in the space Cr(Z,E). Consider an a r b i t r a r y space Z of R-weight A (where w __ A < r), a map g" Z --. E, a factorizing A-spectrum A s -- { E a , p ~ , A } the limit space of which coincides with E, and an index a E A. We wish to construct a C - e m b e d d i n g h" Z --, E such t h a t p a h -- pag. By Proposition 8.3.2, we may suppose w i t h o u t loss of generality t h a t all spaces E a are locally convex AE(0)-spaces of weight A and all limit projections pa are linear 0-soft maps. Moreover, by Proposition 8.3.5, we see t h a t all spaces E a are absolute retracts. Finally, by Proposition 6.3.5, we may assume w i t h o u t loss of generality t h a t all limit projections are (linear and) soft. Let us find an index/~ E A such t h a t f~ > c~ and the projection p~" Et~ ~ E a is a trivial bundle with fiber R )'. P u t a0 -- a. Since the weight of E a o is strictly less t h a n the weight of E and the projection Pao" E --~ E a o is linear and open (Proposition 6.1.26), we conclude t h a t the fiber k e r p a o contains at least two points. T h e n we can choose an index a l > s0 such t h a t ker Paoal also contains at least two points. Suppose now t h a t for every ordinal "r with ~/ < 5, where 5 < A, we have already chosen indexes a~ E A in such a way t h a t the following conditions are satisfied:

324

8. APPLICATIONS

(a) If ~ < u < 5, then c~ < c~v. (b) If v is a limit ordinal, then a~ - s u p { ~ : # < v}. (c) T h e fiber k e r p ~ , contains at least two points (# < v < 5). Now we c o n s t r u c t an index c~ with the desired properties. If 5 is a limit ordinal, then we p u t a~ - sup{c~.y: ~/ < 5}. By the inequality 5 < A and the A-completeness of the s p e c t r u m S, we have t h a t c~ E A. If 5 - ~ + 1, then the index a~ can be c o n s t r u c t e d in precisely the same way as the index c~ - 1. Now consider fl - sup{c~.y: ~, < A} and note t h a t fl E A. Consider the well ordered spectrum , ~ ' = {E.y,- a~+l A}. T h e A-continuity of the s p e c t r u m S g u a r a n t e e s t h a t the limit space of the s p e c t r u m 8 ' is naturally isomorphic to the space Eft, and the limit projection l i m S ' ---+ Eao -- E a is isomorphic to the short projection p~" Ef~ ---+ E a of the s p e c t r u m S. As in L e m m a 8.2.3, we can easily see t h a t the m a p p~ is topologically a trivial bundle with fiber kerp~. It is also easy to verify t h a t the fiber kerp~ is h o m e o m o r p h i c to the p r o d u c t H { k e r , ~r ~

9 ~ < A}.

Consequently, by conditions (a) - (c), the weight of kerp~ is equal to A. By the same conditions we can conclude t h a t even if A -- w, the space is infinite-dimensional (otherwise we get a contradiction to condition (c)). Consequently, by our hypothesis, the space kerp~ is h o m e o m o r p h i c to R x. Moreover, in this case the projection p~ can be topologically identified with the natural projection 71"1: Ec~

X

R A ----+Ec~.

Since the R-weight of the space X is equal to A, there exists a C - e m b e d d i n g i: Z ~ R x. T h e n the diagonal p r o d u c t j -pof/ki:

Z ---+ E ~ • R ~ -

El3

is a C - e m b e d d i n g satisfying the equality p ~ j - p a l . Take an a r b i t r a r y section k: Ef~ ~ E of the limit projection p• of the s p e c t r u m S (recall t h a t pf~ is a soft map). T h e n the composition h -- k j is the desired C - e m b e d d i n g of Z into E. [-I COROLLARY 8.3.7. E v e r y closed c o n v e x body in a locally c o n v e x A E ( O ) - s p a c e o f w e i g h t v > w is h o m e o m o r p h i c

to R r .

PROOF. Let X be a closed and convex b o d y in a locally convex A E ( 0 ) - s p a c e E of weight T > w. Since int X =-~ 0 we may conclude t h a t X - cl(int X ) . In particular, w ( X ) - T. By T h e o r e m 8.3.6, E is h o m e o m o r p h i c to R r and consequently, by Proposition 6.1.8, X is functionally closed in E. By Corollary 6.4.8, X is C - e m b e d d e d in E. By Propositions 6.4.9 and 8.3.5, X is an absolute retract. Corollary 7.1.17 finishes the proof. !-7

8.4. SHAPE PROPERTIES

325

COROLLARY 8.3.8. Every linear G6-subspace of a locally convex AE(O)-space of weight T > w is homeomorphic to R r . PROOF. Let X be a linear G6-subspace of a locally convex AE(0)-space E of weight T > w. P u t Y -- c l X . As in the proof of Corollary 8.3.7, we can conclude t h a t Y is h o m e o m o r p h i c to R r . Consequently it is sufficient to show t h a t X - Y. Indeed, if there exists a point y E Y - X, then the translate X + y would be a dense G6-subspace of Y and (X + y) N Y = 0, which contradicts the Baire property in Y ~ R r . D COROLLARY 8.3.9. The closed convex hall of a compact subset of a locally convex AE(O)-space is compact. PROOF. We only have to observe t h a t each locally convex AE(0)-space, by Proposition 8.3.2, is complete. T h e rest is well known (see [258]). [-1 T h e concept Recall t h a t a Obviously, the a r g u m e n t used

of projector is i m p o r t a n t (continuous) linear map image f ( E ) is a linear in the proof of T h e o r e m

in the theory of linear topological spaces. f:E --~ E is a projector if f2 = f . continuous retract of E. Applying the 8.3.6 we obtain the following s t a t e m e n t .

PROPOSITION 8.3.10. The image of a continuous projector, defined on a product of separable Frgchet spaces, is isomorphic to a product of separable Frgchet spaces.

Historical and bibliographical notes 8.3. All results of this Section are due to the a u t h o r [93]. Corollaries 8.3.7 and 8.3.8 e x t e n d to the non-metrizable case the corresponding results of Slee-Corson and M a z u r - S t r e n b a c h (see [32]).

8.4. Shape properties of non-metrizable compacta 8.4.1. Spectral theorem in Shape category. We start with the following definition. DEFINITION 8.4.1. A n inverse spectrum s -- ( X a , p ~ , A } is said to be homotopically stable with respect to the space Y if for each index a E A, each closed subspace F of X a , and maps f , g : F --~ Y , the relation f p a / p ~ l ( F ) ~_ gpa/p-~l(F) implies the relation f ~_ g. We say that S x is (absolutely) homotopically stable if 3 x is homotopically stable with respect to any metrizable A N R-compactum. LEMMA 8.4.2. There exists a countable collection (Gn: n E w} of metrizable locally compact A N R-spaces such that any inverse spectrum which is homotopically stable with respect to each space Gn, n E w, is absolutely homotopically stable.

326

8. APPLICATIONS

PROOF. As the desired collection, we may take an arbitrary countable open basis of the topology of the Hilbert cube containing unions of its finite subcollections. The verification of this fact is trivial and is left to the reader. W1 LEMMA 8.4.3. Let p: X ---. Y be a surjection and .~ = {Fa} be a closed basis

of the topology of Y containing intersections of its finite subfamilies. an A N R-space and suppose that .for each Fa E jz and any maps f, g: the condition f p / p - 1 (Fa) "" g p / p - l ( F a ) implies the condition f ~_ for each closed subspace F of Y and any maps f , g : F ---. Z, the

Let Z be Fa ~ Z, g. Then condition

PROOF. Since Z is an A N R-space there exists a neighborhood V of F in Y and extensions f , ~ " V ~ Z of f and g respectively. By our assumption, the restrictions f p / p - l ( Y ) and ~ p / p - l ( Y ) to p - l ( F ) are homotopic. Since Z is an ANR-space, we can find a neighborhood U of p - l ( F ) in X contained in p - l ( Y ) such that the restrictions ] p / U and ~p/U are homotopic. Clearly, we may suppose without loss of generality that the set U has the form U - p - l ( w ) , where W is a neighborhood of F in Y. Then the compactness of Y and the corresponding property of ~" imply that there exists an element F~ E $" such that F C Fa C W. By our assumption ] / p - l ( F a ) ~_ ~ / p - l ( F a ) . Consequently,

f~--g.

D

THEOREM 8.4.4. Each w-spectrum, consisting of metrizable compacta, contains a homotopically stable w-closed and co final subspectrum. PROOF. Let S x = {Xa,P~a,A} be an arbitrary w-spectrum consisting of metrizable compacta. Fix an integer n E w. We shall show that the indexing set A of our spectrum contains an w-closed and cofinal subset An such t h a t the corresponding subspectrum s~/A.

=__s .

=

{x~,p~.,A.}

is homotopically stable with respect to the space Gn from Lemma 8.4.2. Let •n

-

{ ( a , ~ ) E A 2" a _~ f~, and for each closed subspace F of X~ and any two maps f , g " F ~ Gn the conditionfp~/p-~l(F) "~ g p a / p ~ l ( F ) implies the c o n d i t i o n f pf~a / ( p ~ ) - l ( f ) ' ~ gpa/(p~)

l(f)}

Let us verify that the existence condition of Proposition 1.1.29 is satisfied with respect to the relation s Fix a E A. Our goal is to show that there exists an index f~ E A such that (a, f~) c s First of all, we shall fix a closed subspace F of the compactum Xa and construct an index a F _> a such that the following condition is satisfied: (*)F for maps f, g" F ~ Gn the condition fpa/p-~l(F) ~_ g p a / p ; l ( F ) implies the condition f p ~ F / ( p ~ F ) - l ( f ) ~-- g p ~ F / ( p ~ F ) - - l ( f ).

8.4. SHAPE PROPERTIES

327

Fix a countable dense subset {hi" i E N } of the space C(F, Gn) (the compactopen topology is considered). Let M denote the set of those pairs (i,j) E N 2 for which the conditions hipa/p'~l(F) ~ hjp,~/p~l(F) are satisfied. For each (i,j) E U we can choose an index a(i,j) >_ ~ such t h a t (see, for example,

[214]): hipg(i,J) / (pg(i,J))-l (F) ~_ hjpg(i'J) / (pg(i,J))-l (F). It follows easily from the construction t h a t for each (i,j) E M and each 9' E A we may suppose t h a t a(i,j) >_ .~. Consequently, well-ordering M in an a r b i t r a r y way, we may assume t h a t {c~(i,j)" (i,j) E M } forms a chain in A. Let us verify t h a t the index a F = m a x { a ( i , j ) " (i,j) E M } is the desired one. Consider two maps f,g" F --, Gn such t h a t f p a / p ~ l ( F ) ~ gpa/p-~l(F). By our assumption, Gn is an ANR-space and, by the construction, {hi" i E N } is a dense subset of the space C(F, Gn). Consequently, there exist integers i,j E N such t h a t f ~_ hi and g "-~ hi. T h e n

hip,/p-~l(F) ~_ fp,/p-~l(F) ~ gp,/p-~l(F) ,.~ h j p , / p ~ l ( F ) . In the other words, (i,j) E i . have

Hence, by the definition of the index a(i,j), we

h i p g F / ( p ~ F ) - l ( F ) _ h.~a(i,j).~aF ~t'a ~'a(i,j)/(P~ F)

--1

(F) ~ hjpg(~'3)paa~i,j)/(pgF)-l(F ) ""

= hjpg~/(p~)-~(F). Then

fp~/(p~F)-l(F)

~_ hip~F/(p~F)-l(F) ~ h j p ~ / ( p ~ F ) - l ( F )

Thus, for each closed subspace F of the c o m p a c t u m Xa we have an index a F satisfying the above condition (*)F. Consider now an a r b i t r a r y countable basis {Fk" k E w} of the topology of X a containing the intersections of its finite subfamilies. For each k E w, fix an index ak -- a F k. Obviously, by Corollary 1.1.28, there exists an index f~ E A such t h a t f~ > c~k for each k E w. It follows from the above construction, and from L e m m a 8.4.3, t h a t (c~,f~) E L:n. T h e verification of the existence condition is completed. T h e verification of the m a j o r a n t n e s s condition from Proposition 1.1.29 is trivial. Finally we must verify the w-closeness condition from Proposition 1.1.29. This means t h a t if we have a countable chain {OLm'm E w} in A and condition (am, f~) E s is satisfied for each m E w and some f~ E A, t h e n the condition (c~, f~) E s where a = sup{c~m" m e w}, is also satisfied. Indeed, since our s p e c t r u m is w-continuous, the c o m p a c t u m X a is naturally O~m-{-1 h o m e o m o r p h i c to the limit space of the inverse sequence {Xa.~,~'am ,W}. Conpa -1 sequently, the sets of the form (a,,.) (Fro), where Fm is closed in Xa~ and

328

8. APPLICATIONS

m E w, form a closed basis of the topology of X a which contains the intersections of its finite subfamilies. Consequently, if we wish to prove the inclusion (c~,~) E s it suffices, by L e m m a 8.4.3, to show that if Fm is closed in Xa.~, m e w, and the maps f,g: F -- (p~,~)-l(Fm) -+ Gn satisfy the condition f p a / p ~ l ( F ) ~ _ gpa/p~l(F), then the condition fp~/(p~a)-l(F)~_ g p ~ / ( p ~ ) - l ( f ) will also be satisfied. Since Gn is an ANR-space, there exist an integer k _> m and maps fl,gl" paok(F) - , Gn such that f ~ flp~)k and g ~- glpg k. Then we have

flpak/p~:(Fk) ~-- glpak/p~:(Fk) where F k - P~k (F). By the assumption, (ak,/~) E s

/

(Fk) "~ glp~k

Consequently, we have

I

(f k).

It only remains to note that ( p ~ ) - l ( F ) = (p~)-l((p~m)-l(Fm)) = (p~k)-l(Fk) and that a

f~

--1

~_ glpgkp~/(p~a)-l(F)

--1

"~ gp~a/(p~)-l(F).

Thus the w-closeness condition is also satisfied. Now, by Proposition 1.1.29, we can conclude that the set An of all s indexes from A is w-closed and cofinal in A. At the same time we note that, by the corresponding definitions, this means that the subspectrum Sn of S x is homotopically stable with respect to the space Gn. Finally, let

A ' - - N{An: n E w}. By Proposition 1.1.27, this set is w-closed and cofinal in A as well. Consequently, the corresponding subspectrum S' -- {Xa,p~, A'} is homotopically stable with respect to each space Gn, n E w. L e m m a 8.4.2 finishes the proof. [-1

Remark 8.4.5. It is easy to see that each homotopically stable spectrum is homotopically stable with respect to every (:W-complex. Let S x - {Xa,p~a, A} be any homotopically stable spectrum and K be a (:W-complex. Suppose that for some index a E A and for some maps f , g : F --, K, where F is a closed subspace of Xa, the condition f p a / p ~ l ( F ) ~_ g p a / p ~ l ( F ) i s satisfied. Then, by compactness of lim S x and by the properties of the topology of C1/Y-complexes, there exists a metrizable A N R-compactum K' lying in K, such that the maps f p a / p ~ l ( F ) and gpa/p-~l(F) are homotopic in g ' . By our assumption, we can conclude that f and g are homotopic in K ' and, consequently, in K as well.

8.4. SHAPE PROPERTIES

329

Let us now recall the definition of the S H A P E category over compacta. T h e class of objects of the S H A P E category is the class of all compacta. T h e morphisms, or more precisely, the elements of the set M O r S H A P E ( X , Y ) are defined to be the maps a" U { [ Y , P ] " P is an A N R }

-+ U { [ X , P]" P is an A N R }

satisfying the following two conditions: . If P is an A N R - c o m p a c t u m , then a([Y, P]) _C IX, P]. . If P and P ' are A N R - c o m p a c t a , r E [Y, P], ~ E [Y, P'] and -y E [P, P'] are such t h a t 7 r = ~, then -ya(r = a ( ~ ) . T h e n a t u r a l (fundamental) functor Jz: H O M O T

~ SHAPE

acts in the following fashion: . ~ ' ( X ) = X for each c o m p a c t u m X. - If ~o E IX, Y], then 9r(~): X ~ Y is given by 9r(~o)(r r E [Y, P] where P is an A N R - c o m p a c t u m . T h e composition of the functors 7-/and ~" is d e n o t e d by Sh: COMP

= r

for any

~ SHAPE

and called the shape functor. It is known [175], [214] t h a t (unlike the h o m o t o p y functor) the shape functor is continuous. LEMMA 8.4.6. Let S x = { X a , p ~ a , A } be an w-spectrum consisting of metrizable compacta and ~: l i m S x --~ Y be a shape m o r p h i s m , where Y is a metrizable compactum. Then there exist an index a E A and a shape m o r p h i s m ~a: X a --* Y such that ~o = ~ a Sh(pa). PROOF. W i t h o u t loss of generality, by T h e o r e m 8.4.4, we may assume t h a t the s p e c t r u m 8 x is homotopically stable. Represent Y as the limit space of some inverse sequence S y = {Yn, qnn+l,w} consisting of compact polyhedra. T h e shape m o r p h i s m ~: X ~ Y, where X = lim 8 x , naturally induces a homotopy class ~([ql]) : X --. ]"1. Denote by gl : X --~ Y1 any representative of this class. Fix an index a l E A and a map gal: X a l ~ Y1 such t h a t gl --- galPal. Now consider the map q2: Y --* ]I2. As above, let g2: X --~ Y2 be any representative of the h o m o t o p y class T([q2])" X --~ ]I2. Since ql - q2q2 and Y1 and Y2 are compact polyhedra, we can conclude, by the definition of shape m o r p h i s m [214], t h a t the condition ~([ql]) - [q12]~o([q2]) is satisfied. Consequently, gl - q2g2. As above, we choose an index a2 >__ a l and a m a p g a 2 : X a 2 -* Y2 such t h a t g2 -- ga2Pa2. T h u s we have c~2

2

2

galPalPa2 -" galPal - gl ~ q g2 = qlga2Pa2" Since Y1 is an A N R - c o m p a c t u m and our s p e c t r u m is homotopically stable, we conclude t h a t g ~ p ~ "~ q21g~2.

330

8. APPLICATIONS

Continuing this process, we get, for each n E w, an index C~n E A and a map dan" X~,~ ---* Yn satisfying the following conditions: (i) an <_ an+l whenever n E w. (ii) ga..Pa,, E ~([qn]) whenever n E w. (iii) Y- a n Paa~n + l ~'~ tin _n+l g a n + l whenever n E w We now let a = s u p { a n " n E w} and define a shape m o r p h i s m ~ " X a --* Y. First, we define the h o m o t o p y classes ~a([qn]) by letting ~oa([qn]) = [ga.Pg.],

n E w.

Suppose now t h a t If]" Y --~ Z is an arbitrary h o m o t o p y class, where Z is any metrizable A N R - c o m p a c t u m . We fix n E w and a map fn" Yn --* Z such t h a t f ~- fnqn. T h e n we put (x

v , ( [ / ] ) = [S~g,~p,,],

n E w.

It follows from conditions (i)--(iii) t h a t the definition of ~oa([f]) does not d e p e n d on the choice of n E w and fn. Therefore, the shape morphism ~a" X a Y is well defined. Finally let us verify the equality ~ -- ~ Sh(pa). For this, it is sufficient to verify the equalities ~([qn]) - ~a([qn])~a], n E w. But these equalities are trivially true, because by (ii) and by the definition of ~ a , =

This completes the proof.

=

=

[::]

Now we are ready to prove the main result of this Section, which is a shape analog of the spectral theorem for usual maps. THEOREM 8.4.7. Let S x -- { X a , p~, A} and S y -- (Ya, q~a,A} be w-spectra consisting of metrizable compacta and having the same indexing set. Let a shape morphism ~" l i m S x ~ l i m S y be given. Then the set of those indexes ~ E A for which there exist shape morphisms ~ " X a ~ Ya satisfying the conditions Sh(qa)~ = ~a Sh(pa) is w-closed and cofinal in A. PROOF. W i t h o u t loss of generality, by T h e o r e m 8.4.4 we may assume t h a t b o t h given spectra are homotopically stable. Let s

=

{(c~,f~) E A 2" a < f~ and there exists a shape morphism ~"

X~ --, Ya such t h a t S h ( q , ) ~ = ~

Sh(p,)}.

T h e validity of the existence condition of Proposition 1.1.29 follows directly from L e m m a 8.4.6. T h e verification of the m a j o r a n t n e s s condition is trivial. Let us verify the w-closeness condition. Consider an arbitrary countable chain { ( ~ n ' n E w} in A, and suppose t h a t (C~n,f~) E s for each n E w and for some f~ E A. We wish to show t h a t (a,f~) E L, where a -- sup{c~n" n E w}. By

8.4. SHAPE PROPERTIES the definition of s there exist shape morphisms ~ " Sh(qan)(P = (P~n Sh(p~), n E w. Let us first show that for each n E w, the equality ~o~,,

331 XZ ~

Xan such that

Sh(qaa: +1 ) ~

is true. Consider an arbitrary map f " }ran ---, Z into some metrizable AN Rcompactum Z. Note that the equality

~o~,~ Sh(p~) = Sh(qa.)~o implies the equality Similarly the equality

~an+l ~ Sh(pz) = Sh(qan+l)~ implies the equality ~'~.+~ ([S " l Q ~ n

J]

~] = ~ \

L

"IQ: n

q~.+~]) = ~([Sq~o])

Consequently,

-- r Since the spectrum S x is homotopically stable, we conclude that ~ a,., (Ill) = ~a,~+~ ([fqaa:+~])" This means t h a t ~o~,, =- Sh(qa,~ -~"+~ )~v~n+~ ~ as desired. By the continuity of the SHAPE functor and by the w-continuity of the spectrum S t , we conclude t h a t in this situation there exists a shape morphism ~ " X Z --, Y~ such t h a t ~v~n = S h ( q ~ ) ~ for each n E w. It only remains to note t h a t the equality ~ Sh(pz) = Sh(q~)~ is also satisfied. The verification of all the conditions of Proposition 1.1.29 is finished. By this Proposition, the set of all /:-reflexive indexes is wclosed and cofinal in A. This completes the proof of our statement. [21 COROLLARY 8.4.8. I f (in the notations of Theorem 8.4.7) ~ is a shape isomorphism, then the set of those a E A for which ~va" X a ~ Ya is a shape isomorphism is w-closed and cofinal in A. PROOF. We use the notation of the proof of Theorem 8.4.7. Consider the shape inverse ~ - 1 : y ~ X, which exists by our assumption. By Theorem 8.4.7, the set A2 of those indexes a E A for which there exists a shape morphism Ca : Ya ~ Z a satisfying the condition Ca Sh(qa) = Sh(pa)~ -1 is w-closed and cofinal in A. Denote by A1 the set of those indexes a E A for which there exists a shape morphism ~oa: Z a ~ Ya satisfying the condition Sh(qa)~ = ~a Sh(pa). By Theorem 8.4.7, this set is w-closed and cofinal in A. By Proposition 1.1.27, the intersection B -- A1 N A2 is also w-closed and cofinal in A. Clearly, for each a E B we have two shape morphisms ~ a : X a ~ Ya and Ca: Ya ~ X a satisfying the above mentioned conditions. The proof will be finished if we show

332

8. APPLICATIONS

that Ca~a = 1 x , and ~aCa = 1y., whenever a E B (1x denotes the identity morphism of an object X in the SHAPE category). Let F" Xa ~ Z be an arbitrary map into a metrizable A N R-compactum. It follows from the above constructions that C a ~ a Sh(pa) -- Ca Sh(q~)~ = S h ( p a ) ~ - l ~ = Sh(pa). Consequently, ~ , ( r = [flip.]. The homotopical stability of the spect r u m $ x implies that ~ a ( r = [f], i.e. r = 1 x . . The verification of the second equality is similar. [7

8.4.2. S h a p e p r o p e r t i e s of n o n - m e t r i z a b l e statement is a direct consequence of Lemma 8.4.6.

c o m p a c t a . The following

PROPOSITION 8.4.9. Let X be a compactum. Then the following conditions are equivalent: (i) For each metrizable A N R-compactum Y , the set [X, Y] is countable. (ii) X has a shape of a metrizable compactura. (iii) X can be represented as the limit space of an w-spectrum, all limit projections of which are shape equivalences. (iv) X admits a surjection onto a metrizable compactum and this surjection is a shape equivalence. PROOF. The equivalence of conditions (i) and (ii) was proved in [310]. (ii) ~ (iii). By Theorem 8.4.4, we can represent X as the limit space of some homotopically stable w-spectrum S x - {X~, p~, A } . By (ii), there exists a shape isomorphism ~" X ~ Y, where Y is a metrizable compactum. By Lemma 8.4.6, there exist an index a0 E A and a shape morphism ~ao" X~o ~ Y such that ~ - ~ao Sh(pao). For each a > a0, define a shape morphism ~a" X a --+ Y by letting ~a ~ao Sh( so)" Clearly, ~ ~a Sh(pa) for each a > a0 Since the c o m p a c t u m X is the limit space of the w-spectrum S' = { X a , p ~ , A o } , where A0 = {a E A" a _> a0}, it suffices to show that each limit projection pc" X --~ X a , a >_ co, is a shape equivalence. This means that for each metrizable A N R c o m p a c t u m Z, the natural correspondence [Xa, Z] --~ IX, Z], induced by the limit projection pa by the rule If] --~ [fpa], is a bijection. Since the spectrum S is homotopically stable, we immediately conclude that this correspondence is injective. Let us show that it is surjective as well. Consider an arbitrary homotopy class [g] E [X, Z]. Then ~-l([g]) e [Y,Z] and ~ a ( ~ - l ( [ g ] ) ) E [Xa, Z]. It only remains to note that

v.(v-~([g]))[p.] v(v-l([g]))= [g].

8.4. SHAPE PROPERTIES

333

(iii) =:~ (iv). As the desired surjection, we m a y take any limit projection of the s p e c t r u m satisfying the properties of condition (iii). T h e implication (iv) ~ (i) is trivial. E] PROPOSITION 8.4.10. Let X be a compactum. Then the following conditions are equivalent: (i) For each closed G~-subset Z of X and each metrizable A N R - c o m p a c t u m P, the set [Z, P] is countable. (ii) X can be represented as the limit space of some w-spectrum, all limit projections of which are hereditary shape equivalences. PROOF. (i) ~ (ii). By T h e o r e m 8.4.4, we can represent X as the limit space of a homotopically stable w-spectrum S x = {X~, p~, A} . Let _

_

{(c~, f~) E A 2" a _< f~, and for each closed subspace F of X a the m a p

~/p~l(F)"

p-~l(F) ~ ( p ~ ) - l ( F ) is a shape equivalence}.

