Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen
1622
Springer
Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Emmanuel Dror Farjoun
Cellular Spaces, Null Spaces and Homotopy Localization
~ Springer
Author Emmanuel Dror Farjoun Mathematics Department Hebrew University of Jerusalem Jerusalem, Israel EMail: farjoun @sunset.huji.ac.il
CataloginginPublication Data applied for Die Deutsche Bibliothek  CIP=Einheitsaufnahme Farjoun, Emmanuel Dror: C e l l u l a r s p a c e s , null s p a c e s a n d h o m o t o p y l o c a l i z a t i o n / Emmanuel Dror Farjoun.  Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; H o n g Kong ; L o n d o n ; Milan ; Paris ; T o k y o : S p r i n g e r , 1995 (Lecture notes in mathematics ; 1622) ISBN 3540606041 NE: GT
Mathematics Subject Classification (1991): 55 ISBN 3540606041 SpringerVerlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. 9 SpringerVerlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Cameraready TEX output by the author SPIN: 10479706 46/3142543210  Printed on acidfree paper
CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
1. Coaugmented homotopy idempotent localization functors Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
A. Local spaces, null spaces, localization functors, elementary facts ..... 1 B. Construction of Lf ................................................ 6 C. Universality and continuity of Lf ................................ .. 17 D. Lf and homotopy colimits, flocal equivalence ..................... 23 E. Examples: Localization according to QuillenSullivan, BousfieldKan, homological localizations, and vtperiodic localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
F. Fibrewise localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
G. Proof of elementary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
H. The fibre of the localization m a p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2. A u g m e n t e d
h o m o t o p y idempotent functors
A. Introduction, Aequivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
B. Construction of C W A , the universal Aequivalence . . . . . . . . . . . . . . .
40
C. A c o m m o n generalization of L f and CWA and model category structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
D. Closed classes and Acellular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
E. AHomotopy theory and universal properties . . . . . . . . . . . . . . . . . . . . . .
53
3. Commutation rules for ~ , L f a n d C W A ,
preservation of fibrations and cofibrations Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
A. C o m m u t a t i o n with the loop functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
B. Relations between C W A and PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
C. Examples of cellular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
D. Localization L f and cofibrations, fibrations . . . . . . . . . . . . . . . . . . . . . . . .
73
E. C W A and fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4. D o l d  T h o m
s y m m e t r i c products and other colimits
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
A. D o l d  T h o m symmetric products as homotopy colimits . . . . . . . . . . . . .
80
B. Localization and cellularization of GEMs . . . . . . . . . . . . . . . . . . . . . . . . . .
87
C. Relation between colimits and pointed homotopy colimits . . . . . . . . . .
92
D. Application: Cellular version of Bousfield's key l e m m a . . . . . . . . . . . .
95
vi
5. General theory of fibrations, GEM error terms Introduction ...........................................................
100
A. G E M a n d p o l y G E M e r r o r t e r m s   m a i n r e s u l t s . . . . . . . . . . . . . . . . . .
101
B. T h e m a i n t h e o r e m o n G E M e r r o r t e r m s . . . . . . . . . . . . . . . . . . . . . . . . . .
104
C. T h e n u l l i f i c a t i o n a p p l i e d t o f i b r a t i o n s a n d f u n c t i o n c o m p l e x e s . . . . .
112
D. L o c a l i z a t i o n w i t h r e s p e c t t o a d o u b l e s u s p e n s i o n m a p . . . . . . . . . . . .
116
E. T h e f u n c t o r C W A
120
and fibrations ................................
F. A p p l i c a t i o n s : A g e n e r a l i z e d Serre t h e o r e m , N e i s e n d o r f e r t h e o r e m
122
6. Homological localization nearly preserves fibrations A. I n t r o d u c t i o n , m a i n r e s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
B. L o c a l i z a t i o n of p o l y G E M s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
C. L o c a l i z a t i o n w i t h r e s p e c t t o M o r a v a K  t h e o r i e s . . . . . . . . . . . . . . . . . . .
132
7. C l a s s i f i c a t i o n
o f nullity and cellular types o f f i n i t e ptorsion
suspension spaces Introduction ...........................................................
135
A. S t a b l e n u l l i t y classes a n d H o p k i n s  S m i t h t y p e s . . . . . . . . . . . . . . . . . . .
135
B. U n s t a b l e n u l l i t y t y p e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
C. U n s t a b l e c e l l u l a r t y p e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
8. vlPeriodic spaces and K  t h e o r y Introduction ...........................................................
144
A. T h e v l  p e r i o d i z a t i o n of s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
B. K  i s o m o r p h i s m s , K  a c y c l i c s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
9. Cellular inequalities A. I n t r o d u c t i o n a n d m a i n r e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
B. T h e h o m o t o p y fibre as h o m o t o p y c o l i m i t . . . . . . . . . . . . . . . . . . . . . . . . .
160
C. T h e w e a k C a u c h y  S c h w a r t z i n e q u a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166
D. E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...............
167
E. A v e r a g e or w e a k c o l i m i t of a d i a g r a m . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172
F. A list of q u e s t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
Appendices HL: H o m o t o p y c o l i m i t s a n d f i b r a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
HC: P o i n t e d h o m o t o p y c o e n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
196
Introduction
In these notes we describe in some detail a certain framework for doing homotopy theory. This approach emerged in the early 1990's but has roots in earlier work of Bousfield about localization and in the big advances made by Mahowald, Ravenel, Devinatz, Hopkins and Smith towards deeper understanding of the role of periodicity in stable homotopy theory. It is natural to look for a similar unstable organization principle. This has not been found. Rather, certain tools have developed that have proved interesting. In addition, these tools are closely related to the above developments, as well as to central developments that occurred in unstable homotopy with the proof by Miller of the Sullivan conjecture and with the fruitful use of Miller's theorem by Lannes, Dwyer, Zabrodsky and many others. During these developments the study of homotopy theory through function complexes has become common and productive. Computation of important function complexes has become possible, especially with classifying spaces as domains. It turns out that it is also very productive to formulate localization theory in terms of function complexes. In particular, the notion of a Wnull space (essentially, a space X for which the pointed'function complex map. (W, X) is contractible) has become central in localization theory. Thus function complexes play a central role in these notes. In fact one can view most of the material as developing techniques that allow better understanding of function complexes not via computing their homology or homotopy groups but directly as spaces. Therefore homotopy colimits become very useful, since it is convenient to have them as domains of function complexes. A typical situation is the decomposition of classifying spaces of compact Lie groups as homotopy colimits by Jackowsky, McClure and Oliver, which allowed a much deeper understanding of hmction complexes between these objects. In this framework we give an exposition of the work of Bousfield and Thompson about unstable localization and relate it to a better understanding of homological localization. In relation to homotopy colimits a new tool that comes into play is that of cellular spaces. We show that these structures are closely related to localization, more specifically to colocalizationshomotopy fibres of the localization map. These structures are treated here as being of interest in their own right. They allow one to write, in some interesting instances, classical constructions as pointed homotopy cohmits. For example, we examine the symmetric product S P ~176in this light. This again allows one to better understand function complexes on these spaces which are decomposed as homotopy colimits.
viii
Introduction
Spaces~ f u n c t i o n c o m p l e x e s : The present notes can be read either in the category of topological spaces having the homotopy type of CWcomplexes, or in the category of simplicial sets. We refer to both as 'spaces' and both categories are denoted by $, or when we talk about pointed spaces as S.. Although it is perfectly possible to carry out almost the whole theory within the category $. of (wellpointed) spaces we do not follow this path, since it is not always the easiest one (see 1.F.7). Rather we mix the discussion of the two categories, pointed and unpointed, trying to avoid the confusion that this might create. The category of simplicial sets is denoted by S S and that of topological spaces by Top. Often we use the notions of cofibrant and fibrant spaces. In Top cofibrant means (wellpointed) CWcomplex while any space in SS is cofibrant. On the other hand, every topological space is fibrant while fibrant in S $ means a simplicial set that satisfies the Kan extension condition [Ql], [Mayl]. Whenever some construction in Top, especially those involving mapping spaces, yields a nonCW space we can and do pull them back to the class of CWspaces via the canonical CWapproximation (compare e.g. (1.B) or (1.F)). By a finite space we mean finite CWcomplex or a simplicial set with a finite number of nondegenerate simplices. Since we make extensive use of function complexes, care must be taken that simplicial sets that serve as ranges in function complexes are fibrant, satisfying the Kan extension condition [Mayl], while spaces that serve as domains are always assumed to be cofibrant. Otherwise the homotopy type of a function complex is not invariant under weak equivalence and has in general no homotopy meaning. When we write map. (X, Y) or map(X, Y) in the topological category we most often use only the underlying weak homotopy type of the space of continuous maps (pointed or unpointed), so there is no need to turn it into an internal function complex having the homotopy type of a CWcomplex. For typographical reasons the notation y X is often used to denote the function complex of maps from X to Y. We denote by a weak homotopy equivalence. Certain constructions though are easier to handle in the category of simplicial sets where map(X, Y) denotes the usual simplicial function complex [Ml]. It is often possible to carry over the necessary construction naturally into topological spaces using the pair of adjoint functors, the realization and singular functors. This is demonstrated in some detail in section 1.F. A n o t e a b o u t c h a p t e r s a n d sections: References within the nine chapters are by sections, such as (B.3.5). When referring to other results or sections outside the current chapter, the number of the chapter precedes that of the section or result, e.g. (1.F.6.1) is a result or a figure from Chapter 1, section F. S o m e d e t a i l s a b o u t t h e contents" In Chapter 1 the basic notions of f local space and flocalization with respect to an arbitrary map denoted L / , are introduced. A special case, when the map f is null homotopic, has particularly
Introduction
ix
pleasant properties and is called nullification, denoted by PA, when the map is A 9. This last functor allows one to introduce an interesting partial order on spaces that is analyzed later on: one says that X 'supports Y' or 'kills Y', denoted by: X < Y, if P x Y ~ *. This is really the same as the implication: For any space T, m a p . ( X , T ) ~ 9 =~ map.(Y,T) ~ * (note, however, the different conventionnotation followed in [B4] where the sense of < is reversed.) We give a list of elementary properties of localization that forms the beginning of a sort of localization calculus, which will allow one to control the behavior of Lf under standard homotopy operations such as suspensions, loops, and homotopy colimits. These functors are universal in two senses: they are both terminal and initial up to homotopy in certain classes of maps. Still we do not know of any inverse limit constructions that present them as initial objects analogous, say, to the BousfieldKan construction of their localizations as an inverse limit. We also begin to note some crucial properties that distinguish the nullification from the general localization. In particular the following seems to be a basic distinction: When PA is applied to the homotopy fibre of the coaugmentation map X P A X one always gets a point up to homotopy: that is, there is a universal equivalence PA(Fib(X ~ PAX)) ~ *. The analogous formula for L l is weaker. Chapter 2 can be seen as an attempt to discuss more carefully the homotopy fibre of the nullification map. We now know that this homotopy fibre when considered as a functor on the pointed category of spaces is an idempotent augmented functor denoted by PA. It is a sort of colocalization. Since PAX is really X stripped of all its 'Ainformation' the homotopy fibre PAX still contains all this information, and in fact map.(A, PAX) is equivalent to map,(A, X). But in general PAX is not the universal space with this property. There is another canonical space denoted by CWAX, which is the universal space having the same function complex from A as X. Furthermore, this space is built out of copies of A and approximates X much in the same way that a classical CWapproximation (which is 'composed of cones on spheres' and extracts the 'spherical information' from X expressed in the usual form of the homotopy groups) gives a 'spherical approximation' to X. Thus we consider here a second partial order, denoted by ~, which, as it turns out, is closely related to <: defined above: namely, X ~< Y if and only if the pointed space Y can be built from the pointed space X by repeatedly applying, say, wedges and homotopy pushouts, possibly infinitely many times. We say that Y is Xcellular in that case and we begin to consider the above cellularization functor and this partial order in this chapter. For example, one shows that a finite product of Xcellular spaces is Xcellular (2.D.16). Notice that if X << Y, and X is acyclic with respect to any homology theory, then so is Y (2.D.2.4). Also we shall see that X << Y always implies X < Y.
x
Introduction
In particular we begin to develop criteria to decide when a given space X is Acellular with respect to another space A, i.e. under what conditions the equivalence
X ~ C W A X , or equivalently A << X, holds. Such criteria are an important concern in these notes. This is handy when, for example, one wants to know under which conditions a Kacyclic space can be constructed by pushouts and telescopes from elementary Kacyclic spaces such as the cofibre of the Adams map, and is therefore the direct limit of its Kacyclic finite subspaces. We then see that we have obtained two seemingly closely related functors. One would like to show, but it is not yet known how to proceed in all cases, that these two idempotent functors, the localization Lf and cellularization CWA, are in fact two facets of a symmetric construction that factors an arbitrary map X ~ Y into a 'cofibration' followed by a 'trivial fibration': One can change the usual notions of weak equivalences in the 'standard model category' of spaces by, say, adding a single map f to the class of weak equivalences, but this will change also the notion of fibre maps along which one should be able to lift weak equivalences that are cofibrations. The localization would then be just factorization of the map X ~ * while cellularizations are factorizations of the map * ~ Y. These observations put the above functors in a reasonable theoretical light. Hirschhorn is developing these directions carefully in a general framework [HH]. We then continue to show that certain standard constructions lead to cellular relations: For example, the cellularity of the third term in a fibration sequence can be predicted when one knows the other two. In Chapter 3 we turn to deeper technical properties of these idempotent functors: the rule of commutation with the loop space functor and in general with taking the homotopy fibre of a map. It turns out that one has general formulas LfFt _ ~LEf and C W A ~ ~ ~ C W E A . These are a fundamental part of the calculus of localization and are used to show, for example, that fibrations over a Wnull base space are preserved by nullification with respect to the suspension of W, but also much more general theorems concerning preservation of fibrations. These formulas also allow us to compute directly, and formally, certain cellular relations, such as the fact that ~ E X , the James construction on X, is always Xcellular. In fact one can show that E X < Y =~ X << Y. Chapter 4 serves two purposes. First, we present a more careful analysis of pointed homotopy colimits and their relations to the usual strict colimits (direct limits) of diagrams of spaces. This allows us to show, for example, that S P ~ X , the DoldThom symmetric product on X, which initially, like the James construction , is defined as a strict colimit via a pointwise construction, can in fact be built by a pointed homotopy colimit starting from the initial space X alone. Since by the DoldThom theorem the infinite symmetric product is a GEM, i.e. a product of
Introduction
xi
EilenbergMac Lane spaces, and is, in fact, the universal GEM associated with X, its expression as a homotopy colimit allows one to understand better the operation of localization and cellularization on generalized EilenbergMac Lane spaces. This paves the way to the second purpose: a cellular version of a 'key lemma' of Bousfield. This version, following an approach taken by Dwyer [Dw2], describes the cofibre of the map from the Borel construction on X to the corresponding strict quotient space. The key lemma is related to a cellular estimate of the cofibre as being E2X cellular. This means, roughly speaking, that in order to build E S P ~ X from EX, exactly one copy of E X is needed, and then only higher suspensions of X. In Chapter 5 we show that if in a fibration sequence F , E * B, PEA kills both the base space and the total space, i.e. P E A B "~ * and P E A E ~ *, it may not kill the fibre F, but it always turns it into an 'homotopy abelian object': we show that P ~ A F is naturally an infinite loop space that is equivalent as such to a product of EilenbergMac Lane spaces with their usual abelian infinite loop space structure. This is done using the results of Chapter 4, and notably Bousfield's key lemma and the 'infinite loop space machine' of Segal, as well as substantial parts of [DFS], extending their results to cofibrations and the cellular approximation functor. This approach leads to a general theorem about the preservation of fibration by the nullifcation 'up to an abelian error term'. We then use the relation between localization with respect to a map and nullification with respect to its cofibre to deduce a general theorem about the localization of arbitrary fibration with respect to any double suspension. It is perhaps worth mentioning here that the fact that the classical Sullivan type of localization preserves fibrations over, say, a 1connected space is a special case of these preservationoffibrations theorems: from the present point of view, the Sullivan localization of a simplyconnected space with respect to a prime p is just the Anderson localization [An], which is in turn simply nullification with respect to EM2(p), the suspension of the twodimensional Moore space. Similar reasoning is then applied in examining the effect of applying C W A to a fibre sequence, with results that are weaker, but similar to the above. We then apply this theory to show two remarkable examples: first, we describe a theorem of Neisendorfer which states that any finite 2connected complex can be recovered, up to pcompletion, from its nconnected cover, for any n > 0. This is much stronger than saying that these spaces must have nontrivial homotopy in infinite dimensions: it shows that somehow this 'infinite tail' has all the information needed to reconstruct the 'lower dimensional information'. Secondly, we show that this Serretype result about homotopy groups in high dimensions generalizes to infinite spaces, too: their Ahomotopy groups must be nontrivial in infinitely many dimensions, as long as these spaces are built by a finite number of Acells from E~A, where i > 1.
xii
Introduction
In Chapter 6 we turn our attention to homological localization with respect to generalized homology theories, such as Morava Ktheory. We essentially reproduce the relevant material from [DFS], showing that short of a small error term these localizations preserve twice looped fibrations. It is reasonable to expect that this is also the case for singleloopspace fibrations. Note, however, that in order to present homological localization in the form L/, for some map ], one needs to take a 'monster map': in general, one must take the union of all E.homology isomorphisms between spaces whose cardinality is not bigger than the coefficients of E.. While theoretically this can be done, it is certainly desirable to replace this 'monster map' with a smaller object. To do that so, one considers a classification ofpossible nullification functors, under some restrictions in Chapter 7. This means roughly the classification of all possible nullification functors with respect to finite ptorsion suspension spaces. The above mentioned nullity classes of spaces define a very rough equivalence on spaces, namely, W and V are nullequivalent or of the same nullity class if, for any pointed space T, one has the double implication: map.(V, T) ~ 9 if and only if map.(W, T) ~ .. This really means that the functors P w and P v are naturally equivalent. The classification of these classes, starting with similar but much easier stable classes, is undertaken in Chapter 7. It turns out, making heavy use of Bousfield's theory of fibrations above, that the stable and unstable classifications are not very different from each other, and the main invariant needed here is the stable one, namely, the 'HopkinsSmith type' related to Morava Ktheories. This possibly is not all that surprising since, if one localizes with respect to a nilpotent selfmap E W * W, one obtains the same results as nullification with respect to W and, by [DHS], there is essentially only one selfmap on the above complexes that is not nilpotent. Using the classification of nullity classes one can also classify the closely related cellular classes of the above suspension spaces. The classification of nullity classes and possible nullification functors can be used to analyse higher periodicity. This is done in Chapter 8 with respect to vl. We follow the work of Bousfield and Thompson in regard to Ktheory localization. But we use the cellular analysis to express the nullification and Klocalization functors in more elementary terms, as a telescope of the Adams map. Thus in this case, modulo some technicalities, the above monster map used to express Klocalization can be replaced by a single map between two Moore spaces. This is a sort of ideal situation that could hold, in general, if a kind of 'unstable telescope conjecture' were true. The basic result is that one can express the function complex map.(M, P v 0 ) E ) as a mapping telescope of function complexes, which inverts vl
Introduction
xiii
in the most elementary way: namely, as
Tel(EM~EEqM+EE2qM.+...) where M denotes the appropriate Moore space. This latter telescope is of course neither idempotent nor coaugmented as a functor, so it cannot replace the localization in general, but it still captures in a direct way the vlperiodic homotopy. In particular its homotopy groups depend only on the action of the vl operator on ~r.E as a graded group. As a corollary one can explain to what extent higher loop spaces on Kacyclic spaces axe still acyclic. In addition one can show that Kacyclic, ptorsion spaces whose loop spaces are also Kacyclic can be built via cofibration sequences from the cofibre of the Adams map vl. In the final chapter, Chapter 9, we develop several tools that allow us to detect and prove interesting cellular inequalities. The basic idea here is the passage from pointed homotopy colimits over arbitrary indexing categories to unpointed homotopy colimits over categories with a contractible classifying space. Explaining ideas from [Ch2] and [DF5], the program here is to show that often the wellknown theorems on connectivity of homotopy constructions such as homotopy fibre can be strengthened to a theorem asserting the cellularity of these constructions. On occasion this gives new connectivity results too. A typical result in this direction is that the fibre of the cofibration quotient map X * X/A is always Acellular. Furthermore the homotopy fibre of a map to connected space can be built by pointed homotopy colimits from the collection of the actual inverses of points (say barycentres) in the base space. We conclude with some applications of this technique: In particular one can show that while there is no easy relationship between the homotopy limit and colimits of diagrams, some inequalities can be proven in general between these constructions. These form a sort of generalization of the elementary inequality E~X
<< ~ E X .
A c k n o w l e d g m e n t s : Several crucial ideas that I have tried to explain here were developed by A.K. Bousfield in his paper [B4] about unstable periodicity. In particular he proved there the key lemma that opens the way to many of the more interesting results about localizations. This work was initiated while I was visiting the University of Geneva and the E.T.H. research institute and I am grateful for their hospitality. A good part of the present work was done while I was visiting Purdue University. I am very grateful for their hospitality and the close cooperation in particular with Jeff Smith with whom [DFS], which is also a part of the present
xiv
Introduction
notes, was written. I am very grateful to W. Dwyer for his help in deciphering the relations of the key lemma to homotopy coends and his proof of (3.C.16). His support in the project of understanding cellular spaces was very useful. It was partly done in the framework of a research grant from the Binational Science Foundation. No less useful was the close cooperation on the initial chapters with Hirschhorn who is now developing the theoretical foundations and model theoretical development in a separate monograph. Dwyer's student W. Chacholsky has taken an interest in these problems and greatly helped in developing ideas about cellularity of homotopy fibres in Chapter 9. The referees' comments were very valuable and helpful. I am also very grateful to TEX specialist Simcha Kojman who was very patient with my changes; she and copy editor Paul Greenberg greatly facilitated the final production.
1. C O A U G M E N T E D H O M O T O P Y I D E M P O T E N T LOCALIZATION FUNCTORS
Introduction
In this chapter basic notions that will be used throughout the present notes are defined, in particular that of flocal spaces and fequivalences between spaces. A list of elementary properties of the basic notions and of localization is given in section 1.A.8 below. These properties mostly follow easily and directly from the definitions and are used in many arguments. A construction of the flocalization functor is given and its property of continuity is discussed. This is useful but not essential in constructing a fibrewise version of localization. Continuity also renders certain induced maps such as aut(X) , a u t ( L / X ) easily understandable, where aut denotes the spaces of selfhomotopy equivalences. We pay some attention to the localization of homotopy colimits. The interested reader can find a brief discussion of these colimits in Appendix HL below. The fact that localization behaves relatively well under homotopy colimitsincluding wedge sum, for exampleis very helpful later on. We then show that wellknown localization functors including e.g. the Quillen plus construction are special cases of this general homotopy localization. Then there is a discussion of fibrewise localization and several approaches are discussed. This discussion is carried out in a bit more general framework of applying homotopy functors fibrewise. We show how to do that using homotopy colimits under mild assumptions on the functor. As a first application of these fibrewise localizations one deduces two very useful properties of the homotopy fibre of the localization map: first we show that if the localization kills the fibre then it preserves the fibration. Then we show that the localization with respect to a null map always kills the homotopy fibre of the localization map. A. Local spaces, null spaces, localization f u n c t o r s , e l e m e n t a r y facts We consider here the notion of flocal space where f : A ~ B is an arbitrary cofibration map between cofibrant spaces (i.e. CWcomplexes if we work in Top). Bousfield in [B2] has already shown how to associate an flocal space L/X with any space X together with a coaugmentation map X ~ L / X . It turns out that in spite of its generality, this localization functor has many useful properties that combine to form a 'calculus of localization'. We examine in this chapter some of these most basic properties. In case the cofibration f is a null homotopic map, the functor L/ has stronger and cleaner properties and is called nullification (with respect to A v B, see below).
2
1. Coaugmented functors
In practice, all the known coaugmented homotopy functors F which are also idempotent (i.e. roughly F F is equivalent to F) have the form L / for a suitable f, so the present framework and result may well apply to any idempotent functor. Notice, however, that BousfieldKan's Roo is not in general idempotent. In this chapter we also consider somewhat more delicate properties of localization, in particular its value on homotopy colimits and on the homotopy fibre of the localization (coaugmentation) map. A.1 DEFINITION : (flocal, Wnull): We say that Y is flocal (where f is a m a p f : A ~ B between cofibrant spaces) i f Y is fibrant and the m a p f induces a weak h o m o t o p y equivalence on function complexes,
map(f, Y ) : map(B, Z) ~ map(A, Z).
In case the m a p is s i m p l y w : 9 ~ W one refers to a wlocal space Y as Wnull; this means that the natural m a p Y
~ 9 map(W, Y ) is an equivalence. Equivalently
one defines these concepts in the pointed category o f spaces (where now all spaces are assumed to be wellpointed): A fibrant space is local i f the corresponding m a p o f function complexes o f pointed maps is a weak equivalence
m a p . ( f , Y ) : map.(B, Y) ~
map.(A, Y).
The fibration map. (V, X) * map(V, X) , X for any cofibrant V over any connected and fibrant X shows that for a connected and fibrant space X the
Remark:
map induced by f, namely map(f, Y), is an equivalence iff the map m a p . ( f , Y) is an equivalence with respect to any choice of ('wellpointed') base points. A.1.1 EXAMPLES: We give examples in sections E and 2.D below. Here we note several quick illustrations: If the map f is the map of the nsphere to a point * , S n, then an flocal connected space is an S%null connected space, i.e. it is a space X whose nth loop space ~/~X is contractible. Thus such a space has no homotopy groups above dimension n  1 and is otherwise arbitrary. Thus it is just an arbitrary Postnikov (n  1)stage. Dually it is easy to see that a space X is nconnected if and only if any EilenbergMac Lane space K ( G , i) for 0 < i < n is Xnull. For a more difficult example, if the map g is the degree p map from the nsphere to itself, then a connected, pointed space X is glocal if the map on its nth loop space raising every loop to its pth power is a weak homotopy equivalence of the underlying spaces, disregarding the loop structure. For n > 1 this means that
1. Coaugmented functors
3
all the homotopy groups above dimension (n  1) are uniquely p divisible, see E.3 below. A.1.2 REMARK: One might ask why we do not define a 'homotopy flocal' space to be a space W for which the induced map on homotopy classes of maps (rather than on the full function complexes), namely the map If, W] : [B, W] [A, W] is an isomorphism of sets. This is a perfectly good definition, but it turns out that it too leads directly to the definition given above. The reason is that given a notion of 'flocal' space we are mostly interested in functors that turn an arbitrary space into an 'flocal' one. Now the following fact shows that as far as functorial constructions are concerned the definition using homotopy classes leads to one in which the full function complexes are used. A.1.3 FACT: For any continuous (or simplicial, see C.8 below), idempotent, coaugmented functor F : {Spaces} ~ {Spaces}, if, for all X, the induced map on homotopy classes If, FX] is an isomorphism of sets, then F X is automatically flocal: m a p ( f , F X ) : map(B, F X ) _7_, map(A, F X ) is an equivalence. This is Corollary (1.3) in [DF4] which was written in view of this and similar questions. Another way of viewing Fact A.1.3 is to notice that it implies that it is impossible to canonically associate a universal 'homotopy flocal' space with every space X. This is best understood by an example (due to G. Mislin): Let f : S 1 ~ *, then a 'homotopy flocal' space is just a simply connected space. We ask: Is there an initial object among all maps of a space, say of R P 2, to 1connected spaces? The answer is NO. To see why, notice that by unique factorization up to homotopy such a space U would need to have H2(U, Z) ~ Z / 2 Z since the nontrivial map R P 2 ~ C P ~ would also have two factors through U, uniquely up to homotopy. But U is 1connected and its second cohomology cannot have torsion. A.2 DEFINITION: A functor F is called coaugmented if it comes with a natural transformation Id ~ F, i.e. for each X E S a natural map j x = j : X * F X . A coaugmented functor F is said to be idempotent if both natural maps: F X ~ F F X , namely both JFX and F ( j x ) , are weak equivalences and are homotopie to each other. We say that the coaugmentation map j x is homotopy universal with respect to maps X + T into flocal spaces T if any such map factors up to homotopy
through X + F X and the factorization is unique up to homotopy.
The next few pages will present a construction of localization functor [B2] [DF2] [CPP]:
4
1. Coaugmented functors
A.3 THEOREM: For any map f : A * B in S (or S,, see remark A.7) there exists a functor L / , called the flocalization functor, which is coaugmented and homotopically idempotent. A n y two such functors are naturally weakly equivalent to each other. The map X * L f X is a homotopically universal map to flocal spaces. Moreover, L I can be chosen to be continuous or simplicial in the sense explained (1.C) below. Proo~ The construction of L / is carried out in section B below. The proofs of claims about L / are in (B.5), (C.1), (C.2), and (C.12) below.
A.4 NULLIFICATIONFUNCTORS P w , NULLITY CLASSES: A special role is played by localization with respect to maps of the form W ~ *, or * ~ W. In that case a pointed and connected space X is (W ~ .)local or, by (A.1), Wnull if and only if map.(W, X) ~ 9 or map(W, X) ~ X. The localization with respect to these null maps deserves a special name due to its much better behavior and common occurrence. One denotes the localization Lw**  L..w by P w ; we call P w the Wnullification functor. Bousfield used the term Wperiodization for P w . It plays a major role in his theory of unstable periodic homotopy, as we shall see below. This notation also emphasizes the affinity of general nullification functors to their early predecessor, the Postnikov section functor Pn that we saw above. Of course, the condition map.(W, X) "" 9 occurs often in homotopy theory especially since the proof by Miller of the Sullivan conjecture, that says in these terms that any finitedimensional space is K(~r, 1)null for any locally finite group It. The concept of trivial function complex plays a major role in the present notes and we use it right away to define a useful partial order on pointed or unpointed spaces. A.5 NULLITY CLASSES, (WEAK) PARTIAL ORDER X < Y: We say that X supports
Y or that Y is Xsupported and denote it by X < Y if any Xnull space is also Ynull. This is a transitive but not antireflexive relation. It is equivalent as we shall see to P x Y ~ * (A.8)(e.9) below. One says that X and Y have the same nullity (class) if X < Y and also Y < X. Thus S '~ < S '~+1 and X V X has the same nullity as X. Notice that [B4] uses the opposite convention in the notation of the partial order A.6 EXAMPLE: Ps~+l is the nth Postnikov section P n X which can be characterized by ~ n + l P n X ~ *. Compare A.I.1 above. An important result of Zabrodsky and Miller [M], a strong version of which is given in (2.D.13) below, says in this notation that for any topological group G one has G < BG. In fact we shall prove the sharper inequality: ~G < B G is always true. See (9.D.4) below.
1. Coaugmented functors
5
A.7 POINTED AND UNPOINTED SPACES: Notice that in Defnition A.1 above we considered the function complex map( , ) of an unpointed map between two unpointed spaces. We have noticed there that by the same token one can consider the notion of flocal pointed spaces in S. where f : A ~ B is a map in S.. As we noticed, if X is a connected pointed space, then X is ]local in S. if and only if it is ]local in S   a f t e r forgetting the base points. One advantage of the unpointed version is that it allows the direct construction of fibrewise localization (F.1). We will see that we get a continuous or simplicial functor (see section C below) and thus it can be applied to each fibre. Since there is no continuous choice of base points across the fibres, we cannot directly use the pointed functor here. If one restricts attention to connected spaces then this is just a convenience because the value of localization functors pointed or unpointed are the same for connected spaces. A.8 ELEMENTARY FACTS CONCERNING fLOCAL SPACES AND L f: The following are almost immediate consequences of the definitions, universality, idempotency and (A.3) above and are used implicitly in many arguments. We will give their proofs in section G below. As usual whenever discussing a function complex one assumes that the range is cofibrant and the target is fibrant. e.1 For any two maps ], g we have Lg ~ L f if and only if any ]local space is glocal and vice versa. e.2 If T is ]local, then for all (wellpointed, cofibrant) X both m a p ( X , T) and m a p . (X, T) are ]local for any choice of base :points. In particular, if T is ]local so is ~ n T for all n _> 0. e.3 More general than (e.2) is the observation that any homotopy limit of ]local spaces is ]local. In particular, the homotopy fibre of a map between two connected ]local spaces is ]local; the product of any family of ]local spaces is ]local. e.4 It follows easily from the exponential law and universality that the natural map L f ( X • Y) ~ L f X • L f Y has a homotopy inverse and thus is a homotopy equivalence. e.5 Let f, g be pointed maps. If L f  Lg and W E S. any space, then L f ^ w L g ^ w . This is because the assumption implies tlTLat any T E S. is ( f A W )  l o c a l iff it is (g A W)local by use of adjunction. e.6 If F ~ E ~ X is a fibration over a connected X, and both F and X are Wnull (A.4), then so is E. This is because~ in general, the induced sequence on function complexes F W ~ E W ~ X W is al~o a fibration sequence, where F W is the fibre over the null component of X W. e.7 A connected space T E S. is local with respect to E ] : E A * E B if and only if (~tT) is ]local. Thus, if in a fibration sequence, as in (e.6), F is ]local and B is Ellocal, then E is also Ellocal. More generally, let X be local with respect
6
1. Coaugmented functors
to the nth suspension of f and let W be any n + 1 connected space. Then the pointed function complex m a p . ( W , X ) is local with respect to f itself. This follows from the adjointness of E and s m a p . ( E f , T) = m a p . ( f , gtT). e.8 If T is flocal, then it is also Ekflocal with respect to any suspension of f , i.e. for all k > 0. e.9 L f X "" * if and only if for any flocal space P one has m a p . ( X , P ) ~ *. e.10 If L f X ~ * then also L r q ( E X ) = L r 4 ( S 1 A X) ___ .. More generally if, in addition, W is any nconnected space, then LE~+~I(W A X ) ~ *. e . l l If f : A * B is a map of nconnected spaces for some n _> 0 and X is also n connected, then so is L f X .
B. Construction
o f Ly
The following construction of the homotopy localization L I can be read simplicially or topologically. In carrying through the construction in the topological category the present construction does not remain inside CWcomplexes. But one can always push it into the C W category in a functorial way as follows, using a natural C W approximation C W ( Y ) ~ Y : Consider the following natural square.
cw(x)
. cw(z
x)
l
l
X
, LfX
We can assume that the coaugmentation map is a cofibration (otherwise we can functorially turn it into one) and so is the induced map on the C W approximations. If L ~ ' X is the pushout then there is a canonical factorization X ~ L ~ X ~ L f X , in which the second map is a weak equivalence. Also, if X is a C W complex then so is the pushout. In this way we have defined a localization map canonically inside the category of C W complexes. We construct L f X for any X as the colimit of a transfinite tower of cofibrations: X = L~
~ L 1 X ~ ... L a X ,, ..
(/3 < .X)
for a certain ordinal ~ = A(A U B). We prove two crucial properties of L f X : First, the easier part if X is flocal then the map X + L f X is a (weak) homotopy: equivalence, see (B.2). Second, for all X we have L f X is flocal, see (B.5). These two will be used to show that L I satisfies the claims of (A.3). (See Section C below.)
1. Coaugmented functors
7
REMARK: The present section can be viewed as a version of section (5) of [B3] which treats homological localizations. The construction is in essence a 'continuous' adaptation of the 'small object argument' of Quillen [Ql]. B.1 SMALL OBJECT ARGUMENT, RECOGNITION OF LOCAL SPACES It might help to put the construction in perspective. Working as we do in the unpointed category one starts with an unpointed criterion for establishing that a map is a weak equivalence. Here we use the basic relative cell or simplex: A[n] A[n], namely the inclusion of the boundary of the nsimplex into the nsimplex A[n] FACT: a map X * Y is a weak equivalence fibre map if and only if for any pair of maps 8 , 5 that renders the following square diagram (strictly) commutative, i.e. with 5 o i  g o 8, there is a diagonal lifting ~ that renders the diagram commutative. /~[n]
~ , X
,1/ l A[nl
. Y
Thus, for example, for n = 0 one has z~[O] = 0, this means that the map g is surjective on the vertices or on points, this together with the condition for n = 1 implies that g induces isomorphism on path components, etc. Given a cofibration f : A ~ B we would like to turn a given space X into an flocal space I,X. Such a space L X would be fibrant and satisfy f*  m a p ( f , L X ) : ( L X ) B . (LX) A is a weak equivalence. Since we assume that f is a cofibration we get the induced map f* is a fibre map [Sp], (assuming L X is fibrant.) So we need that the induced map map(f, LX) would be a fibre map that is a weak equivalence. Thus we use the criterion above to recognize whether the map ( L X ) B , (LX) A is an equivalence. Applying the criterion above, one gets, by a simple application of exponential law of function complex, that a given space Y is flocal if for any pair of maps 81,
8
1. Coaugmented functors
61 in the following diagram:
A • bin]
.:xl
/
. B
x /kin]
B x n[n]
A x A[n]
~1
,Y
There is an extension ~i : B x A[n] ~ Y that renders the whole diagram strictly commutative. Notice that in the above diagram all arrows not involving Y are given apriori by the inclusion/~[n] r A[n] and the cofibration f. Rearranging the spaces and taking union over the upper left triangle we may rewrite the condition on Y to be: Y is flocal if for any map g in the following diagram: A•
U
AX/~[n]
B•
B x A[n]
, L1Y
there exists an extension ~ : B • A[n] ~ Y rendering the upper triangle commutative: g  ~  j. The map j is given by the coherent inclusions. Now for a general Y, ~ does not exists, therefore we change Y into L1Y by taking the (homotopy) pushout of j and g to be L1Y, which is also the homotopy pushout since j is a cofibration. Replacing g by the disjoint union over all possible maps g and then taking the pushout we get the map Y ~ L1Y which is the induction step of constructing LY, a local space associate to Y. One needs to continue this step since L1Y itself will not be local in general. But repeating the step by taking LI(L1Y), LI(LI(L~Y))) etc. transfinitely many times for a cardinal that is large than the 'size' of (A II B) • A[n] the latter become a 'small object' compare to the size of the tower (L~Y)z and the transfinite direct limit leads to an flocal L I Y space as needed. This concludes our rough sketch of the transfinite use of the small object argument. The construction below is essentially a continuous version of this basic small object argument. Let us now carry on with the actual construction of the functor LI. The advantage of the present approach is that it gives a builtin inverse on the function
1. Coaugmented functors
9
space level to the map induced by f, as well as a continuous functor. A version of this approach was first discussed by V. Halperin [HI. Given a map f : A * B we construct, by transfinite induction, for all ordinals fl _> 0, spaces L/~X together with natural transformations j that are all cofibrations (we omit the fixed function f from our notation here):
X = L~
~
... ~
LaX
~
L ~ + I x * . . . .
The functor L a X comes equipped with a natural map of function complexes: s~ : ( L ~ Z ) A * (L/~+Ix) B that will serve in the limit as a (weak) homotopy inverse to the obvious induced map of function complexes: ( L ) ' X ) B ~ (L)~X) A induced by composition from f : A * B. Here as elsewhere we denote, for brevity, by X Y the function complex map(Y, X). The construction of L ~ X is designed to render the following diagram commutative up to homotopy (see (B.4) below) where the maps s~ are defined following diagram (B.1.2) below. This will give, at the limit, the homotopy inverse to f*.
(L~X) B
jB , (L~+IX) B
d B (Lfl+2X) B ' ' "
(B.I.1) (L~X) A
jA
(LZ+IX) A ' ' "
The inductive construction of L~ We define L ~ = X and, if fl is a limit ordinal, we take L a X to be the colimits of L a X for a < ft. Assume that L ~ X is given. Define LZ+IX as L1L~X where L1X is the homotopy pushout in the following square. If one works with simplicial sets then, then, in order to get homotopy invariance, we take L1X to be the Kan complex extension Ext ~ of the following homotopy pushout diagram applied to E x t ~ X rather than to X itself:
A x X A
fllf*, B x
II B x X B AxX B
X A
(B.1.2) X
J
, L1X
10
1. Coaugmented functors
Here the union at the upper left is the obvious identification space which is both the colimit of A x X A ~ A x X B ~ B x X B and its homotopy pushout, since f : A ~~ B is a cofibration. The top map is the obvious map induced by f while ev is the evaluation. The map s~ mentioned above is induced by s t at each stage, where the map Sl : X A ~ (L1X) B in (B.I.1) for ~ = 1 is the adjoint to the map s t in the homotopic pushout square. The homotopy commutativity of diagram (B.1.1) above is immediate from the adjunction in this homotopy commutative pushout square see (B.4) below. We define L I X to be L~X = L ) X for the ordinal ,~ = A(A II B) chosen in (B.3) below. The following two basic facts are immediate consequences of the construction: L s X is a homotopy functor, and it does not change up to homotopy flocal spaces: B.2 PROPOSITION: (i) I f the m a p ~: X ~
Y is a w e a k equivalence then so is
LS(~) (ii) I f X is an f  l o c a l space, then X ~ L } X is a w e a k h o m o t o p y equivalence and thus so is X * L ~ X for all ~. H e n c e for such X
we have a w e a k equivalence X
~
LIX.
In particular
L/(*) ~ *. Proof." Each step of the construction is a homotopy functor i.e. preserves weak equivalences thus the final steep too. It is clear from elementary properties of homotopy pushout that j in (B.1.2) is a homotopy equivalence if f II f* in the pushout square is such. But since the space at the upper left corner is a homotopy pushout, we get immediately that f lI f* is in fact an equivalence if X B ~ X A is an equivalence. By direct inspection L s C _~ C for any contractible space C (but also notice that a contractible fibrant space is flocal for any f).
B.3 THE CHOICE OF )~ )~(A H B). The necessary properties of the colimit follow from the correct choice of an ordinal number ,~, a choice that depends on the cardinality of A I I B . In fact we choose an infinite ordinal ,~ = •(A II B) whose cofinality is greater than the cardinality of A I I B. In the topological case we need to factor maps from the unit interval through a direct limit tower thus we should choose to be bigger than the cardinality of I=[0,1], the unit interval. For example, we can choose A as the first infinite ordinal whose cardinality is greater than that of the product [0, 1] x ( A I I B) where A, B come from the map f. In any case our choice of A is such that guarantees that, for any tower of maps X0 * X1 ~ " ~ X~ ~ X~+I (a < •) of length ,~, there is a canonical equality of function complexes =
map(I x ( A I I B), li_mX~) = li__mmap(I x ( A I I B), X~).