First, let us verify the existence condition of P r o p o s i t i o n 1.1.29. Fix an index a E A and a countable closed basis {Fn" n E w} of the topology of metrizable c o m p a c t u m X a containing the intersections of its finite subcollections. By (i), for each n E w and for each metrizable A N R - c o m p a c t u m P, the set [p~l(Fn)], P] is countable. Consequently, by Proposition 8.4.9, for each n E w the c o m p a c t u m p~l(Fn) can be represented as the limit space of some w - s p e c t r u m all limit projections of which are shape equivalences. By T h e o r e m 1.3.4, there exists an w-closed and cofinal s u b s p e c t r u m with the above p r o p e r t y in every w - s p e c t r u m representing the c o m p a c t u m p-~l(Fn). In particular, this is true for the n a t u r a l l y induced (by S x ) s p e c t r u m {(p~a) - l ( F n ) ,

5 5 -1 (F~),5 >_ ~ _> .}. p~/(v~)

Hence there exists at least one index an _> a such t h a t the m a p v../v-~(Fn)

9V;~(F.)

~

(p."~)-l(F~)

is a shape equivalence. W i t h o u t loss of generality, we m a y assume t h a t {an" n E w} is a chain in A. Let f~ = sup{an" n E w}. Note t h a t for each n C w, the m a p

p/3/p-~l(Fn) " p-~l(Fn) --+ (p~a)-l(Fn) is a shape equivalence. Let us show t h a t ( a , ~ ) E s Let F be an a r b i t r a r y closed subset of X a and f" p ~ l ( F ) --~ P be a m a p into a metrizable A N R c o m p a c t u m . As in the proof of L e m m a 8.4.3, we can find an integer n E w and a m a p f " p~l(Fn) ~ P where F C_ Fn and .f = f ' / p - ~ l ( F ) . By the definition of ~, there exists a m a p g " ( p ~ ) - l ( F n ) --~ P such t h a t f ' ~ g'p~/p~l(Fn). P u t g = g ' / ( p ~ ) - l ( F ) . Obviously, f ~ gp/3/p-~l(F). T h u s we have shown t h a t the n a t u r a l correspondence [(p~)-I(F),P]--~ [p~I(F),P], induced by p z / p ~ l ( F ) , is surjective. Since the s p e c t r u m S x is homotopically stable we conclude t h a t t h e above correspondence is bijective.

334

8. A P P L I C A T I O N S

The verification of the majorantness condition from Proposition 1.1.29 is trivial. Now let {~n: n E w} be a countable chain in A, and let (an, ~) E E for each n E w and for some ~ E A. Let us show t h a t (a, ~) E L, where a - sup{an : n E w}. By the w-continuity of the spectrum S x , the c o m p a c t u m X a is naturally homeomorphic to the limit space of the inverse sequence {Xa~ ,pa~ _a~+l ,w}. Let F be a closed subset of X a and f : p ~ l ( F ) ---. P be an arbitrary map into a metrizable A N R - c o m p a c t u m . Fix a neighborhood V of p-~l(F) in X and an extension f ' : V ~ P of f . W i t h o u t loss of generality, we may assume that V - p ~ l ( U ) , where U is a neighborhood of F in Xa. There exists an integer n E w such t h a t

F c where F n -

c U,

(p~,)(F). Hence c

C Y.

Since ( a n , ~ ) e s there exists a map g" (p~.)(Fn) --* P such that P

--1

P

--1

f / P a ~ (Fn) ~-- g PZ/Pa~ (Fn). Clearly the map g - g ' / ( p ~ ) - l ( f ) satisfies the condition f ~_ gp~/p~l(F). This shows that the natural correspondence [(p~)-l(F),P] --+ [ p ~ l ( F ) , P ] is surjective. The homotopy stability of S x implies that this correspondence is bijective. Thus (a, B) e L. Now, by Proposition 1.1.29, the set A' of L-reflexive indexes of A is w-closed and cofinal in A. It only remains to note t h a t the L-reflexibility of an index a E A means precisely t h a t the limit projection pc: X ---, X a is a hereditary shape equivalence. The proof of the implication (i) ~ (ii) is finished. The verification of the converse implication is trivial. V1 PROPOSITION 8.4.11. Let X be a finite-dimensional 1-UV-compactum. the following conditions are equivalent: (i) X is shape equivalent to a finite polyhedron. (ii) The C ech cohomology groups of X are finitely generated.

Then

PROOF. For metrizable X this s t a t e m e n t was proved in [157]. Therefore, we may assume below that X is a non-metrizable compactum. (ii) ===>(i). Let 8 x - {X~, p~, A} be an w-spectrum the limit space of which coincides with X. By (ii), Proposition 8.4.9 and Theorem 1.3.4, there is an wclosed and cofinal subset A1 _C A such that the limit projection pc: X --+ X a is a shape equivalence whenever a E A1. In particular, the (~ech cohomology groups of X~, a E A 1, are finitely generated [4"7]. For the same reason, X a is 1 - UV for each a E A1. By Theorem 1.3.10, there is an w-closed and cofinal subset A2 C_ A such that d i m X a - d i m X < c~ for each a E A2. Now, by Proposition 1.1.27, the set B -- A1 M A2 is also w-closed and cofinal in A. In particular, B :fi 0. Let E B. Since X a is metrizable, we conclude (see the beginning of the proof)

8.5. FIXED POINT SETS

335

t h a t X a is shape equivalent to a finite p o l y h e d r o n . It only remains to note t h a t X is s h a p e equivalent to X a. T h e implication (i) = ~ (ii) is trivial. [::]

Historical and bibliographical notes 8.4. T h e results of this Section are t a k e n from [99]. T h e equivalence of conditions (i) and (ii) in P r o p o s i t i o n 8.4.9 was proved in [310]. P r o p o s i t i o n 8.4.11 for m e t r i z a b l e c o m p a c t a was o b t a i n e d in [157]. T h e axiomatic description of the s h a p e functor a p p e a r s in [174].

8.5. F i x e d p o i n t s e t s o f T y c h o n o v

cubes

T h e p r o b l e m we are interested in is the following: W h i c h closed subsets of t h e T y c h o n o v cube I r coincide with the set of fixed points of continuous self-maps of I r ? T h e p r o b l e m is m o t i v a t e d by the following two facts. First of all, each n o n e m p t y closed subset of the Hilbert cube I • coincides with the set of fixed points of some self-mapping (actually, a u t o h o m e o m o r p h i s m ) of I ~ [216]. On t h e o t h e r hand, for each u n c o u n t a b l e r there is a closed zero-dimensional subspace K~ of the T y c h o n o v cube I ~ which does not coincide with the set of fixed points of any self-map of I r [197]. T h e first fact says t h a t the Hilbert cube I ~ has the CIP (complete invariance property [308]), whereas t h e second tells us t h a t the T y c h o n o v cube I r (T > w) does not. B o t h results express a p r o p e r t y of t h e a m b i e n t spaces I ~ and I ~ , T > w, respectively. B u t obviously, t h e r e is a n o t h e r dual point of view of this situation. Before we s t a r t a more formal discussion, let us (for simplicity) i n t r o d u c e a p p r o p r i a t e terminology. Let us say t h a t a c o m p a c t u m X can be fixed in a c o m p a c t u m Y if t h e r e is an e m b e d d i n g i" X ---. Y such t h a t i ( X ) = f i x ( f ) for some m a p f " Y ~ Y ( f i x ( f ) denotes the set of all fixed points of f ) . Respectively, we say t h a t X cannot be fixed in Y if t h e r e is no such e m b e d d i n g of X into Y. Of course, if we fix an a m b i e n t c o m p a c t u m Y, t h e n the above properties become topological p r o p e r t i e s of X . Therefore, it does not follow from the above cited result of [197] t h a t the c o m p a c t u m K r c a n n o t be fixed in I r . So t h e r e still is a possibility t h a t any c o m p a c t u m of weight <_ T can be fixed in I r . In this Section we give a c o m p l e t e solution of the above p r o b l e m by providing a s p e c t r a l c h a r a c t e r i z a t i o n of those c o m p a c t a which can (not) be fixed in I ~. T h e main result says t h a t a c o m p a c t u m X can be fixed in I ~ if and only if X can be r e p r e s e n t e d as the limit space of some transfinite s p e c t r u m $ x = { X a , p ~ , T} of length ~- whose short projections p ~ + l . X ~ + I --* X~ are stable and whose first element X0 is metrizable. T h e notion of stability of a m a p p" X ~ Y involves two ingredients. T h e first says t h a t t h e r e is an e m b e d d i n g i" X ~ Y • I ~ such t h a t 7ryi : p, where ~ry" Y • I ~ ---~---~ Y d e n o t e s the n a t u r a l projection. Maps with this p r o p e r t y are exactly m a p s with a m e t r i z a b l e kernel. T h e second ingredient says t h a t t h e r e is a m a p f" Y • I ~ ~ Y • I ~ such t h a t f i x ( f ) -- i ( X )

336

8. APPLICATIONS

and r y f = lry. In other words, i ( X ) can be fixed in Y • I ~ by a map f which acts fiberwise (with respect to Try). Open retractions with metrizable kernels are stable. This allows us to conclude that a retract of any product of metrizable compacta can be fixed in Tychonov cubes of the corresponding weight. Going back to the compacta g r , T > w, we strengthen the above mentioned result of [197] by showing t h a t the compactum K,, 1 cannot be fixed in any non-metrizable ANR-compactum. Representing K,~I as the limit space of a transfinite inverse spectrum consisting of zero-dimensional metrizable compacta, and applying our characterization theorem together with a spectral theorem, we conclude t h a t among the projections of this spectrum, only countably many can be stable. All others serve as examples of surjections between zero-dimensional metrizable compacta which are not stable. Obviously, these maps are not open (open surjections between zero-dimensional metrizable compacta are retractions, and therefore are stable). We also present an example of a stable retraction between metrizable AR-compacta (which is not open).

8.5.1. E x t e n s i o n of h o m e o m o r p h i s m s . The following definition is motivated by the definition of (usual) Z-sets, and is adapted to the case of uncountable products. DEFINITION 8.5.1. Let T > W. We say that a closed subset Z of a space X is a Zr-set in X if for each collection {lilt" t E T } , where Ltt E c o y ( X ) and IT I < T, there is a map f " X ---, X such that f ( X ) M Z = 0 and f is {Lit" t E T}-close to idx. In other words, Z is a Zr-set in X if idx E clc~.(x,x){f E Cr(X,X)"

f ( X ) M Z = 0).

Note that the notion of Z~-set coincides with the usual notion of Z-set. LEMMA 8.5.2. Let X -- Y I { x a : a E A } be a product of compact I ~ - m a n i f o l d s , ]A] -- r > w, and suppose there is an index b E A such that X a is a copy of the Hilbert cube I ~ f o r each a ~ b. Suppose that Z and F are closed subsets of Z such that 7ra(Z) and 7~a(F) are Z - s e t s in X a for each a E A. Then f o r each h o m e o m o r p h i s m g: Z --~ F which is homotopic (in X ) to the inclusion map of Z , there exists an a u t o h o m e o m o r p h i s m G: X ---, X such that G / Z -- g and G ~_ i d x . PROOF. Represent X as the limit space of the standard w-spectrum S x = { X B , lr~,exp,,,A} consisting of countable subproducts X B = Y I { X a ' a E B} of X and natural projections 7r~" X B ~ X c . Consider also the induced w-spectra Sz--

{ZB,p~,exp~A}

and SF = {FB, q ~ , e x p ~ A }

8.5. FIXED POINT SETS

337

where Z B = lrB(Z) , p ~ -- 1r~/ZB, FB -- 7rB(F) and q~ = l r ~ / F B ( C , B E exp,.,A, C C_ B ) . Obviously, Z = lim S z and F = lim SF. By T h e o r e m 1.3.4, the h o m e o m o r p h i s m g is the limit map of some m o r p h i s m g :

{gB: F B -'+ Z B , B E )i~l}: ,SZ/]~I ---+5F/)i~I

consisting of h o m e o m o r p h i s m s gB's, where 1~1 denotes a cofinal and w-closed subset of expwA. Consider an w-spectrum B 5 Z X 51 = { Z B X I , p C X i d i , e x p w A }

where I = [0, 1]. Obviously l i m ( S z x 5x) = Z x I. Fix a h o m o t o p y H : Z x I --, X connecting g and the inclusion map i: Z --, X. Applying T h e o r e m 1.3.4 to the spectra S z x 5 i , 5 x and the h o m o t o p y H, we can represent H as the limit map of some morphism ~'L: { H B : Z B • I ---+ X B , B E ~ 2 } : (SZ • 5 I ) / ~ 2 ~ 5X/]C2 where )E2 denotes a cofinal and w-closed subset of e x p ~ A . By Proposition 1.1.27, -- ~1N)i~2 is again a cofinal and w-closed subset of expwA. It is easy to see t h a t for each B E )E, the h o m e o m o r p h i s m gB and the inclusion map iB: Z B ~ X B are connected by the h o m o t o p y H B. Fix a well-ordering A = { a s : a < T} of A such t h a t a0 = b. Let B0 E /C and a0 E B0 9 By the properties of Z and F , their projections Z Bo and FBo are Z-sets in the Hilbert cube X Bo. T h e h o m e o m o r p h i s m go -- gBo: Z Bo ~ FBo is homotopic to the inclusion map io - i Bo (via the h o m o t o p y Ho -- H Bo). Consequently, by T h e o r e m 2.3.19, go can be e x t e n d e d to an a u t o h o m e o m o r p h i s m Go: XBo --~ XBo which is homotopic (via some h o m o t o p y R0: X S o x I ~ X S o ) to the identity map of XBo. Suppose now t h a t for each/3 < a, where a < T and a > 0, we have already constructed subsets BO of the index set A, h o m e o m o r p h i s m s gf~: Z B , ---* FB~, a u t o h o m e o m o r p h i s m s G~ of X B, and homotopies RZ: X B~ x I ---. XB~ in such a way t h a t the following conditions are satisfied: (a) { a 5 : 6 < ~ } C_ Bfl. (b) B5 C B~ whenever 6
338

8. APPLICATIONS

Suppose now t h a t a = fl + 1. Consider the smallest ordinal "7 < T such t h a t a. r r B~. By (a), "/ _ ft. By Corollary 1.3.17, there exist an element C E e x p ~ ( A - B~), containing a.r, and a h o m e o m o r p h i s m ga =-- gB~oC: ZBf~UC -->

Define the set B a as the union B~ U C. Corollary 1.3.17,

FB~uC.

Obviously, {a/~: fl < a} C__ Ba.

qB~g -- gaPB~ and, consequently, q

By

ga = g/3PB~.

Using T h e o r e m 1.3.16, we may assume without loss of generality that a homotopy Ha: Z B . x I ---, X B o (connecting ga and the inclusion map in: ZB~ ~ X B . ) such t h a t is also defined. By our assumptions, the projections r a(Z) and r a ( f ) are Z-sets in X a for each a E A. Consequently, we may conclude that Z B . and FB.~ are fibered Z-sets with respect to the projection 7rBB; " X Ba -- X B# X X C

~

X B#.

Now, 7rB$ is a trivial bundle with the Hilbert cube X c as a fiber (because b = a0 ~' C). Let t denote the identity m a p of the fiber X c and consider the set T = (G# x t ) ( Z B . ) . Clearly T is a fibered Z-set with respect to zrB$. The homeomorphism r = ga(Gf3 x t ) - l / T :

T ---, FB~

is fiber preserving, i.e. r S $ r = 7rBS$. A fiber preserving homotopy = g a ( ( G ~ x t) x i d i ) - l : T x I ---, X B .

connects r and the inclusion map of T into X B . . By [300], there is a homeomorphism ~" X B . ~ XB~ such that r = r and r B $ ~ = zrB$. Moreover, there is a homotopy O: X B . x I ~ XB~ which connects ~ and the identity map and satisfies the equality rg;O(x,r)

= rS;(x)

for each ( x , , ) E X B . x I.

Now we are in position to define the desired objects. Let G~ = ~ ( G ~ x t) and R a ( x , y, r) = O ( ( R ~ ( x , r), y, r) for each (x, y, r) E XB~ x X C x I = X B . x I. It follows from the construction that G a and Ra are fiber preserving and satisfy all the conditions (a) - (i) formulated for the ordinal a. This completes the inductive step. LetG=lim{Ga: a
8.5. FIXED POINT SETS

339

LEMMA 8.5.3. Let { Z n : n E w} be a countable collection of Z r - s e t s in a prodThen

uct X = I I { x a : a E A } of metrizable compacta X a , where IAI = T > w. there is a collection { A a : ~ < T} of countable subsets of A such that: (i) U{Aa: a < T} = A.

(ii) Aa gl AB = 0 whenever a 7~ ~. (iii) ZrA,(Zn) is a Z - s e t in X A . f o r each a < T and each n E w. PROOF. First we need the following: Claim. Let B C A, IBI < r and C E expoj(A - B ) . F o r / d E c o v ( X c ) there is a countable subset D of A - B , containing C, such that f o r each n E w there is a map Cn" X D ~ X D satisfying the following conditions: (a) ~ b n ( Z D ) N 7 r D ( Z n ) • ~ f o r each n E w and (b) ZrCr D n and lr~ are U-close. Proof of Claim. Consider the collection (b/t" t E T} of open covers of X B from L e m m a 6.5.1. Consider also the collection {Tr-l(/dt)" t E T} U ~r-l(/d) of open covers of X. Since Zn is a Zr-set in X and the cardinality of B is strictly less t h a n T, we can conclude t h a t for each n E w there is a m a p gn" X --+ X such t h a t g n ( X ) f i Zn ---- ~ and gn is ({Tr-l(/dt)" t E T} U {~r-l(U)})-close to the identity map of X. T h e latter condition implies the equality 7rBgn : 7rB, n E w (by the properties of the collection {/dt" t E T}). This is enough to conclude t h a t there is a countable subset D of A - B such t h a t C C D and 7 t o ( Z ) f'? 7rDgn(X) : ~ for each n E co. Consider a section j" X D --~ X of the projection 7rD and define Cn" X D ~ X D as the composition Cn -- 7rDgnj, n E w. Clearly Cn(XD)MZrD(Z) -- $ and IreD Cn is/d-close to lr D (because zrcgn is/d-close to 1re) for each n E w. T h e proof of Claim is complete. Let us consider the following relation ~ B o n e x p ~ o ( A - B):

/:B

=

{ ( C , D ) e (exp~o(A - B)) 2" C C_ D, and for each /d E c o v ( X c ) n E w there is a m a p Ca" X D "-+ X D such t h a t ~ b n ( Z D ) N 7rD(Zn) = and 1 tDe e n and 7r~ are /,/-close}.

We need the following three properties of this relation. Property 1. For each C E e x p o j ( A - B ) , there is a D E e x p ~ o ( A - B ) such that ( C , D ) e E.B. Proof of Property 1. Fix a countable family {/di" i E w} of open covers of X c such t h a t for each open cover/2 of X c there is an index i with the property t h a t /di refines/4. By the Claim, for each i E w there is a countable subset Di of B - A containing C, and maps ~bi,n " X D i ~

XD~ , n E w,

such t h a t r -- 0 and ~rcDi r and ~r~ ~ are/di-close. Let us show t h a t ( C , D ) E 12B, where D -- U{D~" i E w}. Indeed, for any given open cover 34

340

8. APPLICATIONS

of X c , fix an index i such t h a t b/i refines hi. Also fix any section si" X D i ~ X D of the projection r Di* D T h e n the maps ~n

s i ~ i ,nrD~ D . iD ~

I D, n E w

obviously verify the fact t h a t (C, D) E s Property 2. If (C, D) E LB, E E e x p ~ ( A - B) and D C E, then (C, E) E s Proof of Property 2. Fix a section s" X D ~ X E of the projection r E" X E Z D. If an open cover 34 E c o v ( X e ) is given we can, using the fact that (C, D) E s find maps ~n" XD ~ XD, n E w, such t h a t ~ n ( X D ) N T r D ( Z ) -- 0 and ~ n r ~ and r ~ are/,/-close. Let Cn = 8 ~ n r E " X E -~ X E , n E w.

Clearly, C n ( X E ) M r E ( Z n ) -- 0 and r cEC n is U-close to r E, n E w. Property 3. Let (Ci, D) E s and Ci C Ci+l for each i E w. Then (C,D) E s where C -- U{Ci" i E w}. Proof of Property 3. For any open cover /g E c o v ( X c ) , there is an index i e • and an open cover lgi e cov(Xe,) such t h a t ( r ~ , ) - l ( b / i ) refines/4. Since (Ci, D) E s we conclude t h a t there are maps Cn" XD ~ X D , n E w, such t h a t Cn(ZD)NrD(Zn) = 0 and r ~ r is/gi-close to r Ci" D It only remains to note t h a t r cDC n and r ~ are L/-close. We now continue with the proof of our Lemma. By the Properties 1,2,3 and Proposition 1.1.29, the collection ~ B of all C E exp,~(A - B) with the property (C, C) E s is cofinal and w-closed in exp~(A - B). Note t h a t if C E ]~B, then r c ( Z n ) is a Z-set in X c for each n E w. Since IAI = ~- we can write A = { a ~ ' a < 7-}. For a point a0, fix a set Ao E exp~A such t h a t ~rno(Zn) is a Z-set in XAo for each n E w (we use the cofinality of the collection K:B in expw(A - B) with B = 0). Suppose t h a t for each ~ _< c~, where a < T, we have already constructed countable subsets At~ of A so t h a t

{a~" fl S o~} C_U{A~" fl < o~} a n d rA~(Zn) is a Z-set in X A ~ whenever fl <_ a and n E w. Let us construct A a + l . Let 5 denote the smallest ordinal such t h a t c~5 r B = U{A~" ~ _ a}. Clearly, 5 > c~. Note t h a t IB] < T. Since the collection K:B is cofinal in e x p w ( A - B), there is an element A a + l E ]CB such t h a t c~6 E Aa+l. Clearly, A a + l A A~ = 0 for each ~ _< a (because ]~B ~ e x p w ( A - B)) and { a ~ ' ~ _< c~ + 1} C {a~" ~ _< 5} C U{Az" ~ < c~ + 1}. This completes the proof of L e m m a 8.5.3. El

THEOREM 8.5.4. Suppose that the homeomorphism g: Z ---. F between Zrsets of the compact I r -manifold X is homotopic to the inclusion map Z ~-~ X (T > w). Then there is an autohomeomorphism G: X ~ X which extends g and is homotopic to i d x .

8.5. FIXED POINT SETS

341

PROOF. By T h e o r e m 7.1.19, X is h o m e o m o r p h i c to the p r o d u c t of a compact I ~ -manifold and the Tychonov cube I r . Consequently, we can represent X as the product YI{Xa: a E A}, I A I - T, where Xb is a compact I W-manifold for some b E A and all other X a ' s are copies of the Hilbert cube I ~ . Fix a collection {Aa: c~ < r} of countable subsets of A satisfying conditions (i)-(iii) of L e m m a 8.5.3 with respect to the Zr-sets Z and F. If ~ < T and b ~ Aa, choose an a r b i t r a r y point aa E Aa. If b E As, t h e n let a s = b. By the stability of I "~ -manifolds ( T h e o r e m 2.3.10), there is a homeomorphism

II{xo. a

Ao} - , xoo

for each a < ~-. Clearly, h-

x { h a ' c ~ < T)" X ---+ Y

is a homeomorphism, where Y = 1 - I { X a " a < T}. By L e m m a 8.5.3 (condition (iii)), h ( Z ) and h ( F ) are closed subsets of Y such t h a t their projections on each coordinate space Xa~ are Z-sets. Define the h o m e o m o r p h i s m f " h ( Z ) ~ h ( F ) to be the composition f - h g h - 1 / h ( Z ) . Clearly, f is homotopic to the inclusion m a p h ( Z ) --. Y . Consequently, by L e m m a 8.5.2, there is a h o m e o m o r p h i s m f" Y --. Y which extends f and is homotopic to i d y . It only remains to note t h a t the composition G - h - l f h is the desired a u t o h o m e o m o r p h i s m of X. W1 _

A similar a r g u m e n t shows the validity of the following s t a t e m e n t (compare with T h e o r e m 2.3.18). THEOREM 8.5.5. Let X be a compact I T-manifold, T > w. For each collection {/it: t E T} C c o y ( X ) , IT I < T, there exists another collection {)2t: t E T'}, ]T~I < T, such that the following condition is satisfied: (.) For any h o m e o m o r p h i s m h: Z --, F between arbitrary Z r - s e t s of X which is { I t : t E T~}-close to the inclusion map Z ~ X , there is a h o m e o m o r p h i s m H : X ~ X such that H / Z -- h and which is {/it : t E T } - c l o s e to i d x . We conclude this Subsection with the following s t a t e m e n t which allows us to recognize Zr-sets for T > w. PROPOSITION 8.5.6. Let X = 1-I{xa: a E A ) , ]A I = 7 > w, be a product of metrizable compacta such that all X a ' s with a ?~ b (b E A ) are homogeneous. Then the following conditions are equivalent for each closed subset Z of X : (i) r X ) = T for each closed subset F of Z . (ii) Z is a Z r - s e t in X . PROOF. (i) ~ (ii). Suppose t h a t the collection {/It" t E T}, IT I < T, of open covers of X is given. Our goal is the construction of a map f" X --~ X which is {b/t" t E T}-close to i d x and which satisfies the equality f ( X ) M Z - 0.

342

8. APPLICATIONS

Since ITI < T, t h e r e exist a subset B C A of cardinality ~ -

max{w, ]TI} and

a collection {1)t" t e T} of open covers of X B such t h a t PiBl()2t) refines hot for each t E T . W i t h o u t loss of generality we may assume t h a t b E B. Note also t h a t ~, < T. In order to proceed, we need the following s t a t e m e n t .

Claim. There exist a subset D C A such that B C_ D, IDI - ~, and a section i" XB---* XD such that i(ZB) N 7 t o ( Z ) - - 0 . Proof of Claim. Let B0 -- B and let i0 d e n o t e the identity map of X Bo. Suppose t h a t for each fl < a, where a < ~+2, we have already c o n s t r u c t e d subsets Bf~ of A and sections i~ 9 XBo ~ XB~ of the projection rBo satisfying the following conditions:

(a) IBI-

XB~ --+ XBo

,~.

(b) B6 C_ B~ whenever 5 _< ft. (c) B~ = U{B6" 5 < ~} w h e n e v e r fl is a limit ordinal. (d) 7rB6B~if~ = i6 whenever 5 _< ft. (e) i~ = lim{i6" 5 < fl} w h e n e v e r fl is a limit ordinal. For each fl < a, let

v~ = {= e XBo" i~(=) r ~B~(z)). Clearly, V~ is an open subset of Xuo. It follows from the above conditions that" (f) V6 c Vf~ whenever 5 <_ ft. (g) V~ = U{V6" 5 < fl} w h e n e v e r fl is a limit ordinal. Let us c o n s t r u c t the set B e and the section in. If a is a limit ordinal let

Ba=U{B~'fl<

c~} and i n = l i m { i f ~ ' f l <

c~}.

Consider now the case a = f l + 1. Fix any section j" XB~ ~ X of the projection ~B~" X ~ XB~. T h e r e are two possible cases: j(XBa)FI Z = 0 and j(XB~) F1Z :~ O. We consider each case separately. Case 1 (j(XB~) F1 Z = O). In this case there is a finite subset C of A such that

~ c ( j ( x B ~ ) ) n ~ c ( z ) = O. Let Ba

-

B~

U

C and in

--

7rB~jif~.