1. Coaugmented functors
11
This is the only property of A that is used in proving the properties of the flocalization. B.4 HOMOTOPY COMMUTATIVITYOF DIAGRAM(B.I.1). The main technical property of (LAX) is the homotopy commutativity relation among the maps in (B.I.1) above. Since the tower L Z X is defined inductively from the functor L1X homotopy commutativity of (B.I.1) follows directly from that of the following first step: XB
/',
(L1X)B
XA
J~. (L~X)A
(B.4.1)
Now the homotopy commutativity follows directly from the homotopy commutativity of the pushout diagram that defines the functor X ~ L } X  L1X. Thus since the triangle BxX
B
f* ,. B x X
A
(B.4.2) L~X
commutes up to homotopy, it follows immediately by adjunction that X B
(B.4.3)
XA
\1
(L1x) also commutes up to homotopy, and this is precisely the upper left triangle in (B.4.1) By the same reasoning of adjunction, from the homotopy commutative triangle coming from (B.1.2): A x X A
fxid
(B.4.4) L~X
,. B x X A
12
1. Coaugmented functors
we get the homotopy commutativity of the other triangle (low right) in (B.4.1), namely the triangle: Z A
(B.4.5)
/\ (L1X)A ,
r
(L1X)B
Notice that the amalgamated sum in (B.1.2) guarantees that maps X A , (L1X) are the restriction of corresponding maps from X B. To conclude, the construction (B.1.2) guarantees the homotopy commutativity of (B.4.1) and thus of (B.I.1). We now proceed to the main theorem regarding the construction of L f X . B.5 THEOREM: The space L}X = L~X de/~ned in (B.1) above is flocal. B.6 LEMMA: The natural m a p ( L / X ) B ~ ( L / X ) A induces an isomorphism on the set of path components (Tro). Proo~ Referring to (B.3), we have guaranteed by the choice of )~ that taking the direct homotopy limit commutes with taking path components:
~r0 colim(LZX) A ~ c o l i m ~ 0 ( L f l X ) A : colim[A, L~X]. ~<~ ~<~ ~<~ But by the properties of )~ which was chosen to be big enough in comparison with A so that again taking the infinite telescope (the homotopy colimit here) again commutes with taking function complexes with A as the domain, we have also the equation: (L~X) A  (colimL~X)A = colim(LZX) A. ~<~ Z<~ Therefore to consider [A, L/X] and [B, L/X] it is enough to consider colim[A, L~X] Z<~ etc. Since the diagram (B.1.1) above commutes up to homotopy, taking ~0 everywhere renders the diagram (B.1.1) strictly commutative and the lemma follows immediately  because {~0(s~)}~ is now a map of towers of sets; (colim ~0s~) gives 
an inverse to ~0f*. This completes the proof of the lemma. Proof of B.5" By Lemma B.6 above there is an isomorphism between the sets of path components of map(A, L~X) and map(B, LAX). We must show that any
1. Coaugmented functors
13
two corresponding path components are weakly homotopy equivalent, namely the natural map /,:
a A . (L X) o I
induces an isomorphism on the homotopy groups ~r~( , .) with the proper choice of base points. Notice that the map f* is the limit of a map of towers: the towers and
These maps induced by f by composition are all denoted here by f* and they form a map of towers, since the diagram (LBX) B
+/*
1
(L +IX)B
(L~X)A
:
/Z
A
commutes strictly. The main technical difficulty of the proof is that the collection of maps in (B.1.1) s} in the other direction does not form a map of towers since the corresponding ladder, namely the square in (B. 1.1), commutes only up to homotopy. Thus colim s~ is not a well defined map. It is possible, with some effort, to modify (LPX)~ in such a way as to make the corresponding maps s~ into maps of towers that do give a map backward, (LAX) a ~ (LaX) B, which is a homotopy inverse to map(f, LaX) [H]. To get our result more quickly here we only need to show that map(f, LaX) is a weak equivalence, so we get a map backward only on the level of homotopy groups, i.e. a map of groups
(B.6.1)
a
A
a
B
where the subscript denotes pointed connected components for some map ~ : B * LAX. The main technical difficulty in obtaining this map stems from the fact that we are dealing with both nonconnected and unpointed mapping spaces: Even if we had base points in the given spaces X, A, B the two typical components of the mapping spaces we need to compare are unpointed and the diagram (B.I.1) as it stands does not commute up to base point preserving homotopies. To get around difficulties, first notice that by Lemma B.6 proven above, map(f, LAX) induces an isomorphism on the set of path components 7r0. So for each corresponding pair of components we choose base points in (L;~X) A and ( L a x ) B and we must show that, with respect to these base points, our map induces an isomorphism on the homotopy groups of these two corresponding components.
14
1. Coaugmented functors
I N T E G R A L HOMOLOGY ISOMORPHISM We notice that by the same argument as in Lemma B.6 we can conclude:
B.6.2 LEMMA: The natural map ( L f X ) B on the integral homology groups H , (  , Z).
~
( L f X ) A induces an isomorphism
Proof: Integral homology commutes with direct linear limits just as the set of path component. Similarly applying homology to homotopy commutative diagram B.1.1 renders it strictly commutative. Therefore again we colim/~H,(s~, Z) is an inverse to H.(I*, Z). To complete the proof we now notice that each path component of any mapping space map.!W, T) for W a cofibrant space is HZlocal in the sense of Bousfield [Bl], see (E.4) below, since (using elementary fact (1.A.8)(e.3)) it is a homotopy inverse limit of a tower of nilpotent spaces which are local with respect to any map that induces an isomorphism on integral homology: It is easy to see by induction the well known fact that the function complex map.(W, T) is nilpotent if W is finite dimensional. Therefore the map f* when restricted to each corresponding pair of path components is an integral homology isomorphism between two spaces that are local with respect to (maps that induce isomorphism on) integral homology thus it is a weak equivalence and so our space L~X is flocal.  Of course we have just used the theory of homological localization [Bl] together with elementary properties of HZlocal spaces to construct general homotopical localization. This is not the most elementary way to develop general localization theory! Therefore we give below alternative treatments: one of them more elementary, the other more direct. First the direct method. UNPOINTED MAPS OF SPHERES Rather then concluding the proof of (A.5) using integral homology and HZlocal spaces one can proceed using a recent theorem of Casacuberta and Rodriguez [CaR] that asserts that a map between two pointed function complexes m a p . ( B , X ) , m a p . ( A , X ) induced by f is an equivalence if this map induces an isomorphism on unpointed homotopy classes from the unpointed spheres. Since one can restrict to any component of X taking the homotopy fibres of the evaluation maps to the spaces in the tower LZX in diagram (B.I.1) one gets the same diagram but with pointed mapping spaces rather than the unpointed ones in (B.I.1). Now that diagram of pointed function complexes is also homotopy commutative. Therefore it induces in the direct limit isomorphisms on the unpointed homotopy classes of unpointed maps from any space  since we are using finite spaces to map in, we still get the analog of Lemma A.6 above for S '~, the unpointed sphere rather than S ~ Using [CaR] we deduce that our map m a p . ( f , L ~ X ) is an equivalence. Therefore the corresponding unpointed mapping spaces are also equivalent as needed. The proof of the result in [CaR] uses the fact from group
1. Coaugmented functors
15
theory that a map of nilpotent groups that induces surjection on conjugacy classes is a surjective and the fact that the above spaces of pointed maps are always homotopy inverse limits of nilpotent spaces.
ALTERNATIVE, ELEMENTARY,PROOF OF A.5 To get a selfcontained proof we make some preparations (arguing simpliciaUy to ensure that inclusions are coilbrations). We would like to get the diagram (B.I.1) to be homotopy commutative in the pointed category so that it would imply that (B.6.1) exists and is an isomorphism. Let us choose a representative map g : B ~ LXX. Without loss of generality we can assume that g is a limit of maps gz: B * L a X for f~ > 0, ifg~ does not pull back to a map to X but rather to some ga: B , L a X , then modify LZX only for ~>~ .
For each such a choice of g, gz we now modify slightly the spaces L ~ X to get LZX. These will be quotient spaces of the same homotopy type, with the property that they are all pointed; moreover they all come with pointed maps B * I~ZX, and pointed ~ : (I, Z X ) A ~ (I,~X) s. These base point will allow us to compare the homotopy groups of corresponding components. In order to render (B.I.1) homotopy commutative in the pointed category we render (B.4.1) and (B.4.4) homotopy commutative relative to our chosen copy of B and A respectively. The diagram corresponding to (B.I.1) above will now consist of base point preserving maps and will be homotopy commutative in S.. Therefore it gives rise to a corresponding strictly commutative diagram of the corresponding homotopy groups. We now define LZX for a given gX = g: B * L~X for which we have assumed without loss of generality that it pulls back to gO: B * X . We define ]~IX to be the quotient of L1X obtained as follows: In (B.1.2) we first turn, in the canonical way, the map f II f* into a cofibration c ( f I I f * ) by replacing the range B • X A by the mapping cylinder Cyl(f I I f * ) . Second and crucially, in the resulting mapping cylinder we collapse a copy of A[1] • B • gO down to B itself. Notice that this cylinder over B is embedded in Cyl(f I I f * ) since the top map in (B.1.2) is an isomorphism when restricted to B • gO. This does not change the homotopy type of the spaces involved. Finally we take ]~IX to be the strict pushout along the resulting top cofibration c(f I I f * ) , namely, further to step two, we identify points in the mapping cylinder that correspond along the evaluation map ev. Since the top map in (B.1.2), f I I f * , and ev are surjective one gets a certain identification space of:
A[1] x ( A x X A II AxX
BxXB). B
16
1. Coaugmented functors
This collapsing ensures that the two maps of B to the double cylinder ~,IX, the one coming from gO: B ~ X and the other from the point in B • X A, namely {.} • f o gO, are identical and the homotopies given by the homotopy commutative triangles (B.4.2, B.4.4) above are now pointed homotopies that preserve this map that is now taken as a base point in the mapping spaces involved. Now that we have pointed maps and homotopies we can apply homotopy groups to the diagram (B.1.1) above (after changing L to L everywhere) for the appropriate components of the mapping spaces. Since homotopy groups commute with these linear colimits, we get immediately that ~r.f* and ~r.s~ are mutual inverses. Thus f* induces a weak equivalence as needed. This completes the proof of (B.5). 
B.7 CLAIM: Let P be an flocal space. Then for each ordinal/3 >_ 0 the natural map induced by coaugmentation
map(L~X, P) * map(X, P)
is a homotopy equivalence. Proof'. Notice that the tower (L~X in (B.1) is a tower of cofibrations by our definition, so its homotopy colimit is equivalent to its direct limit. We use the basic fact (Appendix HL) that the mapping space functor: m a p (  , P) turns a homotopy direct limit into a homotopy inverse limit. Therefore upon applying this functor to the tower of spaces L a X we see that for any limit ordinal/3 the map in claim B.7 is a map between two homotopy (inverse) limits
holim~map(L ~X, P) * holim~map( X, P) = map(X, P). Again it follows from the basic properties of homotopy limits that if for each a the above map is an equivalence, then so is the induced map on the homotopy limits. It follows that it is sufficient to prove the claim for nonlimit ordinals and by the inductive definition therefore it is sufficient to consider the case LIlx for an arbitrary space X. Referring to the definition of L } X in (B.1.2) above as a homotopy pushout, one again uses the observation that taking function complexes with P as a range turns any homotopy pushout square into a homotopy pullback square. Because in such squares pulling along a weak equivalence gives a weak equivalence we only need to verify that the map: map(C, P) * m a p ( B x X A, P)
1. Coaugmented functors where C = A x
X A
II
AxX B
17
B
X X B
is the space in the left hand corner of (B.1.2), is a
homotopy equivalence. Once again we use the fact that C is the homotopy pushout by definition. But it is immediate from the assumption on P that map(B, P) map(A, P) is a homotopy equivalence. Then notice that by the exponential law for function complexes, it follows from the assumption on P that map(B • X W, P) ~(A x X W, P) for any (cofibrant) space W. Now C being a homotopy pushout, we show that map(C, P ) ~ map(A x X A, P) is a homotopy equivalence, and thus so is map(B x X A, P) ,map(C, P). To accomplish this final step consider the diagram that presents C as the homotopy pushout with the well defined map to B x X A that follows from the strict commutativity of the square (appendix [HL]):
AxX B
(B.7.1)
Axf*
f•
/
/\
AxX A
B x X B
C
fxid
B x X A
The diagram is commutative, since C by definition is the (homotopy) pushout. Since both maps f x id induce homotopy equivalences: map ( f x id, P), we get by elementary pullback that map(A x f*, P) is a homotopy equivalence and thus so are the maps map(B • X A, P) , map(C, P) , map(A z X A, P). This completes the proof of (B.7).  We now address the claim that L f is idempotent. In fact we saw above (B.5) that L f X is always flocal, so we combine this with (B.7) and address idempotency and universality of Lf.
C. Universality and continuity of L I We now collect the results of Section 1.B above and conclude the proof of Theorem A.3 about the existence of the flocalization functor Lf. This leads naturally to a discussion of the universality properties of L I or, more correctly, universality properties of the coaugmentation X ~ L f X . The discussion above leads to the characterization of the functor L I in terms of three basic properties. (i) The space L I X is flocal. (ii) The coaugmentation map Y , L f Y is a weak equivalence when applied to an flocal space Y.
18
1. Coaugmented functors
(iii) For any flocal space P the map induced by the coaugmentation a, namely map(a, P), is a weak equivalence.
C.1 THEOREM: For any map X * P into an flocal space there exists a factorization X * L f X . P which is unique up to homotopy.
The above proposition says that X ~ L f X i s initial among all maps of X to flocal spaces. Below we will see that the same map is also a terminal map in a certain class of maps out of X called flocal equivalences, or Lf equivalences.
Remark:
C.2 COROLLARY: Consider the two maps u, v : L f X * L f L I X
where u : Lf(a)
and v = a L f (i.e., u is L f applied to the coaugmentation and v is the coaugmentation on L f X ) .
Then v ,,~ u and both are h o m o t o p y equivalences.
P r o o f o f C.1:
To get the faetorization we consider the square X
'~ , L f X
p
b , L/P
We know from (B.2) above that for the flocal space P the coaugmentation b is a weak homotopy equivalence. Now since P is local, we saw in (B.7) above that
map(LfX, P) * map(X, P)
is a homotopy equivalence. Looking at the induces isomorphism on the set of path components we see that there is exactly one homotopy class L I X ~ P that pulls back to the given map X ~ P.  Before deriving the corollary we formulate a general observation that will be used repeatedly: h
C.3 PROPOSITION (DIVISIBILITY):For any two maps g,h: L I X ~ P
into any f g local space P , one has h ~ g if and only i f the compositions h o a ,,~ g o a with the coaugmentation a : X ~ L f X are homotopic. Proo~ As above this follows immediately from the equivalence of function complexes (B.7) above.
1. Coaugmented functors
P r o o f o f Corollary C.2: the commutative square:
19
Both maps pull back to the same map X * L f L f X in
X
a
* LfX
1o LfX
Ls(~)
, L/LfX
Thus by divisibility we get the desired homotopy v ,,~ u. We saw above that a L i = v is a homotopy equivalence, thus so is u. Notice a further useful consequence of universality (idempotency) C.4 PROPOSITION (NO ZERO DIVISORS): L e t W be a retract o f L I X . I f the com~ W is null homotopic then W "~ *. In particular Lf(*) "~ *
position X ~ L f X
and Lf(nullmap) is a null map. Proo~ First notice that by (B.2) above Lf turns a contractible space into a contractible one. Since a null map factors through a contractible space L I turns a
nullhomotopic map into a nullhomotopic map. The first claims then follow immediately by applying Lf to the composition. Since W is a retract of a local space it is local. By idempotency (C.2) we get that L f ( X * L I X * W ) is just the retraction map L / X ~ W, so the retraction is null homotopic since the composition before applying localization  and thus also after  is null. Thus W is contractible. UNIVERSALITY CONTINUED, fLOCAL EQUIVALENCE. The discussions in (C.1) and (C.2) above prove the universal properties of L f claimed in (A.3). The proof that the above construction leads to a continuous (or simplicial) functor Lf is given in (C.11) below. However, the functor L / is, like any other coaugmented idempotent functor, universal also in another sense: The coaugmentation map is also terminal among a certain class of maps out of X. This class is interesting for its own sake, namely the class of flocal equivalences, sometimes called L/equivalence: C.5 DEFINITION AND PROPOSITION: A m a p g : X ~ Y is called an flocal equivalence or Llequivalence if it satisfies one o f the two equivalent conditions: (i) For all flocal T the induced map map(g, T ) is an equivalence. (ii) The map g induces a h o m o t o p y equivalence Lf(g) : L f X * L f Y . In this definition, i f X or Y are not cofibrant one should first replace them by weakly equivalent cofibrant spaces. Proo~
This is an easy consequence of (B.7), (C.1) and (C.2) above.
20
1. Coaugmented functors
C.5.1 EXAMPLE: Of course the principal example of an flocal equivalence is the coaugmentation map, see (B.7) above. C.6 PROPOSITION: For any map g : X + Y which is an Llequivalence there is an extension X * Y ~ L I X that is unique up to h o m o t o p y with f . g ~ a : X * L I X . Proof" We get g by using the homotopy inverse to Lf(g). Uniqueness of [g] follows from divisibility above (C.3).
C.7 PROPOSITION:
(LI is
terminal) The coaugmentation m a p X ~ L I X is term/nal, up to homotopy, among a11 flocal equivalences e : X + Y : For any such e there exists a map g : Y + Z / X , unique up to homotopy, such that g o e is homotopic to the coaugmentation a. This follows directly from (C.6) and the fact (C.5.1) that the coaugmentation is always an flocal equivalence.
Proof."
CONTINUITY OF Iaf: A convenient property of the functor L I is 'continuity'. This means naively that the setfunction given by functoriality: map(X, Y ) + m a p ( L : X , Z / Y )
is a continuous map when the usual 'compactly generated' topology is assigned to these sets of maps. Here we find it more convenient to regard our category of spaces 8 as a simplicial category with a morphism set map(X, Y) a simplicial set whose nsimplices are maps A[n] • X + Y. From this point of view a continuous functor is just a simplicial functor, namely one which preserves the simplicial structure given on hornsets: So it induces a simplicial map map(X, Y) + m a p ( F X , F Y ) , natural and respecting composition. Even the very existence of a simplicial map is not guaranteed by the functorial properties of F  see examples in (C.9.1). Given a simplex in map(X, Y), namely a map ~ : A[n] x X + Y, we want to assign to it a map A[n] x L I X ~ L I Y in m a p ( L : X , L / Y ) . One first considers L/(~) : L/(A[n] x X ) + L I Y but then one needs a map:
x[,q • D x
+
• X).
This discussion motivates our definition in the category of simplicial sets: A similar definition can obtain in the topological category with simplicial function complexes.
1. Coaugmented functors
21
C.8 DEFINITION: A coaugmented functor Id J, F : S ~ S will be called simplicial if for all X , K E S there exists a map:
(r : K x F X ~ F ( K x X)
which is natural in both K and X . and, in addition, satisfies the following: (i) If* is the onepoint space then the following composition is the identity: FX ~,*xFX ~
~,FX
(ii) It renders the following natural diagram commutative for any X , K, L: KxX
KxFX
J
, F(KxX)
, F(KxFX)
(iii) It renders the following natural diagram commutative for any X , K, L:
KxLxFX
KXO"
KxF(LxX)
F(K x L x X) C.9 PROPOSITION: Any simplicial functor F preserves the simplicial structure of S, inducing a map as above between function complexes. ~ r t h e r , this map is natural
in both variables and respects compositions. Proof'. (Compare [B3], [HH]) This is a straightforward consequence of the naturality of ~ in both variables and properties (i)(iii) that are designed precisely to render the necessary diagrams commutative. C.9.1 A COUNTEREXAMPLE TO CONTINUITY The coaugmented functor X +
W G X does not preserve the simplicial structure, even t h o u g h here as often it can be slightly modified to preserve that structure [DKl], [Ql]. F o r another example, let A be any space and define a functor W A = W : S, ~ S, as a large wedge;
22
1. Coaugmented functors
WAX
= WX
=
V A* X
A, namely we take one copy of A for each g : A * X. This
is certainly functorial as the copy corresponding to g (as above) maps by identity under f : X ~ Y to the copy corresponding to f o g. But W is not continuous. If we take A = S ~ and X = S ~ Y = I, the unit interval, then map(S ~, I ) = I and W S ~ = S ~ while W I = I * ( I * is I with the discrete topology). The function = map.(S~ ~) in not continuous. Of course the map.(S ~ I) ~ map(WS~ above counterexamples are functors that do not preserve weak equivalences. A functor that preserves weak equivalences has been shown to be 'essentially' continuous by the work of Dwyer and Kan about computing function complexes using only the model category structure via simplicial localizations: [DKl]. C.9.2 EXAMPLE: Given any space T E 8 the functor m a p ( T ,  ) is simplicial: (r(k, f ) = k x f ~here k : T ~ K is the constant function (A[n] x TP~ A[n] ~ K). C.10 PROPOSITION: L e t F be a n y /  d i a g r a m
o f f u n c t o r s and natural trans[orma
tions b e t w e e n t h e m . I f for~each a 6 I the f u n c t o r F~ : S * S is simplicial, then so is hocolim F. I
This follows from naturality of the defining map a (C.8). The map a is assembled from the corresponding maps on each space in the diagram using the naturality. Proof
C. 11 CONTINUITY OF L]. We now address the question of continuity of the functors L f for f : A * B. Since L f is the (homotopy) colimit of (L~)z<~ and L~ +1 = L f ( L f ) , it is sufficient to show that L} is a continuous functor. But L~ itself is a homotopy colimit of three continuous functors: the identity, A • X A II B • X B AxX B
and B • x A ; it follows that X * L~X is continuous: The map a is assembled from the corresponding maps from the above three functors via the hocolim functor. In particular, it can be seen by direct inspection that the diagram: KxX
, KxL~X
(ca2) L ~ ( K x X)
. L ~ ( K x L~X)
is commutative for/3  1, it follows that it is so for all/3 < ~. The same argument works for the other conditions of (C.8). Thus we can conclude:
1. Coaugmented functors
23
C.13 THEOREM: The functor L : is simplicial (C.8).
D. Lf and homotopy colimits, flocal equivalence In this section we analyze some relations between L / and homotopy colimits (i.e. homotopy direct limits) both pointed and unpointed. We would like to find a relation between the localization of the homotopy colimit of a commutative diagram of spaces and the homotopy colimit of the localizations of individual spaces in the diagram. While in general localization does not commute with homotopy colimits, these colimits preserve flocal equivalence (D.2) and (D.3) below. In a special case nullification with respect to a finite space W does commutes with linear homotopy colimits (Telescopes), see (D.6) below.
In Appendix HL the reader can find a short account of some relevant material about homotopy (co)limits. In fact we will work here mostly in S. and S/, but everything we say holds with the obvious changes in S and S I. D.1 SIMPLICIAL AND DISCRETE DIAGRAMS In general we will work with a diagram over a small category I. But sometimes it is more natural and useful to consider a diagram over a simplicial category, i.e. a category whose morphism sets are enriched by a simplicial structure [Mac], for example, when considering a topological or simplicial group acting on a space. Here we simply notice that homotopy colimits over a (small) simplicial category ~, i.e. a simplicial functor X : G ~ S., can be expressed as a composition of homotopy colimits over setvalued categories; see for example [DK1]. One simply takes first the homotopy colimits over the small categories G~ and then puts them all together by realizing the resulting simplicial space. The main results of this section are: D.2 PROPOSITION: Let g : X * Y be a map of (pointed) Idiagrams of cofibrant spaces inS. Assume that g~ : X~ * Y~ is an L:equivalence for f : A * B. Then hocolim g : h o c o l i m X ~ hocolimY is also an L/equivalence. (The same is true I
~
I
for pointed hocolim..)
D.2.1 REMARK : One of the main examples of Proposition D.2 above is the localization map on any diagram of spaces X ~ L / X . Here we use the fact that L / is defined as a functor on the category of spaces and not just on the associated homotopy category. One gets a map of commutative diagrams such that, for each c I, the map is an flocal equivalence. Now using the equivalence of the two
24
1. Coaugmented functors
conditions that define flocal equivalence we get as a corollary the following commutation rule for localizing homotopy colimits where the commutation map claimed can be constructed using the simplicial structure on Ly: D.3 THEOREM: Given a small category I and a diagram over it X : I ~ S. of cofibrant space, the natural map obtained by applying localization to the coaugmentation: Ly(a) : L f hocolim, X * L f hocolim. L y X , is a weak homotopy equivI
~
I
alence. Moreover, there is a natural map c (comparison or commutation map) e : hocolim. L y X + L f hocolim. X such that Ly(c) is an inverse to the above L f ( a ) I
~
I
and thus it is a homotopy equivalence. (The same is true for unpointed hocolim.) I

Proof of (D.2): This follows immediately from writing the function complex out of a hocolim as a homotopy inverse limit: Let P be any flocal space; we must show by Definition (C.5) of L fequivalence above that the induced map map.(hoc~limg, P) : map.(hocolimY, P) * map,(hoc?limX, P ) i s a weak equivalence. But by th~ fundamental property of hocolim it follows that map.(hoc~lim X, P) is weakly equivalent to the homotopy inverse limit of function complexes: holim m a p . ( X , P), where the homotopy limits and colimits are taken over the categories I and I ~p see Appendix HL below. Now by assumption, for each a E I the induced map map. (gn, P) is a weak equivalence, therefore taking homotopy limits we still get a weak equivalence as needed. D.4 EXAMPLE: A basic example of L fequivalence is E.equivalence for some generalized homology theory E.. In this case, in order to express this equivalence as an L fequivalence we take f to be the (pointed)union of all pointed E.equivalences An * Bn with cardinality An H B~ limited by E.(pt): we can take the first infinite cardinality bigger than E.(pt). D.5 EXAMPLE: If g : X ~ Y is any (pointed) Lfequivalence of cofibrant spaces then it is not hard to check from the general result above about homotopy colimits that so are the following: (i) W x X : W x X   ~ W x Y . (ii) Eg : ~ X ~ ~Y. (iii W A g : W A X + W A Y (for a pointed W). (iv) For any pointed spaces X , Y one has L f ( X v Y) ~ L f ( L f X V L I Y ) . The same goes for smash products of two pointed spaces. (v) In any cofibration sequence A , X g* X / A , if L f A ~ * then Lf(g) is a weak equivalence. Under special but still useful assumptions, taking homotopy colimit does commute with taking localization.
1. Coaugmented functors
25
D.6 PROPOSITION: L e t Xo  + X l + X 2 + 9 9 9 ~ X o o b e a telescope of cofibrations with Xoo the infinite telescope or h o m o t o p y colimit o f the tower. I f W is a finite space then the natural m a p hocolimPw(Xi)
~ * P w X ~ is an equivalence.
Proo~ (Arguing simplicially) Using (D.3) it is enough to show that the left hand side is Wnull. But W being a finite space, taking function complex out of W
commutes with taking infinite telescopes: map.(W, Yoo) ~ hocolimi map.(W, Yi) for any tower (Y/). We can conclude that m a p . ( W , h o c o l i m P w ( X i ) ) is given as a homotopy colimit of contractible spaces hence it is contractible as needed. D.7 EXAMPLE As a final example of an fequivalence consider the following. If A is a space such that L I A "~ 9 for some map f then for any X the nullification , P A X is an fequivalence. This can be seen either directly from the map X construction of PA via a transfinite telescope where the assumption leads directly to the result that each step in it is an fequivalence, or using for example Theorems H.1 and H.2 below. Since the fibre F of X * P A X is killed by PA and L / kills , P A X becomes an A, Lf kills also that fibre F. therefore by (H.1) the map X equivalence upon applying L / t o it as claimed, the map D.8 EXAMPLE: fEQUIVALENCEIN FIBRATIONSWe saw above in (D.5) that sometimes fequivalence respects cofibration. Here is an example where it interacts with fibrations. This will shed light on fibrewise localization in Section F below. Claim. Given any principal fibrations ladder over base spaces B, B: G
,E
,B
G
bE
~B
(D.8.1)
If both g and h are fequivalences then so the map k. The point is that for a principal fibration one can construct the base space as the barconstruction: It is a homotopy colimit of a diagram consisting of various products of the group and the total space: B  B ( G , E )  II(G x G . . . z G x Ell , where I I  I] denotes the realization of simplicial space, an important example of homotopy colimit [S], Appendix HC. It now follows from (D.5(i)) and (D.3) above that B ( G , E ) , B ( G , E ) is an fequivalence as claimed since it is such before taking realizations. As a corollary we get another comparison: If (D.8.1) is now a ladder of general fibrations and with equal connected bases B = B, and if g is an fequivalence then so is h. This follows easily by backing up the fibrations to get principal fibrations with equal groups: f~B  ftB.
26
1. Coaugmented functors
The above claim is definitely not true for general nonprincipal fibration sequences. In particular it is possible for both g and h above to be homology isomorphisms without k being so. D.9 COUNTEREXAMPLE: On the other hand, the homotopy inverse limits of weakly L/equivalent diagrams X ~ Y most often are not L fequivalent. Consider the fact that if one applies the loop space functor to a map that is an E.equivalence for some homology theory, or even an HZequivalence, then one will not in general get an E.equivalence.
E. E x a m p l e s : L o c a l i z a t i o n a c c o r d i n g t o Q u i l l e n  S u l l i v a n , B o u s f i e l d  K a n , homological localizations, and vlperiodic localization In this section we present some of the more useful examples of localization functors. They all depend on the proper choice of a map f : A ~ B. E.1 POSTNIKOV SECTION. This is perhaps the oldest homotopically idempotent functor although the usual construction does not present it as a continuous or simplicial functor. The above construction presents a simplicial (continuous) version of it. One takes the map to be U~+l : * ~ S n+l. Then P n X = Ps~+l = L ~ + I since m a p . ( S ~ + l , X ) ~ ~2~+lx and ~ + I X ~ * is equivalent to ~r~X _~ 0 for i > n + 1. Many of the properties of the general nullification functor with respect to suspensions are generalizations of analogous properties of this case. E.2 SULLIVANQUILLENLOCALIZATIONS: This functor inverts in a natural way a set of primes P by tensoring ~r.X with Z I P 1] for all n :> 2 and X 1connected. To get this effect we can choose the map f as the wedge f : V $2 ~ pEP
V $2, of pEP
all the maps of degree p : S 2 + S 2. Notice, however, that Lp is now defined for an arbitrary space X and is equivalent to the 'fibrewise Sullivan localization' of the fibration f ( ~ X ~ K(~rlX, 1). E.3 BOUSFIELDKAN LOCALIZATIONS AND COMPLETIONS: These localizations agree with Sullivan's for 1connected spaces of finite type and subrings of the rationals. Although they are always well defined and simplicial, these functors are not localizations since , in general, the BousfieldKan functor R ~ is not idempotent. In any subcategory on which Ro~ is idempotent it agrees with L f for a well chosen f . Thus if one takes the wedge of degree p maps over a set P of primes: f : V $1 vp ~ V $1, this will render all the homotopy groups uniquely pdivisible for P
P
all p E P. It will also render them divisible by certain elements in the group ring of the fundamental group as was shown by Peshke, see [CaP]. This is closely related
1. Coaugmented functors
27
to the work of Baumslag [Baum1]. For nilpotent space X the localization L f X with respect to this map of circles is still nilpotent and unique divisibility implies Lp L flocal. So one gets a new localization when localizing nonsimply connected spaces with respect to the degree p map on the circle S 1 . One gets a space for which the degree p map on the loop space is a homotopy equivalence (disregarding the loop structure). In terms of homotopy groups this is a stronger condition than unique divisibility. It is equivalent to the condition that all elements in the group ring of the fundamental group of the form 1 + ~ + ~2 + ... + ~p act on the higher homotopy groups by automorphism. For other rings such as Z/pZ, one can use other maps (see [DF4]). The main advantage of Roo over L I is the existence of a manageable (Adamslike) spectral sequence converging to ~r~map(W, R~X). E.4 BOUSFIELD HOMOLOGICALLOCALIZATION: Let h, be any generalized homology. It follows from the discussion [Bl] [DF4] that if a space X is flocal for any map f : A1 ~ A2 that induces an h,isomorphism between two spaces of cardinality not bigger than the cardinality of h,(S~ then X is h,local i.e. it is local with respect to any map that induces an isomorphism on h,. Therefore, in order to invert all h,equivalence we choose f : A * B to be the wedge over 'all' h,isomorphisms between small enough spacestaking one copy of each homotopy type, of course. The resulting functor will be homotopy equivalent to Bousfield's Lh,. This applies in particular to integral homology. The advantage of the present construction is that L f here is simplicial (compare [B3]). Also, we will see that our ability to consider L~f, where E f : EA * EB is the usual suspension of the above 'monster' map, is very useful in proving fibre lemmas for Lb,.
E.5 QUILLEN'S PLUS CONSTRUCTION: This functor X ~ X + can be expressed as
PA, where A is a large acyclic space with respect to integral homology. Namely we take A to be the wedge of all acyclic spaces whose cardinality (i.e. of the set of cells) i s t h e same as H0(*, Z). Since any H , (  , Z)acyclic space is a direct limit of its countable acyclic subcomplexes, it is not hard to see, using the commutation rules for homotopy colimits, that P A X "~' X +. In fact, one can use this interpretation to define a version of the plus construction for any homology theory E , simply by taking XE + to be PA(E)X, where A(E) is a large enough wedge of E,acyclic spaces. Theorem 1.H.2 below says that the homotopy fibre of the map X ~ XE + is always E,acyclic. E .6 PERIODICITY FUNCTORS: Mahowald, Ravenel and others consider algebraic and geometrical vlperiodic families and spaces. Let M'~(Z/pZ) denote the Moore space with a top cell at dimension n. Homotopically we can consider the modp homotopy group [M"(Z/pZ), X] = ~,~(X, Z / p Z ) and the vlperiodicity operator
28
1. Coaugmented functors
induced by the Adam's map Vl : Mn+q(Z/pZ) ~ M'~(Z/pZ) (q = 2 p  2 for an odd prime p,[CN]). A vlperiodic space, naively speaking, is a space for which vl induces an isomorphism on Z/pZhomotopy. The functor Lvl, localization with respect to Vl, turns every space into a vlperiodic one. In Chapter 8 below we give an approach to this functor largely following Bousfield and Thompson. E.7 FURTHER EXAMPLES: Notice that if we localize with respect to any nullhomotopic map u = A ~ B, we get an (A V B)null space. If we localize with respect to a nilpotent selfmap E k w * W , we get the same result as localization with respect to the null map, thus we get the functor Pw. We will later see that properties of P w are closely related to the HopkinsSmith theory of selfmaps EkW ~ W for finite ptorsion spaces [Ho]. F. F i b r e w i s e l o c a l i z a t i o n
From a general point of view any functor F : S * $ that preserves weak homotopy equivalences can be applied fibrewise to fibre maps p : E ~ X by 'applying F to each fibre'. It is easier to construct a version fibrewise application of a functor that is natural only up to homotopy compare [B4] and sometimes that is the only way we present here (3.E.2). This means that the construction is functorial in the homotopy category. In our present applications that will suffice. Still, it is nice to have a functorial, 'rigid' fibrewise localization that will have universal properties in the category of spaces over B similar to those of L / i n the category of spaces. This can be done both for simplicial sets and in the topological category. It takes a bit of pushing and pulling to define the fibrewise application of a functor functorially in the category of simplicial sets, and this is done below. By further homotopy pushouts and pullbacks along weak equivalences using the singular and realization functors, we show below how to induce a functorial fibrewise localization from simplicial sets to topological spaces (F.6), thus constructing these localizations in the latter category. In (F.7) we give a rough outline of another direct functorial fibrewise localization in a topological setting without using (strict) continuity. Continuity 'up to homotopy' is always there for homotopy functors. An extended and careful treatment of these issues is given in [HH]. So first we formulate in a special case an existence theorem for fibrewise application, up to homotopy, of simplicial functors.
1. Coaugmented functors
29
F . 1 THEOREM (FIBREWISE APPLICATION OF COAUGMENTED HOMOTOPY FUNCTORS): Let Id X ~
E ~
. F : S ~ S be a simplicial coaugmented functor. Every fibration
B o f connected spaces can be mapped via a h o m o t o p y commutative
ladder (F.1.3) to a fibration over B o f the form: F X ~ E ~ B. I f X fequivalence then in this ladder the m a p E
* F X is an
, E is an fequivalence.
REMARK: In fact it is enough to assume t h a t our functor F is a h o m o t o p y functor, i.e. t h a t it preserves weak equivalences. In t h a t case the theory of Dwyer and K a n [DK1] shows t h a t it is essentially a simplicial functor. Notice t h a t once one constructs the ladder the last s t a t e m e n t in (F.1) follows at once from example (1.D.8). We use the simplicial structure of le: This Proo~
implies t h a t for any space there is a m a p a u t X * a u t F X from the space of h o m o t o p y selfequivalence of X to t h a t of F X . This m a p is the restriction of the corresponding m a p on the spaces of all selfmaps to the components of selfequivalences. Now for a connected X there is always a universal fibration X
, Baut'X
,
B a u t X where a u t ' X is the monoid of pointed selfequivalences of X with respect to any choice of base point * C X . T h a t fibration comes from the evaluation m a p and every fibration with fibre X is, up to fibre h o m o t o p y equivalence, a pullback from this fibration [DFZ1]. Now notice t h a t for any fibration X * E * B of connected spaces we have a c o m m u t a t i v e d i a g r a m in which the n a t u r a l m a p from E to the h o m o t o p y pullback is a h o m o t o p y equivalence.
Thus, up to homotopy, the following is a h o m o t o p y
pullback diagram: E
(F.:.:)
: B
* Baut~
1 . BautX
This presents our fibration up to h o m o t o p y as a pullback from a canonical fibration t h a t depends only on the choice of base point * E X . Notice t h a t the m a p out of B is d e f n e d by the given fibration only up to homotopy. In order to complete the construction we need also a canonical m a p B a u t ' X
. Baut'FX.
Choose a
base point 9 E F X by composition with the coaugmentation. This guarantees t h a t the m a p of spaces of selfequivalences restricts to a corresponding m a p of pointed
30
1. C o a u g m e n t e d functors
selfequivalences. Thus there is a c o m m u t a t i v e diagram: X
, FX
1 1
(F.1.2)
1 1
Baut~
, Baut~
BautX
, BautFX
Once we have the above (F.I.1) pullback presentation, we can compose the b o t t o m m a p with the m a p B a u t X * B a u t F X o b t a i n e d by t a k i n g the classifying space functor on the m a p above. We now can define E as the pullback from the universal fibration with fibre F X using (F. 1.2). This gives us a h o m o t o p y c o m m u t a t i v e d i a g r a m claimed by the theorem:
X
~ E
~ B
FX
~ E
~ B
(F.1.3)
The following proposition is now clear from the construction. F.1.4
COROLLARY: If the fibration X
B + B a u t X , then F X
~
E
~
B is classified by the map
+ E + B , where E is the fibrewise application o f F
to E over B , is classified by B * B a u t X + B a u t F X ,
where the second m a p is
induced by the simplicial structure o f the functor F.
F . 2 RIGID AND FUNCTORIAL FIBREWISE LOCALIZATION, UNIVERSAL PROPERTIES. We will now use a slightly more sophisticated technique, b u t still a straightforward one, to construct a fibrewise localization in a functorial way by literally applying our functor to the inverse image in E of each simplex in B. This technique is useful also for other purposes as will be seen in C h a p t e r 9. For simplicity, one m a y assume t h a t B is connected and t h a t is all we need below. But this is by no means a necessary restriction. In fact this can be done with every functor F t h a t preserves weak equivalences. This can also be done in the framework of 'simplicial localizations' [DK1] t h a t guarantees t h a t 'up to equivalence' there is always a m a p Horn(X, Y) *
1. Coaugmented functors
31
Hom(FX, F Y ) since one can recover the homotopy type of Horn(X, Y) from the collection of weak equivalences. In our more elementary framework of simplicial sets this can be done more concretely by considering the total space E up to homotopy equivalence as a homotopy colimit (unpointed) of a diagram of the fibres over the indexing category that can be taken as the category of simplices in B with morphisms being simplicial face and degeneracy maps. The basic idea is simple and appeared among other places in works about homotopy theory of small categories [LTW]: One first decomposes the base space B into a diagram of its simplices. This way B appears as a colimit of a diagram FB whose shape (i.e. the underlying small category) is determined by B itself and where the spaces in this diagram are the simplices A[n] for n = 0, 1, 2 , . . . which together give B as a space (simplicial set or complex). Now one can consider the inverse images of the simplices of B in E. The collection of all these inverse images forms a diagram over F B consisting of spaces that are homotopy equivalent to the fibres over the corresponding component; in this diagram all maps are weak homotopy equivalences. Now since our functor F preserves every weak equivalence, after applying the functor to the diagram of inverse images we still get a diagram of weak equivalences. Taking the homotopy colimit of this diagram we obtain the desired new total space up to homotopy equivalence. The only problem is that, as it stands, it maps naturally to the classifying space of FB, a space that is homotopy equivalent to B, and it does not map to B itself. Now we use some pushing and pulling to bring it, in a natural way, over B. We will need a few general lemmas about decomposing fibrations. This information is essentially contained in Quillen's Theorem B [Q2] or in V. Puppe's theorem [Pu] about homotopy colimits of fibrations; see Appendix HL at the end. All this is a straightforward procedure. It is essential, though, for this approach that our functor will be defined in the unpointed category of spaces.
F.3' THEOREM: Let F : S * S be any coaugmented functor that sends contractible spaces to contractible spaces and preserves weak equivalences. Let g : E ~ B be a fibre map. There is a natural c o m m u t a t i v e diagram over B: Xo
~
E
~
B
=
F Xo E B
where Xo is a h o m o t o p y fibre over any component o f B and F X o the corresponding h o m o t o p y fibre.
32
1. Coaugmented functors
Example: The above construction has been used fruitfully for functors other than localization. For example, Dwyer in [Dw1] applies such a theorem for the case where F is the function complex with the domain being any space W, F w = F X = map(W, X), that is coaugmented by the constant maps. Thus for every fibration X + E * B we get an associated fibration map(W, X) ~ E * B, and a corresponding commutative diagram. Proof." We use the discussion in Appendix HL regarding the decomposition of fibre maps into free diagrams. Given the map E * B we get a map of diagrams over the indexing diagram I  F(B), namely E * B, where F(B) is the small category of all the simplices of B with maps beingthe s~mplicial maps. We also have a natural diagram of weak equivalences, where I is a shorthand here for the diagram F(B), i.e. I = F(B): hocolimE
~
colim E = E
hocolimB
~
colimB = B
(F.3.1)
We now apply the functor F objectwise to both diagrams E and B to get the diagrams maps E * F E and B ~ FB. Taking homotopy colimits we obtain a natural commutative square (with I = F(B)): hocolimE
(F.3.2)
I
~
J.
hocolimFE I
J. ge
hocolimB I
)
~
b ~ hocolimFB I
Now since F(pt) is contractible and F respects weak equivalences, the diagram F B is a diagram of contractible spaces. In other words, the map of diagrams over F(B), namely B : B ~ FB, induces a weak equivalent on each object a E FiB ), therefore upon taking homotopy colimit we get a weak equivalence, which is the map b. Now let E1 be the homotopy pullback in (F.3.2) of hoc~)limFE along b. We have a fibre map gl : E1 * hocolimB whose homotopy fibres over each component are the same as those of gF in (F.3.2) above. We now combine the above, composing gl with the bottom map in (F.3.1) to get a diagram of spaces over B:
1. Coaugmented functors
33
E (F.3.3)
1
B
~~
hocolimE
=
B
I
1
~
E1
1
=
B
where the m a p l is o b t a i n e d by the strict c o m m u t a t i v i t y of (F.3.2) as usual, see A p p e n d i x HL. The weak equivalence on the t o p left in (F.3.3) is the same one as in (F.3.1). We now finally define E to be the h o m o t o p y pushout of the t o p arrows in (F.3.3). Since there is a pushout along a weak equivalence, the n a t u r a l m a p E1 ~ E is a weak equivalence and its h o m o t o p y fibres over B are weakly equivalent to those of E1 * B. Since these are equivalent to F ( g  l ( a ) ) for any a E B, we get the result claimed. Since all the steps were natural, the resulting m a p s E * E a n d E ~ B are functorial in g : E * B. This completes the proof. In fact the argument in the proof of (F.3) proves a bit more:
F.4 THEOREM: Let E ~ B be a fibre m a p a n d let a : E ~ E be the fibrewise
localization obtained by applying L f fibrewise. Then a is an flocal equivalence. Proof'.