Clearly, i O ~ ( X S o ) N 7rB,~ ( Z ) ~- 0. Consequently, D = B a and i = in verify the claim in this case. Case 2 ( j ( X B ~ ) N Z ~ 0). Choose a point z G j ( X B , ) N Z and let y = ~B,(Z). Note t h a t y e 7rB,(Z), and hence y r Vz. Consider the set ~B,(Y). Since IB~[ = a < T, the c h a r a c t e r of y in X B , is <_ a. Consequently, r <_ a < T. By (i), ~rB,(y) is not c o n t a i n e d in Z. Choose a point z' 9 ~ B , ( Y ) - Z. T h e r e exists a finite subset C of A such t h a t r c ( z ' ) r ~ c ( Z ) . Let B a = B~ U C. Clearly 2~+ denotes the smallest cardinal greater than ~.

8.5. FIXED POINT SETS

343

yl - l r s = ( j ( y ) ) -Tt lrBo (z') -- Y2 and I r s , ( y l ) -- y ---- lrB~(Y2). Consequently Since X C - B ~ is homogeneous (as a p r o d u c t of homogeneous compacta; b r C - Bt~ ) there is a h o m e o m o r p h i s m

h" X C - B ~ ~ X C - B ~ such t h a t h ( l r V - B ~ ( Y l ) ) = 7rC-B~(Y2). Let

H - - i d x a • h. T h e n H is an a u t o h o m e o m o r p h i s m of X B . such t h a t H (Yl) = Y2 and lrB$ H -~ r ~ . Let

ia -- H~rB~ji~" X B o ~ X B ~ . It is easy to see t h a t V~ is a proper subset of Va (y E V a - Vt~). This completes the inductive step. Thus, subsets B a and sections ia satisfying the above conditions exist for each a < a+. T h e family {Va" a < a + } of the open subsets of X Bo described above is increasing and has length n. Since the weight of XSo is a, it follows t h a t the collection {Va" a < a} must stabilize, i.e. there must be an index/~ < a+ such t h a t Va - V~ for each a > ~. Because of our construction, this is possible only when V~ = XBo. Let D -- Bt~ and i -- i~. T h e proof of the claim is complete. We now complete the proof of our proposition. Fix a subset D and a section i as in the Claim. Let j" XD ----> X be any section of the projection 7tO" X ~ XD. T h e n f -- jiTrB" X --+ X is {Llt" t E T}-close to i d x and f (X) N Z -- 0. (ii) ===~ (i). This is straightforward and so is omitted. [El COROLLARY 8.5.7. Let T > w, and let X be a closed subset of a compact I ~ -manifold Y . If d i m X < c~ or w ( X ) < T, then X is a Z r - s e t in Y . PROOF. Suppose t h a t X is not a Z~-set in Y. By Proposition 8.5.6, there must exist a closed subset F of X such t h a t r Y) < T. In this case, F d e p e n d s on fewer t h a n T coordinates and, consequently, contains a copy of I r . In either case this is impossible. [El

8.5.2. F i x e d p o i n t s e t s o f T y c h o n o v c u b e s . T h e main result of this Subsection ( T h e o r e m 8.5.9) gives a complete characterization of c o m p a c t a which can be fixed in Tychonov cubes as limit spaces of transfinite inverse spectra whose first elements are metrizable c o m p a c t a and all short projections of which are stable in the sense of Definition 8.5.8. Here is a simple scheme illustrating how this result might be used. Suppose t h a t X is a c o m p a c t u m of weight wl. Consider any w-spectrum S x whose limit space coincides with X . Converting S z into a transfinite spectrum, we may assume from the outset t h a t S x -- { X a , p ~ +1, wl}

344

8. APPLICATIONS

is a transfinite spectrum of length wl consisting of metrizable compacta. We now investigate the short projections of S x . Theorem 8.5.9 says that if for "many" a's, the projection p~+l is stable, then X can be fixed in the Tychonov cube of weight wl. This is in some sense the obvious part. The more important fact is t h a t if for "many" a's, the projection aa+l is not stable, then X cannot be fixed in t h a t cube. Let us emphasize this point again. The spectrum 8 x is chosen arbitrarily (there are many others representing X). Nevertheless, the behavior of the short projections (from the point of view of stability) of a given spectrum allows us to conclude the existence or non-existence of an embedding of X into I W1 whose image is the fixed point set of some self-mapping of I ~1 . For example, any (uncountable) product of metrizable c o m p a c t a can be represented as the limit space of the standard spectrum consisting of the corresponding subproducts and natural projections between them. Any closed subspace X of such a product is then the limit space of an induced spectrum ,~x 9 If, additionally, X is a retract of that product, then we can conclude that "many" projections of 8 x are open retractions (because the projections of the spectrum representing the whole product are obviously open retractions). Therefore they are stable (Proposition 8.5.13) and X can be fixed in the Tychonov cube of the corresponding weight. This proves Proposition 8.5.14. DEFINITION 8.5.8. W e s a y t h a t a m a p p: X embedding i : X ~ ryf

-- r y

Y • I~

and fix(f)

~

Y

is stable if there is an

a n d a m a p f : Y • I ~ ---+ Y • I ~ such t h a t r y i -- p,

= i(X),

where ~ry: Y • I W ---+ Y is the projection.

THEOREM 8.5.9. L e t r > w. T h e f o l l o w i n g c o n d i t i o n s are equivalent f o r a n y compactum X :

(a) (b) (c) (d)

X

can be fixed in the T y c h o n o v cube I v .

X

can be fixed in a n y c o m p a c t I v - m a n i f o l d .

X

can be fixed in s o m e c o m p a c t I v - m a n i f o l d .

X

can be r e p r e s e n t e d as the l i m i t space o f s o m e t r a n s f i n i t e i n v e r s e spec-

trum Sx

-

( X a , p a a+l, v } w h o s e s h o r t p r o j e c t i o n s p~+l.a X~+I --+ X~

are stable and w h o s e f i r s t e l e m e n t X o is m e t r i z a b l e .

PROOF. (a) ==~ (b). Any c o m p a c t / r - m a n i f o l d Y contains a copy of I v . Fix a r e t r a c t i o n r : Y ~ I v and a m a p f : I r ---, I v such that f i x ( f ) - X . Then fix(g)X , where g - f r. (b) ==~ (c). Trivial. (c) ===> (d). Suppose that Y is a compact I v -manifold and f i x ( f ) - X for some map f : Y --+ Y. Represent Y as a product Y = 1-I(Ya: a E A } of compact IW-manifolds, where all Y a ' s with a ~ b are copies of the Hilbert cube I ~ (b E A) and A is a set of cardinality r. Fix a well ordering ( a s : (~ < T} of A such that a0 = b. By Theorem 1.3.16, there is a countable subset A0 of A, containing a0, and a map f0: YAo ~ YAo such that 7rAo f -- foTrAo. Let us show that 7rAo(X ) -- f i x ( f o ) . Indeed, if x E 7rAo(Z),

8.5. FIXED POINT SETS

345

then there is a point y E X such t h a t 7rAo(Y ) -- x. Therefore

So(~) = s0(~Ao(y))= ~Ao(S(Y)) = ~Ao(Y)= and XAo C f i x ( f o ) . Conversely, if x E f i x ( f o ) , then the same equality 7rAof -fOzrAo shows t h a t f(TrAlo(X)) C 7r-l(x) Being a topological copy of I • the fiber 71-Ao - 1 (x) has the fixed point p r o p e r t y (note t h a t b r A - A0). Therefore there is a point y E 7r-l(X)Ao such t h a t f ( y ) = y. T h e n y E f i x ( f ) = X . It only remains to note t h a t x = 7rAo(Y) e 7rAo(X). Therefore 7rAo(X) = f i x ( f o ) . Suppose now t h a t for each/~ < a, where a < T, we have already c o n s t r u c t e d a subset At~ of A and a m a p f~: YAm ---* YAm in such a way t h a t the following conditions are satisfied: (i) At~ C_ A~+x and [A~+I - A~[ _< w. (ii) At~ = U { A 6 : 5 < fl} whenever ~ is a limit ordinal. (iii) {a~: 5 < fl} C_ A~ whenever ~ > 0. (iv) 7rAm.f = :~TrAa. --

(v)

Am 7r A , 5 f fl =

An

Ao

f 57rA,5 whenever 5

"

_
(vi) f# = l i m { f 6 : 5 < Z} whenever fl is a limit ordinal. (vii) f i x ( f / 3 ) = 7rAa(X ). Let us construct the set A s and a m a p .fa: YA,~ --* YA,,, with the corresponding properties. If a is a limit ordinal, let Aa = U{At~: fl < a} and f a = lim{f~: ~ < a}. Suppose now t h a t a = f l + l . Let 3' denote the smallest ordinal such t h a t a.y ~ A~. By T h e o r e m 1.3.16, there is a countable subset C~ of A - A~, containing a.r, and

a map

fa: YAmwC~ ~ YAmuC ~ such t h a t 7rAmuC~f = f,~lrAmuC~. Let A s = AZ U Ct~. This obviously completes the inductive step of our construction. Consequently, the corresponding objects are defined for each c~ < T. Let X a --- 7 r A , . . ( X ) and Pa--a+l = 7rAA:+I/Xa+I for each a < T. Obviously, the limit space of the transfinite inverse s p e c t r u m S x = { X a p,~+l T} coincides with X. It only remains to note t h a t ~rlg = ~rl and f i x ( g ) = X a + l , where 7 r l : X a x I C '~ - A '~ ----, X a

denotes the projection onto the first c o o r d i n a t e and

This shows t h a t paa + l is stable in the above sense. (d) ==~ (a). Since for each a < T the short projection p ~ + l : X a + l --* X a is stable, the c o m p a c t u m X a + l can be identified with the subspace of the p r o d u c t

346

8. APPLICATIONS

X a • I ~ so t h a t paa+l coincides with the restriction of the projection ~r~" X a x I w --+ X a and f i x ( f o , + l ) - Xo,+l for some m a p f a + l " X a x I ~~ --+ X a x I ~~ with r ~ f a + l = r ~ . E m b e d X0 into the Hilbert cube QO and choose a m a p go" Q0 ~ Q0 such t h a t f i x ( g o ) = Xo (see [216] for the possibility of choosing a homeomorphism such as go; compare with the next Section). We proceed by transfinite induction. Suppose t h a t for each f~ < a, where 0 < a < T, we have already made an identification of X t~ with a subspace of the Tychonov cube Q~ and have constructed a map gt~" Qt~ _~ Qf~ such that"

(i)~/i~(g~) = x~. (ii)/3 p~ = lr2/X/3 whenever 6 < f~ (here 7r~" Qa _+ Qt~ denotes the s t a n d a r d projection onto the corresponding subproduct).

(iii)~ lr~hgf~ = g~lr~5 whenever 6 < f~. (iv)f~ gt~ = lim{g~" 6 < f~}. Let us carry out the inductive step. If a is a limit ordinal, let ga = lim{gz" f~ < a}. Conditions ( i i ) a - - ( i v ) a are trivially satisfied. Let us verify condition ( i ) , . T h e identifications m a d e above allow us to represent X a as the limit space of an inverse s p e c t r u m ,~a - {X/3,p~ +1, f~ < c~}. Consequently, if x E X a , then

g~(~) = A { g ~ ( ~ ) .

~ < ~}.

But g/37r~(x) e Xf~ and, therefore, by (i)f~, gzTr~(x) - r ~ ( x ) . This shows t h a t ga(x) -- x and X a C_ f i x ( g a ) . Conversely, if x r X a , then for some ~ < a we have r ~ ( x ) r X~. Again, by (i)t~ , this means t h a t g ~ r ~ ( x ) ~ r ~ ( x ) . Therefore x ~ ga(x) and x tg f i x ( g a ) . This finishes the verification of condition (i)a. Suppose now t h a t a = f~+ 1. Since r ~ is a trivial bundle (with the Hilbert cube as a fiber) there exists a m a p ga" Qa ~ Qa which extends fa" Xf~ • Q ~ Xf~ x Q (i.e. g a / ( x / ~ x Q) - fa) and which c o m m u t e s with g:~ (i.e. ~r~ga = gt~r~). T h e n X a = f i x ( g a ) . Indeed, since ga is an extension of f a and X a - f i x ( f ~ ) , it follows immediately t h a t X a C_ f i x ( g a ) . Conversely, suppose that x ~ f i x ( g a ) . T h e n x -- ga(x) and, consequently

~(~) = ~(~(~)) = ~(.~(~)). This means t h a t ~r~(x) ~ f i x ( g ~ ) - X~. Therefore x e Xt~ x Q. But ga and f a coincide on this set and we have x ~ f i x ( f a ) = X a . This completes the inductive step. In this situation, it can be easily seen t h a t X , as a subset of I ~ , can be fixed there by the map g = lim{ga" a < T}. W! LEMMA 8.5.10. Suppose that the compactum X can be fixed in I r , where T > w a n d v > w ( X ) T h e n X can be fixed i n I x w h e r e ~ - - m a x { w ( X ) , w }

8.5. FIXED POINT SETS

347

PROOF. T h e s t a t e m e n t is non-trivial w h e n T > n. It follows easily from T h e o r e m 1.3.4, for n-spectra, t h a t there exists a subset A of T such t h a t IAI -- n and the restriction 7 r s / X " X --. l r B ( X ) of t h e p r o j e c t i o n ~rs" I r --. I B is a h o m e o m o r p h i s m for each B with A C B C T. Let f " I r ---, I r be a m a p such t h a t f i x ( f ) -- X . Using T h e o r e m 1.3.4 for n - s p e c t r a (with respect to the s t a n d a r d s p e c t r u m consisting of n - s u b p r o d u c t s of I r and n a t u r a l projections) we can represent f as the limit m a p of some m o r p h i s m J= = { f c " x c --* I C ; c ~ lC}

w h e r e / C is a cofinal and n-closed subset of eXp~T. Consequently, t h e r e exists C E K: such t h a t A C C. Let us show t h a t ~ r c ( X ) (which is copy of X ) is fixed in I c by the m a p f c , i.e. f i x ( f c ) = ~rc(X). W h a t we need is the e q u a t i o n 7rcf = fcTrc, which follows from the fact t h a t $" is a self-morphism of the above m e n t i o n e d s p e c t r u m . Let y E ~rc(X), and fix a point x E X such t h a t ~rc(x) = y. Then fc(y)

= fc~c(~)

= ~cf(~)

= ~ c ( ~ ) = y.

Consequently, r c ( X ) c f i x ( f c ) . Conversely, if y e f i x ( f c ) 7 r c ( X ) , then, using the equality 7rcf - fcTrc, we m a y conclude t h a t f ( r c l ( y ) ) c_ r c l ( y ) . B u t ~rcl(y) is, topologically, the T y c h o n o v c u b e I r (because ~rc" I r --* I c is the n a t u r a l projection onto the c o r r e s p o n d i n g s u b p r o d u c t ) . Therefore t h e r e is a point x e r c l ( y ) such t h a t f ( x ) - x. T h e last equality implies t h a t x e X . On t h e o t h e r h a n d x r X (because y q~ r c ( X ) ) . This c o n t r a d i c t i o n shows t h a t fix(fc)r c ( X ) - 0 and consequently ~ r c ( X ) = f i x ( f c ) . We now need only to recall t h a t r c ( X ) is h o m e o m o r p h i c to X . V1 PROPOSITION 8.5.11. There is a zero-dimensional compactum K of weight wl which admits an embedding into every non-metrizable A N R - c o m p a c t u m but cannot be fixed in any of them. PROOF. Let K d e n o t e the zero-dimensional closed s u b s p a c e of I ~1 c o n s t r u c t e d in [197]. Recall t h a t w ( K ) -- w~ and t h a t K c a n n o t be r e p r e s e n t e d as the set of fixed points of any continuous self-map of I ~1 . Since every n o n - m e t r i z a b l e A N Rc o m p a c t u m contains a copy of the T y c h o n o v cube I "~, we see t h a t K a d m i t s an e m b e d d i n g into any n o n - m e t r i z a b l e A N R - c o m p a c t u m . S u p p o s e t h a t K can be fixed in some A N R - c o m p a c t u m . Since each A N R - c o m p a c t u m is a r e t r a c t of some c o m p a c t I ~" -manifold, we conclude t h a t K can be fixed in a c o m p a c t I r -manifold (for some T _ wl) aS well. B y T h e o r e m 8.5.9, K can be fixed in I r . By L e m m a 8.5.10, K can be fixed in I ~1. Since K is zero-dimensional, by Corollary 8.5.7, all copies of K in I ~1 are Z ~ - s e t s . Therefore, by T h e o r e m 8.5.4, all e m b e d d i n g s of K in I ~ are equivalent. This c o n t r a d i c t i o n finishes the p r o o f [:] COROLLARY 8.5.12. There is no non-metrizable compact absolute retract with the complete invariance property.

348

8. APPLICATIONS

Applying T h e o r e m 8.5.9 to the c o m p a c t u m K from Proposition 8.5.11, we may conclude (see the beginning of this section) that there is a surjection p: X ---+ Y between zero-dimensional metrizable c o m p a c t a which is not stable in the above sense. On the other hand, the following s t a t e m e n t provides a wide class of stable maps. PROPOSITION 8.5.13. L e t f : X ~ Y be an open r e t r a c t i o n between c o m p a c t a . S u p p o s e also t h a t X 7ry/X

is e m b e d d e d in the p r o d u c t Y x I W so t h a t the r e s t r i c t i o n

o f the p r o j e c t i o n lry : Y x I ~ ---+ Y

c o i n c i d e s with f . T h e n there exists a

m a p g: Y x I W --~ Y x I W s u c h t h a t the f o l l o w i n g c o n d i t i o n s are satisfied:

(i) f i x ( g ) = X . (ii) 7ryg : Try. PROOF. By T h e o r e m 6.3.1, the general case reduces to the case when Y (and, consequently, X ) is metrizable. For any point r 9 [0, 1], define a homotopy H r : [0, 1] • [0, 1] ---* [0, 1] by the formula H r ( x , t ) = ( 1 - t ) x + f t . Note t h a t (i) g r ( x , O ) = x for any x 9 [0, 1]. (ii) g r ( x , 1) = r for any x 9 [0, 1]. (iii) g r ( x , t) --fix whenever t > 0 and x :/: r. (iv) The homotopies g r and g s are close (as maps of [0, 1] 2 into [0, 1]) whenever the points r and s are sufficiently close. Suppose now that p - {pi: i 9 w} is a point in the Hilbert cube I W . Define the homotopy Hp: I "~ x [0, 1] ---. I 0~ as follows: H p ( { X i : i 9 w } , t) = { H p ( x i , t) : i 9 w}.

As above, note that: (v) H p ( x , O ) = x for any x e I W . (vi) H p ( x , 1 ) = p for a n y x e I W. (vii) H p ( x , t) ~ x whenever t > 0 and x # p. (viii) The homotopies Hp and Ha are close (as maps of I "~ x [0, 1] into I W ) whenever the points p and q are sufficiently close (in I W ). Fix any section j : Y --~ X of the retraction f ( i.e. f j - i d y ) . Also fix metrics rl and r2 on Y and I "~ , respectively. On the product Y x I ~ , consider a metric d bounded by 1 and equivalent to the metric v/r 2 + r22. It follows directly from the above listed properties t h a t the h o m o t o p y H : Y • I W • [O, 1]--*Y •

defined (ix) (x) (xi)

W

by the formula H (y, x, t) -- (y, H i ( y ) ( x , t)) has the following properties: H ( y , x , 0 ) = (y,x) for any (y,x) e Y • W. H(y,x,1)= ( y , i ( y ) ) for any ( y , x ) e Y x I W . g (y, x, t) =J= (y, x) whenever t > 0 and x ~ i ( y ) .

8.5. FIXED POINT SETS

349

Define the m a p g as follows: g(y,x) = H(y,x,d(x,f-l(y)))

for any point ( y , x ) e Y x I ' .

T h e openness of the m a p f implies t h a t g is continuous. It only remains to note t h a t 7ryg : Try and f i x ( g ) - X . [q PROPOSITION 8.5.14. A retract of any product of an arbitrary family of metrizable compacta can be fixed in the Tychonov cube of the corresponding weight. PROOF. Apply T h e o r e m 8.5.9 and P r o p o s i t i o n 8.5.13.

V1

Finally, we present a simple example of a stable non-open retraction p: X Y. Let Y be the closed segment [-1, 1]. Consider the subspace X = [-1,0] x [0, 1] U [0, 1] x {0} of the p r o d u c t Y x [0, 1] and let A -- [0, 1] x {0} U {0} x [0, 1]. T h e m a p p coincides with the restriction of the projection ~rl: Y x [0, 1] ~ Y onto X . Obviously, p is a non-open retraction. Define a deformation H : [0, 1] x [0, 1] --~ [0, 1] by H ( t , s ) = t(1 - s), (t, s) e [0, 1] 2. Let g: Y x [0, 1] ~ Y x [0, 1] be defined as follows:

g(y,t) =

(y,H(t,d((y,t),A))), (y,t),

if (y, t ) e [0, 1] x [0, 1] if ( y , t ) E [-1, 0] x [0, 1].

Here d denotes the s t a n d a r d metric on the p r o d u c t [0, 1] x [0, 1]. Clearly 7rig = ~rl and f i x ( g ) = X . E m b e d the s e g m e n t [0,1] into the Hilbert cube I ~ and fix a retraction r: I ~ --, [0, 1]. Finally, let f=g(idy

•

Y x I w --~Y •

W.

Obviously, 7ryf - Try (where 1ry: Y x I w ~ Y is the projection), ~ r y / X = p and f i x ( f ) = X . This shows the stability of f .

Historical and bibliographical notes 8.5. T h e results of this Section originally a p p e a r e d in [107].

350

8. APPLICATIONS 8.6. C o m p a c t

g r o u p s a n d fixed p o i n t sets

In this Section we investigate spaces with the complete invariance property with respect to homeomorphisms, shortly, CIPH. A space X has the CIPH if for each non-empty closed subset F of X there is a homeomorphism h F" X ---+ X such t h a t F = {x C X" x = hF(x)}. Recall that if, in the above definition, hF is only required to be a continuous map, then we say that X has the CIP (complete invariance property; see Section 8.5). Closed surfaces, even-dimensional disks, positive-dimensional spheres and the Hilbert cube are all known to have CIPH (see [268], [269] and [216] respectively). In certain situations the existence of a free action of a "nice" group G on a given space X allows one to draw an even stronger conclusion: some closed subset F C X is the fixed point set of each non-trivial homeomorphism g e G (see [311], [182]). 8.6.1. S t r u c t u r e t h e o r e m s for c o m p a c t a b e l i a n g r o u p s . In this Subsection a general structure theorem for compact abelian groups is proven and this is used as a basis for a self-contained development yielding an explicit structure theorem for finite-dimensional compact abelian groups. These results are used to show t h a t a nondegenerate compact metrizable group has the CIPH iff it is infinite. Furthermore, it is shown that the product of two metrizable spaces has the C I P H if one of the factors is a positive-dimensional compact group. We assume as known the duality between the category of compact abelian groups and that of discrete abelian groups according to Pontryagin and van Kampen. For two topological abelian groups A and B we shall always denote by Horn(A, B) the topological abelian group of all continuous group homomorphisms from A to B equipped with the compact open topology. We use the notation I~ for the locally compact additive group of reals and "ll"= N / Z for the circle group. We write G = Hom(G,'IF) for the group of characters X" G ~ T (continuous in the category of compact abelian groups). The duality theorem states that the evaluation morphism r/G" G ~ G, r/a(g)(x) -- x(g) is an isomorphism in all cases. One identifies l~ with R in a natural way. DEFINITION 8.6.1. For a compact topological group G we write

L(G) a~=IHorn(N, G) ~ Horn(G, I~) and note that L(G) is a real topological vector space with the topology of uniform convergence on compact sets. We set exp" L(G)-, a, exp X = X(1). All one-parameter subgroups of G are of the form X = (r ~-~ exp r - X ) " ]~ --~ G. We note that Hom(G, IR) --- Hom(]R | G, N) is the vector space of all linear functionals on the real vector space IRNG, i.e., the algebraic dual (R| given the weak-* topology. The only isomorphy invariant is the cardinal

d de___ld i m N ( R | G) = d i m Q ( Q | G) = rank G.

8.6. COMPACT GROUPS AND FIXED POINT SETS

351

Therefore, L ( G ) is algebraically and topologically isomorphic to ]~d. PROPOSITION 8.6.2. For a compact abelian group G there is a compact zerodimensional subgroup A such that the homomorphism r

A • L ( G ) --~ G, r

satisfies the following conditions: (i) r is continuous, surjective and open, i.e., is a quotient morphism. (ii) kerr is algebraically and topologically isomorphic to D de=f e x p _ l ( A ) ' and D is a closed totally disconnected subgroup of L ( G ) . In particular, it does not contain any nonzero vector spaces. (iii) r • L(G)) -- exp L ( G ) is dense in Go, the identity component of G. A

PROOF. By Zorn's Lemma, the abelian group G contains a free subgroup F of maximal rank. Then E dej G//F is a torsion group. We set A = F • (the annihilator of F in G). Then, by duality, A --~ E. Since E is a torsion group, A is a totally disconnected subgroup. Clearly, r A • L ( G ) ~ G defined by r ---- d e x p X -- dX(1) is a well defined continuous homomorphism. Further, T dej ~ , as the character group of a direct sum of copies of Z, is a product of copies of T ~ Z, i.e., a torus. In particular, exp" L ( T ) ~ T is surjective. Let p" G ~ T denote the quotient morphism which identifies G / A with T. The morphism L(p)-

Hom(]~,p)" L ( G ) = Hom(]~, G)--~ Horn(R, T ) = L ( T )

is an isomorphism since nom(incl, R)" n o m ( G , I ~ ) ~ H o m ( F , R ) is an isomorphism (as H o m ( E , ] R ) = {0}). Proof of (i). The surjectivity of r The exact sequence 0---. F inr ~ q ~ E ---* 0 gives an exact sequence of compact groups 0 --~ A

incl~ G

P--~ T--~ O.

Since L(p) L(T)

L(G)

exp T

eXPG

G

P

,T

352

8. APPLICATIONS

is c o m m u t a t i v e , p ( e x p c L(G)) = T. Hence A exp L(G) = ( k e r p ) e x p L(G) = G. T h u s r is surjective. The openness of r For every s u b g r o u p F ~ of F with finite index we get exact sequences

O..__~F' i_~ ~ q_~ E' _..+O and

O----~ A ' i-~ G ---~ p' T ~ ---.0, and A ~ has finite index in A. We note t h a t 9 the family of all A t intersects in {1}, 9 the a r g u m e n t showing A exp L(G) --- G above shows in exactly the same fashion t h a t A ~exp L ( G ) = G. Since F / F ~ is finite, we can write F = F1 9 F2 such t h a t F ~ -- F1 @ F~ with

F2/F~ ~- F / F ' . Accordingly, we have T = T1 x T2, T ~ = T1 • T~ (with a n a t u r a l identification), Horn(F, ]R)---- Horn(F1, R ) @ H o r n ( F 2 , R), and thus L ( T ) - - L ( T ) I @L(T)2. Correspondingly, L ( G ) - L ( G ) I @ L(G)2. We write

p!