A p p l y example (1.D.7). Alternatively, directly by construction, the m a p
E * E is h a m o t o p y equivalent to the m a p h o c o l i m E * h o c o l i m L / E , with I = I
F(B).
~
/
But by T h e o r e m D.2 above this is an f  l o c a l equivalence, since for each
object a E F ( B ) we get a localization m a p which is always an f  l o c a l equivalence (C.5.1) (D.2.1). F.5 UNIVERSAL
PROPERTIES: Fibrewise localization can be easily seen to have the
a p p r o p r i a t e universal properties: T h e m a p E ~ E as a m a p over the space over X is universal among all m a p s of E to spaces over X t h a t have an f  l o c a l space as a (homotopy) fibre. F.6
F I B R E W I S E LOCALIZATION IN THE T O P O L O G I C A L CATEGORY:
In order to pull
back the above construction and in fact other constructions such as the localization in the absolute case, from simplicial sets (denoted $ $ ) to topological spaces (Top) over a fixed space B, we use the pair of adjoint functors ]  [ = r e a l i z a t i o n : S S * Top and Sing, the singular functor, going in t h e other direction. There is a n a t u r a l t r a n s f o r m a t i o n which is a weak equivalence u : [Sing]  ~ Id a n d we use it to construct the localization in Top using the fibrewise localization in $ S . Let E * B be a fibre m a p in Top. T h e n Sing(E) ~ Sing(B) is a fibre m a p in 8 5 . We a p p l y to it fibrewise localization in the category of simplicial sets and get Sing(E) ~ Sing(B). One then can take the realization of the localization m a p over Sing(B) and compose it into B by the above t r a n s f o r m a t i o n to the identity u. So one obtains a pushout d i a g r a m where the t o p left arrow is a cofibre m a p and the right one is a weak equivalence:
34
1. Coaugmented functors
[Sing(E)l ,r176 [Sing(E)[ (F.6.1)
1 ]Sing(B)l , =
= , E
I ISing(B)l
I =,
B
m
Now the pushout, denoted say by E of the top two arrows in (F.6.1) (along the cofibration cof), is the desired space over B. This space is of course weakly equivalent to the realization of the fibrewise localization in the simplicial category SS. F . 7 A N O T H E R APPROACH TO FIBREWISE LOCALIZATION WITHOUT CONTINUITY
Given L f : ,.S ~ ,S associated to f: V , W, we can extend it to fibrewise localization by the following method without assuming a simplicial structure on the given functor Lf. It was shown in [HH] for pointed spaces this approach can work only for a fibration with connected fibre since it localizes all the components of the fibre. We start by choosing a base point in the base space B of the given fibration sequence F *E , B. One applies Lf only to the fibre over the base point, then one turns, naturally, the resulting map into a fibre map. Again apply L f to the fibre over the base point (.), etc., doing this A times for A = A(V II W) to give fibrewise localization. Here is an outline of an argument: Let L f be a given localization functor in the category of spaces. Given a fibre map E ~ B one first applies Lf to the inverse image of the base point : Define E ' to be the pushout along cofibration of L f F 9 F , E . Thus E ' maps naturally to B see [HL]. Now define E1 to be the functorial space that turns the map E ' ~ B into a fibre map so that E1 ~ B is a fibre map. Now the fibre of this map over the base point 9 E B is no longer flocal so we repeat the last two steps again, and by a transfinite induction we construct a long telescope. We will need to carry on until a well chosen transfinite ordinal as in (B.3) above. Now the transfinite homotopy colimit F~ is a long direct limit of flocal spaces, since every other place in the telescope, namely the spaces F~', is f local and taking fibres commute with this colimit (HL), and for this ordinal taking mapping spaces commute with direct limits too,see (B.3) above. Furthermore, all the maps in the telescope of inverse images of the base point in B are fequivalences. This follows directly from (D.3) above and the expression of the homotopy fibre as a homotopy colimit of the inverse images of simplices in the base (9.B.1 below): Every other map in this tower is localization so certainly an fequivalence and for the other family one uses Proposition F.7.1 below:
1. Coaugmented functors
35
F.7.1 PROPOSITION: Let X +Y be any m a p over a pointed space B: X
g * Y
B
=,B.
I f for any cr E B the m a p g induces an ]equivalence L l ( P  ! ( a ) ) ~ L / ( q  l ( a ) ) then g is an fequivalence, moreover the m a p g induces an fequivalence on the h o m o t o p y fibres L/Fib(p) ~ LfFib(q). Proof'. This is clear from the fact that hocolim preserves ]equivalence see Section D above and from the construction of the homotopy fibre and of the total space as an (unpointed) homotopy colimit of the diagram of the inverse images of points or simplices in the base indexed by the small category FB associated with B see (F.2) and Appendix HL. 
G. P r o o f o f e l e m e n t a r y f a c t s We give short proofs of the properties listed in section A.8 above. e.1 If any flocal space is glocal and vice versa, then L f and Lg satisfy the same universality condition. If Lf ~ LI, then an flocal space X satisfies X ~ L f X ~ L g X so it is also glocal. e.2 This can be seen by adjunction: map(A, map(X, T))  map(X, map(A, T)). Since by assumption T is flocal one can replace in the latter A by B and get the desired (unpointed) result. For the pointed case one argues similarly or use for any choice of base point in T the fibration:
map. (X, T) * map(X, T) *T.
Since we can assume without loss of generality that T is connected, the statement follows by (e.3). e.3 Let X : I * S be a diagram of ]local spaces. We must check that the map induced by f : A ~ B on holimiX namely
map(f, holimiX) : map(B, holimiX) * map(A, holimiX),
is a weak equivalence. But this map is holimimap(f, X) by the basic property of homotopy limits. Now by assumption this map o~ the diagram induces a
36
1. Coaugmented functors
weak equivalence on X(i) for each i E I. By the fundamental property of homotopy limits [BK], Appendix HL at the end, we get the desired weak equivalence. e.4 We construct maps as follows, for which, by universality (divisibility), one gets immediately that they are homotopy inverse to each other:
t : LI(X • Y) ~LfX
e.5
e.7
e.8 e.9 e.10
e.ll
• L / Y : r.
To get l we extend by universality the product of the two coaugmentations X • Y ~ L f X • L f Y , since by (e.3) the product of flocal spaces is f local. To get r we start with the coaugmentation X • Y ~ L / ( X • Y). Adjunction gives a map X ~ map(Y, L f ( X z Y)). Since the range is flocal by (e.2), this map factors through L I X . By adjunction again we get a map Y * m a p ( L f X , L / ( X • Y)), which again factors through L : Y , yielding by another adjunction the desired map. For any map f : A ~ B of pointed spaces, f A W : A/~ W ~ B A W satisfies m a p . ( f A W, T) ~ map.(f, m a p . ( W , T ) ) . Now use (e.1), (e.2) to get the conclusion. For W = S 1 this follows directly form adjointness of suspension of loop functors. For a general CW complex one argues by induction on skeleta. In order show that map.(W, X) is flocal, one uses induction on the skeleton of W. Since W is nconnected, only (n +/)dimensional spheres for i >_ 1 are used in the inductive construction S '~+~ ~ W ~ W t. Now use the usual theorem about mapping of a cofibration into a space (Appendix HL below) to get map. (W ~, X) as the homotopy fibre of maps between two flocal spaces. The infinite union is treated similarly, since the needed function complex appears as the homotopy (inverse) limit of flocal spaces (by Appendix HL), so by (e.3) it is also flocal. This follows directly from (e.2) and (e.7). This follows directly from universality. This follows from (e.7) for W = S 1, which is the first case. In general, in order to show an equation L : X "~., it is sufficient by universality to show that for any flocal space T the pointed function space map, (X, T) is contractible. Let T then be a En+lflocal space. Then m a p , ( X A W , T ) ~m a p . ( X , m a p . ( W , T ) ) "~ 9 since, by the connectivity of W, it follows that m a p . ( W , T ) is flocal (e.7). This is immediate from (e.2): Simply use T a low dimensional EilenbergMac Lane space K ( G , i ) for i < n any any G. See (A.I.1). Such a space is flocal and so for any G and above i we have * ~ map(X, K(G,i)) ~m a p ( L / X , K(G, i)). Hence the claimed connectivity.
1. Coaugmented functors
37
H. T h e f i b r e o f t h e l o c a l i z a t i o n m a p In this section we apply the notion of fibrewise localization in order to prove two central properties of flocalizations: H.1 THEOREM: I f F * E p X is a fibration and L f F ~ ,, then L / ( p ) : L I E * L f X is a h o m o t o p y equivalence.
H.2 THEOREM: Let P w = L w ~ . any space W . PwX
be the Wnullification functor with respect to
Let X be a pointed connected space. Then P w P w X
~ *, where
is the h o m o t o p y fibre o f the nullification X ~ P w X .
H.3 THEOREM: Let X be a pointed connected space. Then L f L ~ I X
~ * where
L a X denotes the h o m o t o p y fibre o f the localization m a p X * L g X .
Remark:
Notice that (H.3) gives a 'weaker statement' t h a n (H.2). B
If one fibrewise localize E over X, one gets a map E ~ X whose fibre L f F ~ * is contractible, thus L / E ~ X. But L / E ~ L / E by (F.4), so L I E ~ L I X as needed.
P r o o f of H.l:
P r o o f o f H.2:
We may assume that the map X + P w X
is a fibre map. Consider
the following fibrewise localization diagram:
F
(H.2.1)
, X
l PwF
~ ~ PwX
1/1: ~X
~ * PwX
By (A.3) above X is in fact Wnull and by universality one has a map a with ~ a o I. Since g ocr o g ~ g o ~ ,~ id o g we have by uniqueness of factorization (C.3) g o a ,~ id. Therefore a is a section of the fibre map g. This means, by the usual long exact sequence of fibration, that P w F * X induces a onetoone map on pointed homotopy classes [V, P w F ] . ~ IV, X ] . for any space V and, in particular, for V = F. Since F * X factors through the base space P w X of the fibration, it must be null homotopic and thus F ~ P w F is also null hom0topic. Idempotency of P w (C.4) now gives P w F ~ *, as claimed.
38
1. Coaugmented functors
The proof is a slight variation on that of (H.2) above: Consider the fibration sequence in which all the maps are the natural ones:
Proo[ of Theorem H.3:
F
(H.3.1)
* X
I LIF
t , L~fX
1/1 , X
~ , L~IX
By elementary fact e.7 in (A.8) above the total space X is ~flocal. By the universality property of the map X ~ L ~ f X , the map X ~ X factors through it up to homotopy. This gives a map a. But this means that the fibration X ~ L ~ f X has a crosssection. Notice that by uniqueness of factorization we shall get l o a ,~ id from l o a o l ,,, l, but a o l ,~ b and lb , l by construction (compare proof of H.2). Thus the map L f F * X induces a onetoone map on pointed homotopy class [W, L f F ] ~ [W, X] for any space W, in particular for W ~ F. Notice, however, that the map F ~ X factors through the base of the fibration L ~ I X , and therefore it is a null homotopic map. But this means that F ~ L f F is null homotopic, and therefore the idempotency of L f gives that L f F ,,~ ,, as claimed. H.4 REMARK. It is essential in (H.3) that L f F "~ 9 rather than L ~ f F "~ 9 holds. The latter is, in general, not true as can be observed by taking f to be the degree pmap between two 2spheres E f = p : S 2 ~ S 2. If X = S 2 itself IrlF will be Z[~]/Z, a ptorsion group. But it is clear from the construction of L ~ f that ~ r l L ~ f F = 7rlF ~ O. However, if we had taken f = p : S 1 ~ S 1, then with the same F we would have L f F "~ ,. H.5 REMARK. An inductive application of (H.1) gives for W = K ( Z / p Z , n) and any positive natural numbers n and k:
P w K ( Z / p k Z , n) "~ (*).
2. A U G M E N T E D
HOMOTOPY
IDEMPOTENT
FUNCTORS
A. Introduction, Aequivalence In this chapter we treat certain augmented functors F, namely functors equipped for each space X with a natural map F X , X. I n fact these turn out to be kind of colocalization functors   they are closely related to homotopy fibres of localization maps. Just as one can introduce localizations by first defining flocal spaces, and flocal equivalences, we start here with a class of maps with respect to which the new functors are universal. It is interesting to notice that assuming idempotency these augmented functors, as opposed to coaugmented ones, are nontrivial only on pointed spaces as will shortly be seen (A.3.4). So in the present chapter we work only in the category S. of pointed spaces. Typical examples of these functors are universal and nconnected covers for n > 1. The first definition attempts to capture the concept of 'the information in a space X that can be detected by maps from a fixed space A.' A.1 DEFINITION: A pointed m a p g: W ~ X of fibrant spaces is caned an Ahomotopy equivalence or simply Aequivalence, where A E S . is cofibrant, if it induces a (weak) h o m o t o p y equivalence on the pointed function complex
map,(A, g): map, (A, W) ~ map,(A, X).
A.1.1 REMARK: In case the spaces involved in g are not fibrant, we ask for a weak equivalence on the function complexes of the associated realization or associated fibrant objects. It is pointless to consider the unpointed version of an Aequivalence since it would be, in general, a weak homotopy equivalence. Any space in S is a retract of map(A, S) if A ~ 0 and a retract of an equivalence is an equivalence. A.2 DEFINITION: We say that a functor T: S . ~ 8 , is h o m o t o p y idempotent and augmented if it comes with a natural augmentation a = a x : T X ~ X and the two maps T(a), aT : T 2 X " * T X are homotopic to each other and both are h o m o t o p y equivalences. A.3 EXAMPLES. 1. A well known example of an augmented functor is the universal cover functor X ~ X. More generally, we have the nconnected functor cover X ( n ) , X for X c S.. Notice that the latter is an $,~+1 equivalence in the sense of (A.1).
40
2. Augmented functors
2. Let A E S, and let P A X + X be the fibre of the coaugmentation X from (1.H.2) above. We claim that this is a homotopy idempotent augmented f u n c t o r PA : S , ~ S , . Idempotency is an immediate consequence of (1.H.2) above, since P A P A "~ * m e a n s P A P A "~ P A X . Now it follows directly from this that the map P A X + X is universal among maps W ~ X with P A W "~ *, i.e. any such map factors up to homotopy through P A X uniquely up to homotopy. The acyclic functor i.e. the homotopy fibre of Quillen's plus construction belongs to this class of examples [DF6]. We are not aware of a similar characterization of the fibre L s X of X ~ L s X . Certainly LS is not homotopy idempotent for f: S n ~ S n (n > 1), for example (1.H.4). PAX
3. Let Top, be the category of pointed (general) topological spaces. Then one has a functor C W : Top, ~ Top, that associates to each X E Top a CWcomplex in the sense of Whitehead (section B below). This is again an augmented homotopy idempotent functor. 4. UNPOINTED AUGMENTED FUNCTORS: In fact the CWapproximation of a topological space T can be defined in the unpointed category as ISingX], i.e. the realization of the singular complex. One can show that any homotopically idempotent augmented functor F X * X that preserves weak equivalences, i.e. is equivalent to a simplicial or continuous functor in the unpointed topological category, is either the empty set F X = 0 or weakly equivalent to ]SingX[, i.e. weakly equivalent to the identity. To see why we assume that our functor F is continuous or simplicial. This is not a strong assumption since we assume that F is a homotopy functor (1.C.11). It follows directly from idempotency that F ( , ) is contractible if it is not empty. Now consider the map X x 9 ~ X as a continuous family of maps of * to X. Since the functor is unpointed we get a family of induced maps F ( , ) ~ X. This is the adjoint to the simplicial or continuous function on mapping spaces: map(,, X) ~ m a p ( F ( , ) , F X ) . Now using the coaugmentation one gets that X is a retract of F X . By idempotency one get that X is equivalent to F X as needed.
5. Universal Aequivalence: Given any space A E $, there is a universal Aequivalence C W A X ~ X where CWA: S, ~ $, is an augmented homotopy idempotent functor. It plays a major role in these notes. See (B) below. Examples (1) and (3) are special cases of (5), but example (2) is not.
B. Construction of CWA, the universal Aequivalence Given A E S, we construct a continuous (or simplicial, see (1.C.8)), homotopy idempotent augmented functor CWA: S, ~ S, such that the map C W A X ~ X
2. Augmented functors
41
is a universal Aequivalence: For any other Aequivalence T * X there is, up to homotopy, a factorization C W A X ~ T ~ X of the augmentation above which it is, moreover, unique up to homotopy in $.. It will shortly turn out that this universal Aequivalence is also the (co)universal map of 'Acellular spaces' into X, see (D.2.1) below. The construction of C W A X is given here as a colimit of a transfinite telescope C W A X = colim~<~C~A X = C ~ X for large enough A = A(A). The construction is very similar to that of L a X in (1.B.1) above, so we will omit details. Here is the inductive step: Given a map h V ~ X in S., we construct a space C V and derive maps V ~ C V c(0 X. The idea is that (CV) A is a 'better approximation' to X A than V A. First define C V as the pointed homotopy pushout in
A>~V A
~A , A ~ x A
1
1
V ~
*
CV
 , C V
c(o
X
Since the strict pushout of ev and ~A maps into X , so does the homotopy pushout in a natural way. Define C V as the factorization C V * ~ C V ~ X of the given map C V ~ X into a trivial cofibration (w.e.) followed by a fibration. Thus C V is fibrant too. (It seems as if this construction 'feeds' the desired function complex X A into the inductively existing one, V A, until eventually the latter becomes equivalent to the former.) We now consider the map * ~ X given by V = *, the base point, and define CO(.) = , and, by induction, C ~ + I V = C ( C ~ V ) , while for limit ordinal we take the (homotopy) colimit. B.1 PROPOSITION: For )~
=
A(A) the m a p C~(*) = C W A ~
X i8 a universal
Aequivalence.
First notice that by construction, if a given map V ~ X is an Aequivalence, then V ~ C V is a homotopy equivalence. The collection of maps C~A(*) * X gives a map of towers {C~A(.)} ~ {X}, with a collection of 'homotopy sections' {s} in the ladder of mapping spaces: Proob
(,)A X A
,
(c(,))A
(CU,))A
X A
X A
,
''"
X A
~ X
42
2. Augmented functors
where s is the obvious adjoint to A • X A + C ( , ) that comes with the homotopy pushout which defines C; see (D.4) below for a discussion of halfsmash product )~. By construction the triangles involving s in the ladder commute up to homotopy. We now need to prove weak equivalence of function complexes. Since we work in S,, we can consider homotopy groups and we get that the maps induced by (s~)z<~ and the fibrations l: CZ(,) ~ X induce in the limit an isomorphism on the homotopy groups of all components 1fox A "~ ~r0(C~(,)) A. Hence the result. B.2 UNIVERSALITY AND IDEMPOTENCY OF CA ~ AND C W A . Idempotency is immediate since if for all/3 > 0 W ~ W' is an Aequivalence, then so is C~A(W) *
As for universality it follows by a similar argument to (1.C) above: If T ~ X is an Aequivalence, i.e. T A ~ X A is a weak homotopy equivalence, then we have by the above C W A T ~ C W A X , and thus for any map t: T ~ X that is an Aequivalence, the augmentation C W A X * X factors through T via C W A X T ~ X. Since the map C W A X * T is equivalent to the map C W A T ~ T, it is determined uniquely (up to homotopy) by T ~ X. Thus we have shown: B.3 THEOREM: The functor CWA: S , 4..4 S , is a homotopy idempotent, augmented continuous functor, with C W A X * X universal among all Aequivalences into X . C. A c o m m o n g e n e r a l i z a t i o n o f L f a n d C W A a n d m o d e l c a t e g o r y s t r u c tures The functors X ~ L : X and C W A X ~ X are special cases of one relative functor that, for a given cofibration f : A ~* B, gives a factorization of g: X ~ Y into X ~ C L f ( g ) * Y , with C L f ( X * ,) = L / X and CL,.A(, * X) = C W A X . The functor C L : ( g ) can be characterized by the following universal property: Its value o n a map g : X ~ Y is universal up to homotopy among decomposition, X ~ W ~ Y, such that
map(B, W) ~ map(A, W) • map(A,Y) map(B, Y).
For any such factorization via a space W with this property one has a map C L / ( g ) W that commutes with the obvious maps in sight, and that map is unique up to homotopy. Similarly we can easily consider the opposite universal property of CL/(g); we will not do it.
2. Augmented functors
43
C.1 CONSTRUCTION OF CLf(g). This proceeds as in the cases of C W A X and L I X . Here the inductive step is a bit more involved: Given a map V ~ W one defines CLI(g) ~ CL}(g) as the homotopy pushout of
Bx
VB
A•
II
"..• B
V
,
eLl(g)
where ev is obtained by projection and evaluation. This homotopy pushout has a natural map to W. (Again, as in the proof of (B.1) above, we repeatedly add the desired function complex to the inductively given one until after transfinitely many times they agree.)
C.2 LOCALIZINGA MODELCATEGORYSTRUCTURE. The extended localization functor C L f can be sometimes understood in the context of localizing or colocalizing the usual model category structure on simplicial sets or topological spaces. This is proved in special cases in [N], [Sm2] and [HH] and it is not really straightforward. Here only a rough outline of what one would like to do, and has been done in special cases, is given: Given a model category AA and any map f : A * B that is a cofibration in AA, we can change the model category in two opposite directions. In one direction we add weak equivalences by enriching the class of trivial cofibration, thus reducing the class of fibration and, in particular, of fibrant objects. In the other direction we use f to enrich the class of trivial fibration, thus reducing the class of cofibration and cofibrant objects. In more detail one defines M r , M f as being equal to A4 as categories, but with a different class of weak equivalences. AAf is obtained from A4 by adding the cofibration f: A ~ B to the class of trivial cofibrations and taking the consequences in terms of fibrations. This reduces the class of fibrations to a smaller one that includes only maps E ~ X that are fibration in A4 and have the homotopy lifting property with respect to all cofibration maps in AA that are fequivalence. Namely fibrations are maps E * X that satisfy in A4 the following equation for any cofibration that is an fequivalence V , W: (This equation is called the Homotopy Right Lifting Property  HRLP.) (HRLP)
map(W, E) "~ map(< E) x map(V,x) map(W, X)
44
2. Augmented functors
In particular, fibrant objects are flocal objects. The classes of trivial fibrations and cofibration do not change. Thus ?~4f is obtained from ~4 by enriching the class of trivial cofibration, cutting down the class of fibration, leaving unchanged the class of cofibrations and trivial fibrations. NOTICE that in the above one uses a whole class of maps to define the new notion of fibration in AAf. It is far from obvious that using this definition one could actually construct a factorization of an arbitrary map as needed in a model category. This is explained in detail in [HH]. Now, the other proposed model category A4 f would, in general, be obtained dually by enriching the class of trivial fibrations to be the class of the fibrations in ~4 that have the lifting property HRLP as above with respect to the given map f. This includes all the fibration in A/t I defined above but also some more fibre maps in A~. The class of cofibration is appropriately reduced. The other two classes do not change. Taking the appropriate smaller class of cofibrations here means that only maps that are repeated extensions of the initial object by f are cofibrations: Thus the map with which we started the discussion, f: A ~* B, is the typical cofibration and coff = BUCA is the typical cofibrant object. Fibrations and trivial cofibrations remain the same as in the initial category ~4. NOTICE For a general nonnull map f the above proposal for model category on A/t f does not work as it stands [HH]. As far as we now know only 2~4f works well for a general f. It is still unknown whether there exists a modified version of the proposed model category ~4 f that does satisfy Quillen's or similar axioms in general. Nofech's thesis [No] shows that it works for the map 9 , A, see Section E below. C.3 FACTORIZATION. Now the point is that, given a map V ~ X in AA, it can be viewed either as a map in Ad I or as a map in A~ f. One would like to understand factorization of a given map in both categories. The relative localization V
CLf(V, X) * X is a factorization into trivial cofibration followed by a map with RHLP in AAf which may or may not be a fibration in A~f. If f is null map the same factorization is also one into cofibration followed by a trivial fibration. other factorizations in A4f and AJ f are the usual ones in ,~4. The abstract closed model category discussion above can be considered concretely in Top. or 8.. In Top. this leads one to consider nonCW spaces
then The very that
are still 'cellular' or cofibrant with respect to some 'locally complicated' space. We have not pursued this path. For simplicity we will restrict attention to S. and consider only the map f: 9 ~ A. We would like to develop tools that allow one to prove that a space X E 8. is cofibrant with respect to * * A or is Acellular. This will shed new light on such topics as the HiltonMilnor decomposition theorem, the James construction and the symmetric products of ThomDold. It is also crucial for understanding the deeper properties of Lf and PA. In the foregoing section we
2. Augmented functors
45
set some basic notions about those Acellular spaces. Further discussion is carried on in Chapter 9 below. D . C l o s e d c l a s s e s a n d Acellular s p a c e s
In this section we discuss certain full subcategories of S, called closed classes. The main example of such classes is the class C'(A) of all Acellular spaces with C W A X ~ X for a given pointed A. Another important example of a closed class is the class of spaces Y that 'map trivially' to all finitedimensional spaces (i.e. for which the function complex map,(Y, K)  9 for any finite complex K is trivial) or to any other class of spaces. D. 1 DEFINITION: A full subcategory of pointed spaces C" C S, is called 'dosed' if it is closed under weak equivalence and arbitrary pointed homotopy colimits: namely for any (pointed) diagram of spaces in C" (i.e. a functor X: I + C') the space
hocolim, X is also in C'. We prove several closure theorems for any closed class C'. The most important ones are: 1. C is closed under finite product. 2. If X E C" and Y is any (unpointed) space, then X >~Y = (X • Y ) / * x Y is in C'. 3. If F * E * B is a fibration sequence with B connected and F, E in C', then B is also in C'. 4. If A ~ X ~ X U C A is any cofibration sequence and A is in C', then so is the homotopy fibre of i. 5. C" is closed under retracts.
(A retract can be obtained as an infinite direct
limit of the retraction followed by the inclusion.) D.2 EXAMPLES OF CLOSED CLASSES D.2.1 THE CLASS C'(A) of Acellular spaces and the partial order X << Y on spaces.' This is the smallest closed class that contains a given pointed space A. It can be built by a process of transfinite induction by starting with the full subcategory containing the single space A and closing it repeatedly under arbitrary pointed hocolim and weak equivalences. In Section E.5 below we give a 'cellular' description of spaces in up to homotopy equivalence. We refer to members of C (A) as Acellular spaces or sometimes simply as Aspaces. Notice that by (D.8) below the smash product of A with any pointed space is in this class as well as the halfsmash with any unpointed space. W h e n Y is X cellular we sometimes denote it by X << Y. This notation emphasizes that this relation is a (weak) partial order on the class of (pointed) spaces, as can easily be seen, and that the space Y is 'bigger' than X in the sense that it is obtained by assembling several copies of X together
46
2. Augmented functors
along some hocolim scheme. For connected spaces the order does not depend on the choice of base point. Thus X << Y iff g'(Y) G g ' ( X ) . Put otherwise X << Y iff Y can be built by 'assembling together many copies of X ' in a pointed fashion. D.2.2 THE CLASS E'(A ~f B) ~ E'(f). Here we start with any map (or a class of maps) f E S, of pointed spaces and consider all spaces X such that the induced map on pointed function complexes
map, (X, A) * map, (X, B)
is a (weak)~homotopy equivalence of simplicial sets. Since map,(hocolimXa, A) "~ I
holim map,(X~, A), it is immediate that g ' ( f ) is a closed class. If the map f is a /op null map, we get the class of space with contractible function complex to A • B. Otherwise we don't know many interesting instances of this class which may often be the class of ,cellular spaces, i.e. of spaces weakly equivalent to a point. But this class for, say, B = 9 onepoint space is exactly the class of all spaces X with PAX "" * by elementary fact 1.A.8 (e.9). It turns out that this class is close but not equal to the class of Acellular spaces, see (3.B) below. It is easy to see that this class contains all the Acellular spaces using (1.D). D.2.3 CLASSICAL CWCOMPLEXES. If we consider the above concepts within the category of general topological spaces and take 'weak equivalence' to be a homotopy equivalence, then the smallest closed class that contains the zerosphere S O is the class of all spaces that are homotopy equivalent to CWcomplexes. D.2.4 MILLER SPACES. This is a specially interesting case of (D.2.2) above: the class of spaces X that map trivially to all finitedimensional spaces K, namely with m a p , ( X , K ) ~ *. This includes K(~r, 1) for a finite group ~rby Miller's theorem. If we replace finite dimensional by pcompletion of a finite nilpotent spaces then we get a different closed class that includes for example all connected infinite loop spaces that have torsion fundamental group. [McG, T h m 3]. It is also interesting to replace the finite group ~r with a compact Lie group G, see [Dw1, 1.2, 1.3].
D.2.5 E,ACYCLIC SPACES. Since pointed homotopy colimits of acyclic spaces with respect to any generalized homology theory are again acyclic, it follows that the class of E,acyclic spaces is closed. To check the statement about homotopy colimits of acyclic spaces it is enough e.g. to consider arbitrary wedges and any homotopy coequalizer of two maps X ==t Y since by definition any homotopy colimit is a composition of these operations.
2. Augmented functors
47
D.2.6 nCONNECTED SPACES. Since any pointed homotopy colimit of nconnected spaces for any fixed integer n is again nconnected, the class of nconnected spaces is closed. In fact this class, when intersected with spaces having the homotopy type of CWcomplexes, is simply the class C'(S n+l) generated by the n + 1sphere. Similarly, the class of spaces with vanishing homology with given coefficient up to a given dimension is also closed. D.2.7 UNPOINTED HOMOTOPY COLIMITS. We shall see in Chapter 9 that given any closed class of spaces, if we further close it under unpointed homotopy colimits indexed by any contractible indexing small category, the class will still be closed; in fact it will not change at all. D.3 POINTED AND UNPOINTED HOMOTOPY COLIMITS. Let A be a pointed space.
We have considered C"(A), the smallest class of pointed spaces closed under arbitrary p o i n t e d hocolim, and homotopy equivalence, which contains the space A. Notice that if we consider classes closed under arbitrary nonpointed hocolim, we get only two classes: the empty class and the class of all unpointed spaces. This is true, since a class closed under unpointed hocolim that contains a contractible space, contains all weak homotopy types, since every space is weakly equivalent to the free hocolim of its own simplices (see Appendix HL and 1.F above). Notice also that if A is not empty, then C'(A) contains the onepoint space * ~ hocolim.(A ~ A ~ A * ...) where all the maps in this infinite telescope are the trivial maps into the base point 9 E X. In general, given a pointed/diagram X we can consider its homotopy (inverse) limit in either the pointed or unpointed category. By definition, these two homotopy limits have the same (pointed or unpointed) homotopy type. They have in fact the same underlying space. On the other hand, the homotopy colimits of X will generally have a different homotopy type when taken in the pointed or unpointed category: If * is t h e /  d i a g r a m of base points in X, then by the very definition of homotopy colimits, pointed and unpointed, we have a cofibration, with N I = the classifying space (or the nerve) of the category I:
N I ~ freehocolim X ~ pointedhocolim X.
See [BK, p. 327 & p. 333]. D . 3 . 1 COROLLARY: / f the classifying space of the indexing category I is con
tractib]e, then for any pointed Idiagram y we have a homotopy equivalence freehocolim y ~_ pointedhocolim y . I
~
I
48
2. A u g m e n t e d functors
Remark:
tower 9~
Thus over the usual pushout d i a g r a m 9 ~ 9~ 9~ . . . . . .
9and over the infinite
hocolim takes the same value in the p o i n t e d and unpointed
categories, b u t not, e.g., over a discrete group. Thus
for any small category I
we have h o c o l i m , { , }
=
{,},
while
freehocolim{,} = B I = N I is the nerve (or the classifying space) of I . D.4
I HALFSMASHES
AND PRODUCTS
IN C L O S E D
CLASSES.
We
now
show
that
a
closed class C' is closed under halfsmash with an a r b i t r a r y unpointed space (i.e. C is an ideal in S , under the operation C" ~ C" ~ Y), and under internal finite Cartesian products. But first: D . 5 GENERALITIES ABOUT HALFSMASH. Recall the n o t a t i o n X x Y = ( X x Y ) / 9
x Y and X ~< Y = ( X • Y ) / X
• , , where X is pointed and Y is unpointed space.
This gives a bifunctor S , • S ~ $ , . There is another bifunctor S x $ , ~ $ , given by map(Y, X ) where Y is unpointed and X pointed and where m~p(Y, X ) is the space of all m a p s equipped with the base point Y ~ * ~ X . Thus the underlying space of map(Y, X ) is the same as t h a t of the free m a p s while the underlying space of X ~ Y is different in general from t h a t of the base point free p r o d u c t X x Y. There are obvious adjunction identities: (i) m a p . ( A >~ Y, X ) = m a p . ( A , map. (Y, X ) ) , (ii) m a p . ( A x Y, X ) = map(Y, m a p . ( A , X ) ) . T h e first identity (i) says t h a t for each Y E S the functor  x Y: 3 . * S . is left adjoint to map(Y,  ) , whereas identity (ii) says t h a t for each A E $ . the functor A >~  : S * $ . is left adjoint to m a p . ( A ,  ) , where this m a p p i n g space is taken as an unpointed space. In particular, we conclude from the general properties of left adjoints: D.6 PROPOSITION: For each A E S . and Y E S, the functors  >4 Y and A >4 c o m m u t e with colimits and hocolimits. D.6.1 NOTE: To say t h a t A )~  : S * S . commutes with hocolim involves c o m m u t i n g pointed hocolim, i.e. the hocolim in S . with unpointed hocolim in S. Explicitly: For any base point free d i a g r a m of space y : I * S we have an equivalence:
A ~ (free hocolim y ) _~ pointed hocolim, (A >~ y ) . I
~
I
D . 7 LEMMA: I[ Y is any unpointed space then for any indexing diagram I the functor  >~Y: S . * S . commutes with hocolim, and i[ X is any pointed space the I
functors X >4  : S * S . and  A X : $ . * S . c o m m u t e with hocolim. I
We have just considered X >~. Similarly  A Y is left adjoint to m a p . (Y,  ) and again commutes with colim and hocolim.
Proofi
2. Augmented functors
49
D.8 THEOREM: I f X is in any closed class C then: 1. For any (unpointed) space Y the halfsmash X >~Y is in C'. 2. For any (pointed) Bcellular space Y and any Acellular space X the smash X A Y is an (A A B)cellular space. D.9 REMARK: Notice that, X being a retract of the halfsmash, one can also read (D.8(1)) backward: If X >4 Y is in C then so is X for Y nonempty. Proof:
space.
To prove (1) we start with an example showing that X ~ S 1 is an Xcellular In fact it can be obtained directly as a pointed hocolim of the pushout
diagram: fold
XVX
~
X
X
'
XNS
1
This diagram is obtained simply by halfsmashing X with the diagram that presents S 1 as freehocolim of discrete sets:
{0,1}
~
{0}
{1}
~
S1
By induction we present S ~+1 as a pushout * ~ S ~ ~ * which gives by induction X >~ S T M as a pushout along X ~ X >~ S" * X, that arises, since (D.6) (X >~  ) commutes with freehocolim on the right (smashed) side. Since the filtration of Y by skeleton Y0 C Y1 C .. presents Yn+l = Y,, U ( C I J S ~) we get upon halfsmashing with X a presentation of Y >~ X as a pointedhocolim. D.9.1 REMARK: Here is a 'global' formulation of the above proof using (D.6): Present the space Y as freehocolim{,}, where FY is any small category whose FY
nerve is equivalent to Y, see (1.F) and Appendix HL. Further, by {*} we denote the FYdiagram consisting of the onepoint space for each object of FY. Now by (D.6) above:
X >~ Y = X
>4 f r e e  h o c o l i m ( * ) PY
~"
=
pointedhocolimX >~ {*}. FY
Thus X x Y is directly presented as a pointed hocolim of a pointed diagram consisting solely of copies of the space X itself. Now to prove (2) one just notices that X A Y = ( X x Y ) / X x {pt}, so X A Y is certainly an Xcellular space. Now since any pointedhocolim commutes with
50
2. Augmented functors
smashproduct we work by double induction: First we show by induction on the presentation of X as an Aspace that X A B is an A/~ Bspace, and then by induction on the presentation of Y as a Bspace that X A Y is an A A Bspace as needed.
D.10 FIBRATIONS AND CLOSED CLASSES. While closed classes are defined using pointed homotopy colimits and, in particular, cofibrations, the more interesting results relate the cellular structures of members in fibration sequences. This is harder to come by since e.g. the homotopy fibre of a map is not easily related by homotopy colimits to the base and total spaces. A deeper look into these matters is taken in Chapter 9. Here we confine ourselves to results needed in the coming developments and a few other examples. D.11 THEOREM: Let F ~ E ~ B be any fibration of pointed spaces with connected B. I f F and E are members o f some closed class C" then so is B. We shall see later (3.E.1) that this implies: D.12 COROLLARY: In a fibration, if the base and total spaces are EAcellular then the fibre is AceBular.
The following consequence immediately implies the generalized formulation of a lemma of Zabrodsky and Miller given in [M, 4.6], compare also [B4, 4.7]: D.13 COROLLARY: Let F * E * B be any fibration sequence over a connected base B. I f both the fibre and the total space have a trivial pointed function complex to a given pointed space Y , then so does the base space B. The second corollary follows immediately by observing (D.2.2) that the class of spaces with a trivial function complex to a given space is closed. We define a sequence offibrations Fi ~ Ei * B by Eo = E, Fo = F, Ei+l = Ei U CFi and Fi+l is the homotopy fibre of an obvious map Ei+x ~ B. All Ei, Fi are naturally pointed spaces. Proo~
F
)
FI
,
F2
)
E
l 1
E U CF
)
=
E1
E1 U C F1 = E2
1 Eoo "~ B
B
~
B
)
B
2. Augmented functors
51
By Ganea's theorem [G] (but see also Appendix HL at the end) F~+I ~ Fi * f i b _~ E(Fi A ~tB) and therefore connectivity of F~+I is at least i, since F0 is (1)connected. Notice that by the above closure properties of closed classes, since E0, F0 are in g' spaces so are E~, F~ for all i. But since conn F~ + oo, we deduce that hocolim E~ = B. Therefore B is also in g', as needed. We now turn to the somewhat surprising closure property of closed classes (D.l(4)) that will be treated more fully only in Chapter 9. D.14 THEOREM: For a n y m a p A * X o f p o i n t e d spaces, t h e h o m o t o p y fibre F o f X ~ X U C A satisfies P A F ~ *. I n p a r t i c u l a r PA(A) ~ *. M o r e o v e r , F is Acellular. Outline of Proof:
The proof uses the following diagram: A
F
~ X
c ~ XUCA
1 IYl PAF
, X ~
X u CA
where the vertical arrows are given by the fibrewise localization (1.F.1) of the top row. Thus the fibre map ~ is induced from the composition X t3 C A ~Baut F * Baut P A F , where by B we denote here the classifyir~g space functor. Taking F to be the usual path space we have a well defined map i ~ : A * F of A to the homotopy fibre. Since map(A, P A F ) ~ *, by construction of P A F the composition A * X + X factorizing through P A F is nullhomotopic, where the null homotopy comes from the cone A ~ F ~ F t3 C A * P A F that defines P A F . This null homotopy gives a well defined map c I : X tA C A ~ X rendering the diagram commutative. Therefore the fibration ~ is a split fibration having c~ as a section. Also, since F ~ X factors through X t3 C A it is a null homotopic map. But the splitting of implies from the long exact sequence of the fibration that the map P A F ~ X is injective on pointed homotopy class [ W ,  ] . for any W E $.. And since F ~ P A F ~ X is null homotopic we conclude that F ~ P A F is null. Now idempotency of PA implies P A F ~ * as needed. A complete proof that F is Acellular is longer and given in Chapter 9 below. See (9.A.10). D.15 CLOSURE UNDER PRODUCTS. Many of the pleasant properties of C W A depend on its commutation with finite products. This commutation rests on the following basic closure property of any closed class.
52
2. A u g m e n t e d functors
D.16 THEOREM: A n y closed class C" is dosed under finite products: I f X , Y E C" then so is X x Y . D . 1 6 . 1 REMARK: It is well known t h a t an infinite p r o d u c t of S l ' s does not have the h o m o t o p y t y p e of a C W  c o m p l e x , i.e. the class of all C W  c o m p l e x e s in Top. in E x a m p l e D.2.3 above is not closed under a r b i t r a r y products. Also notice t h a t an infinite p r o d u c t of H Z  a c y c l i c spaces m a y not be acyclic, thus the closed class of all H Z  a c y c l i c s is not closed under a r b i t r a r y products. D . 1 6 . 2 REMARK: If A = E A r and B = E B r where A, B E C', then A x B is easily seen to be in C" via the cofibration
A r * B r + E A r V E B r * E A r
x
E B r.
Since A r * B '  E A ' A B r one uses (D.8) above. P r o o f olD.16:
We owe the proof to Dwyer. A n independent proof can be e x t r a c t e d
from [Bl]. Proof." We filter Y by its usual skeleton filtration Y,~+I  Y~ U en+l .... We may assume X, Y are connected. For brevity of notation we add one
pointed cell at a time but the proof works verbatim for an arbitrary number of cells. Let P(n) be the subspace of X x Y given by
P ( n ) = {.} • YU X x Y,.
Clearly the tower P ( n ) "+P ( n + 1) is 'cofibrant' and its colimit X x Y is equivalent to its h o m o t o p y colimit. Since C" is closed under hocolim it is sufficient to show, by induction, t h a t P ( n ) E C" for all n > 0 . For n = 0, we have P ( 0 ) = X V Y clearly in C'. Now P ( n ) is given as a h o m o t o p y pushout diagram: X X S n1
U * xD ~
1
X x D '~
 ,
X : ~ S n1
~ '
X >4 D "~
...., { * } x Y
1
coming from the presentation of Y~ as a pushout
'
UXxY,,_I
t
P(n)
over a pointed diagram:
Y~I + S ~1 ~ D n. Since the upperleft corner is equivalent to the halfsmash X >~ S '~, it is in C' by L e m m a (D.8.1) above. Notice t h a t all the m a p s are pointed. Therefore P ( n ) is a h o m o t o p y pushout of members of C" as needed.