0--+ A' i-~l a ---+ T1 ~T~ --,0. Suppose t h a t p'(g) E T1. T h e n t h e r e is an X1 E L ( G ) I such t h a t exPTz L(p')(X1) -- p'(exp G X1) = p'(g). Thus g expc(-X1)

E kerp ~ = A ~. Therefore, A ' expG L ( G ) I ---- ( p ' ) - I ( T 1 ) .

Hence A' e x p a L(G)I is closed in G. Let U x V be a p r o d u c t z e r o - n e i g h b o r h o o d of A x L(G). T h e n U contains a s u b g r o u p A ~ of A which is open in A and has finite index in A and V contains a vector subspace L ( G ) I of L(G) with dim L(G)/L(G)I < oc. It is no loss of generality to assume t h a t U = U A ~ and V ---- VL(G)I, i.e., t h a t V and Y are " s a t u r a t e d " . We observe (A x L(G))/(A1 x L ( G ) I ) ----A / A 1 x L ( G ) / L ( G ) I is a Lie group with finitely m a n y c o m p o n e n t s and r induces a surjective h o m o m o r p h i s m of Lie groups r (A x L(G))/(A1 x L ( G ) I ) G / A ~exp L(G)I. By the O p e n M a p p i n g T h e o r e m for locally c o m p a c t groups, r is open. T h e following d i a g r a m is c o m m u t a t i v e :

8.6. COMPACT GROUPS AND FIXED POINT SETS

quot A x L(G)

353

AxL(G) ~- A'xL(G)I

!

G

quot

G *- A exPG L(G)I

Since U x V is s a t u r a t e d we may conclude t h a t r x V) is open in G. P r o o f of (ii). We have k e r r = { ( d , X ) 6 A • L(G) : d e x p X = 1}. T h e map X H ( e x p - X , X ) : D ---, k e r r is therefore bijective. This m o r p h i s m has the inverse ( d , X ) ~-, X : k e r r ---, D. Hence it is an algebraic and topological isomorphism. T h e projection prA: A x L ( G ) ---+ A induces an injective m o r p h i s m j" ker r ---, A. Let C dej (ker r denote the identity c o m p o n e n t of ker r T h e n j (C) is a connected subgroup of the totally disconnected group A. It is therefore singleton and thus C is singleton, i.e., ker r and thus D are totally disconnected. P r o o f of (iii). This s t a t e m e n t is known. Here is a short argument: Set H = exp L(G). The inclusion j : g ~ G induces an isomorphism n ( j ) : L ( H ) ~ L(G) in view of the definitions. Since L(.) = Hom(]~, .), the following sequence is exact:

0--* L ( H ) L2) L(G)---, L ( G / H ) - - - , Ext(R, H ) . But H is divisible as the underlying group of a connected c o m p a c t abelian group. Hence H is injective and thus E x t ( ~ , H ) -- {0}. Hence L ( G / H ) = {0}. Since every nondegenerate compact connected abelian group contains a n o n d e g e n e r a t e o n e - p a r a m e t e r subgroup, (G/H)o -- {0}. Therefore H = Go. ['7 To the best of our knowledge Proposition 8.6.2 is new in the generality stated. It would be more valuable if we knew the totally disconnected closed subgroups of a topological vector space R d for any cardinal d. In finite dimensions, this m a t t e r is no problem at all as we shall record in the following. PROPOSITION 8.6.3. (Dixmier, see [122]) For any compact abelian group G the subgroup exp L(G) is exactly the arc component of 0 in G. We can write ~r0(G) = G~ e x p L ( G ) , and Dixmier has shown t h a t ~0(G) -~ E x t ( G , Z). COROLLARY 8.6.4. The morphism r A x L(G) ~ G maps arc components A (algebraically). onto arc components and ~ o ( G ) = AnexpL(G) THEOREM 8.6.5. The following conditions are equivalent for a compact abelian group G and a natural number n:

354

8. APPLICATIONS A

A

(i) rank G = d i m Q ( Q | G) = n. (ii) There is an exact sequence 0 ~ tor(G) ~ G ~ Qn ~ E ---+ 0 with the torsion subgroup tor(G) and some torsion group E . (iii) There is an exact sequence 0---+ Z n ---+ G ---+ E ---~ 0 with some torsion group E .

(iv) There is a compact zero-dimensional subgroup Z of (Q)n and an exact sequence 0 -~ z ~

(Q)~ -~ a -~ a/ao

- ~ o,

where Go is the identity component of G. (v) There is a compact zero-dimensional subgroup A

of G and an exact

sequence

O ~ A - - , G ~ Tn ~ 0 . (vi) d i m L ( G ) = n.

(vii) There is a compact zero-dimensional subgroup A of G and quotient homomorphism r

A x ]~n --~ G which has a discrete kernel. In particular,

r yields a local i s o m o r p h i s m of A x ]~n and G.

(viii) The identity of G has a neighborhood oasis each m e m b e r of which is h o m e o m o r p h i c to D x C n with some totally disconnected compact space D and an n-cell C n.

PROOF. In the theory of abelian groups, conditions (1), (2), and (a) are all known to be equivalent to the s t a t e m e n t t h a t the (torsion free) rank of G is n. By duality, (4) is equivalent to (2), and (5) is equivalent to (3). Condition (6), saying dim Horn(R, G) = n in view of Uom(R, G)

TM

Uom(G, R)

TM

U o m ( Q | G, ~ ) ~

~dimQ(Q|

is equivalent to (1). (1) ==, (7). By (1) we have L ( G ) -~ I~n. By Proposition 8.6.2, we obtain r A x R n - , G as asserted, because ker r ~ e x p - l ( A ) and this closed subgroup of L ( G ) -~ R n does not contain vector subgroups, hence is discrete. (7) - - 5 (8). I m m e d i a t e consequence of (7). (8) ~ (6). Let U be an identity neighborhood of G and h" D x C n --, U a h o m e o m o r p h i s m with a closed n-cell C n. We let p" G - , T be as in the proof of Proposition 8.6.2 so that L(p)" L(G) --, L ( T ) is an isomorphism. Let K = k e r ( L ( p ) e x P T ) = k e r ( p e x p a ) as before. We may assume that A = ker p is such that there is a compact identity neighborhood V CUwithVA=V. We recall the free subgroup F of G and let ( e j ) j e j be a basis of F. Set p0" IRJ --* Hom(F,]~), p o ( ( r j ) j e j ) ( ~ j e j r j , ej) = Y ~ i e j e j ( r j ) . Then p0 is an m

8.6. COMPACT GROUPS AND FIXED POINT SETS

355

isomorphism of topological groups, and since H o m ( F , R ) ~ Hom(R, T) ~- L ( T ) we have an isomorphism of topological groups p" ]~g ~ L ( T ) . Moreover, if we write F

-- ~ j e j Z . e j

and, accordingly T =

1-IjejZ.ej

O"

--~ "IFJ, t h e n

exp ~ - a exPT p" ]RJ ~ T J has the kernel Z J. It follows t h a t K = ker exPT L(p) maps isomorphically onto Z J under pL(p). Now p ( V ) is an identity neighborhood of T -~ T J. T h e n we find an 89 > r > 0 so t h a t Sr = L ( p ) - l p - l ( [ - r , r] J) satisfies S 2 r N 7/~J -- {0} and p(exPG St) = exp T L ( p ) ( S r ) C_ p ( Y ) . T h e n p exPG maps Sr homeomorphically into T and t h e n a f o r t i o r i e x p a maps Sr homeomorphically into G. But V -- V A = p - l p ( V ) C U. Hence expG Sr C U, and e = exPG ISr" Sr --* U is a h o m e o m o r p h i s m onto the image. T h e n h - l e 9 Sr --~ D x C n is a h o m e o m o r p h i s m onto the image. If (d, c) = h - l ( 1 ) , then h({d} • C n) is the connected c o m p o n e n t of 1 in U, and thus h - l e ( S r ) is a h o m e o m o r p h i c copy of Sr contained in the n-cell {d} • C n. B u t Sr is h o m e o m o r p h i c to [ - r , r] J hence to [-1, 1] J. Since [-1, 1] J contains [-1, 1] m for m = 0, 1, 2 , . . . , lYl, this entails IJI _< n by the invariance of domain. def

T h u s m -- dim L(G) = IJI _< n. T h e n by " ( 6 ) = ~ (8)", there are arbitrarily small identity neighborhoods h o m e o m o r p h i c to D ~ • C m with a totally disconnected compact space D ~ and an m-cell C m. Thus, by hypothesis (8), an n-cell must be contained in an m-cell, and this implies n _< m by invariance of domain. El DEFINITION 8.6.6. Let G be a compact abelian group. Then we set d i m G = d i m Q ( Q | G) and call this cardinal the dimension of G. I f dim G is finite, then G is called finite-dimensional and otherwise infinite-dimensional. C O R O L L A R Y 8.6.7. I f for a compact abelian group G there is a natural number n such that the equivalent conditions of Theorem 8.6.5 are satisfied, then n -dim G.

Suppose t h a t DIM is a dimension function defined for compact topological spaces such t h a t 9 it assigns to a product D x C n of a c o m p a c t totally disconnected space and a compact n-cell C n the dimension n, 9 it assigns to a homogeneous space containing such a subspace with n o n e m p t y interior the dimension n, and 9 it assigns to a compact space the dimension c~ if it contains a homeomorphic copy of [-1, 1] ~. T h e n we will have DIM(G) = {dimac~

if G is finite-dimensional, if G is infinite-dimensional.

PROPOSITION 8.6.8. (Pontryagin) For a compact abelian group G one has w(G) --IGI, where w(G) denotes the weight of the space G. A

356

8. APPLICATIONS

From Theorem 8.6.5 for an infinite compact group we note that

IG/tor(G)[ = IQ|

GI = max{w, d i m G } .

We derive PROPOSITION 8.6.9. For an infinite compact abelian group G,

w ( a ) = w ( a / a o ) max{w, dim G}. In particular, a finite-dimensional connected compact abelian group is metric. More generally, a finite-dimensional compact abelian group is metrizable iff w(G/Go) <_ w. For the proof of the next proposition we need some facts on abelian groups. LEMMA 8.6.10. (Pontryagin) Let A be a countable torsion free group and assume that every finite rank pure subgroup is free. Then A is free. PROOF. The claim is clear if rank A < oo. We now assume rank A = w. Let {el, e 2 , . . . } denote a maximal free set. Define Pj to be the free pure subgroup generated by { e l , . . . ,ej}. Inductively define a free set {fl, f 2 , . . . } such t h a t f l , . . . ,Ij(n) is a basis of Pn for a suitable sequence of natural numbers j ( n ) . Indeed, if f l , . . . , fj (n) is a basis of Pn, then t n + l is free and the pure subgroup Pn is a direct s u m m a n d (in view of the elementary divisor theorem). Hence we can complement the basis of Pn to a basis f l , . . . , fj(n), f j ( n ) + l , . . . , fj(n+l). This completes the recursion. The span of the free s e t / 1 , . . , contains all P,~ and thus all e,~, and thus is A. [3 Now let A be an abelian group. Let K: denote the set of all subgroups K o f an A such that A / K is free. Then K: is a filter basis; for if K1, K2 6 1C then K = K1 N K2 is the kernel of the map a ~ (a + K l , a + K2) ---* A / K 1 x A / K 2 . The image of this homomorphism is a subgroup of a free group and thus is free by the Schreier Subgroup Theorem. We can form

Koo = Koo(A) = N I C . Then all morphisms into free groups factor through A --+ A / K o o and the homomorphisms A / K o o ~ Z separate the points. In particular, A/Koo is torsion free and tor(A) C Koo. Notably, Koo is a pure subgroup. For a subgroup H of a torsion free abelian group A, the group [HI def {a 6 A" 3n 6 N such that n . a 6 H } is the smallest pure subgroup containing H. LEMMA 8.6.11. / f goo(A) = {0}, then every finite rank pure subgroup of A is free.

8.6. COMPACT GROUPS AND FIXED POINT SETS

357

PROOF. Let P be a finite rank pure subgroup of A and F a m a x i m a l rank free subgroup contained in P. T h e n P = IF]. Since Koo(A) = {0}, there is a subgroup K E K: such t h a t F M K = {0}. It follows t h a t P M K = {0}, for if p E P M K t h e n there is an m E l~l such t h a t m . p E F M K -- {0}, whence p -- 0 since A is torsion free. T h e m a p x ~-. x + K : P ~ A / K is therefore injective. But A / K is free by the definition of E, and thus P is free by the Schreier Subgroup Theorem, as we wanted to show. Wl LEMMA 8.6.12. (Main Lemma) Let A be an abelian group such that A / K c ~ ( A ) is countable. Then A contains a free subgroup F such that A = F ~ Kc~(A). Moreover, Kc~(A) does not have any nondegenerate free quotients. PROOF. T h e group A / K o o ( A ) is torsion free, countable, and the m o r p h i s m s into free groups separate the points. Hence g o o ( A / g ~ ( A ) ) -- {0}. T h u s from L e m m a 8.6.11 we know t h a t every finite rank pure subgroup is free. Then, by L e m m a 8.6.10, the quotient A / K ~ ( A ) is free. Since free groups are projective, this implies the existence of F as asserted. Again any free quotient of Kc~(A) splits, so Koo(A) -- F' @ K with a free F ' isomorphic to the free quotient. B u t t h e n F ~ F ' is free and thus K E K:. It follows t h a t K ~ ( A ) C K and that, as a consequence, F ' is degenerate. WI THEOREM 8.6.13. (i) If T is a torus subgroup (a product of circle groups) of a compact abelian group G, then there is a (not necessarily unique) subgroup C such that (t,c) H t c : T

xC--~G

is an isomorphism of compact groups. (ii) Every compact abelian group G contains a (fully characteristic) unique smallest closed subgroup M -- M (G) containing all circle subgroups, and M• goo(G). The subgroup M ( G ) is always locally connected. (iii) If M (G) satisfies the first (and hence the second) axiom of countability, i.e., is metric, then M ( G ) is a torus and G contains a closed subgroup C not containing circle groups such that ( m , c ) ~ - ~ m c : M (G) x C ~ G is an isomorphism of compact groups, i.e., G is a direct product of a metric torus and a torus free compact subgroup. (iv) Every metric compact group G is the direct product of a fully characteristic maximal torus subgroup M (G) and some torus free closed subgroup. PROOF. P a r t (i) is restating the fact, by duality, t h a t in the category of abelian groups the free groups are the projectives and t h a t a h o m o m o r p h i s m onto a projective splits. Proof of part (ii): We let M be the closure of the group generated by the union of all circle groups. This is the smallest closed subgroup of G containing

358

8. APPLICATIONS

all circle groups and is, therefore, fully characteristic. The product of two tori is a torus (It is clearly a compact subgroup, and the first factor splits by (i); the complementary factor is a homomorphic image of the second factor and is, therefore, a torus). Hence the set of all tori in G is upwards directed, and M is the closure of its union. Dually, the annihilator of M is the intersection of all T • as T ranges through all tori. By duality, the T • are exactly the members of K:(G). Thus M ( G ) • = K ~ ( G ) . In particular, M ( G ) ~- G / g o o ( G ) . Thus by L e m m a 8.6.11, in the character group of M (G) every finite rank pure subgroup is free. By a theorem of Pontryagin [253, II, Satz 48, Seite 33], this property characterizes locally connected compact abelian groups. (iii) Now suppose that M ( G ) is metric. Then G / M ( G ) • ~- G is countable. By (ii) we have M (G) • -- K ~ ( G ) . Now Lemma 8.6.12 implies G = F @ M (G) • with a (countable) free group F. We set C -- F • C G and obtain G ~ M (G) x C as stated in (iii). Since M (G) contains all circle groups, C does not contain any circle groups. (Equivalently: Kcc(G) does not have any free quotients by Lemma 8.6.12.) (iv) If G is metric, then, in particular, (iii) applies and proves the assertion. WI It is known, t h a t for metric compact connected groups, arc connectivity, local connectivity, and being a torus group are equivalent properties. The character group G of Z • is connected, locally connected, but not arcwise connected, let alone a torus group. But for this G we have G - M ( G ) . This shows that metrizability in (iii) is essential. Dixmier [122] has observed (using the Axiom of Choice) that there is an abelian group A containing a subgroup Z isomorphic to Z such that A / Z Z ~ and that Z does not split. If g E Z is a generator and if there existed a homomorphism f" A ---, F into a free group F with f ( g ) ~ O, then If(Z)] -~ Z and F -- F1 @ [f(Z)]; hence there would exist a morphism r A ---+ Z with r - Z, yielding A -- H (9 Z and thus contradicting the fact that Z does not split. Hence Z c K ~ ( A ) . Since A / Z ~- Z W, the free quotients separate the points of A / Z whence Z - K ~ (A). One notes that every finitely generated pure subgroup of A is free. As a consequence G clef ~ is a compact connected locally connected (not arcwise connected) group for which G / M ( G ) is a circle group. I.e., M ( G / M ( G ) ) need not be zero in general! One notes that, in the absence of metrizability, this is the starting point of a transfinite ascending recursion process which we shall not pursue in this paper. However, in view of Theorem 8.6.13(iv), in the metric case, the hypothesis t h a t G be torus free is frequently no restriction of generality. PROPOSITION 8.6.14. Let F be a compact finite-dimensional abelian group. We have the following conclusions: (i) F ~- ][~n x G with a unique maximal torus subgroup T m x {1}, and some torus free compact n-dimensional group G, m + n - dim F.

8.6. COMPACT GROUPS AND FIXED POINT SETS

359

(ii) There is a compact zero-dimensional subgroup A of G and a quotient homomorphism r

]~n • A ~

G with a discrete kernel isomorphic to a

lattice Z p with p <_ n - dim G. (iii) The arc component of 1 in G is Ga = exp L ( G ) : r components of G are the sets d = Ga = Gad = r 2 1 5

n • {1}) The arc {d}) = d e x p L ( G )

and r maps the arc component l~ n • {d} continuously and bijectively onto dGa.

(iv) I f F / F 0 is metric, then we m a y endow ]1~dimF x A : ]~m X ]1~n X A with an invariant product metric d = do • d l with a Hilbert space metric do on R dimF and an ultrametric dl on A , and give F the quotient m e t r i c dr for the h o m o m o r p h i s m (I)" ] ~ d i m F X A --+ F ,

( X , Y , d ) ~-+ ( X + Z n , r

9 R m • R n • A --+ ,][,m • G ~ F.

Then (F, dr) is locally isometric to ]~dimr • A u n d e r (~.

PROOf. (i) By P r o p o s i t i o n 8.6.9, the c o m p o n e n t F0 is metric, hence M (F) -M ( F 0 ) is metric. Thus T h e o r e m 8.6.13(iii) proves F -- M ( F ) x G with a torus M (F) and a torus free compact group G. We have dim F = dim M (F) -F dim G. This proves (i). (ii) From T h e o r e m 8.6.5 we may identify L ( G ) with ~ n and we get r L ( G ) x A --, G, r = d e x p X , a local isomorphism. (Recall from the proof of T h e o r e m 8.6.5 t h a t totally disconnected closed subgroups of ~ n are discrete lattices.) (iii) From Corollary 8.6.4 we know t h a t r m a p s arc c o m p o n e n t s as stated. If exp X = 1, t h e n exp R . X is a circle group or is trivial. B u t G does not contain any circle groups. Hence Z = 0 and r m a p s ]I(n x {1} injectively. T h u s (iii) is proved. (iv) is a straightforward consequence of (i) and (ii).

[:]

DEFINITION 8.6.15. A solenoid is a 1-dimensional compact connected group. A proper solenoid is a solenoid which is not a circle group. PROPOSITION 8.6.16. A compact abelian group G is a solenoid iff its character group is isomorphic to a nonzero subgroup of Q. It is proper iff this subgroup is not free.

PROPOSITION 8.6.17. There is a 2 - d i m e n s i o n a l compact connected group G which is not a product of two solenoids.

PROOF. P o n t r y a g i n (see [253, I, Beispiel 15, Seite Seite 39]) has exhibited subgroups A of Q2 with Z 2 c_ n o n d e g e n e r a t e direct s u m m a n d s . For such a group, G c o m p a c t connected group which is not a p r o d u c t of two

48 and II, Beispiel 68, A C_ Q2 which have no -- A is a 2-dimensional solenoids. [:] A

360

8. APPLICATIONS

A continuum is a compact connected space. It is called decomposable if it is the union of two proper subcontinua; otherwise, it is indecomposable. The proper solenoids are known to be indecomposable metric continua. The following proposition shows that this situation cannot occur for compact groups having dimension greater than one. THEOREM 8.6.18. For a compact group G the following statements are equivalent: (i) G is a proper solenoid. (ii) The underlying space of G is an indecomposable continuum. PROOF. We just noted (i) ==:# (ii) and observe that the circle is decomposable. Therefore we must prove that dim G > 1 implies the decomposability of G. This is certainly the case if G is disconnected, so we assume G to be connected. (a) We preface the proof by the following remark. Assume that N is a compact and connected normal subgroup of G and t h a t p" G ~ G I N is the quotient morphism. If X c G I N is connected, then p - l ( x ) is connected. Thus, if G I N is decomposable, then G is decomposable. (b) Assume next that G is semisimple. The structure theory of compact connected semisimple groups says that there is a family of simple compact connected Lie groups {Gj" j E J} and a totally disconnected central subgroup D such that G ~ (l-IjeJ G j ) / D (see, for example, [49, Chap. 9]). Identify G with this factor

rljEj_{i}

group. If G is not degenerate, pick an i E J. Then N dej GjD/D is a compact connected normal subgroup of G and G I N ~- G i / ( N M Gi) is a simple connected Lie group, and thus is a compact manifold of dimension at least 3. Hence it is decomposable. From part (a) of the proof we derive that nondegenerate compact connected semisimple groups are decomposable. (c) The c o m m u t a t o r subgroup G ~ of a compact connected group G is closed and semisimple [49]. Each compact connected group is a semidirect product of its (closed!) c o m m u t a t o r subgroup G I and an abelian subgroup A which is isomorphic to GIG ~ [17'2]. Thus if G is not abelian it has a decomposable direct topological factor by part (b). We may therefore assume for the remainder of the proof that G is abelian. (d) Assume t h a t G is a compact connected abelian group. Then G is a torsion free group whose rank is at least 2. Hence we find a pure subgroup P of rank 2. (Indeed let F be a free subgroup of maximal rank. Write it in the form F1 @ F2 with rank F1 = 2 and set P = {X E G" n ' x E F1 for some n E 1~}.) Let A

C = P • the annihilator of P in G. Then C -~ G / P and G / C ~ P. Since P is pure, G / P and thus C is torsion free. Hence C is connected. In view of part (a) of the proof, G is decomposable if G / C is decomposable. Furthermore, rank P = 2 implies d i m G / C -- 2. In order to complete the proof it therefore suffices to prove that a 2-dimensional compact connected abelian group is decomposable. By Theorem 8.6.13(iv) we may assume t h a t G is torus free since G is clearly

8.6. COMPACT GROUPS AND FIXED POINT SETS

361

d e c o m p o s a b l e if G contains a circle g r o u p as a factor. Using P r o p o s i t i o n 8.6.14, we o b t a i n a c o m p a c t z e r o - d i m e n s i o n a l s u b g r o u p A a n d a closed e-ball n e i g h b o r h o o d B of t h e origin in ]I(2 such t h a t r m a p s A • B h o m e o m o r p h i c a l l y onto an identity n e i g h b o r h o o d W of G. As in P r o p o s i t i o n 8.6.2 and its proof we set D = e x p - l ( A ) a n d note t h a t e x p X = r = 5 e x p Y with 5 E A means 5 = e x p ( X - Y ) and thus X = y + d w i t h d E D. T h u s , identifying L ( G ) w i t h R 2 we see t h a t exp: ]~2 ---, G m a p s D + B bijectively a n d c o n t i n u o u s l y onto W M e x p L ( G ) . Let U = I n t B , t h e manifold interior of B. By P r o p o s i t i o n 8.6.2(ii), t h e closed s u b g r o u p D of ~2 is a discrete lattice a n d therefore is countable.

T h u s D + U is a c o u n t a b l e disjoint union of o p e n

disks in the plane. Therefore its c o m p l e m e n t

E dej

]~2 _

{D + U} is c o n n e c t e d .

B y P r o p o s i t i o n 8.6.2(iii), t h e set e x p L ( G ) is dense in G and t h e c o m p l e m e n t exp L ( G ) - W -- exp E is dense in t h e c o m p l e m e n t A de=__fG - r

• U). N o t e

t h a t A is a c o m p a c t subset of G since r is an o p e n m a p and t h a t A A = A. Also, A, being the closure of a c o n n e c t e d set, is connected. Every arc c o m p o n e n t of W is of t h e form 5 exp B, 5 E A, and its intersection with A is 5 e x p 0 B . Let K1 be a p r o p e r c o m p a c t open s u b g r o u p of/X a n d set K2 -- K - K1. Set C1 = A U r • B ) and C2 = A U r x B). Since each arc c o m p o n e n t of t h e c o m p a c t set r • B), j = 1, 2 intersects t h e c o n t i n u u m A, each of t h e sets C1 a n d C2 is a c o n t i n u u m . Therefore, since C1, C2 are p r o p e r s u b c o n t i n u a of G a n d G = G1 U G2, t h e space G is d e c o m p o s a b l e as claimed. [--1 8 . 6 . 2 . C o m p a c t M e t r i z a b l e G r o u p s a n d C I P H . A flow on a space X is a c o n t i n u o u s function (f" X • R ~ X such t h a t (f(x, 0) = x a n d (f((f(x, s), t) ( f ( x , s + t ) for all s , t E R a n d x E X T h e m a p (ft" X ~ X defined by (ft(x) = ( f ( x , t ) is a h o m e o m o r p h i s m since ( f - t = (ft-1 9 A point p E X is an

invariant point of (f if (ft(p) = p for all t E I~, a n d t h e set N{fix((ft)" t E ]1~}, called t h e invariant set of the flow, is a closed subset of X if X is a H a u s d o r f f space. DEFINITION 8.6.19. A flow (f : M • ]~ ~ M on a metric space (M, d) is called u n i f o r m if it satisfies the following conditions: (i) d(x, (f(x, t)) <_ Cit I for some positive C and all x E M , t E I~. (ii) For all x E M , t E R either (f(x, t) = x iff t = 0 or there exists a positive real number p such that (f(x, t) = x iff t E pZ. PROPOSITION 8.6.20. Let ( M , d ) be a compact metric space with a u n i f o r m flow (f. Then every n o n e m p t y closed subset of M is the fixed point set of an orbit-preserving a u t o h o m e o m o r p h i s m of M . In particular, M has the CIPH. PROOF. In t h e case where p > 0 it suffices to consider p - 1. T h e f o r m u l a t i o n here is a m a t t e r of technical convenience for later proofs. If 8.6.19(ii) is satisfied w i t h p > 0, define a new m e t r i c D and a new flow r by D ( x , y) = ( p C ) - l ( d ( x , y)

362

8. APPLICATIONS

and r

t) = ~o(x,pt). Then D(x,r

=

p-~d(x,~o(x,pt))

<_ (pC) -1 . CIpt] = Itl

a n d r satisfies 8.6.19(ii). W h e n p = 1 there is a free circle action on M a n d the p r o p o s i t i o n follows from [216, 2.3]. Now s u p p o s e we have the case c o r r e s p o n d i n g to p = 0 a n d A is a n o n e m p t y closed s u b s e t of M . Let r(x) = ~-ucd(x,A) for x E M a n d define a m a p p i n g h: M ~ M by h ( x ) = ~o(x, r(x)) for x e M . Clearly, f i x ( h ) = A. To see t h a t h is one-to-one, s u p p o s e t h a t h(x) = h(y). T h e n x and y m u s t lie in t h e s a m e orbit a n d for s o m e real n u m b e r t, y -- ~o(x, t). T h e n

~(~, r(~)) = ~(~,(~, t), ,-(y)) = ~(~, t + ,-(y)). T h u s t = r(x) - r ( y ) .

T h e t r i a n g l e inequality applied to x, y, A implies

d(x,y) > [ d ( x , A ) - d ( y , A ) l . Since CIt I >_ d(x,y), we have C[t I > 2Clr(x ) - r ( y ) l . r(x) = r(y). T h e n we have

v(v(~, ~ ( ~ ) ) , - ~ ( ~ ) )

T h e r e f o r e t = 0 and

= v(v(y, ~ ( y ) ) , - ~ ( y ) ) .

Consequently, x = ~p(x, 0) = ~o(y, 0) = y and h is one-to-one. To c o m p l e t e t h e proof t h a t M has the C I P H , it suffices to show t h a t h is onto. To see this, let y E M . For s o m e b E JR, the m e t r i c d is b o u n d e d by b. Let x d e n o t e the u n i q u e point in the orbit ~o({y} x N ) such t h a t ~o(x, b) = y and let [x, y] d e n o t e the arc in ~o({y} x R) w i t h e n d p o i n t s x a n d y. Since hi[x, y] is an orderp r e s e r v i n g h o m e o m o r p h i s m from Ix, y] onto h([x,y]) such t h a t h(x) < y and h(y) > y, it follows t h a t y E h([x, y]). Consequently, h is onto as required, g-I LEMMA 8.6.21. If (Xj, dj), j = 1,2, are metric spaces and ~o is a uniform flow on (X2, d2), then X = X1 x X2 is a metric space with respect to the metric D defined by !

!

!

!

D ( ( x l , x 2 ) , (Xl,X2)) = max{dl(Xl,Xl),d2(x2, x2) }, and the flowO on X given by O ( ( x l , x 2 ) , t ) = (xl,~o(x2, t)) is uniform on ( X , D ) . PROOF. N o t e

D((Xl, X2), •((Xl,X2),t)

= D((Xl,X2), (Xl,qO(x2, t)) = d2(x2, qo(x2, t)).

C o n d i t i o n 8.6.19(i) follows readily and 8.6.19(ii) is s t r a i g h t f o r w a r d .

V1

PROPOSITION 8.6.22. If (M, d) is a compact metric space with a unifortn flow and X is a metrizable space, then M x X has CIPH.

8.6. COMPACT GROUPS AND FIXED POINT SETS

363

PROOF. By L e m m a 8.6.21 there is a metric p for M • X and a flow ~ which is uniform on (M • X , p ) . T h e proof of Proposition 8.6.20 can be applied to M • X. Since M • X need not be compact, the one a d j u s t m e n t required is to use the compactness of M to show t h a t the m a p p i n g h is a closed m a p p i n g and hence an a u t o h o m e o m o r p h i s m of M • X. [7 Recall each compact metrizable group and each abelian metrizable group G has metrics d satisfying d(gxh, gyh) = d(x, y). Such metrics will be called invariant. T h e y are completely characterized by the function g ~-, IIgll" G ---, R defined by Ilgll = d(1, g) from which d is recoverable by d ( g , h ) = Iig-lhll. T h e function II" II satisfies IIx-lll = IIxll, IIghll <_ IIgll + IIhll, IIhgh-lll = IIhll and IIgll = 0 iff g - 1. We call a function II-II an invariant n o r m on G and r e m e m b e r t h a t the invariant norms are in bijective correspondence with the invariant metrics on a group. (See e.g. [50, Chap.IX, pp. T G IX.24] where this is discussed for abelian groups; the a r g u m e n t s work for invariant metrics quite generally.) LEMMA 8.6.23. Assume that G there is a one-parameter subgroup positive number C. Suppose that N ing a ( ~ ) . Then there is a metric D is uniform on ( G / N , D ) .

is a group with an invariant norm II " II and a" R --~ G such that Ila(t)ll <_ c I t I for some is a closed normal subgroup of G not containon G / N such that the flow (Ng, t) ~-, N g a ( t )

PROOF. We define IINgIIN = inf{llngll, n E N } . T h e n II" IIN is an invariant n o r m on G / N . T h e topology defined by the metric D associated with this n o r m is the quotient topology. We must verify (i) and (ii) of Definition 8.6.19 for the flow ~ on G / N defined by ~ ( N g , t ) = N g a ( t ) . Regarding ( i ) w e c o m p u t e

D ( N g , ~ ( N g , t)) = D ( N g , N g a ( t ) ) =

[INa(t)l[N

= inf{llna(t)ll, n ~ N } __ Ila(t)l I _< c ( t ) by hypothesis on a. Thus 8.6.19(i) is satisfied. Regarding (ii) we note t h a t ~ ( N g , t ) = N g means g N a ( t ) = N g a ( t ) = N g g N , i.e., a(t) E N. Since a(IR) is not contained in N, the inverse image a - l ( N ) a closed proper subgroup of R and thus is of the form p Z with a nonnegative p R. If p = 0 t h e n ~ ( N g , t) -- Hg holds iff t = 0. If p is positive, then ~ ( N g , t) H g holds iff t E pZ. T h u s 8.6.19(ii) holds and the proof is complete. V1

= is E

=

PROPOSITION 8.6.24. The Cantor set and the space of irrationals have CIPH. PROOF. Note t h a t the space of irrationals is h o m e o m o r p h i c to Z ~ ([8]) and the Cantor set is h o m e o m o r p h i c to I[])~ where ]I]) denotes the two-point discrete group ([56]). We shall only deal here with the C a n t o r set. Fix a metric d for ]I])~ and let A be a n o n e m p t y closed subset of ]I])~. T h e n the complement I[])~ - A, as an open subspace of ]I])W, can be written as a countable (finite or infinite) disjoint union C1U C2 U . . . of closed and open subspaces of ID~

364

8. APPLICATIONS

each of which is h o m e o m o r p h i c to ID~. Since translation by a nonzero element in an abelian group determines an a u t o h o m e o m o r p h i s m of the group, it follows t h a t there is a sequence of fixed point free a u t o h o m e o m o r p h i s m s hi, h 2 , . . , of C1, C 2 , . . . respectively. Moreover, we can choose each hi so t h a t d(hi(x), x) < 71 for x E Ci. It only remains to observe t h a t the function h: D W ---. ]I}~ defined by

h(~) =

is an a u t o h o m e o m o r p h i s m of ~

x

ifxEA

h~(~)

if x e c~

with fix(h) = A.

[-7

LEMMA 8.6.25. Every compact metrizable totally disconnected infinite group is homeomorphic to the Cantor set and hence has CIPH. PROOF. T h e proof follows from [56] and Proposition 8.6.24.

[::]

THEOREM 8.6.26. In every infinite compact metrizable group G each nonempty closed subset is the fixed point set of some autohomeomorphism. If G is not totally disconnected, then G admits a metric d and a uniform flow on (G, d) defined by the action of some one-parameter group of G on the right. In particular, every nonempty closed subset of G is the fixed point set of an orbitpreserving autohomeomorphism of G. PROOF. By Mostert's Cross Section T h e o r e m [173, 1.14], every compact group G possesses a zero-dimensional compact subset Z homeomorphic to G/Go such t h a t

(z,g) ~

zg : Z x Go ~ G

is a homeomorphism. Since G is infinite then at least one of G/Go or Go is infinite. If Go is not degenerate, in view of L e m m a 8.6.21 it suffices to prove the t h e o r e m for Go. If Go is singleton, then G is totally disconnected and the assertion follows from L e m m a 8.6.25. T h u s the proof will be completed if we establish the claim for G connected which we shall assume for the r e m a i n d e r of the proof. Since every compact group is the projective limit of Lie groups, there is a closed normal subgroup M such t h a t G / M is a Lie group. (See e.g. [49, Chap. IX, p. LIE IX.99, Corollaire 1].) T h e identity c o m p o n e n t K of M is a compact connected normal subgroup of G. T h e r e exists a compact connected normal subgroup H of G such t h a t g O g is zero-dimensional and central (see e.g. [173, p.299, 2.5]). Since H / ( K N H ) ~- G / M , the group H is finite-dimensional. T h e map q: K • ---. G, q(k, h) -- kh is a surjective continuous morphism. T h u s G ~ ( g • H ) / N with N -- ker q. In view of L e m m a 8.6.23 it suffices to prove the required d a t a on K x H. T h e n by L e m m a 8.6.21 it suffices again to prove the claim for H. Thus we may assume now t h a t G is c o m p a c t connected and finite-dimensional. Now G is the semidirect product G' • A of the closed normal c o m m u t a t o r subgroup

8.6. COMPACT GROUPS AND FIXED POINT SETS

365

G' a n d an abelian s u b g r o u p A isomorphic to G / G ' (see [172]). Since G' is finitedimensional and semisimple, it is a Lie group. By L e m m a 8.6.21 it suffices to prove the assertion for each of the two factors separately.

Case 1: G is a semisimple Lie group. We claim t h a t the t h e o r e m holds for every c o m p a c t connected Lie group G. Every such G contains a m a x i m a l t o r u s (see [49, p . L I E IX.8]) and has an invariant R i e m a n n i a n metric d which induces on T a R i e m a n n i a n invariant metric and thus induces on each circle group in T an invariant R i e m a n n i a n metric. However, an invariant R i e m a n n i a n metric on P~/Z is a positive multiple of the metric given by d(s + Z, t + Z) -- Is - t]. T h u s we find a o n e - p a r a m e t e r s u b g r o u p a : ]1( -+ G such t h a t Ila(t)ll <_ CIt ] with the invariant n o r m associated with d. T h e n L e m m a 8.6.23 yields a proof of the assertion in this case with N = {1}. Case 2: G is a finite-dimensional abelian group. T h e n G = T • A with a u n i q u e m a x i m a l torus and a torus free factor by T h e o r e m 8.6.13(iv). By L e m m a 8.6.21 the proof is accomplished if it can be done for each of the factors separately. In the proof of Case 1 we took care of the case of a Lie group. T h u s t h e r e r e m a i n s only the case t h a t G is a torus free c o m p a c t c o n n e c t e d abelian group. Now l e t / 3 : ]I( --+ G denote any n o n d e g e n e r a t e o n e - p a r a m e t e r subgroup. One such exists since G is assumed to be n o n d e g e n e r a t e . Since G is torus free we have (8.6.1)

f~(t) = 0 iff t = 0.

In p a r t i c u l a r , / 3 ( 1 ) -fi 0. T h e c h a r a c t e r s of G s e p a r a t e the points, hence t h e r e is a c h a r a c t e r X: G -~ T = ~ / Z such t h a t X(f~(1)) ~ 0. T h e m o r p h i s m X o/3: ~ -~ R / Z lifts to a m o r p h i s m )~: l~ --+ R such t h a t ~( o t3 = p o A with the q u o t i e n t (covering) m o r p h i s m p" R -~ ~ / Z . (This is elementary. This also follows from a topological fact since R is simply c o n n e c t e d and p is a covering map.) E v e r y continuous e n d o m o r p h i s m A of ~ is of the form )~(t) = rt for some r E 1~. Since p(r) -- X~ ~ 0, we certainly have r ~ 0. We t h e n have X(~(t)) = r t + Z and define a" ~ ~ G by c ~ ( t ) = f~(t). T h e n (8.6.1) implies (s.6.2)

.(t) = o i~ t = o

a n d we have (8.6.3)

X o a = p : ]~ ---+ ~ .

Now we set K = ker X and define (8.6.4)

v: K • ~-,

a,

~ ( k , t) = k + ~ ( t ) .

Claim: ~ is surjective. For a proof let g C G. Pick t C ]Rsuch t h a t x(g) = t + Z and set k = g - a ( t ) . T h e n x(g - c~(t)) = x(g) - X(c~(t)) = x(g) - (t + Z) = 0 in view of (8.6.3). Hence k E K and thus g = k + c ~ ( t ) = ~ ( k , t ) . This proves t h e claim.

366

R.

8. APPLICATIONS

Now L e m m a 8.6.23 shows t h a t it is sufficient to prove the claim for K • T h e n by L e m m a 8.6.21 it suffices to prove the claim for R for which it is

obvious.

[3

COROLLARY 8.6.27. I f G is a positive-dimensional compact metrizable group and X is a metrizable space, then G x X has CIPH. PROOF. A p p l y T h e o r e m 8.6.26 and P r o p o s i t i o n 8.6.22.

El

COROLLARY 8.6.28. A compact metrizable group has the C I P H iff it is either degenerate or infinite. PROOF. An a u t o h o m e o m o r p h i s m of a n o n d e g e n e r a t e finite discrete group c a n n o t fix all points but one. [3

COROLLARY 8.6.29. Every compact metrizable group has the CIP. PROOF. Every n o n e m p t y subset of a finite discrete group is the fixed point set of a retraction. [3 In [235, p.134] a space Y is defined to have property W ( s t r o n g ) if it admits a h o m o t o p y H : Y • [0, 1] ~ Y such t h a t H ( y , t ) -- y iff t = 0. We note t h a t a metric space has p r o p e r t y W ( s t r o n g ) if it a d m i t s a uniform flow since then we can set g ( x , t )

) ---- ~ ( x , t ) (or ~o ( x,t ~

for p > 0). It is shown in [218, 3.14]

t h a t if M is a connected m e t r i z a b l e space with p r o p e r t y W (strong) and X is a zero-dimensional c o m p a c t (Hausdorff) space, then M x X x X has CIP iff X is metrizable. In the next result we use these facts and the concept of C I P H to give a c h a r a c t e r i z a t i o n of the c o m p a c t zero-dimensional spaces which are metrizable PROPOSITION 8.6.30. I f (M, d) is a compact connected metric space admitting a uniform flow and X is a compact zero-dimensional space, then M • X • X has C I P H iff X is metrizable. PROOF. Suj~ciency. If X is metrizable, then X x X is metrizable and M x X x X has C I P H by P r o p o s i t i o n 8.6.22. Necessity. Since a space having the C I P H has the CIP, the necessity follows from the remarks preceding the proposition. WI We note t h a t if X is a n o n d e g e n e r a t e c o m p a c t zero-dimensional metrizable space, t h e n X ~ x X ~ is m e t r i z a b l e iff X'x is metrizable iff )~ is a c o u n t a b l e cardinal. Moreover, when A is u n c o u n t a b l e we have X ~ x X ~ - X ~. Consequently, we can apply P r o p o s i t i o n 8.6.30 to o b t a i n the following result. COROLLARY 8.6.31. I f (M, d) is a compact connected metric space admitting a uniform flow and X is a nondegenerate compact zero-dimensional metrizable space, then M x X ;~ has the C I P H iff )~ is a countable cardinal.

8.6. COMPACT GROUPS AND FIXED POINT SETS

367

We note t h a t if a Hausdorff space D x M has the CIPH, where D is a totally disconnected space and M is a connected space, then M has the C I P H since, given a n o n e m p t y closed subset A of M , an a u t o h o m e o m o r p h i s m h of D x M with fix(h) -- D x A must preserve c o m p o n e n t s and thereby d e t e r m i n e an a u t o h o m e o m o r p h i s m of M whose fixed point set is A. Since an n-manifold having a compact b o u n d a r y c o m p o n e n t with a nonzero Euler characteristic does not have the C I P H ([216, 3.1]), it follows t h a t ]I])~ x I A, I - [0, 1], does not have the C I P H if A is an odd integer. If G is a compact metrizable group which is not totally disconnected, then Corollary 8.6.27 shows t h a t G x I A has the C I P H if A is a countable cardinal. However, no nonmetrizable A N R - c o m p a c t u m has the CIP ([107, 3.4]). Since a compact Lie group is an A g R - s p a c e , its p r o d u c t with a Tychonov cube is a nonmetrizable A N R - c o m p a c t u m and we o b t a i n the following result. PROPOSITION 8.6.32. I f G is a compact Lie group, then G x I A has the C I P H iff A is a countable cardinal. R e m a r k 8.6.33. We note t h a t Corollary 8.6.31 and Proposition 8.6.32 provide examples which contrast the metrizable and nonmetrizable cases regarding the CIPH. In what follows, let A be an u n c o u n t a b l e cardinal. (i) Let G be a compact Lie group and consider the Hilbert cube Q - I ~. By Proposition 8.6.32, G x I A -- G x QA does not have the CIPH. However, since G has the C I P H by T h e o r e m 8.6.26 and Q has the C I P H by [216, 12.4], G x QA is a p r o d u c t of homogeneous spaces each of which has the CIPH. This is in contrast to the metric case where it is not known if every homogeneous locally contractible space has the CIPH. (ii) Let G be a compact connected metrizable group and let C - ]I])~. T h e n G x D A - G x C A does not have the C I P H by T h e o r e m 8.6.26 and Corollary 8.6.31. By T h e o r e m 8.6.26, G x C A is a p r o d u c t of c o m p a c t groups each of which has the CIPH. This is in contrast to the metric case since, by Corollary 8.6.28, a metrizable p r o d u c t of compact groups has the C I P H iff it is either degenerate or infinite.

We remark also t h a t the hypothesis of compactness in Proposition 8.6.20 can be relaxed so t h a t the conclusion of the proposition is valid for free ~-actions. However, one cannot conclude t h a t connected metrizable groups must have C I P since there exists a connected subgroup of the plane ]I~2 which fails to have CIP [218, 2.2]. Finally, we end this section by noting t h a t any metrizable space M a d m i t t i n g a flow satisfying condition (ii) of Definition 8.6.19 admits distinct flow s t r u c t u r e s corresponding to the closed subsets of M . PROPOSITION 8.6.34. Let M be a metrizable space admitting a flow satisfying condition (ii) of Definition 8.6.19. I f A is a closed subset of M , then there is a flow on M whose invariant set is A.

368

8. APPLICATIONS

PROOF. In [28] it is shown t h a t every closed subset of a metrizable space X is the invariant set of a flow on X iff X a d m i t s a flow having an e m p t y invariant set. T h e proof follows since a flow ~ on M satisfying condition (ii) of Definition 8.6.19 has no invariant points. V1 By L e m m a 8.6.21 a p r o d u c t of two m e t r i z a b l e spaces will a d m i t a uniform flow if one the factor spaces does and thus we obtain the following result.

COROLLARY 8.6.35. L e t M

be a m e t r i z a b l e space and let G be a c o m p a c t

m e t r i z a b l e p o s i t i v e - d i m e n s i o n a l group.

T h e n e v e r y closed subset o f G x M is the

i n v a r i a n t set o f s o m e f l o w on G x M .

8 . 6 . 3 . C o m p a c t N o n - m e t r i z a b l e G r o u p s a n d C I P H . Unlike the metrizable case t h e r e are n o n m e t r i z a b l e c o m p a c t groups which do not have CIP. T h e p r o d u c t ]I)~1 x ~ serves as a c o r r e s p o n d i n g e x a m p l e (see Corollary 8.6.42 ). In this section we show t h a t G r has C I P H for each n o n d e g e n e r a t e metrizable c o m p a c t group and each T > w. U n d e r Axiom of Choice every infinite cardinal r is an aleph, i.e. ~" -- wa for some ordinal c~. Recall t h a t w E < w~ whenever ~ < a and t h a t w~+l -- w+, where ~+ denotes the smallest cardinal larger t h a n ~. Also, if a is a limit ordinal, t h e n wa = sup{w E" ~ < c~}. T h e cofinality of T, d e n o t e d by cf(~-), is the smallest cardinal ~ such t h a t ~" has a cofinal subset of cardinMity ~. Clearly cf(wa+l) - - w a + l and in general Cf(T) ~ T. Let T be an u n c o u n t a b l e cardinal. If T = wa+l (i.e. if T is a regular cardinal), t h e n D r can be represented as the limit space of the continuous inverse s p e c t r u m {(I~)e,r~+l,Z where r ~ +1" (I[~ ")~+1 +

< cf(r)}

(D~=)8 denotes the projection (D ~ ) 8

x D W"

(D~o)~. If T is a singular carrdinal, then T -- sup{w~," Z < Cf(T)} where w _ w ~ < T for each fl < cf(~'). Consequently, in this case, D r can be represented as the limit space of the continuous inverse s p e c t r u m lim{I~,,

7r~+1, z < cf(T)}

where 7r~+1" I~W~+ ~ --, I~W~ denotes the projection onto the corresponding subproduct. C o m b i n i n g these to cases we see t h a t I~r = lim,.q where

S - {Yz,~+1,~ < cf(r)}. Here { ( D w~)o Y~ =

I~

ifT=wa+l if T = sup{wa~" /3 < cf(z)}

By 7r~" I~~ ~ Y~ we denote t h e / 3 - t h limit projection of the s p e c t r u m S.

8.6. COMPACT GROUPS AND FIXED POINT SETS LEMMA 8.6.36. Let r > w and X

369

be a z e r o - d i m e n s i o n a l c o m p a c t space of

weight <_ r . T h e n X is h o m e o m o r p h i c to the limit space of a c o n t i n u o u s i n v e r s e spectrum S x = {X e, p { + l , 5 < c f ( r ) } such that dim X/~ = 0 and

~(xe) <_{'~

r 0~8

i f T --- W o L + l

if r = s u p { w a , "/~ < c f ( r ) }

f o r each/5 < c f ( r ) .

PROOF. Since dim X = 0 and w ( X ) <_ T we m a y assume t h a t X C D r. Let S be a s p e c t r u m defined above. We let X/~ = r r / ~ ( X ) a n d p ~ + l = r r ~ + l / X ~ + l . T h e n the s p e c t r u m S x = {X/~, p~+l,/3 < Cf(T)}

has all the required properties.

I--1

We s t a r t w i t h t h e following s t a t e m e n t which in the case when r = w can be viewed as a p a r a m e t r i c version of P r o p o s i t i o n 8.6.24. THEOREM 8.6.37. Let T > w, X be a z e r o - d i m e n s i o n a l c o m p a c t u m of weight < T and F be a n o n e m p t y closed subset of the product X x D r. T h e n there exists a h o m e o m o r p h i s m h" X x D r ~ where 7rl" X x D r ~ X

X x D r such t h a t r r l h = rrl and fix(h) = F ,

denotes the p r o j e c t i o n onto the first coordinate.

PROOF. We shall carry out the transfinite i n d u c t i o n with respect to w ( X ) . C a s e 1 (T = w). I f F is open and closed in X x D ~~ t h e n the s t a t e m e n t is trivial. T h e r e f o r e we may assume t h a t the c o m p l e m e n t X x ]]}oa- F can be w r i t t e n as a c o u n t a b l e disjoint union C1 U C2 U - - - U Cn U . . . of closed a n d o p e n subsets of X x ]]}~. T h e restriction r r l / C n : Cn --+ rrl(Cn), n E w, is an o p e n m a p between zero-dimensional metrizable c o m p a c t a whose fibers

(~/c~)-~(~)

= ~ f l ( x ) n c ~ = ({z} • D ~) n c ~ , ~ e ~1(c~),

being n o n e m p t y closed and open subsets of D ~', are all h o m e o m o r p h i c to ]I~. Consequently, by L e m m a 8.1.7, the restriction r r l / C n : Cn --+ 7rl (Cn) is a trivial bundle with fiber D ~ for each n C w. This m e a n s t h a t for each n E w t h e r e is a h o m e o m o r p h i s m tn: rrl(Cn)• w ---+ C n such t h a t 7 1 " 1 t n - - - 71n, where rrn : 7rl(Cn) x D ~ -+ rrl(Cn) denotes the projection onto the first coordinate. Here is t h e corresponding diagram

370

8. APPLICATIONS

c.

~

,,,

t.

~l(c.)

/

x

I[Y"

n

71"1(Cn)

Let d be a m e t r i c on X x ]D)~. C h o o s e a fixed point free h o m e o m o r p h i s m fn" ]I)• --+ I~~ so close to the i d e n t i t y t h a t the h o m e o m o r p h i s m gn" Cn ---+ Cn defined as the 1 for c o m p o s i t i o n gn - tn(id~rl(c,,) x fn)t-~ 1 satisfies the c o n d i t i o n d ( g n ( x ) , x ) < -~ each x E Cn. N o t e t h a t gn preserves the first c o o r d i n a t e , i.e. 7rlgn - r l / C n for each n E w. C o n s i d e r a h o m e o m o r p h i s m hn" X x I[]P' ---) X x ]]~, n E w, defined as follows: h(x) = ~x

I

gn(x)

if x E F ifx ECn.

Clearly, h preserves the first c o o r d i n a t e (i.e. 7rlh -- ~rl) a n d fix(h) -- F . This finishes the p r o o f of T h e o r e m in t h e case when T - w. C a s e 2 (T > w). S u p p o s e t h a t T h e o r e m has b e e n proved in the cases when w ( Z ) < ~ w h e r e w <_ ~ < r a n d consider the case w ( X ) -- T. R e p r e s e n t D r as the limit space of the continuos s p e c t r u m S = {Yf~,zr~+l,fl < cf(T)} which was defined above. By L e m m a 8.6.36, X - lim,.qx, where S x = {Xz, p ~ + l , ~ < cf(T)} is the s p e c t r u m from t h a t l e m m a . O b s e r v e t h a t t h e limit space of the s p e c t r u m S X • S = { X ~ • y~,p~+X • 7r~+l,j3 < c f ( r ) }

coincides w i t h the p r o d u c t X x D r. Let F~ --- (p~ x r ~ ) ( f ) for each f~ < Cf(T). D u r i n g the c o n s t r u c t i o n the p r o j e c t i o n X f~ • Yf~ ---+ X f~ will be d e n o t e d by ~r~. Since w ( X o ) <_ w ( Y o ) < T a n d dim Xo -- 0, by the inductive h y p o t h e s e s , t h e r e exists a h o m e o m o r p h i s m h0" X0 x Y0 ---+ X0 x Yo such t h a t lr~ -- Ir ~ and f i x ( h 0 ) - F0. S u p p o s e t h a t for each f} with f~ < -7, where -y < cf(~-), we have a l r e a d y c o n s t r u c t e d h o m e o m o r p h i s m s h E 9 X f~ x Yf~ ---+ X/~ x Yf~ such t h a t

8.6. COMPACT GROUPS AND FIXED POINT SETS

371

(i)~ ~r~h~ = ~r~ for each fl with fl < "r.