2. Augmented functors
53
D.17 COROLLARY: For any two Acellular spaces X , Y their product X x Y is an Acellular space. Proof: Consider the class C' (A). By the theorem just proved it is closed under finite product, therefore the product of any two Acellular spaces is Acellular.
E. AHomotopy theory and universal properties In this section we describe some initial elements of Ahomotopy theory which is a special case of (C.2) above. This will allow us to better grasp properties of the functor C W A . In this framework one replaces the usual sphere S O in the usual homotopy theory of CWcomplexes or simplicial sets by an arbitrary space A. It can be considered in the framework of general, compactly generated spaces where A can be chosen to be any such space. We will, however, restrict our discussion to A E S., a pointed space. It turns out that there is a model category structure on 8. denoted by S A, where a weak equivalence f: X + Y is a map that induces a usual weak equivalence
map. (A, f): map. (A, X) * map.(A, Y) of function complexes, and Afibre maps are defined similarly. Cofibrations are then determined by the lifting property IN]; see (C.2) above. The cofibrant objects, i.e. the CWcomplexes, are Acellular spaces. The natural homotopy groups in this framework are Ahomotopy groups
r i ( X ; A) = [ ~ A , X]. = ri map.(A, X, null) = [A, fliX]..
The classical Whitehead theorem about CWcomplexes takes in the present context the form: E.1 THEOREM (AWHITEHEADTHEOREM): A map f: X ~ Y between two pointed connected Acellular spaces has a homotopy inverse (in the usual sense) if and only if it induces a homotopy equivalence on pointed function complexes
(*)
map.(A, X) % map.(A, Y),
54
2. Augmented functors
or equivalently, iff f induces an isomorphism on the pointed homotopy classes:
(**)
[A >~S n, X]. ~ [A :~ S '~, Y],
for all n >_ O. If the two pointed function complexes are connected, i.e. 1to(X; A) ~ 7to(Y; A) ~ 9 or irA = ~A' is a suspension, then a necessary and sufficient condition is that it induces an isomorphism on Ahomotopy groups:
Ir.(X; A) ~ 7r.(Y; A). Proos It is sufficient to show that under (.), for every W E C ( A ) , we have that map.(W, X) % map(W, Y) is a homotopy equivalence. This can be easily shown by a transfinite induction on the presentation of W as a hocolim of spaces in C (A). Namely, one needs only to show that the class of spaces W for which map.(Y, f ) is a homotopy equivalence is a closed class. But this is the content of (D.2.2) above. Since by assumption it contains A, it follows that it contains also C' (A) and therefore, by our assumption, it contains both X and Y. Thus we get a homotopy inverse to X ~ Y by taking Y = W. This shows that in fact (*) implies a homotopy equivalence.
A more careful argument is necessary to show that looking at pointed homotopy classes as given by (**) is a sufficient condition for a weak homotopy equivalence of the function complexes and thus by (*) for f being a weak equivalence. The difficult point is that when one expresses that isomorphism (.) in terms of pointed homotopy classes one must keep a fixed map of A to X and Y fixed throughout the homotopy while in (**) we keep only the base point fixed. This was proved in
[CaR]. E.2 HALFSUSPENSIONS ~nX. A basic building block for C W A is the halfsmash S n  S ~ x A U D ~+1 x {.} with the base point {.} x {.}. We denote these spaces by ~nA, and call them half nsuspensions of A. Just as a homotopy class a E 7r~map.(A, X; null) in the null component is represented by a pointed map EnA ~ X, so does a map &: ~ A * X represent an element in r ~ m a p . ( A , X; f ) of the fcomponent where f: A + X is any map. The map f is obtained from & by restricting & to * x A C_ ~nA. Notice that if A itself is a suspension A = EB, then ~'~A ~ E~A V A [DF4] but in general such a decomposition does not hold. Thus for suspension A = EB, an element 5 as above is given simply by a pair (a y f): EnA V A ~ X. In that case, of course, all the components of map.(A, X) have the same homotopy type. A x
E.3 ELEMENTARY CONSTRUCTION OF CWAX. Let co: CoX = V ~ e I ~ A % X be the wedge of all the pointed maps ~]~A * X from all halfsuspensions ~iA to X.
2. Augmented functors
55
Clearly the map co induces a surjection on the homotopy classes [~iA,] for every i > 0. We now proceed to add enough 'Acells' to Co, so as to get an isomorphism on these classes. We take the first (transfinite) limit ordinal A = A(A) bigger than the cardinality of A itself (= the cardinality of the simplices or cells or points in A). The ordinal A = A(A) clearly has the limit property:
Given any transfinite
tower of spaces of length A
ro+ri+...~Yn...,rw+Yw+l+...~Ya
~...
(~ < A)
every map Ei A * lim Y~ factors through Ei A * Y~ for some ordinal 13 < A.
o,<),
This is clear for every individual cell of EiA, and since the number of these cells is strictly smaller than the cofinality of A, it is true for EiA. We proceed to Proof:
construct a Atower of correction Co = C o X ~ C 1 X ~ C 2 X ~ . . . * C z X ~ . . . to our original map Co * X :
(E.4) Do = VKo ~ i A
Co V I o E i A
X=
D1 = VK1 ~ i A
. C1X .
X=
.
D~ = V K , E i A . . .
.
.
CzX
. . . . (Z 5 A)
X ....
Since Co ~ X is surjective on the Ahomotopy of all components of m a p , (A, X), we proceed to kill the kernel in a functorial fashion. In order to preserve functoriality we kill it over and over again: First notice that any element &: E n A * X representing an Ahomotopy class in the component &l{* } x A = f : A * X is null homotopic in that component iff ~ can be extended along the map
(E.4a)
E~A = S '~ x A U D ~+1 x {,} ~* D '~+1 x A.
Now let k0: Do ~ Co be the wedge of all maps g: E i A ~ Co with a given extension as (E.4.1) of Coog (the space Do being a point if there are no such extensions). Thus Do ~ Co captures every null homotopic map EiA ~ Co * X many times. The
56
2. Augmented functors
map Do ~ Co is given by g. We define C I X as the pushout along the extension to D n+l x A:
VE~A Ko
~ V D+lxA Ko
1 Co
1 ~C1 = C1X
In this fashion we proceed by induction. The map C1X * X is given by the null homotopies in the indexing set of Do = V E~A. Taking limits at limits ordinal we k0
define a functorial tower C/3X for/J < A. We now define C W A X = C~,X. This is the classical small object argument [QI]. Since co induces a surjection on Ahomotopy sets [F.iA, X] for i > 0 on all components we get immediately that so does cz for all/3 < A. The limit property of ,~ = A(A) now easily implies that C),X + X is injective in ~r~( , A; f ) for any f: A ~ X. Since every null homotopic composition EiA * C),X ~ X factors through EiA * C~X * X for some/3, a composition that is also null homotopic by commutativity, therefore this map is null homotopic in C~+IX and thus in C:,X, as needed. E.5 A SMALLER NONFUNCTORIALACELLULAR APPROXIMATIONCall be built by choosing representatives in the associated homotopy classes. But it is clear that, in general, even if A, X are of finite complexes C W A X may not be of finite type, since C W s 2 (S 1 V S~) ~ Voo s~ and this construction is just the universal cover of
s1V S~. E.6 COROLLARY: Let A be a finite complex. Then for any countable space X we have the following form:
CW AX
= (V 2iA)O,p, C~fl A U~ C~,i~A 9. . U ~ CE ~ A U .  .
where the 'characteristic maps' ~ are de~ned over Ei~A for 0 < ~ < oo, and therefore C W A X is also a countable cell complex. E.7 REMARK If A is a finite suspension space A = EB of pointed B, we have F,iA ~ EiA V A and therefore, in order to kill the kernels of C~ ~ X, it is sufficient to attach cones over the usual suspension of A: E~A * C a. Thus in this case the Acellular approximation to X has the usual form
ow,,
= (V
2. Augmented functors
57
which is just the usual C W  c o m p l e x for A = S 1 = E S ~ and where X is any connected C W complex. As is usual in homotopy theory, any map X * Y can be turned into a cofibration X r
X ~ * Y where X "* X ~ is an Acofibration, i.e. X ~ is obtained
from X by adding 'Acells' and X ~ * Y is a trivial fibration, i.e. in particular it induces an isomorphism on Ahomotopy groups (compare (C.2) above). Thus if Y _~ 9 we get X ~ ~ P A X , since m a p . ( A , P A X ) ~ m a p . ( A , * ) and X   * P A X is an Acofibration. If, on the other hand, we take X ~ *, the factorization becomes * * C W A Y * Y where C W A Y now appears as the Acellular approximation to X with the same Ahomotopy in all dimensions. E .8 UNIVERSAL PROPERTIES.
We now show that r: C W A X ~ X has two universal properties: (U1) [B2, 7.5] The map r is initial among all maps f : Y ~ X with m a p , ( A , f ) a homotopy equivalence. Namely for any such map there is a factorization ] : CWAX
v , X
Y
and such ] with f o ] ~,, r is unique up to homotopy. (U2) The map r is terminal among all map w: W ~ X of spaces W E C'(A) into X. Namely for every w there is a ~o: W , C W A X with r o ~ ,, w unique up to homotopy. Proof'. Both (U1) and (U2) are easy consequences of the functoriality of C W A when coupled with the AWhitehead theorem. Thus to prove (U1) consider
CWA(f): C W A Y
This map is an Aequivalence between two Acellular spaces, therefore it is a homotopy equivalence. Uniqueness follows by a simple diagram chase using naturality and idempotency of C W A . To prove (U2): One gets a map A * C W A X by noticing that C W A W ~ W , So C W ( w ) gives the '* C W A X .
unique factorization. Furthermore, uniqueness of factorization implies that each one of these universal properties determines C W A X up to an equivalence which itself is unique up to homotopy. This proves (U1) and (U2).
58
2. Augmented functors
E.9 PROPOSITION: The following conditions on pointed spaces A and B are equivalent:
(1) (2) (3) (4)
For any space X there is an equivalence C W A X C'(A) = C'(B).
~_ C W B X .
A m a p f : X * Y is an Aequivalence i f and only i f it is a Bequivalence.
A ~ C W B A and B ~ C W A B .
Proof'.
These equivalences follow easily from the universal properties of C W A X
X. (1)r CWAX
Since the members of C ( A ) are precisely the space X for which
~ X , this is clear from universality.
(1)~=~(3) Clearly map(B, f ) is an equivalence and C W B f is a homotopy equivalence. But since by (1)~=~(2), C W B f ~ C W A f and we get (3). (2)r This is immediate from the definitions. E.10 THEOREM: For any A , X , Y E S there is a h o m o t o p y equivalence
~]~: C W A ( X x Y ) +C W A X x C W AY. Proof'.
There is an obvious map
g: C W A X
x CWAY
+ X • ]I.
It is clear that g induces a homotopy equivalence map(A, g) and therefore the map in the theorem induces the same equivalence map(A, q2). But by Corollary D.17 the range of fig is an Acellular space. Thus by the AWhitehead theorem fig is a homotopy equivalence. E.11 LEMMA: I f Z ~_ C W A X
and Y is a retract o f X , then Y ~ C W A Y .
The retraction r : X ~ Y implies that the map C W A Y ~ Y is a retract of the homotopy equivalence C W A X ~ X . But a retract of an equivalence is an equivalence. ProoD.
3. C O M M U T A T I O N R U L E S F O R ~, L f A N D C W A , PRESERVATION OF FIBRATIONS AND COFIBRATIONS Introduction In this chapter we begin to consider the behaviour of fibration and cofibration under L f and C W A . As in the case of localization at a subring of the rationals or pcompletion, the main technical property of both L f and C W A is that, under some relatively mild restrictions, they 'nearly' preserve fibration sequences. In particular, they nearly commute with the loop functor: We first prove the equivalences L f ~ X ~ ~ L ~ f X and C W A ~ X ~ ~~CWEAX (notice E f ) and draw a few quick consequences. We then consider several cases where L f and C W A preserve fibration sequence. The most general results are obtained in Chapter 5 below. Here we con*X * PEA. tinue by relating C W A and PA via a fibration sequence C W A X We show that this is in fact a fibration sequence whenever the composite is null homotopic: This in particular implies that if P E A X ~ *, then X is Acellular for any A, X E S.. In section 7.C below we give several applications showing that certain naturally arising spaces have Acellular structure, e.g. certain E.acyclic spaces can be built from finite ones. A. C o m m u t a t i o n w i t h t h e l o o p f u n c t o r In the first section we prove that both L f and C W A for arbitrary f, A E S. 'commute' with the loop functor in the following sense (compare [B4]): A.1 THEOREM: Let f: A , B be any map in S. and X C S, a connected space. There is a natural homotopy equivalence
Lf~X
~ ~L2fX.
A.2 THEOREM: Let A, X E S. be pointed and connected spaces. There is a natural homotopy equivalence C W A ~ X ~_ ~'~CWEAX. Remark: Thus both L f ~ X and C W A g I X have a natural loop space structure where the corresponding classifying spaces are L ~ f X and C W E A X . By induction we get a similar result for L f ~ n X and C W A g I ~ X . Still, it is not known whether for any other W E S. the space L f map, (W, X) is a Wfunction space, i.e. has the form map.(W, Y) for some Y. Certainly one cannot simply take Y = L w ^ f X .
60
3. Commutation rules
A.3 LEMMA: f i X is a connected pointed space, then there is a natural loop space structure on L f f ~ X with respect to which the coaugmentation map f~X LynX is homotopic to a loop map. ~
The lemma says in other words that there is a space Y with gtY _~ Lf~tX, and this space Y is natural in X and comes with a loop map YtX * ~Y. Notice that the proof only uses two properties of L f, namely L f preserves products up to homotopy equivalence and is coaugmented.
Remark:
Proof: To equip L f F t X with a loop structure one uses Segal's 'machine' to identify loop spaces. According to this machine IS], [A], [Pu] a pointed space V is a loop space if it can be embedded in a 'special' simplicial space W. = {W~}~>0 with W0 ~ * and W1 ~ V with the crucial condition being: for any n the space W~ is homotopy equivalent to the nth power W~' = W1 • W1 • " • W1 of the space in dimension one W1 where the equivalence is given via the structure maps. Namely, W~ "~ W~' _~ V ~ for n > 1, and where these homotopy equivalences W~ , W~'
n h",, where hi: (0,1) are given as a product of maps rL1 , ( 0 , . . . , n ) is given by hi(0) = i  1 and hi(l) = i; the usual simplicial operators map A ~ to A~ p. Together, they give maps Wn 9 W1 that combine to form an equivalence 1IW(,~i):
Wn
* W 1 X "''
x W 1.
If such a simplicial space (i.e. a functor A ~ 9 S.) induces a group structure on ~r0W1, then Segal's theorem says that there is a natural homotopy equivalence ~[W.[ "~ W1 where IW.I is the usual realization of W. (i.e. its homotopy colimit as a diagram of spaces). Now, given X we present ~ X as a 'Segal loop space' by taking a monoidic version of ~ X , say ~ X , the Moore loop space of maps [0, a] ~ X, so that we get a precise simplicial space made out of the monoid operations and projections into * Top be given by G,~ = (~tX)'~; this factors. Let the simplicial space G.: A ~ has the right properties so that ~ I G , I "~ G1 = ~tX. Now the idea is to apply the functor L f to the diagram of spaces given by G. ( = a simplicia! space). Since L f is a functor we still get a simplicial space and since L f commutes with finite products up to homotopy it is easy to see that one still gets a 'special' simplicial space, that gives the desired loop space structure on the space Lf~tX. In more details, take the simplicial space L f G . with ( L f G . ) ~ = L f ( G ~ ) , the localization of the product (~tX) ~. Since the homotopy equivalence
=
. (al)
:
(ax)
3. Commutation rules
61
is in this case the identity map given by a product of projection, this same projection ;~i gives on LfG, a product map L f ( ( ~ X ) '~) , ( L / ~ X ) '~, which we know by elementary fact (1.A.8) (e.4) above to be a homotopy equivalence. Therefore L f G . is a special simplicial space, satisfying the said conditions. Notice that in dimension one we have 7r0LfG. = rcoLff2X, which is clearly a group, since the equivalence L I ( X • X) ~ L f X • L f X is natural and since L f does not change the set of components when applied to a nonconnected space because f is a map of connected spaces. Therefore f~lLfG.I  LfG1 ~ L f ~ X , presenting L f ~ X as a loop space. In fact, since G. , LfG. is a simplicial map between simplicial spaces, it induces IG*I
~ X *x
, ILfG.h a map which gives a loop map
~la, I
, ~[L.fa, I  ,
Lf~X.
This equivalence is of loop spaces, so they combine to show that ~ X
, L f ~ X is
a map of loop spaces. A.4 PROOF OF A . I : We will construct two maps t? : L f ~ X '   , ~"ILEfX; each will be given by the appropriate universality conditions on localizations. The uniqueness part of universality will then imply that they are homotopy inverses. First notice that by elementary fact (e.7) in (1.A.8) above, the map f~X , ~ L E y X , which is the loop of augmentation X , LEfX, is a map of ~2X to an flocal space. Therefore, by universality, it factors uniquely up to homotopy through a map l, f~X j , LIYIX l, ~Ly, f X . To construct the map the other way we may proceed by using the result above that flX , L f X is a loop map, and so we can classify it to get X , ITVI2X ,
WLff~X. In fact we may use IL/G.I as a model for W L f f l X . Then the map in the other direction will be given as a loop map if we construct a map of spaces L~IX , WLII2X, by looping down and composing with the obvious maps. Again we use elementary fact (1.A.8) (e.7) to notice that W L f f l X is Z flocal, and therefore by universality of L~I it is sufficient to find a map X , WLff~X. This can be taken as the composite that was constructed above: m
X
, W~X
_
_
w e WLIY~X"
62
Now
3. C o m m u t a t i o n rules
we have a diagram of maps
f~X
=
~ ftX
I jl h LI~/X
I aj2 * 9 ~'/LzlX: r
which commutes up to homotopy on both sides, since the b o t t o m arrows were found by universality. This means r o 12j2 ~ jl or r o ~ o j l ~ Jl, but by uniqueness of factorization through the universal L f ~ X we get that r o ~ is homotopic to the identity. Similarly one gets ~ o r ~ Id. This completes the proof.  A.5 PROOF OF (A.1): The proof proceeds in line with(A.3), (A.4), since C W A preserves products up to homotopy equivalence, except that here we have augmented rather than coaugmented functors. So since CWA above to conclude
preserves products we can use the same argument as in (A.3) that CWAIIX is still a loop space and the natural map
CWAglX ~ IIX can be taken to be a loop map. Now that we have shown that CWA~X has loop space structure compatible with that on lIX we can use universality properties of CWA to get the desired equivalence. First take ~'~JE: ~'~CWEAX
" ~'~X
to be the loop of the structure m a p for C W E A . We consider the factorization of the last m a p that gives us r:
CWA~ X
= , CWA~'~ x
t2CW~AX
nj~ , IIX
(A.5.1)
To get the factorization first note that (12jz) induces a homotopy equivalence on m a p , ( A ,  ) . Therefore, by universality of the m a p (jg/) we get the m a p r in (A.5.1) which is unique up to homotopy. To get the m a p ~ in the opposite direction we first construct a m a p gt = 'Wg'
~": C W E A X
~' W C W A ~ " ~ X
3. Commutation rules
63
where W is the classifying functor otherwise denoted by B  . Here we use crucially the fact proven above that C W A ~ 2 X , f i X is a loop map. One deloops this map , X . We lift the structure map C W ~ A X , X to get a map Wj~I: W C W A ~ X across W f l 2 to W C W A ~ X to get the desired map g. Again this lift exists by universality of C W ~ A X (E.8, U.1) since W ( j ~ ) is easily seen by adjunction to induce homotopy equivalence on the pointed mapping space from A: i.e. map,(A, Wj~t) is a homotopy equivalence. Since these two maps were defined by universality, it is easily checked as in (A.4) above that these are mutual inverses up to homotopy. This completes the proof of (A.2).
B. R e l a t i o n s b e t w e e n C W A a n d PA In this section we prove several results that put C W A X and P A X in an 'ideal' relation to each other. Intuitively C W A X contains all the 'Ainformation' on X available via the function complex map,(A, X) while P A X contains what remains of X after all that Ainformation was destroyed by the nullification functor. Thus CWAX , X should morally be the homotopy fibre of X , P A X . This is 'almost' the case but, as we shall see, not precisely. B.1 PROPOSITION: For a11A, X E $, one has P A C W A X "" * and C W A P A X ~ *.
Proo~ The second equivalence is clear since map, (A, P A X ) ~ * and so * , P A X is an Aequivalence, inducing a homotopy equivalence on m a p ( A ,  ) . Thus C W A turns that map into a homotopy equivalence. To see the first equivalence notice that C W A X is built out of the space A E S, via a sequence of pointed homotopy colimits. Now in view of the commutativity relations (1.D) with homotopy colimits, if PAX(O0 ~ * for each a E I then PA hocolim, X ( a ) ~ P A hocolim, PAX(a) ~ PAhocolim,(pt) ~ PA(pt) ~ (pt) where (pt) is the one point space. Thus for any closed class C'(A) for A E $, one has P A W ~ * for all W E g'(A). But C W A X C C ' ( A ) hence the conclusion P A C W A _~ ,. 