(ii)[3 fix(ht~ ) = F~ for each fl with fl < "r. (iii)[3 (p~ x 7r~)h~ = h~(p~ x r ~ ) for each 6, fl with 6 < fl < 7. (iv)~ ht~ = lim{h6: V < fl} for each limit ordinal fl with fl < -),. Let us define the h o m e o m o r p h i s m h.r: X. r x Y'r --* X'r x Yr" If 3' is a limit ordinal, then let h. r = lim{h~: fl < 3'}. P r o p e r t i e s (i)t~ - (iv)E , fl < -)', imply t h a t h. r is a h o m e o m o r p h i s m such t h a t

(i), ~ h , = ~ . (~), fix(h.)= F,. (iii). r (p~ x lr~)h, r = h6(p~ x r ~ ) for each 5 with 6 < "r. If 7 = fl 4- 1, then we proceed as follows. Consider the fibered p r o d u c t Z~ of spaces X~ x Yfl and X t~+l with respect to the maps 7r~" X fl x Y~ ---, X~ and p ~ + l . Xt~+ 1 ---, X~. By ~o" Zfl ---, Xfl x Y~ and r Zfl ---, XZ+I we d e n o t e the corresponding projections of the above fibered product. Here is the induced diagram

~1~+1 X~+ 1 X

'~Xfl+l

pflfl+l

Z~

x ~ x ~'~

~

~x~

Since ~r~ is a trivial bundle with fiber Y~ and r is the canonical projection of the fibered product, we see t h a t r also is a trivial bundle with fiber Y~. Moreover, since lr~ +1 is a trivial bundle with fiber YZ+I, we conclude t h a t in the described situation the m a p X = (P~ + i x 7r~+1)A ~r~+1 is a trivial bundle. F i b e r of this bundle is I~~ where /

= ~wa

[ w,~+~ - w ~

if 7 -- Wa+ 1 if T = s u p { w ~ : ~ < cf(~)}.

Clearly, w(Z~) <_ w ( X z x Y~ x X f l + l ) . Therefore

372

8. APPLICATIONS

w(Zf~)

f < ~wa

if

- [~,+~

if ~ = s u p { ~ , .

T -- Wc~+I

~ < cfff)}.

Since there exists, by the inductive hypothesis, a homeomorphism ff~+l : Xf~+l • Yf~+l ---+ Xf~+l x Yf~+l such that f ~ + l X = X and fix(ft,+l) = F~+I. gz: Z~ --* Z~ defined by letting g~ = h ~ A r

Consider the homeomorphism Finally, let h~+l = f ~ + l ( g ~ x

idD(,,,,,~+ 1 -,.,,~ ) ).

A straightforward verification shows that ~f~+l~ _~+1 9 "1 r ~ f ~ + 1 ~- 711 " /, f~+l.. ~+1~-~+1 ~+1 9 ~t,f~

• lrf~

)n/3+1 = h/3(p

x ~r

).

9 fix(hE+l ) ----F~+I This completes the construction of homeomorphisms h~ for each ~ < Cf(T). Then the homeomorphism h = lim{he" ~ < cf(r)}" X x D r -+ X x D r preserves the first coordinate and fix(h) -- F. Proof is complete. Obvious changes of the above proof, coupled with the results of Subsection 8.6.2 allow us to prove the following. THEOREM 8.6.38. Let T > w, X be a c o m p a c t u m of weight << ~" and F be a n o n e m p t y closed subset of the product X x G r where G is a p o s i t i v e - d i m e n s i o n a l c o m p a c t m e t r i z a b l e group.

T h e n there exists a h o m e o m o r p h i s m h" X x G r ---+

X x G r such that 7rlh = 7rl and fix(h) = F , where 7r1" X x G r ---+ X denotes the projection onto the first coordinate.

Theorems 8.6.37 and 8.6.38 imply the following. COROLLARY

8.6.39. Let ~" > w and G be a n o n d e g e n e r a t e compact metrizable

group. T h e n G r has the C I P H .

Slightly modifying the proofs of Theorems 8.6.37 and 8.6.38 we obtain the following result. THEOREM 8.6.40. Let X and Y be compact spaces and f : X ---+Y be a principal G r - b u n d l e , T >_ w, where G is a n o n d e g e n e r a t e c o m p a c t metrizable group. Let w ( Y ) <_ T and a s s u m e that one o f the following two conditions is satisfied:

(a) d i m X = d i m Y = d i m G = 0 , (b) d i m G > 0. Then X

or

has the C I P H . Moreover, each n o n - e m p t y closed subset of X

fixed by a fiber-preserving a u t o h o m e o m o r p h i s m of X .

can be

8.6. COMPACT GROUPS AND FIXED POINT SETS

373

T h e o r e m 8.6.40 tells us, in p a r t i c u l a r , t h a t for any c o m p a c t u m X of w e i g h t T > w t h e p r o d u c t X x ~ has t h e C I P H . C a n we have less t h a n 7- copies of a n d m a i n t a i n t h e s a m e conclusion? [218, P r o o f of T h e o r e m 3.12] shows t h a t if T > w a n d n < wl, t h e n t h e p r o d u c t ]I}r x T ~ does not even have t h e CIP. O u r n e x t result shows t h a t t h e c a r d i n a l i t y of t h e set of T factors in T h e o r e m 8.6.40 c a n n o t be reduced. LEMMA 8.6.41. Let w ( X ) = T and w ( Y ) C o n s i d e r the following (i) There exists a

w <_ n < T and X and Y be compact spaces such that = n. Let F be a closed subset of the product Z x Y . two conditions: m a p h" X • Y ---+ X • Y such that

(a) f i x ( h ) = F , and (b) l r z . h = ~rz , where ~ z

"X x Y projection onto the first factor. (ii) F is a G ~ - s u b s e t of the product X • Y . T h e n (i) implies (ii).

---+ X

denotes the natural

PROOF. R e p r e s e n t X as t h e limit space of s o m e n - s p e c t r u m S - { X a , P~a, A}. Obviously, t h e s p e c t r u m S ' = { X a x Y,P~a • id, A } is also a n - s p e c t r u m w i t h l i m S ' -- X x Y. By t h e s p e c t r a l t h e o r e m for n - s p e c t r a , t h e r e exists a n-closed a n d cofinal s u b s e t A ' in A such t h a t for each a c A', t h e r e is a m a p ha" X a x Y --+ X a x Y satisfying t h e following two c o n d i t i o n s :

(~)~

h ~ . (p~ x i d ) = (p~ x id). h.

(b)a r x ~ " h a = ~ x ~ , w h e r e r x ~ " X a x Y --~ X a d e n o t e s t h e n a t u r a l project i o n o n t o t h e first factor. Let a E A ' a n d x a E (pc x i d ) ( F ) . F i x a p o i n t x E F such t h a t (pc x i d ) ( x ) = x a . By (a), x = h ( x ) . C o n s e q u e n t l y , by (a)a, x a = (pc x i d ) ( x ) = (Pc x i d ) h ( x ) = h a " (pc x i d ) ( x ) = h a ( x a ) . This shows t h a t (pc x i d ) ( F ) C_ f i x ( h a ) . Let

Z = (pc x i d ) - l ( ( p a x i d ) ( F ) ) . Clearly, F C Z. S u p p o s e t h a t Z - F ~ 0 a n d consider a point y e Z - F . Let y = ( x , s ) e X x Y a n d h ( y ) = ( x ' , s ' ) e X x Y . B y (b), x = x'. Since (pc x i d ) ( F ) C f i x ( h a ) , we see t h a t ( p a ( x ' ) , s ' ) -- (pc x

i~)(x', ~') = (p~ x i~). h(~, ~) = h ~ . (p~ x i~)(~, ~) = h~(p~(~), ~) = (p~(~),

~).

T h e r e f o r e , s -- s' a n d y - h ( y ) which c o n t r a d i c t s c o n d i t i o n (a). C o n s e q u e n t l y F -- Z, a n d since any closed s u b s e t of t h e p r o d u c t X a x Y has c h a r a c t e r <_ n ( b e c a u s e w ( Z a x Y) -- n), we see t h a t F is a G ~ - s u b s e t of t h e p r o d u c t Z x Y. [-I

COROLLARY 8.6.42. Let T > w, n >_ w and let G be a nondegenerate c o m p a c t connected metrizable group. I f n >_ T, then the product ]I}T x G ~ has the C I P H . I f T > n, then the product ]~T x G ~ does n o t have the CIP. PROOF. T h e first p a r t of our s t a t e m e n t follows from T h e o r e m 8.6.40. S u p p o s e now t h a t ~" > n. Let F be a closed s u b s e t of t h e p r o d u c t ]I}r x G ~ such t h a t r]i)~(F) = ]I}r a n d F is n o t a G ~ - s u b s e t of t h a t p r o d u c t . S u p p o s e t h a t F -- f i x ( h ) for s o m e m a p h" IDr x G ~ --~ ]]])r x G ~. By L e m m a 8.6.41, t h e r e is a p o i n t (x, s) e]I} ~ x G ~ - F such t h a t x ~ x', w h e r e (x', s') = h ( x , s).

374

8. APPLICATIONS

Consider t h e c o n n e c t e d set S ' = zrI[])~(h({x } x G~)). Clearly, x' e S t. On t h e o t h e r h a n d , since 7 r i ~ ( F ) -- D r we see t h a t x E S ~. Therefore, S I is a c o n n e c t e d (as a c o n t i n u o u s image of G ~) and zero-dimensional (as a subspace of IDr) set c o n t a i n i n g two distinct points. This c o n t r a d i c t i o n shows t h a t there is no m a p h w i t h F = f i x ( h ) . T h e r e f o r e the p r o d u c t IDr x G ~ does not have CIP.

[3

Historical and bibliographical notes 8.6. T h e results of Subsections 8.6.1 a n d 8.6.2 are t a k e n from [103].

8.7. G r o u p actions In this Section we investigate actions of n o n m e t r i z a b l e groups. Of course, each principal G - b u n d l e g e n e r a t e s a canonical free G - a c t i o n (and vice versa for G a Lie group). Consequently, one m i g h t ask: 9 W h a t sort of p r o p e r t i e s of a space X , in t h e presence of a free action of a "nice" group G on X , g u a r a n t e e t h a t X has t h e C I P H ? This q u e s t i o n r e m a i n s o p e n even for p a r t i c u l a r l y chosen groups. On the o t h e r hand, if G -= 71" (or G - R) the question is answered for any metrizable space [216]. T h e l a t t e r fact suggests t h a t the difficulties involved in e x t e n d i n g our result, c o n c e r n i n g principal G - b u n d l e s (see T h e o r e m 8.6.40), to free G-actions are p e r h a p s of a s e t - t h e o r e t i c a l n a t u r e . For instance, we show (Corollary 8.7.3) t h a t there is no n o n - m e t r i z a b l e (locally) c o m p a c t group G, acting freely on any locally c o m p a c t ANR-space (note t h a t the t o t a l space of any principal T r - b u n d l e is n o t an A N R ) . This answers a n o n - m e t r i z a b l e version of R . D . E d w a r d s ' s question 3 [140] asking w h e t h e r t h e r e is a c o m p a c t Hilbert cube manifold a d m i t t i n g a free action of t h e g r o u p Ap 7.

8.7.1. Spectral representations o f g r o u p a c t i o n s . By an action of a topological group G on a space X we m e a n a c o n t i n u o u s m a p A: G x X --~ X satisfying the following two conditions: 9 A ( g , A ( h , x ) ) = A(gh, x) w h e n e v e r g,h e G and x e Z . 9 A(e,x) = x for each x E X , where e denotes t h e unit element of t h e g r o u p G. We prefer to use t h e l a n g u a g e of diagrams. If #: G x G --* G is a continuous m u l t i p l i c a t i o n o p e r a t i o n on G, t h e n the validity of the first condition is equivalent to the c o m m u t a t i v i t y of t h e following square diagram:

3To the best of our knowledge, Edwards's question is still open. Obviously it is a weaker version of the well-known Hilbert-Smith conjecture which asks whether a locally compact group acting effectively on an n-manifold must be a Lie group.

8.7. GROUP ACTIONS

375

ida x A GxGxX

,.- G x X

# x idx

A

GxX

,~X

T h e second condition is j u s t the equality

~/({~} where 7rx" G

x

x x)=

~x/({~} x x)

X ~ X denotes the p r o j e c t i o n onto the second coordinate.

THEOREM 8.7.1. Let r >_ w and A" G x X ~ X

be an action. Suppose that 8 x = { X s , p ~ , A } is a r-spectrum, the limit space of which is h o m e o m o r p h i c to X , and SG = { G s , s ~ , A } is a T-spectrum consisting of topological groups and continuous h o m o m o r p h i s m s , the limit of which is isomorphic to G. I f X and G are both locally compact and Lindelbf, or if both are AE(O)-spaces (topologically), then A is induced by a cofinal and T-closed m o r p h i s m of "level actions"

{~.

a ~ x x ~ ~ X~; B}" 8 c / B

x Sx/B ~ 8x/B.

Moreover, if X and G are both locally compact and Lindel6f and A is an effective 4 action, we m a y assume that all actions As, a E B , also are effective. PROOF. T h e conditions imposed on G and X are needed to ensure t h a t not only the given s p e c t r a 8 a and S x , b u t also the s p e c t r u m

s~ x &

= { a . x x . . ~ x p~. A}

is factorizing. A p p l y i n g T h e o r e m 1.3.6 to the m a p A (which m a p s the limit space of the s p e c t r u m S a x ,~x into the limit space of the s p e c t r u m 8 x ) , we conclude t h a t t h e r e exists a cofinal and T-closed subset B of the indexing set A a n d m a p s

As" G s X X s --* X s ,

a E B

such t h a t A = lira{As}. Since A is an action, we easily see t h a t each m a p As, a E B, is an action of t h e group G s to the space X a . Indeed, let e s d e n o t e the unit e l e m e n t of G s a n d

x s E X s . Take a point x E X such t h a t p s ( x ) = x s . T h e n

a~(~., ~.)

= a~(~(~),v~(~))

= a~((~

• p ~ ) ( ~ , ~ ) ) = v ~ a ( ~ , ~) = , ~ ( ~ )

= x~.

4Recall that an action A" G x X --~ X is effective if for each 9 G G, g r e, there exists a point x E X such that g(x) :/: x.

376

8. APPLICATIONS

This proves the commutativity of the first diagram from the definition of an action. The second diagram from that definition appears as the bottom of the following cubic diagram: idc

GxGxX It X i

x h

~GxX

sa X sa X pa

GxX

,~X

sa x Pa

Pa

sa

x pa

Ga

x

/

Ga

idc~ x ha

x Xa

*-Ga x

Xa

/ , " ~ a x idxo

ha

Ga x X a

,-Xa

The commutativity of the top (h is an action) and of all the sides ({ha} is a morphism) of the above cube allow us to conclude the commutativity of the ! bottom. Indeed, let (ga, g a , x a ) e. Ga x Ga • X a . Choose a point (g,g',x) E ! G • G x X such t h a t s a ( g ) - ga, s a(g ~) = g,~ and p c ( x ) = x a. Then

Aa((idco xAa)(ga, ga, ' xa)) = A a ( ( i d c , xAa)((sa x sa • p a ) ( g , g ' , x ) ) ) =

x p.)((idc

p.(

((idc

p c ( h ( ( # x i d x ) ( g , g ' , x ) ) ) - Aa((sa x Pc)((# x i d x ) ( g , g ' , x ) ) ) h a ( ( , a X i d x . ) ( ( s a x sa x p a ) ( g , g ' , x ) ) ) -

h a ( ( , a x idx~)(ga, g'~,xa)).

Therefore, ha is an action for each a E B. This proves the first part of theorem. In order to prove the second part of theorem consider the following relation LCB2: L--

{ ( a , f ~ ) e B 2" a__f~ and for each g e G ~ - k e r ( s ~ ) there exists a point x e X~ such that hE(g, x) ~ x}.

Let us verify conditions of Proposition 1.1.29 for the above relation L. Existance. Let a E B and g E G - ker sa. Since A is an effective action, there exists a point xg e X such that h(g, xg) ~ xg. Let O[ and O~ be disjoint open neighborhoods in X of the points h(g, xg) and xg respectively. The continuity of h implies that the set h -1(O~) is open in the product G • X. Consequently there exist open subsets Ug _C G and Vg c_ 02 such that (g, xg) E U~ x Vg c_ h - l ( O ~ ) .

8.7. GROUP ACTIONS

377

Clearly A(Ug x Vg)M Vg = O. This shows t h a t A(g',x') -~ x' for each (g',x') E Ug x Vg. Observe t h a t the open cover {Ug: g E G - k e r s o } contains an open refinement {Ut: t E T} of cardinality at most T, i.e. ITI _< T (to see this note t h a t the restriction s o / ( G - k e r so): ( G - k e r so) ~ ( G o - { c o } ) i s a proper m a p and t h a t the space Go - {co} can be w r i t t e n as an union of at most T c o m p a c t subsets of the group Go). If t E T, then Ut C_ Ug for some g E G - k e r s o . By Vt we denote the set Vg constructed above. W i t h o u t loss of generality we m a y assume t h a t for each t E T there exist an index ~t E B and open sets UEt C GEt and VEt C_ XE~ such t h a t Ut = sE-tl(UEt) and Vt = p-~(VEt). T h e r e exists an index f~ E B such t h a t / 3 _ c~ and f~ >_ ~t for each t E T. Let U~ - (s~t)-l(UE~) and V~ = (p~t)-l(VEt) for each t E T. PE- l ( v t ~ ) = Vt for each t E T.

T h e n we see t h a t s~l(Ut ~) = Ut a n d

Since AE(sf~ x PE) = PE A we conclude t h a t

)~E(Ut~ • V~)M Vt~ -- 0. Finally observe t h a t G E - ker(s~) = U { u t ~ . t E T}. It only remains to note t h a t in this situation for each g E G E - ker(s~) there is a point x E Xt~ such t h a t AE(g,x) ~ x. This proves t h a t ( a , ~ ) E L. Majorantness. Let (a, fl) E L and ~, _> f~ for some 7 E B. consider an element g E G ~ - ker(s~). Clearly s~(g) E G E - ker(s~). Since (c~,f~) E L, there exists a point x E X E such t h a t )~E(s~(g),x) ~ x. Consider a point y E X.y such t h a t p ~ ( y ) - x. T h e n )~(g, y) -~ y. T h u s (c~, ~,) E L. T-closedness. Let P = {a~: -~ < T} be a chain of element of B of cardinality at most T. Suppose also t h a t (c~.y,f~) E L for each ~, < T and for some element E B. We need to show t h a t (c~,/?) E L, where a = sup P. Let g E G E - k e r ( s ~ ) . Since the s p e c t r u m SG is a Y-spectrum, the space Go is canonically h o m e o m o r phic (even isomorphic in our situation) to the limit space of the s p e c t r u m S c / P . Therefore {co} -- N{ker(saa~): "), < T}. This allows us to find an ordinal ~ < T such t h a t g E G E - k e r ( s ~ ) . Since (c~.y,f~) E L, there exists a point x E X E such t h a t ~E(g,x) ~ x Thus the relation L satisfies all three conditions of Proposition 1.1.29. Consequently, by t h a t Proposition, the set of all L-reflexive indexes, denote it by C, is cofinal and T-closed in B. It only remains to note t h a t L-reflexibility of an index a means t h a t the action Ao is effective. This completes the proof of theorem. [::] Recall t h a t for a given action A: G • X --~ X and a point x E X the closed subgroup Gx -- {g E G: g(x) - x} is called a stabilizer of x. An action ~ is said to be semi-free if the stabilizer of each point of X coincides with G or is trivial. If only the second possibility occurs for every point x E X , then A is said to be

free. COROLLARY 8.7.2. If an AE(O)-group G acts semi-freely on a non-metrizable (locally) compact A N R-space X , then (a) ~ is the trivial action or

378

8. APPLICATIONS

(b) ~(a)< ,~(x). PROOF. Let w ( X ) = T > w and suppose that w(G) > T. First consider the case w(G) > T. By Lemma 8.2.1, G is isomorphic to the limit space of a factorizing r - s p e c t r u m SG = { G , , s ~ , A } consisting of AS(O)groups of weight T and 0-soft limit homomorphisms. Applying Theorem 8.7.1 to this situation, we find an index a E A and an action

Ao~" Go~ • X --~ X such that

A-

(8.7.1)

Aa(sa • i d x ) .

Since w(G) > T -- w(Ga), we see that kers~ contains at least two elements (otherwise sa" G ---, Ga is an isomorphism). Let g 6 kersa be a non-trivial element of G. Then, by (1),

for each point x 6 X (here ea denotes the trivial element of the group G~). This shows that A is the trivial action. Next consider the case w(G) - T. The action A is trivial if (and only if) the set X c -- {x 6 X" Gx - G} coincides with X. Obviously, X c is closed in X. Therefore, we only need to show that X G is dense in X. By Lemma 8.2.1, G is isomorphic to the limit space of a factorizing w-spectrum 8G = {Ga, s ~ , A } consisting of Polish groups Ga and 0-soft limit homomorphisms sa. Similarly, by Proposition 6.3.5 and Remark 6.3.6, X is the limit space of a factorizing w-spectrum S x - {Xa, p~, A} consisting of locally compact A N R - s p a c e s (with countable bases) X a and proper and soft limit projections pc. By Theorem 8.7.1, we may assume, without loss of generality, that A is the limit of "level actions" As" Ga • Xa --, Xc~, a 6 A. Let U be a non-empty open Fa-subset of X. Since the spectrum S x is an w-spectrum, there is an index a 6 A such that U - p ~ l ( p a ( U ) ) . As above, there is a non-trivial element g 6 ker sa. Let xa 6 pc(U). We are going to show that the map Ag' X --~ X (defined by letting Ag(x) -- A ( g , x ) ) preserves the fiber p ~ l ( x a ) of the limit projection pc. Indeed, take a point x 6 p ~ i ( x a ) . Then

Therefore, Aa(p-~l(x,)) C_ p ~ l ( x , ) . The softness and properness of the projection pa guarantees that the fiber p ~ l ( x a ) is an AR-compactum and, hence, has the fixed point property. Therefore there is a point z 6 p-~l(xa) such that g(z) = z. Since A is a semi-free action, the latter means that z E X G. It only remains to note that, by the choice of the index a, z 6 U. Therefore,

unxa#O.

[3

8.7. GROUP ACTIONS

379

COROLLARY 8.7.3. There is no non-metrizable A N R - c o m p a c t u m admitting a non-trivial semi-free action of any non-metrizable compact group. PROOF. Let A: G • X --~ X be a non-trivial semi-free action of a c o m p a c t group G of weight T > w onto a n o n - m e t r i z a b l e c o m p a c t u m X of weight a > w. Since A is non-trivial, we have, by C o r o l l a r y 8.7.2, T < a. R e p r e s e n t X as t h e limit space of a factorizing T - s p e c t r u m S x -- { X s , p ~ , A } consisting of A N Rc o m p a c t spaces of weight T and soft limit projections. By T h e o r e m 8.7.1, we m a y assume t h a t A is the limit of a m o r p h i s m {As: G • X s --~ X s ; A } of "level actions". Since A is non-trivial action, t h e r e is a pair ( g , x ) E G • X such t h a t A(g, x) -~ x. Take an index a E A such t h a t

(8.7.2)

p.(~(g, ~)) # p,(~).

We claim t h a t the action Aa: G x X s ~ X s is also non-trivial and semi-free. Indeed, since p s A - A s ( i d a • p s ) , we see t h a t , by (2),

This shows t h a t the element g E G moves the point p s ( x ) and, hence, As is a non-trivial action. Next we show t h a t As is semi-free. Take any point x s E X s , and s u p p o s e t h a t for some non-trivial element g E G we have As(g, x s ) = x s . Take any point x E p-~l(Xs). T h e n we have

This shows t h a t the m a p Aa: X --~ X (defined, as above, by l e t t i n g Ag(x) -A ( g , x ) ) m a p s t h e fiber p ~ l ( x s ) over the point x s into itself. T h e softness of the limit p r o j e c t i o n p s of the s p e c t r u m S x g u a r a n t e e s t h a t this fiber is an A R c o m p a c t u m . Therefore, t h e r e is a point y E p ~ l ( x s ) such t h a t y = Ag(y) A(g,y). Since g is a non-trivial element of G, the stabilizer Gy of the point y E X is non-trivial. Since A is a semi-free action, this implies t h a t G y -- G, i.e. A(g ~, y) = y for each element g~ E G. B u t t h e n

~.(g'. ~ . ) = ~.(idG • p.)(g', y) = p.~(g', y) = p.(y) =

~..

This shows t h a t the stabilizer G x . of any point x s E X a coincides w i t h G whenever this stabilizer contains at least one non-trivial element. This m e a n s t h a t As is a semi-free action. It only remains to observe t h a t the e q u a l i t y w ( G ) = T = w ( X s ) , coupled w i t h Corollary 8.7.2 (applied to t h e non-trivial semi-free action As: G x X s -~ X s ) , leads us to a contradiction. [:]

380

8. APPLICATIONS

Remark 8.7.4. Note that the assumption of non-metrizability in Corollary 8.7.2 is essential. Indeed, the group A n acts on the Hilbert cube I ~ fixing a single point and acting freely off that point [140]. Hence Ap acts freely on the non-compact Hilbert cube manifold I ~ - {pt} ~ I ~ x [0, 1). We also remark that the product M x [0, 1) has the CIPH for each/W-manifold M [216]. 8.7.2. E x a m p l e s . PROPOSITION 8.7.5. Let f : X --, Y be a principal R-bundle between metrizable spaces. If F is a nonempty closed subset of X , then there is a fiber preserving autohomeomorphism of X whose fixed point set is F (in particular, X has the CIPH). PROOF. Let [-[ denote the usual norm metric on the real line R and let d be a metric for X such that d ( x , y ) - - I x - y [ whenever x , y 9 Z and f ( x ) = f ( y ) (see [295] or [300]). Given a nonempty closed subset F of X, set r ( x ) - 89 Consider (we maintain the notation in the proof of Theorem 8.6.40) a collection {Ui: i 9 I} of open subsets of Y, a collection

hi: U i x R--+ f - l ( U i ) ,

i 9 A,

of fiber-preserving homeomorphisms and a collection

g i,j: U i M U j___,R, i , j 9 of continuous maps such that

h i ( x , t ) = h J ( x , t +gi'J(x)) for each (x,t) e (V i M U j) x R and i , j 9 A. Here the plus sign represents the usual addition operation in the additive group R and A stands for an indexing set. Let

Hi(x)-

hi(Tr~((hi)-l(x)),Tr~((hi)-l(x)) + r ( x ) ) whenever x e f - l ( U i ) , i E A.

Here 7rl and 7r~ denote the natural projections of the product U i x R onto the first and the second coordinates respectively. Note that if x E f - l ( U i M UJ), i , j E A, then

Hi/f-l(ui

M U j) = H J / f - I ( u

i M UJ).

Therefore, the map H : X --, X defined by letting H ( x ) -- H i ( x ) for each x E f - l ( U i ) , i E A, is well defined. Obviously, f H -- f and the set of all fixed points of H coincides with F. A straightforward verification (using a slight modification of the argument presented in the proof of [216, Theorem 2.2]) shows that H is a homeomorphism. V] Another important class of spaces with the C I P H spaces.

is the class of normed linear

8.7. GROUP ACTIONS

381

PROPOSITION 8.7.6. Any normed linear space has the CIPH. PROOF. Let (X, I1" II) be a n o r m e d linear space. By d we denote the metric generated by the norm, i.e. d(x, y ) = ]ix - y i [ for each x, y E X. Let A be a closed subset of X. W i t h o u t loss of generality we may assume t h a t 0 E A. Define a m a p h: X ~ X as follows:

h(x) = ~ x + {d(x, A ) . I-~x

(

for x :~ 0, for x = 0.

0

Obviously, f i x ( h ) = A. Let us show t h a t h is a h o m e o m o r p h i s m . Let x, y E X and h(x) = h(y). T h e n 1

x + -~d(x,A).~

x

1

= y + -~d(y,A).

Ilvll

Consequently,

1

1

(1 + 21lxlld(X,A))llxll = (1 + Then

1 I1~11+ -~d(x,A)

2IlyiI)

d(y,A))llYll.

1

=

Ilyll + -~d(y,A)

and

IlY-~11- II1~11- IlYlII = I~1 (d(x, A) - d(y , A)I 9 On the other hand, the triangle inequality applied to x, y and A implies

Ily - ~11 ~ Id(x,A) d(y,A)l. Id(x,A)- d(y,A)l = 0 and x -- y. This shows -

Therefore, t h a t h is an injection. We now show t h a t for each point x E X , there is a point y E X such t h a t h(y) = x (i.e. h is a surjection). Since, by definition, h(0) = 0 we may assume t h a t x -~ 0. Since 0 E A by our assumption, we conclude t h a t Ilxi[ >_ d(x, A). Choose a real n u m b e r t such t h a t t _ I i x i I - d ( x , A ) . T h e n d(x,A) <_ I i x i I - t (in particular, tx 0 < t < [IxiI). T h e triangle inequality applied to iix-]-~,x and A implies t h a t

tx

tx

d(]-~,A) <_ I1~-i1-~11 +d(~,A)=

tx

Ilxll-II I1-~11 +d(~,A)=

II~ll-t+d(~,A)<_

2(11~11- t). Therefore l d ( tx t + ~ ~]~-~,A) < I1~11. On the other hand, tx

IIh(ll-~)ll = I1~ + ~d(ll-~,A)" iii~11

ii-~lixll + d(~-,

I1~11

382

8. APPLICATIONS i tx t + -~d ( ~-~ , A ) .

Therefore [ [ h ( ]t x~ ) [I-< [[x ll -- []h(x) ]!. In this situation, the connected set h({rx I-~ --- r ___ 1}) (i.e. the image of the segment with end points ~t x and x) is contained in {rx : r E R} and, as was shown above, contains the point x. Therefore there is a point y E [ ~ , x ] such t h a t h ( y ) = x . This shows t h a t h is a surjection and hence a continuous bijection. It only remains to show t h a t the inverse function h - l : X ~ X is also continuous. Let {xi} i=~ be a sequence of points of X such t h a t the sequence {h(xi) }i=li=~176 converges in X to the point y 9 In order to show t h a t {xi }i=1 i=o~ also converges, we need the following observation. Claim. If x , y E X , then 89 x - Yll -< lih(x) - h(y)ll. Proof of Claim. If at least one of the given points coincides with 0, then the s t a t e m e n t follows from the definition of h. Hence we may assume t h a t x ~ 0 ~ y. T h e triangle inequality, applied to the triples (x,y,h(y)) and (x,h(x),h(y)), implies I1~ - yll _< lib(y) - yll + I1~ - h(y)li

1

_< lib(y) - yll + IIh(~) - ~11 + IIh(~) - h(y)ll =

1

1

-~(d(y,A) +d(x,A)) + lih(x) - h(y)li <_ ~(llYll+llxll)<_ ~llY- Xll + llh(x) - h(y)ll. Therefore 89 - Yl] -< ]lh(x) - h(y)l] as required. T h e claim is established. Using the above claim we see t h a t ]]h-l(y)-xil] <_ 2]]y-h(xi)]l. Consequently, X i=oo i}i=l converges to the point h-l(y). V1

Historical and bibliographical notes 8.7. T h e results of this Section are due to the author.

8.8. B a i r e i s o m o r p h i s m s In this Section we show t h a t the spectral t h e o r e m is valid not only for continuous maps but for Baire maps as well. While it is possible to cover the general case of a r b i t r a r y realcompact spaces, in order to simplify the situation, we consider below only spectrally complete spaces, i.e. spaces which can be represented as the limit spaces of factorizing w-spectra consisting of Polish spaces and surjective projections (see Section 6.2). First of all, recall some definitions. Let X be a space and C(X) the set of continuous real-valued functions on X. Let Bo(X) -- C(X), and inductively define Ba(X) for each ordinal a _ wl to be the set of pointwise limits of sequences of functions in U { B # ( X ) " fl < a}. Let B~(X) be the set of b o u n d e d functions in Ba(X). T h e functions in

B~I(X ) = U { B , ( X ) " c~ < wl}

8.8. BAIRE ISOMORPHISMS

383

are called Baire functions (or Borel functions if X is perfectly normal). T h e Baire sets (or Borel sets if X is perfectly normal) of X of multiplicative class a, d e n o t e d by Z a ( Z ) , are defined to be the zero sets of functions in B~ (X). Those of additive class a, denoted by C Z a ( X ) , are defined as the c o m p l e m e n t s of sets in Z a ( X ) . Clearly, for each a < or1, Z,~(X) is closed under countable intersections and finite unions, and CZ,~(X) is closed under countable unions and finite intersections. It is well-known t h a t 9 M e m b e r s of Z,~+I(X) are countable intersections of m e m b e r s of CZ,~(X). 9 M e m b e r s of CZ~+I (X) are countable unions of m e m b e r s of Z ~ ( X ) . 9 If a is a limit ordinal, then m e m b e r s of Z a ( X ) are countable intersections of countable unions of m e m b e r s of

9 If a is a limit ordinal, then m e m b e r s of C Z ~ ( X ) are countable unions of countable intersections of m e m b e r s of

9

Let a E wl. A map f" X ~ Y is called a Baire map of class a (or Borel map of class oL if X and Y are perfectly normal) if

:-~(Zo(V) c_ z . ( x ) . We say t h a t f" X ~ Y is a Baire map (or Borel map if X and Y are perfectly normal) if

:-~(Zo(V) c_ z ~ (x). Obviously Baire maps are Baire maps of class wl (and conversely). A bijection f" X --~ Y is called a Baire isomorphism of class (a,/~) if f is a Baire map of class a and f - 1 . y ~ X is a Baire map of class ~. Baire isomorphisms of the class (wl, wl) are, for simplicity, called Baire isomorphisms. Finally a bijection f" X ~ Y is called an a-th level Baire isomorphism, 0 _ a <_ wl, if f ( Z a ( X ) ) = Z a ( Y ) . T h e proof of the following s t a t e m e n t follows the proof of T h e o r e m 1.3.4 (or T h e o r e m 1.3.6). THEOREM 8.8.1. Let 0 <_ a, fl <_ wl. Each Baire map .f" lira S x ~ lim 8y of class a between the limit spaces of factorizing w-spectra ,-qx = {Xa, P~a,A} and

s y = {v~, q~, A } , co~i~ti,,g of Polish ~pac~, i~ ~ n d ~ d by a mo~phi~m (co,,sisting of Baire maps of class a) of cofinal and w-closed subspectra. If f is a

B~

~omo~ph~m of ~la~ (~,~) ( r ~ p ~ t ~ l y , an ~-th l~.~l i~omo~ph~m), ~

may assume that the above morphism consists of Baire isomorphism of class

(~,~) ( ~ p ~ t i ~ d y , of ~-th l~.~l Bai~ ~omo~phi~.~).

384

8. APPLICATIONS

Below we give several applications of Theorem 8.8.1 . PROPOSITION 8.8.2. If X and Y are first level Baire isomorphic compacta, then dim X - dim Y. PROOF. Our s t a t e m e n t is true for metrizable compacta [259]. Consequently, without loss of generality we may assume that w ( X ) - w ( Y ) > w. Let f" X Y be a first level Baire isomorphism and dim X _< n. Represent X and Y as the limit spaces of (factorizing) w-spectra 8 x = { X a , p ~ , A } and 8 y - {Ya, q~,A} respectively. By Theorem 1.3.10, we may additionally assume that dim Xa _< n for each a E A. By Theorem 8.8.1, f is the limit of the morphism

{f~" X~ ---+Ya;n}" 8 x ---+8 y consisting of first level Baire isomorphisms. By the above cited result from [259], dim Ya <_ n for each a E A. Therefore, by Theorem 1.3.10, dim Y _< n as desired. D Let us recall (see, for example, [146]) that the transfinite dimensions ind and Ind are the ordinal valued functions obtained by extending the usual concept of small and large inductive dimensions. The transfinite dimensions of separable metrizable spaces are always countable ordinals. It is also known (W. Hurewicz) that the small transfinite dimension ind X of a Polish space X is defined if and only if X is countable dimensional (i.e. X can be represented as the union of countable many zero-dimensional subspaces). The last property is an invariant of first level Baire isomorphisms in the class of metrizable compacta [259]. Let ]C denote the class of compacta admitting zero-dimensional maps onto metrizable compacta. PROPOSITION 8.8.3. Let X and Y be first level Baire isomorphic compacta. If X E ]C and the transfinite dimension ind X is defined, then Y E ]C and the transfinite dimension ind Y is defined. PROOF. Let f" X ~ Y be a first level Baire isomorphism. Without loss of generality we may assume that w ( X ) = w(Y) > w. Represent X and Y as the limit spaces of (factorizing) w-spectra S x = { Z a , p ~ , A } and ,S'y = {Ya, q~,A} respectively. By Theorem 8.8.1, f is the limit map of the morphism

{fox" Xa --+ Ya;A}" ,3x ~ S y consisting of first level Baire isomorphisms. Since X E/C, by [76, Theorem 5], there is a zero-dimensional map g" X ~ K onto a countable-dimensional metrizable c o m p a c t u m K. Represent K as the union of countably many zero-dimensional G6-subspaces Ki. Clearly x

=

9 i e

and for each i E w the subspace g - l ( K i ) , denoted by Xi, is zero-dimensional, Lindelhf and Cech-complete. Applying Theorem 1.3.10 to each of the subspaces

8.8. B A I R E I S O M O R P H I S M S

385

Xi, we may assume without loss of generality that X~ is countable-dimensional for each a E A. Note also that (again, without loss of generality) each limit projection pa of the spectrum S x is zero-dimensional. To see this observe t h a t the map g" X ---, K can be factored through some index a E A, i.e. there is a map go" Xa ~ K such that g = g,~pa. Therefore, the fibers of pa are subsets of the corresponding fibers of g. This shows the zero-dimensionality of pc. By [259], each Y~ is countable dimensional. Next observe that the fibers of the limit projection q~ of the spectrum S y are first level Baire isomorphic to the corresponding fibers of the limit projection pa of the spectrum S x (this follows from the existence of the above indicated morphism). By Proposition 8.8.2, the fibers of the limit projection qa are zero-dimensional. Consequently, Y E K: and, by [76], the transfinite dimension of Y is defined. [:] PROPOSITION 8.8.4. Let T > 2 •. D r+ and expDr+.

Then there is no Baire isomorphism between

PROOF. Consider the Cantor cube D r+ of weight T +. limit space of the T-spectrum

Clearly D r+ is the

S -- {D A, p~, exprT + } where exprT + denotes the collection of all subsets of T + of cardinality _ T and p~ is the natural projection of D B onto D A. Observe that all limit projections PA of S are homeomorphic to the natural projection of D r+ onto D r. Consider also the T-spectrum

e x p 3 -- {expD A, e x p ( p ~ ) , exprT + } whose limit space is homeomorphic to e x p D r+. Suppose now that there exists a Baire isomorphism f 9 D r+ ---+ expD r+ . By Theorem 8.8.1, f is the limit map of a Baire morphism

{ f A" DA ---* expDA; exprT+}" S ---+expS. This simply means that f APA ~- e x p ( p A ) f for each A C exprT +. Consequently, for each point F of expD A, the fiber e x p ( p A ) - l ( F ) is Baire isomorphic to the corresponding fiber p A l ( f A l ( F ) ) , which in turn is homeomorphic to D r+. Let T be a subspace of D A such that T is discrete in the relative topology and ITI -- T. Let F denote the closure of T in D A. Obviously, F is a point of expD A and there exists a pairwise disjoint collection of cardinality T of open subsets of F. One can easily check t h a t the fiber e x p ( p A ) - l ( F ) also contains a pairwise disjoint collection of cardinality T of open subsets. T h e n we can conclude that there exists a pairwise disjoint collection of cardinality T of Baire sets in p A l ( f n l ( F ) ) -- D r+. Since each Baire set is the union of functionally closed sets, we see that there exists a pairwise disjoint collection of cardinality

386

8. APPLICATIONS

r of functionally closed subsets of D r+. r > 2 ~. V1

This is impossible by the inequality

COROLLARY 8.8.5. A s s u m e wl = 2 ~. between D ~3 and e x p D ~3.

Then there is no Baire isomorphism

We conclude this Section by proving the following two statements. PROPOSITION 8.8.6. Let X be a compact space and Y a Baire subset of the Stone-C ech compactification ~ Y . I f X and Y are first level Baire isomorphic, then Y is a-compact. PROPOSITION 8.8.7. Let X be a Baire subset of its Stone-Cech compactification ~ X and Y a Suslin subset of f l y (generated by the collection of Z o ( ~ Y ) ) . I f X and Y are Baire isomorphic, then Y is a Baire subset of ~ Y . Both statements are known to be true for metrizable X and Y (see, for example, [260]). Our goal is to illustrate how Theorem 8.8.1 can be applied in order to get the general statements. The proof of Propositions 8.8.6 and 8.8.7 are similar. For this reason, below we sketch them simultaneously. Sketch of proof of Propositions 8.8.6. It can be shown, using the standard argument (see [260, Theorem 5.8.7]), that Y embeds as a closed subspace of the product I T • B of the Tychonov cube I T and a Baire subset B of some Polish space. Consider the images of Y under the natural projections of this product onto the subproducts of the form I g • B, where K is a countable subset of T. Obviously, each of these images is closed in the corresponding subproduct, and the restrictions of the indicated projections onto Y are proper. Therefore, we can conclude that Y is homeomorphic to the limit space of the inverse spectrum ,.qy - {Y~, q~, A) consisting of the indicated images and restrictions of natural projections. It is also easy to see that this spectrum is an w-spectrum. Furthermore, the closedness of Y in the normal space ! T • B guarantees that the spectrum 8 y is factorizing. W i t h o u t loss of generality we may also assume t h a t the compactum X is the limit space of a factorizing w-spectrum $ x = { X a , p ~ , A } with the same indexing set. Let f" X ---, Y denote a Baire isomorphism of the first level. Applying Theorem 8.8.1 to our situation, we see that the set A contains a cofinal and w-closed subset consisting of indexes c~ such that faPa - q ~ f , where f a : Xa --- Y~ is a Baire isomorphism of the first level. Since the spaces X a and Ya are separable and metrizable, we conclude that Ya is a-compact. Since the limit projection qa: Y ~ Ya is proper we see that Y is also a-compact. This completes the proof of Proposition 8.8.6. Sketch of proof of Proposition 8.8.7. According to [260, Theorem 5.8.9], Y can be identified with a closed subspace of the product I T • B of a Tychonov cube I T and an analytic subspace B of a Polish space. Repeating the above argument, we easily conclude t h a t Y is the limit space of a factorizing w-spectrum $Y = {Ya, q~, A} consisting of analytic subspaces of Polish spaces and proper

8.9. DOUBLE SPECTRA

387

limit projections.

Similarly, X is the limit space of a factorizing w-spectrum consisting of Baire subspaces of Polish spaces and proper limit projections. By Theorem 8.8.1, there exists at least one index a E A such that the spaces Xa and Ya are Baire isomorphic. This obviously implies that Y~ is also a Baire subset of a Polish space. Then Y is homeomorphic to a closed subspace of the product of a c o m p a c t u m and a Baire subspace of a Polish space (here we have used properness of the limit projection q~). This is enough (by Theorem 5.8.7 of [260]) to conclude that Y is a Baire subset of its own Stone-Cech compactification. This finishes the proof of Proposition 8.8.7.

S x -- { X a , p ~ , A }

Historical and bibliographical notes 8.8. Theorem 8.8.1 and all other results of this Section were proved by the author [87], [82], [95]. Corollary 8.8.5, without assuming the Continuum Hypothesis, and also a related result (stating that there is no Baire isomorphism between D ~2 and expD W2) was independently obtained by L. B. Shapiro. Propositions 8.8.6 and 8.8.7 solve the corresponding questions from [260].

8.9. D o u b l e s p e c t r a 8.9.1. D i r e c t s p e c t r a . We consider direct spectra S x = {Xa, i~, A} all projections i~" X a ~ XZ, c~ <_ ~, of which are closed embeddings. By l i m S x and by is" Xa ~ lira S x , we denote the direct limit and c~-th limit projection of the direct spectrum S z , respectively. We say that S z - {Xa, i~, A} is T-continuous if 9 i ~ ( Z z ) = c l ( U { i a ( X a ) ' a e B}) for each chain B in A with I B I< T and supB = ~. Further we call a direct spectrum S z - {X~, i~, A} a T-spectrum if the following conditions are satisfied"

(~)_~

~(x~)

< ~ for ~ c h

~ e A.

(ii)__. The spectrum S x is T-continuous. (iii)__. The indexing set A of the spectrum S x is T-complete. By a morphism between two direct spectra 8 x = { X a , i ~ , A } and 8 y = {Yc~,j~,A} we understand a collection $" = {fa" ~ e A} of m a p s / a " Z,~ ---. Ya, a E A, such t h a t f~i~ = j ~ f a whenever a < ~, a , ~ c A, i.e. the diagrams of the following type commute

388

8. APPLICATIONS

~Y~

X~

X~

f~

,Y~.

By the (direct) limit map of such a morphism we understand the map lim ~'" lim S x

~ lim S y

such that lim ~ ' - i a = j a . f~ for each a E A. Next we show t h a t continuous maps between the limit spaces of (direct) rspectra can be realized as the limit maps of morphisms in the above sense. PROPOSITION 8.9.1. Let f" l i m S x

~

limSy

be a map between the limit

spaces of two direct T-spectra S x = {Za, i ~ , A } and S y = { Y a , j ~ , A } . Then the collection A' = {a E A" f i a = j a f a } is cofinal and w-closed in A. PROOF. W i t h o u t loss of generality we may identify the spaces Xa and Ya with their images ia(Xa) and Ja(Ya) in lim S x and lim S y respectively. Consequently, X a c lim S x and Ya c_ lim S y for each a E A. Consider the following relation on the setA" L=

{ ( a , ~ ) E A 2" a _ < ~

and f a ( X a ) C_Y#}.

We are going to verify the conditions of Proposition 1.1.29. Existence condition. Let a E A. Since d(Xa) < T, Xa contains a dense subset Ba = {bt" t E T} of cardinality r, i.e. IT[ = T. For each t E T, there is an index a t E A such that f(bt) E Yat. By Corollary 1.1.28, there exists an index fl E A such that ~ > at, t E T. We may assume without loss of generality that ~ >_ a. Obviously,

f ( X a ) = f ( c l l i m S x Ba) C cllimS Y ( f ( B a ) ) C_ Cllim8 Y Y~ = Y~. This shows that (a, ~) E L. The majorantness and T-closedness conditions are trivial. Therefore, by Proposition 1.1.29, the set of all L-reflexive elements is cofinal and T-closed in A. It only remains to note that the index a E A is L-reflexive (i.e. (a, a) E L) if and only if fo~(Xa) C Ya. VI

8.9. DOUBLE SPECTRA

389

8 . 9 . 2 . E - p r o d u c t s . Let P be the Hilbert space (or any other Polish A R space) and q* E P. Let T be an uncountable indexing set with ITI = T > w. For e a c h t E T, let Pt = P a n d q ~ = q*. We denote the product y I { P t . t E T} by p T . For a point x ---- {xt" t E T} E p T let us set

supp(x) = {t E T" xt ~ q*}. The subspace E(T, q*) = {x E P T. Isupp(x)l <_ w} c_ p T is called the E-product (of cardinality T) of P with the base point q*. If A C B C T, then rA" p T .__+p A and r ~ " p B __~ p A denote the natural projections onto the corresponding subproducts. These retraction have naturally defined sections iA" p A ___+p T and i~" p A ___+p B , where

iA({Xt" t E A}) = {xt" t E A} • {q~" t E T -

A}

and

i~({xt" t e A}) = {xt" t E A} • {q~" t E B - A}. For each A C B C T we use the following notation:

EA(T, q*) = E ( r , q * ) M ( H { P t " t E A} • {q;" t E T - A}). We

also define the natural retractions

rA" ~(T, q*) ~ ~A(T, q*) and

rl~" EB(T, q*) --+ EA(~', q*) as follows:

rA({Xt" t E A}, {xt" t E T - A}) = ({xt" t E A}, {q~" t E T - A}) and

rB({xt" t e A}, {xt" t E B - A}) = ({xt" t E A}, {q~" t E B - A}). If {ta" a < T} is a well-ordering of the set T, then we use slightly different notations. Namely, for each a < T we let Ea(T, q*)-- ~(T, q*)CI

z <

• {qt*~~

>

a}).

Definitions of the corresponding retractions

ra" E(T, q*) --~ Ea(T, q*) and r~" EZ(T, q*) --~ Ea(T, q*) are as follows:

ra({xt6" 6 < 7-})= ({xt~" ~ <_ a}, {qt*~" 6 > c~}) and

r~({xt," 6 <_ ~}, {qt*6" 6 > f l } ) = ({xt," ~ ~ ce}, {qt*~" 6 > ce}).

390

8. APPLICATIONS

Straightforward verification shows that the following conditions are satisfied: (a)E r.(~, q*) = u ( r ~ . ( ~ , q*). ~ < ~}. Ea(T, q*) C Et~(T, q*), whenever a _
{r~.(~, q*), ~."+~, ~ < ~}. Ea+l(T,q*) = Ea(T,q*) • Pt.+, and the map raa+l" E~+I(T,q*) Ea(~', q*) is a trivial bundle with fiber Pt.+l, a < T. IEa(T, q*) is a fibered Z-set in Ea+~(T, q*) with respect to the projection r~ +1, whenever a < T.

8.9.3. L i m i t s o f d o u b l e s p e c t r a a n d t h e i r p r o p e r t i e s . In this Subsection we consider double spectra S x - { X a , p ~ , i ~ , A } , i.e. systems that are inverse spectra and direct spectra simultaneously. This means, by definition, that a double spectrum consists of an indexing set A, spaces Xa, c~ E A, retractions p~" Xt3 ~ Xa, a < /3, and their sections i~" Xa ~ X~, c~ w. Consider the system

S~ -- ( P A , r ~ , i1~, expwT }, where the retractions r ~ and their section i1~ are defined as in Subsection 8.9.2. Obviously, S 2 is a double w-spectrum. Its inverse limit coincides with the product p T and its direct limit coincides with the E-product E(T, q*). We suggest that the reader keep this basic example in mind, and analyze the situations described in various statements below in light of this example. First of all we show that for a double spectrum S x - { X a , p ~ , i ~ , A } the direct limit lim S x can be identified with a certain canonically defined subspace --..--4

of its inverse limit lim S x . As noted above, for each a < /3 the map i~" Xa --* X;~ is a section of the retraction p~" X~ ~ Xa. In other words, p ~ . i~ - i d x . whenever c~ < Z. Let

8.9. DOUBLE SPECTRA is = li__m{i~" f~ _ a}. projection Pc" lim S x of l i m S x 9 Let i ' =

Obviously, is" X a ~

391

li+___mSx is a section of the limit

-~ X a . Thus i a ( X a ) is a closed subspace (even a retract)

U{ia(X,~)" a ~ A}.

LEMMA 8.9.3. Let S x -- {Xa, p~a, i~, A} be a double T-spectrum. Suppose that F is a subspace of X ' = U{ia(Xs)" a E A } such that each of the intersections f M i a ( X s ) , a E A, is closed in i a ( X a ) . Then the collection

AF = {a e A" p c ( F ) = p s ( F M i a ( X a ) ) } is cofinal and T-closed in A. PROOF. First let us show t h a t A F is a cofinal subset of A. Let c~ E A. We need to find f~ E AF such t h a t f~ _> a. D e n o t e c~ by c~0 and consider an open base /go of cardinality _< T of the space Xao. Let 1)0 -- {U E/go" U Mpa o (F) ~ 0}. For each U E V0, pick an index c~u >_ s0 such t h a t Pao-l(U)NFNisu ( X a u ) ~ q}. T h u s we have a collection {c~u E A" U E 1)o} of cardinality _< v of indexes in A. By Corollary 1.1.28, there is an index c~1 E: A such t h a t c~1 >_ c~u for each U E 120. If an index c~k E A has already been constructed, the next index a k + l _ c~k can be c o n s t r u c t e d as above. Namely, take an open base/gk of cardinality _ T of the space Xak and let ])k -- {U C/gk" U M Pak ( F ) ~: 0}. For each U C Vk, pick an index a u _> ak such t h a t Pak-l(U) M F M iau ( X s u ) ~ O. Finally, let ak-{-1 E A be an index in A such t h a t ak+l >_ a v for each U E ~)k. Now consider the countable chain {ak" k E w} a n d let ;3 = sup{ak" k E w}. We claim t h a t /~ E A F. Indeed, if this is not the case, t h e n there is a point x e p/3(F) - p / 3 ( f M i/3(X~)). Recall t h a t the intersection F M i/3(X/3) is closed in i~(Xf~). Therefore, there is an open n e i g h b o r h o o d G of x in Xf~ such t h a t G M p/3(F M i/3(X/3)) = q). Since $ x is T-continuous and v >_ w, the space X~ is canonically h o m e o m o r p h i c to the (inverse) limit space Y of the inverse sequence {Xsk,PakSk+~,W}. Consequently, for some ak (constructed above) and for some U E /gk we have x E ( p ~ k ) - l ( u ) C_ G. But, by the choice of t h e point x, p ~ l ( x ) M F ~ O. Therefore, -1 -1 and Psi-l(U) M F :fi 0. In other words, U E ])k. This in t u r n implies (see the construction of ak+l ) t h a t there is a point y E p-~I(u) M F M isk+~(X,~,,+~). Since ak+l < f~, we have pf~(y) E p~(F M i/~(Xf~)) " Also p~(y) - i f~ pa k (y) Sk ( i ~ ) - l ( U ) C_ G. Thus, p~(y) ~ G Mp~(F M i~(X~)). This contradicts the choice of the n e i g h b o r h o o d G. Cofinality of A F is proven. Suppose now t h a t {c~" ~/ < T} is a chain of cardinality <_ T in A consisting of elements of A F. Let (~ -- s u p { ( ~ " ~/ < T}. By T-completeness of A, c~ ~ A. Let us show t h a t a ~ A F. T h e r - c o n t i n u i t y of the inverse s p e c t r u m S x {Xa, p~, A } g u a r a n t e e s t h a t the space X a is canonically e m b e d d e d into the limit space Y of the inverse s p e c t r u m ( X a ~ , paean+l, T}. A s s u m i n g t h a t c~ ~t AF we can find a point x E p c ( F ) - pa(F M i a ( X a ) ) . Therefore, there exists an ordinal

392

8. APPLICATIONS

-y < T such t h a t

pg~(x) ~ p ~ ( p a ( F Mia(Xa)) -- pa~(F Mia(Xa)). Next observe t h a t since a~ E A F a n d since

p a ~ ( F ) - - p a ~ ( F M i a ~ ( X a ~ ) C_pa~(FMia(Xa)), we may conclude t h a t p ~ (x) ~ pa~ ( f ) . This c o n t r a d i c t s the choice of x. Therefore A F is T-closed in A. I'-! COROLLARY 8.9.4. Let ,~x -- { X a , p ~ , i ~ , A ) be a double T-spectrum. Then the direct limit lim,~x coincides with X ' - - U { i a ( X a ) " a e A). PROOF. Obviously lim S x a n d X ' coincide as sets. Therefore we have only to show t h a t the topology induced on X ' from lim S x is the weak topology with r

respect to the collection (ia(Xa)" a E A). Consider a subset F C X ' such t h a t F M ia(Xa) is closed in ia(Xa) for each a E A. We need to show t h a t F itself is closed in X ' . Consider the closure cl(F) (taken in l i m S x ) of the set F and suppose t h a t there is a point x E cI(F) M X ~. Since x E X ~, there exists an index a E A such t h a t x E ia(Xa). Since x E cl(F) we see t h a t p.y(x) e p.y(cl(F)) C clx~(p~(F)) for each -y >_ a. By L e m m a 8.9.3, there is an index f~ _ a such t h a t cIx~(p/3(F)) = clx~(p/3(f M i/3(X/3)) = p/3(F M i/3(X/3)). Therefore x E F as desired. [-1 COROLLARY 8.9.5. The direct limit l i m S x

of a double T-spectrum S x ----

PROOF. Let F1 and F2 be two disjoint closed subsets of lim S x 9 By L e m m a 8.9.3, the sets

AF: -- {a e A" p c ( F 1 ) ----pc(F1 M ia(Xa))} and

AF2 ---- {a E A" pa(F2) -- pc(F2 M i a ( X a ) ) ) are b o t h cofinal and T-closed in A. Therefore, by P r o p o s i t i o n 1.1.27, the intersection A F1 M A F2 is cofinal and r-closed in A. In particular, AFt M A F2 ~ 0. Take an index a from this intersection. T h e n , as is easy to see, the sets pc(F1) = pc(F1 M ia(Xa)) and p,(F2) -- pc(F2 M ia(Xa)) are disjoint closed subsets of the n o r m a l space X a . Let G1 and G2 be disjoint open subsets of X a such t h a t pa(Fk) C_ Gk, k -- 1,2. Obviously the sets Vk - p~l(Gk)M l i m S x , __..r

k -- 1, 2, are disjoint and open in lira s k--1,2

D

9 It only remains to note t h a t Fk C_ Vk,

8.9. DOUBLE SPECTRA COROLLARY 8.9.6. Let S x = {Xa, P~, i/3, be a closed subspace of l i m S x . Then

393

be a double T-spectrum and F

S F = {pa(F),p~a,i~,AF} is a double T-spectrum such that

li__.mS F = clli m S x (F) and li_.__mSF PROOF. Apply Lemma 8.9.3

= F.

[-1

We call a double T-spectrum S x = { X a , p ~ , i~, A} factorizing if the corresponding inverse spectrum S x = { X a , p ~ , A} is factorizing in the sense of subsection 1.3.1. By a Polish double Spectrum we understand a double w-spectrum S x - {Xa, p~, i~, A} consisting of Polish spaces X a , c~ E A, and open retractions p~" Xf~ ---+ Xa, a __ f~. LEMMA 8.9.7. Let S x = { X a , p ~ , i ~ , A } be a Polish double spectrum. lim S x is the Hewitt realcompactification of lim,5'x 9

Then

PROOF. Inverse limits of w-spectra consisting of Polish spaces and open (surjective) bonding maps are perfectly ~;-normal. This guarantees that every dense subspace of l i m S x is z-embedded (Proposition 1.1.21). w-completeness of the ,i---indexing

set A, coupled with the fact that all limit projections p,~" lim S x

~ Xa

are surjective, shows that lira S x is G6-dense in lim S x 9 Consequently, every continuous real- valued function ~o" lira S x ~" l i m S x t-----

~R.

~

R has a continuous extension

V!

THEOREM 8.9.8. Let f" l i m S x

-~ l i m S y

be a map between the direct limits

of two double Polish spectra S x = {Xa,p~a, i~a, A} and Then the set

Sy

=

{Ya, q~a,J~a, A}.

A f = {a E A" there is a map fa" X a ~ Ya with faPa = qaf a n d j a f a = f i a } is cofinal and w-closed in A. PROOF. By Lemma 8.9.7, l i m S x and l i m S y are the Hewitt realcompactifications of lim S x and lim S y respectively. Therefore, the map f" lim S x

--,

------4

limSy

can be extended to the map v f " l i m S x ~

--+ l i m S y . 4-----

Now apply Propo-

sition 8.9.1 and Theorem 1.3.6 to the maps f and v f respectively. Finally, use Proposition 1.1.27. [-1

394

8. APPLICATIONS

8.9.4. Closed subspaces of E-products. a guide, we give the following definition.

Using properties ( a ) 2 -

(h)p. as

DEFINITION 8.9.9. We say that a space X is the limit space of a well-ordered E - s y s t e m 8 -- { X a , p~, r } consisting of closed subspaces X a of X and retractions p~" X I~ --+ X a , defined whenever ~ < fl < T, if the following conditions are satisfied: (a)x x = u{x~: a < r}. (b)x X a C X ~ whenever a <~ fl < r. X t 3 - c l x ( U { X a : a < fl}, whenever fl < T is a limit ordinal. X o is a Polish space. (f)x

I f fl < r is a limit ordinal, then the diagonal product / k { p ~ . a < fl} is an embedding of Xt3 into the limit space of the spectrum ,913(X) =

(g)x

Retraction r~ +1" X a + l ~ X a has a Polish kernel whenever a < T.

{ X a , p g +1, a < [J}.

Obviously the collection Sp.(r,q*) - { E a ( r , q * ) , r ~ , T } is a well-ordered Esystem whose limit space coincides with the E - p r o d u c t E(T, q*). Our next result characterizes the closed subspaces of R r ( ~ p T with IT[ = T) e m b e d d a b l e into E(T,q*) as the limit spaces of well-ordered E-systems (of length < T). We recall t h a t the c o m p a c t spaces e m b e d d a b l e into E ( r , q * ) for u n c o u n t a b l e T are known in the literature as Corson compacta. In [227, T h e o r e m 6.1] these c o m p a c t a were characterized as those which have point-countable, s e p a r a t i n g collections of open Fa-subsets. Our result covers the non-compact case and, moreover, is given in totally different terms, and this allows us to use the spectral technique. THEOREM 8.9.10. Let T > w. The following conditions are equivalent for any space X : (i) X is homeomorphic to a closed subspace of R r embeddable into the Eproduct E('r, q*). (ii) X can be represented as the limit space of a well-ordered E - s y s t e m of length T. PROOF. D u r i n g the proof we will use the following notations: E - E ( r , q*) and E a = E a ( r , q*) for each a < T. (ii) ~ (i). Suppose t h a t X is the limit space of a well-ordered E-system Sx

=

For each ordinal a < T we are going to construct a closed e m b e d d i n g f a : X a ---* E a so t h a t the following conditions are satisfied: (i) f a p ~ - r~f/3 for each a < T, whenever a < fl < r. (ii) f t ~ - l i m { f a : a < fl} whenever fl < r is a limit ordinal. (iii) f a / X a - - f / 3 / X a whenever a < fl < T.

8.9. DOUBLE SPECTRA

395

Since E0 is the Hilbert space R ~ , there exists a closed embedding f 0 : X 0 ---* Eo. Suppose now t h a t the closed embeddings f a : X a ~ Ea, satisfying the conditions (i)-(iii) for appropriate ordinals, have already been constructed for each c~ < /~, where ~ < T. We now construct a closed embedding f~: X~ --. E~. C a s e 1 (~ -- a + 1). By (g)x, there is a Polish space g such t h a t X a + l C X a x K and raa+l = ~rl/Xa, where lrl" X a x K ~ X a denotes the first projection. Since K is a Polish space, there exists a closed embedding g: K ---. Pt,~+l. Clearly the map f a x g: X a x K ~ Ea x Pt~+~ is a closed embedding. Therefore its restriction f a + l = (fa x g ) / X a + l : X a + l ~ E a + l also is a closed embedding. By the construction, r~+lfa+l = fap~ +1. Changing f a + l if necessary (and preserving the first coordinate) we may additionaly assume, by (h)r~, t h a t

f ~ + ~ / x ~ = f~. C a s e 2 (~ is a l i m i t o r d i n a l ) . In this case, simply let f - l i m ( f a : c~ < g ) . Condition (i) implies t h a t f maps the limit space of the spectrum {:Xa, p~+l, ~ ) into the limit space of the spectrum ( E a , r~ +1, ~). Since, by our constructions, all fa's, c~ < g, are closed embeddings, we conclude that f is also a closed embedding. By (g)x, f ( X z ) c_ l i m S ~ ( E ) . Recall also t h a t E~ c_ l i m S ~ ( E ) . Further, suppose t h a t x E tJ{Xa : a < ~ ) and choose an index c~ < ~ such t h a t x E Xa. By condition (iii), one can easily conclude t h a t f ( x ) E Ea. Therefore, t2(Xa: a < ~ ) C_ U{:Ea: c~ < ~). By (c)x and (c)~, we have / ( x ~ ) = f ( d x ~ ( U { X ~ : ~ < ~))) c_ d ~ , ~ ( f ( u { x ~ : ~ < ~ ) ) =

~l~,~(u{f(x~). ~ < Z}) c_ ~ , ~ ( u ( r ~ .

~ < Z } ) = r,~.

Therefore the map fF -- f/X/3 satisfies all the required conditions. This finishes the construction of the maps fa for each c~ < T. Now consider the inverse spectra $ x -- { Z a , p~, T} and $r~ = {Ea, r~, T}. By condition (i), the limit map g - l i m { f a : t~ < T} maps l i m S x into limSr.. Also note t h a t rag = faPa for each a < r (here pa : lim S x ~ X a and ra : lim St. Ea denote the a - t h limit projections of these spectra). It can easily be seen t h a t X C_limSx and E C _ l i m S r . . It also follows from the construction t h a t g(X) c_ E. It only remains to note t h a t the map f = g / X : X ~ E is the desired closed embedding. (i) ~ (ii). Suppose now t h a t X is a closed subspace R r contained in the Eproduct E. We are going to define closed subspaces X a , a < T, and retractions r~" X/3 ---* Xa satisfying conditions (a)x - ( g ) x . Let us call a subset A of T admissible if

~A ( x ) = ~A ( x n r~A).

396

8. APPLICATIONS

Applying L e m m a 8.9.3 to the double spectrum S~, and its closed subspace X, we see that the collection of countable admissible subsets of T is w-closed and cofinal in expwT. Now we proceed as in the final part of the proof of Theorem 6.3.1. Since ITI-- T, we can write T -- {ta" c~ < T}. Since the collection of countable admissible subsets of T is cofinal in expwT, each point ta E T is contained in a countable admissible subset Ba C T. Let As -- U{Bf~"/3 _< c~}. Finally, let Xa -- X n EA,~ and pg+l _. rA:+l. It follows from the properties ( a ) ~ - ( h ) ~ , and the definition of admissible subsets that the spectrum

sx = has all the required properties. This completes the proof of the Theorem.

El

In particular we have the following. PROPOSITION 8.9.11. Let T >_ w. Then the following conditions are equivalent: (i) X is a Corson compactum of weight <_ T. (iX) There is a T - s y s t e m S x = {Xa,p3a,T} consistin9 of closed subspaces X a of X , a < T, and retractions p 3 . X 3 ~ X a , defined whenever a < 13 < T, with the following properties: x = u{x... <

(b)

X a C X 3 whenever a < 3 < v. (c) X 3 - c l x ( U { X a " a
Proposition 8.9.11 has several consequences. below.

Some of them are mentioned

COROLLARY 8.9.12. Every separable subspace of a Corson compactum has a countable base.

COROLLARY 8.9.13. A subset F of a Corson compactum X is open in X if and only if its intersection F M K is closed in K f o r every metrizable compactum KCX.

COROLLARY 8.9.14. Let T > w. Then the Tychonov cube I T is not a Corson compactum.

8.10. SKELETOIDS IN TYCHONOV CUBES

397

The following statement follows from the proof of Theorem 8.9.10. COROLLARY 8.9.15. If X is the limit space of a E-system of uncountable length (in particular, if X is a Corson compactum), then d(X) = w ( X ) . PROOF. Obviously always d(X) < w(X). Assume that d(X) < w ( X ) and let Y be a dense subset (in X) of cardinality A < T -- w(X). It follows from the properties of admissible subsets t h a t Y c X a for some a < r. Therefore X - - c l Y C c l X a = Xa. But, again from the construction, it follows that w(Xa) < T. This is a contradiction. [:]

Historical and bibliographical notes 8.9. The concept of double spectra was first presented in [101]. E-products have been extensively studied from different points of view. Variations of Lemma 8.9.3 appear in [30], [160]. They arise, in particular, in functional analysis. The topological characterization of Corson compacta was obtained in [227]. Corollary 8.9.15 is well known (see, for example, [241] where many other results in this direction are also presented). It is known [9] t h a t for any Corson compactum X, the space Cp(X) of all continuous realvalued functions on X with the topology of pointwise convergence is Lindeltif. Measures on Corson compacta are investigated in [201]).

8.10. S k e l e t o i d s in T y c h o n o v c u b e s We have seen above that several i m p o r t a n t model spaces have naturally defined pseudo-boundaries (and pseudo-interiors). In most cases they have been constructed as absorbers (skeletoids). Moreover, in some cases these pseudoboundaries have even been characterized topologically. W h a t is the situation in the non metrizable case? For instance, do we have a satisfactory understanding of what exactly (if any) should be called a skeletoid in the Tychonov cube? Below we consider one of the possible approaches. As a guide we take the property (.) from Theorem 2.2.22. Namely, recall t h a t a countable collection {A~" i E N} of Z-sets in the Hilbert cube I ~ is a Z-skeleton (and the union U { A i ' i C N } is a Z-skeletoid) if for every Z-set K C I ~ , every index i C N and every e > 0, there exist an index j > i and an embedding f" K --~ I "~ such that

(.) d(f, idK) < e, f / ( K NA~) -- id and f ( K ) C Aj. Of course, here d stands for a metric on I "~ (more formally, for a metric on C (I ~ , I ~ )) and we measure closeness of functions in the compact-open topology.

398

8. APPLICATIONS

This topology works perfectly well when we deal with the Hilbert cube, but not with the Tychonov cube. In the non-metrizable case we need a finer topology on the space of functions (compare with Theorem 7.2.8). In Subsection 6.5.1 we have already introduced a topology on function spaces which depends on a cardinal T and is finer for larger cardinals. It turns out t h a t this topology works well in the present situation as well. Let us, for simplicity, consider only the Tychonov cube

of weight wi. Every metrizable c o m p a c t u m K C I ~1 is a Z~-set in I ~1 (see Definition 8.5.1 and Corollary 8.5.7) with respect to the topology C , , ( I " ~ , I ' ~ ) . Consider the E-product E associated with I "~, i.e. the collection of those points of I ~ only countably many coordinates of which differ from 0 -- ( 0 , . . . , 0 , . . . ). Obviously we can represent E as the union

where each Ea is a copy of the Hilbert cube (we keep the notation of Subsection 8.9.2). In particular, the sets Ea satisfy properties ( a ) 2 - (j)E from that Subsection. Let us show that the collection {Ea" a < wi} has a property similar, in some sense, to the property (.) stated above. Consider a metrizable c o m p a c t u m A C_ I ~1, an ordinal c~ < wi and a neighborhood U of the inclusion map i" A C I ~1 in the space C,.,(A, I "1). Translating into the spectral language we may, without loss of generality, assume that for some "), < wi we have

U = { f E C,.,(A,I"I) 9 f ~

= i~,y}

where

~.

i~

= (i ~

)~, _~ ( i ~ )~

denotes the projection onto the corresponding subproduct. We are going to show t h a t there exist an index f~ >_ c~ and an embedding f" A ~ Ef~ such that

f/(AAE,~)=id

and f E U.

Since A is a metrizable compactum, there exists an index 6 < wi such that the restriction

~5/A" A --, 7rs(A) is a homeomorphism. Let fl = max{a,-),, 6}. Let if~" (I ~ )f~ ---+ I wl denote the section of the projection nf~ determined by the point 0 - (0, . . . , 0 , . . . ) (see Subsection 8.9.2). Observe that the restriction

r ~ / A " A --~ rf~(A)

8.10. SKELETOIDS IN TYCHONOV CUBES

399

is also a homeomorphism. Finally let f-

i ~ r ~ / A " A --~ E~.

By the choice of the ordinal fl, f is an embedding. Since f~ _> a we see t h a t f / ( A M Ea) = id. Since f~ _ q, one can easily observe that f E U. Thus we have the following statement. PROPOSITION 8.10.1. The collection {Ea" a < wl} has the following property: (*)(w,w~) For each compactum A C_ I W~ of weight <_ w, f o r each ordinal a < wl and f o r each neighborhood U of the inclusion A ~ I W~ in the space C ~ ( A , I W~), there exist an ordinal ~ > a and an embedding f " A --~ I W~ such that f/(AMEa)--id

and f E U.

It is easy to see t h a t E is a countably compact space. Therefore E is not realcompact and, consequently, by Proposition 6.1.7, cannot be an A N R - s p a c e . Nevertheless the following statement shows t h a t E is an absolute extensor with respect to the class of metrizable compacta. PROPOSITION 8.10.2. Let B be a metrizable compactum and let a map f " A be defined on a closed subspace A of B .

Then f

can be extended to a map

f'B----~E.

PROOF. The image f ( A ) , being a metrizable compactum, is contained in Ea for some countable ordinal a. But Ea is the copy of the Hilbert cube I W . Therefore there exists an extension

].B--+E.c_E off.

!"7

The next statement shows t h a t E has some sort of strong universality property with respect to the class of metrizable compacta. PROPOSITION 8.10.3. Let A be a closed subspace of a metrizable c o m p a c t u m B and f : B --+ E be a map such that the restriction f / A : A ~ E is an embedding. Let U be a neighborhood of f in the space C w ( B , E). T h e n there exists an embedding g : B ---+ E such that g / A -- f / A

and g E U.

PROOF. Since E is pseudocompact and a dense subspace of the Tychonov cube I W1, it follows easily that I W~ serves as the Stone-(Tech compactification of E, i.e. I W~ - j3E. This in turn guarantees t h a t any neighborhood U of f in the space C w ( B , E) uniquely determines a neighborhood 0 of the same map f in the

400

8. APPLICATIONS

space C ~ ( B , I ~ I ) . Further, we may assume without loss of generality that the n e i g h b o r h o o d D has the form D = {~ ~ C~(B,•

9~

= f.~}.

As above we can find a countable ordinal f~ such t h a t f ( B ) max{f~, 7}. Since the projection

C Eft. Let a =

r g + l . E~+I --+ E~ is the trivial bundle with fiber I ~ (see the property (i)~ and Ea is a fibered Z-set in E a + l with respect to r aa+l (see the property (j)~) we immediately conclude t h a t there is a map g" B ~ Ea-F1 such t h a t g/A =//A

and r ~ + l g = r~+l f .

It only remains to observe t h a t g E U (which is g u a r a n t e e d by the inequality a >__~ and the construction of g). I-l It is not hard to see t h a t above s t a t e m e n t s 5 remain valid for E-products associated with the representation I ~ = (I ~ )~. Do these properties characterize E-products? At least as subspaces of the corresponding Tychonov cubes? There is a stronger version of this question" Is there a corresponding theory of skeletoids in Tychonov cubes? More precisely, we ask" PROBLEM 8.10.4. A r e any two skeletoids of I r equivalent as subspaces of I r , i.e. if A and B are skeletoids in I r , is there a h o m e o m o r p h i s m f " I r --~ I v (as close 6 to the identity as we wish) such that f ( A ) =

B ?

PROBLEM 8.10.5. Let K be a m e t r i z a b l e c o m p a c t u m in I v and U be a neighborhood of i d l r

in C r ( I ~" , I v ). Does there exist a h o m e o m o r p h i s m h" I v ---+I r

such that h is U - c l o s e

to i d I r

and h ( E U K ) - - E .

In particular, we ask the following.

PROBLEM 8.10.6. I n the n o t a t i o n of P r o b l e m 8.10.5, are E U K and F. h o m e omorphic ?

5Recall that everywhere in this Section r denotes an uncountable cardinal. 6Of course, closeness of f and id must be measured in the space C~.(I r , I ~ ).

8.10. SKELETOIDS IN TYCHONOV CUBES PROBLEM 8.10.7. L e t U be a n e i g h b o r h o o d o f i d l r Does there exist an embedding that f (I r)

f" I r

---, I r

401

i n t h e s p a c e C r ( I r , I r ).

w h i c h is U - c l o s e

to i d i r

and such

C_ I r - ~ ?

All of these questions remain open at this point. The next possible step in this direction is the investigation of various types of skeletoids of Tychonov cubes. For example, are there n-dimensional skeletoids in I r ? In particular, what about the easiest (perhaps) case n = 07 Of course, it is interesting to investigate the complements of the above skeletoids, i.e. versions of pseudo-interiors of Tychonov cubes. The situation is not trivial at all. For instance, let us ask the following question: is the complement I r - E homeomorphic to R r ? The answer is negative. Moreover, the complement I r - ~ is not even realcompact. Indeed, assuming that I r - ~ is realcompact, we conclude, by Proposition 1.1.24, that E contains a functionally closed subset of I r . Since every functionally closed subset of I r contains, in turn, a copy of the Tychonov cube I r , we see that a copy of the Tychonov cube must be contained in E. This implies that I r is a Corson compactum, which is an obvious contradiction (see Corollary 8.9.14). Thus I r - E ~ R r . This suggests the following problem. PROBLEM 8.10.8. F i n d a t o p o l o g i c a l c h a r a c t e r i z a t i o n

of I r -~.

Of course, the positive solution of Problem 8.10.4 would imply the topological homogeneity of the space I r - ~ . But at this point the following question remains unanswered. PROBLEM 8.10.9. I s t h e s p a c e I r - E t o p o l o g i c a l l y h o m o g e n e o u s ? A positive solution of Problem 8.10.7 would guarantee that the space I r - E contains a copy of the Tychonov cube I r and, consequently, that I r - E is universal with respect to the class of spaces of weight _< T. A more delicate problem is: PROBLEM 8.10.10. C h a r a c t e r i z e t h e c l o s e d s u b s p a c e s o f I r - ~ .

H i s t o r i c a l a n d b i b l i o g r a p h i c a l n o t e s 8.10. This concluding Section consists of questions suggested for the reader. Of course, there are several others closely related to those presented in the text. We hope that the interested reader will formulate (and solve!) them independently.

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Subject Index

absolute (neighborhood) extensor 38,228 in dimension n 38, 228 in integral cohomological dimension n 118 retract 42, 228 action free 377 semi-free 377

compactum cell-like 45 Corson 394 Dugundji 231 locally setwise homogeneous 177 Menger 127 of trivial shape 45 U V '~- 45 complete invariance property 335 with respect to homeomorphism 350

Baire function 383 isomorphism 383 a - t h level 383 of class (a, ~) 383 map 383 of class a 383 property 2 set 383

diagram bicommutative square 239 n-soft commutative 239 soft commutative 239 diameter of a map binary (n, k)-dimensional 100 n-dimensional 100

Borel function 383 map 383 of class a 383 set 383

Estimated Extension Property 55 exactness axiom 110 filtration 112

Bott periodicity 111 fine homotopy equivalence 43 boundary 171 functor fundamental 329 shape 329

chain 9 clean I ~ -manifold 171

H-cogroup 110 cofinal subspectrum 13 Hewitt realcompactification 8 cohomology theory generalized 110 reduced 110 unreduced 111

homotopy groups of ends epimorphism 132 monomorphism 132

419

420

SUBJECT INDEX isotopy displacement 63 reflection 62 limit inverse 12 projection 12 of morphism 16

near-homeomorphism 36 n-homotopy domination 133 near c~ 176 n-homotopy kernel 163 non degenerate value point 72

manifold strictly Y-stable 61

n-tameness at c~ 171

map approximately invertible 115 n-soft 45 polyhedrally n-soft 45 soft 43 bonding 12 cell-like 45 characteristic 159, 239 convex 41 functionally open 236 closed 236 many-valued 40 n-full 193 n-invertible 236 n-soft 46, 232 non-stretching 103 polyhedrally n-soft 140 semi-continuous lower 40 upper 40 soft 46, 232 stable 342 strongly .A~,n-universal 189 T-approximatively n-soft 305 soft 305 U V ~ - 45 U V n - 45 with metrizable kernel 257 with Polish kernel 249 (Z, n)-invertible 117 (Z, n)-soft 117

perfect collection 55

metric convex 86 invariant 86

product reduced 61 ]E- 389 smash 112 projector 325 projection 12 pseudo-boundary 198 of I ~ 56 pseudo-interior 198 of I w 56 polyhedral n-dimensional 190 universal n-dimensional 190 reduced cone 110 suspension 110 reflective isotopy property 62 reflexive element 10 regular averaging operator 234 R-weight 28 sequence Cauchy 1 inverse 12 Mayer-Vietoris exact 111 selection 40 set

morphism bicommutative 241 of inverse spectra 15 n-clean Menger manifold 171

absorbing 222 admissible 252 cofinal 9 deformable 179 directed 9

SUBJECT INDEX (F,/C)-absorbing 52 strong Z- 49 r-closed 9 r-complete 9 thin 55 Zn-47 Z- 48 Zr- 334 shape category 329 skeleton 54 space adjunction 61 almost 0-dimensional 178 classifying 111 complete 1 countable dimensional 384 strongly 182 finitely n-dominated near c~ 176 fK- 237 Frgchet 41 homogeneous with respect to pseudocharacter 289 Moore 112 perfectly x-normal 6 Polish 1 realcompact 4 spectrally complete 245 strongly 7~-universal 66 7~r,n-universal 274 universal NSbeling 189 spectrum direct T-cintinuous 387 double 390 factorizing 17 Haydon 259 homotopically stable 323 homotopically stable with respect to space 325 induced 13 inverse 12 Polish 247 T-continuous 19 T- 19 transfinite 19 stabilizer 377 stable homeomorphism 180 standard embedding 247

stearing function 64 Sullivan Conjecture 115 thread 12 r-near-homeomorphism 306 topology limitation 33 of d-uniform convergence 35 of uniform convergence 35 transfinite dimension 384 U-close maps 33 //-homotopic maps 39 U-map 33 uniformly locally connected collection 40 unitary group 111 z-embedded subspace 5

421

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