We now formulate the main relation between the nullification and cellularization with respect to A. B.2 THEOREM: Consider the sequence
CWAX
~, X
r, PEAX
for arbitrary pointed connected spaces A, X . This sequence is a fibration sequence if (and only if) the composition r o g is null homotopie. Moreover, the same conclusion holds if [A, X]  .. Remark: Notice that if C W A X ~ *, namely m a p . ( A , X )  *, the conclusion of the theorem is obvious since in that case X is also ~Anull, so the map r is an
64
3. Commutation rules
equivalence. In examples (C.9) and (C.11) below we will see that the composition is not always null nor is the sequence always a fibration sequence. We first prove the theorem under the special assumption: P~AX  *, which of course is a special case of the theorem.
Proof"
B.3
PROPOSITION: For
any
i f P EA X ~ * then C W A X
two
connected
CWcomplexes
A, X
in
8,,
"", X is a h o m o t o p y equivalence.
Proo~ In the following fibre sequence one shows t h a t the homotopy fibre F must be contractible. Since we assume that X is connected, this implies that the aug
mentation is an equivalence.
(B.3.1)
f~X
9F
. CWAX
, X.
In order to show that F ~ 9 one proves: (1) m a p . ( A , F ) "~., (2) P A F "" *. Clearly any space Y that satisfies (1), i.e. is Anull, does not change under PA, thus (1) and (2) imply F ~ *. The fibration (B.3.1) implies that m a p , ( A , F ) is the homotopy fibre of map,(A, C W A X ) , map,(A, X) over the trivial component. But by the definition of C W A the latter map is a homotopy equivalence, thus its fibre is contractible and (1) holds. To prove (2) we use Theorem 1.H.1 above, with respect to the fibration sequence ~ X , F , C W A X . Since both X and A are connected so is C W A X . First notice that by (3.A.1) P A ~ X "" Y t P E A X which is, by our assumption, contractible. But now Theorem 1.H.1 means that P A F ~, P A C W A X is a homotopy equivalence. Theorem B.1 above now implies P A F ~ *, as claimed in (2). This completes the proof of the proposition.  We now proceed with the proof of Theorem B.2. Let Y be the fibre of X * P E A X . By Theorem 1.H.2 we deduce that P~.AY  * and therefore, by the proposition just proved, we deduce C W A Y ~, Y is a homotopy equivalence. The following claim now completes the proof: CLAIM: C W A Y
"~ C W A X .
Proof: The map Y , X gives us a map Y ~ C W A Y , C W A X . Since both spaces are Acellular, it suffices by the AWhitehead theorem (2.E.1) to prove that we have a homotopy equivalence:
map,(A, C W A Y ) ~ map,(A, C W A X ) .
3. Commutation rules
65
Since for any space W the map C W A W . W is a natural Aequivalence, by the universal property of C W A (E.8.U.1), it suffices to show that map.(A, Y) map. (A, X) is a homotopy equivalence of function complexes. Consider first the set of components: By definition of Y as a fibre we have an exact sequence of pointed sets:
(B.3.2)
[EA, PEAX]
 [A, Y]
. [A, X]
, [A, PzAX].
Now we claim that it follows from the assumption r o g _~. in our theorem, that the rightmost arrow is null. This is because, by the universal property of C W A (Theorem E.8 (U.2)), every map A * X factors (uniquely up to homotopy) through
CWAX ~ X, therefore by the assumption of the theorem ( r o e ~ *) its composition with r: A , X r , P z A X must be null homotopic. Now consider the pullback sequence: map.(A, Y)
9
* map. (A, X)
, map,(A, PzAX;null) ~ *
We just saw that ~ = map.(A, r) carries the whole function complex to the null component of map. (A, PEAX). Therefore we can and do restrict the lower right corner of the square to the null component. But the component of the null map in map.(A, PEAX) is contractible. This is true, since this component is connected and its loop ~tmap.(A, P~AX; null) is by adjunction just map.(EA, Pr, AX) ~ *, as needed. Now a pullback square with two lower corners contractible must have a top arrow that is an equivalence, as needed. Under the assumption [A, X] ~ 9 we get, of course, the same conclusion since the proof above works as well since the rightmost arrow in (B.3.2) above is again null.
C. E x a m p l e s o f cellular spaces Using the adjunction relations (A.2) and Theorem (B.2) one can prove that certain spaces are Acellular with respect to an apPropriate space A. In particular, many wellknown constructions in classical homotopy theory lead to Acellular space. C. l JAMES FUNCTORJ X . James, in his thesis [J], gave a combinatorial construction of a space that, under mild conditions, is homotopy equivalent to the loops on the
66
3. C o m m u t a t i o n rules
suspension of the given space. In spite of the combinatorial n a t u r e of the original definition it is not h a r d to see b o t h directly and indirectly t h a t : CLAIM: For any X the space J X ~ ~IEX is an Xcellular space, in a formula:
CWxJX Proof:
~ J X .
F i r s t we give an explicit construction of J X as a h o m o t o p y colimit of
Xcellular spaces [BI1, 3.5]. We have a filtration
X C "'" C J n X
C Jn}i x
C "'" C J X
We first define an intermediate space T ~ X as a h o m o t o p y pushout of pointed spaces along the obvious inclusion maps:
JniX
. JniX • X
1
i
&X
. TnX
and t h e n complete the inductive construction of J ~ + l X as a h o m o t o p y pushout of ' p o i n t e d spaces below, where the m a p s on T ~ X are induced by the corresponding m a p s from J n X and J , ~  l X x X t h a t agree on J n  l X in the above pushout square; see comments on pushout squares in A p p e n d i x HL below: T,~X
. J,~X x X
l
1
J~X
. &+iX
This gives an inductive definition of J ~ X . Since by T h e o r e m 2.D.16 above a p r o d u c t of two Xcellular spaces is an Xcellular space, we get by induction t h a t J ~ + l X is an X  c e l l u l a r space. Therefore J X = hocolim J n X is also an X  c e l l u l a r space. I
This gives an explicit construction of J X as a pointed hocolim s t a r t i n g with X , i.e. as a m e m b e r of C" (X). But the a b s t r a c t fact t h a t ~2EX E C ( X ) can be o b t a i n e d directly from (3.B.2) above: Simply c o m p u t e C W x ~ E X
~ ~ t C W z x E X
= ftEX.
COUNTEREXAMPLE: It goes without saying t h a t if X is connected then ~ X is never an Xcellular, since it has a lower connectivity t h a n X itself (2.D.5).
3. Commutation rules
67
C.2 HILTONMILNORJAMES DECOMPOSITION. The famous theorem of Milnor and Hilton that followed a similar result by James provides a decomposition of ~ E X for an arbitrary pointed X as a wedge of smashpowers of E X and X itself. Thus it gives an explicit description of E ~ E X as an Xcellular space. (Any smashpower of W is Wcellular (2.D.8).) Using the adjunction relation (A.2) yields immediately that E ~ E X is in fact a EXcellular without, however, saying anything about the nature of the decomposition. Using (C.1) above one computes: m
CW~x ~2~X
" W C W x~2~fW,,X "~ W ~ 2 ~ 2 ~ X ~ ~,,~2~X.
Here we used (C.1) in the second equivalence, so that applying twice the James functor to X still gives an Xcellular space. In fact, the claim follows also immediately from (C.1) above: given that ~ E X is Xcellular it follows immediately that E ~ E X is EXcellular as claimed. Notice, however, that for nonsuspension C W y E ~ t Y ~ EY/Y. In fact E~tY is not a Ycellular space, rather the other way around: As we shall see X is E ~ X cellular (C.6)(C.7). For example, E~tK(Z, 3) ~ ECP ~ is not K(Z, 3)cellular since any K(Z, 3)cellular space must have vanishing reduced complex modp Ktheory [AnH] and ECP ~ is not Kaeyclic. Similarly ~ n S n X is also an Xcellular space. C.3 THEOREM: For any X the D o l d  T h o m functor S P ~ 1 7 6 is an Xspace. In Chapter 4 below a direct presentation of the symmetric products, finite and infinite, will be given as a pointed homotopy colimit of a diagram made up of finite powers of X, so the theorem will be proved more generally with more elementary means. Here we indicate a proof using a technique of Bousfield.
Remark:
This follows from a more general observation about arbitrary 'convergent funetors' of [BF], or Fspaces of [S]. Let F C Sets. be the full subcategory of the objects n + = {0,... ,n} with base point 0 C n +, for n _> 0. A Fspace is a functor U: F , S. that assign the point to 0 +. It is special if the canonical product map: U ( n +) ~ U(1 +) x ... x U(I+), of the maps that send all elements except one to 0 E 1+, is an equivalence and very special if the induced monoid on ~r0U(1+) is an Proof:
abelian group. Each Fspace determines a functor U: S.
* 8. with U X = diag(UX.).
where ( U X k ) . is the space associated by the Fspace U to the set of ksimplices Xk of X . Thus every, very special Fspace h: F , ,S. determines a reduced homology theory ~r.hX =_ h . X .
68
3. Commutation rules
C.4 PROPOSITION: For any Fspace U and any X E S. the space U X is an X cellular space. Proof: Almost by definition U can be written as the 'tensor product' of F~ with Fspace [B4, 6.4], [B3, 6.1]:
U X ~_ / f
x A U(.)
where f f denotes the homotopy 'coend' (coequalizer) ([Mac] and Appendix HC below) over F. Notice that X : F ~ * space is X n+ =Xx...• X ~+  map ,(n +, X), F~
(n + 1) times; this gives a functor;
9 spaces.
Now since by (2.D.16) X ~+ is an Xspace and by Lemma 2.D.8 above X n+ AY is an Xspace for any Y, we get that U X is a pointed hocolim of Xspaces and therefore an Xspace. In order to deduce Theorem C.3 above it is enough to show that S P ~ 1 7 6 is equivalent to U X for some Fspace U: F , 8.. But [B4, 6.2] shows that choosing the discrete Fspace Z to be Z(n +) = Z 9 "." 9 Z ntimes, and regarding Z as a discrete Fspace, gives Z X ~_ S P ~ 1 7 6 Therefore S P ~ 1 7 6is an Xspace. By the same token ~~176176is also an Xspace since ~~176176can be obtained as a diagonal of a Fspace. Further examples of Acellular spaces can be derived from the following: C.5 PROPOSITION: Let V(n) denote a finite ptorsion space of HopkinsSmith type n, where in particular V(n) is acyclie with respect to the Morava homology theory K(n). There exists an integer k = k(n) such that, for all i >_ O,
K(G, i + k) = CWv(n)K(G, i + k). In particular, K ( G , i + k) is a homotopy colimit of finite K ( n )acyclic subcomplexes. Proof: The point is that one can show that Pr, v(n)K(G, n + k) ~ 9 for large k and then use (B.3) above. To show this equivalence one proceeds by induction. Consider n = 1. It follows from the usual cofibration sequence that the modp homotopy groups of the EV(1)nullification must be vlperiodic (i.e. Vllocal) (compare Chapter 8). But we will see (Chapter 4) that t h e nullification of any
3. Commutation rules
69
EilenbergMac Lane space is a product of (at most two) such spaces (4.B.4.1). But a product of nontrivial EilenbergMac Lane spaces cannot be periodic with respect to the vl map unless the function complex from both the range and the domain are contractible since all composition operations that change dimensions on such a space must vanish. Moreover, a product of two EilenbergMac Lane spaces cannot have nontrivial homotopy groups in infinite number of dimensions. So modp homotopies must vanish from the dimension d for which the Adams map vl : Md+q(Z/pZ) * M d ( Z / p Z ) exists, namely, if p is odd, for d > 3 [CN]. This proves that odd primes, P n v ( 1 ) K ( Z / p Z , 3) ~ *. This gives the result for n = 1. For higher dimensions one proceeds similarly by induction on the construction of V ( n + 1) out of V(n). C.6 CLASSIFYING SPACES. It is not hard to see directly that Milnor's classifying space construction leads to a description of BG, for any groupspace G, as a Gcellular space, i.e. B G E C'(G). But this fact is a direct corollary of (A.2) above. In fact, one can prove: PEGBG ~ *. To this end use (A.1) to get P ~ G B G = W P G ~ B G ~ W P G G ~ W { . } = {.}. Therefore B G = C W G ( B G ) as needed. In particular, K(G, n + k) is a K(G, n)space for any k > 1. Moreover, B G is always a EGspace since (using A.2)
C W ~ G B G ~ B C W G ~ B G = B C W G G = BG.
From this observation we get also C.7 COROLLARY: A n y connected space X in S., is in C'(E~X). In fact it is not difficult to write for a group object G 6 S., the classifying space W G is EGcellular. We use the pointed Borel construction of the suspension of the group G. While G has no fixed point with respect to the selfaction by left multiplication, the suspension has two fixed points and so we can take the pointed homotopy colimit of the suspension EG. It takes a bit of technique to see that we get again the classifying space of G:
W G = E G ~(v EG.
(see (9.D.3) below).
C.8 PROPOSITION: For all n, k >_ O, K(G, n + k) is a K(Z, n)cellular space. Proo~ Since K(G, m + 1) = B K ( G , m) it is sufficient to prove that K(G, n) is always K(Z,n)cellular. For n = 1 it is clear, since any connected X 6 ,9. is
70
3. Commutation rules
SLcellular. For n > 1, the group G is abelian and thus we have a fibration
K(F, n) ~ K(F', n) ~ K(G, n) with F and F t free abelian groups. Therefore, by (2.D.11) above we can assume that G is a free abelian group. But then we can write K(F, n) as a homotopy limit of an increasing sequence of K(Fc~, n) for free abelian subgroups Fa. So we are done by a transfinite induction argument using (2.D.17) to deduce that, if K(F, n) is K(Z, n)cellular, then so is K ( F @ Z, n) ~" K(F, n) x K(Z, n). The following example shows that the sequence in (B.2) is not always a fibration sequence: C.9 EXAMPLE.
CWK(z/pZ,1)K(Z/p2Z,1) = K(Z/pZ, 1).
Proo~ Consider the map g: Z/pZ * Z/p2Z of abelian groups 1 * p. This is a generator of Hom(Z/pZ, Z/p2Z) ~: Z/pZ, and it induces a map on the classifying spaces: Bg: K(Z/pZ, 1) ~ K(Z/p2Z, 1). Since the source is clearly a K(Z/pZ, 1)cellular space it is sufficient to show that Bg induces a homotopy equivalence, namely map.(K(Z/pZ, 1), Bg), on the pointed function complexes. But the pointed function complex is homotopically discrete with
map,(K(Z/pZ, 1), K(G, 1)) = Hom(Z/pZ, G). Therefore the above map Bg gives us the correct CWAapproximation for A K(Z/pZ, 1). In view of (1.H.5) we see that the sequence in (B.2) is not always a fibration sequence. C.10 COROLLARY: IrA = K(Z/p~Z,n) and X = K(Z/p~Z,n), then
C W A X =
A
if k < 2,
X
if k >~.
Proo~ This is clear using the above together with the fibration theorem (2.D.11). Thus the fibration
K(Z/p2Z, n) xp K(Z/p2Z, n)
, K(Z/pZ, n) x K ( Z / p Z n + 1)
by (C.8) and (C.9) above presents the space K(Z/pZ, n) as a K(Z/p2Z, n)cellular space.
3. Commutation rules
71
C.11 EXAMPLE Let X = M~+l(p ~) and A = M'~+l(p) be two Moore spaces, with Hn(M~+I(pe), Z) = Z/peZ. Then C W A X is a fibre in:
F
while E X
= CW
, X
, K ( Z / p e  l Z , n),
A ~X.
Proof: Compare [B12, 3.1] To compute the fibre of the composition X , K(TrnX, n) , K ( Z / p e  I Z , n) as C W A X we consider the pointed function complex of M~+l(p) into the fibration. Since map,(Mn+l(p), K(Z/peZ, n)) ~ Z / p Z is homotopically discrete by cohomological computation, we first notice that the fibre has the correct function complex from M~+l(p). We then must show that the fibre is a M'~+l(p)cellular space. But the fibre is a ptorsion space so it has a HiltonEckmann cell decomposition M ~+1 (p) U C M ~+2 (Hn+ 1(F), n + 1) U . . . where all the attaching maps can be taken to be pointed maps. Now we can use Lemma C.12 below and (B.3) above to conclude that the Moore space M~+2(H,~+I(F, Z)) is cellular with respect to M ~+1 (p) = M TM (Z/pZ). This gives a direct representation of F as M~+l(p)cellular since clearly C.12 LEMMA: For any pgroup G the Moore space M'~+J(G,n + j) for j >_ 2 is an M ~+l (p )space.
Proof." Use Proposition B.3. Notice that P~M~+I(p)Mn+J(G,n + j) ~ ,
since the localization is an nconnected ptorsion space with all maps from M n+l (p) being null, hence this localization is contractible. C.13 E,ACYCLIC SPACES. The fibration (B.2) relating PEA x and C W A X can be used to show that certain E,acyclic spaces are V(n)cellular, where V(n) are the spaces introduced by [Sm1],[Mit] (see also [R]). One can use the following observation:
C.14 OBSERVATION: Let A be a finite complex with E , A ~ O. Then for all X the space C W A X is the direct limit of its finite E,acyclic subcomplexes. C.14.1 REMARK: By definition there is a construction of the given acyclic space from A by repeated homotopy colimits. Recall that any pointed homotopy colimit of E,acyclic spaces is again E,acyclic. The point of the observation is, however, that here there is a direct system of finite E,acyclic spaces whose homotopy colimit is equivalent to the given space C W A X .
72
3. Commutation rules
Proof." Recall the construction of C W A X . For a finite A the limit ordinal A(A) is the first infinite ordinal w. Therefore in that case
C W A X = lira (X1 ~~ X2 ~* Xi "+), i~oo
where Xi are all subcomplexes of CWAX. But now, by induction, we can show that each Xi is the limit of a finite E.acyclic subcomplex. Notice that if A is any E.acyclic space then so is the halfsuspension ~ n A = S ~ >4A = S n • A / S '~ • {*} by a MayerVietoris argument. Now if by induction X j = limA(i), where A(i) are c~
finite E.acyclic, then since Xj+I is a pushout along a collection of maps from ~,nA, Xj+I is again l i m A ~ ( j + 1) where A ( j + 1) are all finite. This completes the proof. Our principal tool to detect whether an E.acyclic complex X is the limit of its finite E.acyclic subcomplexes is the following: C.15 PROPOSITION: Let A be an E.acyclic t~nite complex. Then X is the timit o[ its finite acyclic subcomplexes i f P ~ A X "~ *, Proo~
This is immediate from Theorem B.2 and (B.3) and the lemma above.
It is possible to apply the proposition to the spaces A = V(n) of type n + 1. Thus V(0) is S 1 Up e 2, and for every prime p and n > 0 there exists a finite ptorsion space V(n) of type n. This means ~ [ ( m ) . V ( n ) = 0 for all m < n and [ ( ( m ) . V ( n ) ~ 0 for all m > n, where K ( n ) denotes the nth Morava Ktheory (compare discussion in [MT], [B4], [R] and [FS]). In Chapter 8 we apply this proposition to detect under what conditions an acyclic space can be constructed from elementary (and finite) ones using [B4, 9.14 and 13.6]. C.16 REMARK: Using similar techniques one can show (compare 8.B): In the following cases every E.acyclic space is in C ( V ( n ) ) for an appropriate n > 0: (1) For all n there exist m > n with K ( G , m + j ) E C'V(n) for all j, and all ptorsion groups G. (2) If I ( c ~ X ~ 0 then X 9 C ( Y ( n ) ) for any ptorsion 2connected X, see (C.5) above. (3) For every n > 1 there exists N _> n so that if X is Nconnected, ptorsion and S(n).~Nx = 0, then X E C ( V ( n ) ) where S(n). is the homology theory from
[B4]. We will not give the proofs here.
I
3. Commutation rules
73
D. L o c a l i z a t i o n Lf a n d cofibrations, f i b r a t i o n s In this section we will consider some simple cases where L f preserves fibrations and cofibrations. Later on, in Chapter 5, we will consider a more general theory where fibrations are 'almost' preserved by L~f. In general there is little hope that L f will preserve cofibrations since the cofibre of a map between tw o flocal spaces is very rarely flocal. This happens more often in the stable category, where sometimes homotopy colimits of E.local spaces are E,local for certain ('smashing') homology theories E.. Nevertheless the following is often useful: D.1 THEOREM: Let L f be the localization with respect to a map f E S.. A ~, X J, X U C A b e a c o f i b r a t i o n s e q u e n c e . (i) I f L f A " 9 then L I ( j ) is a weak homotopy equivalence. (ii) I f L f ( i ) is a weak equivalence then L f ( Z U CA) ~ *.
Let
Proof." This really follows directly from the general formula for the localization of homotopy colimits (1.D.3) above, but it may be worthwhile to give a direct argument in this simple case. Consider the cofibration A "* X , X / A . We have a factorization of the map into a colimit of the diagram C A ~ A ~ X given by X / A , LfX/LfA , L f ( X / A ) with the composition being the coaugmentation of Lf. Now we assume that L f A * L f X is a homotopy equivalence so that L f X / L I A is contractible and X / A , L f ( X / A ) is null homotopic. By the idempotency of L f we get L y ( X / A ) ~  . , as needed. Now assume that L I A _ .. Then we have an equivalence L y X ~, L f X / L f A . Therefore the natural map X / A , L f ( X / A ) factors through a new map X / A , L f X . By universality we get a map L f ( X / A ) , L f X , which is easily seen to be a homotopy equivalence as required. D. 1.1 FURTHER CASES: The above results do not treat the more difficult case when we assume L f X ~_ L f X / A ~ 9 and try to conclude something about L f A . As in the analogous case of fibration this can be shown, for nullification with respect to suspension, to be manageable. See (5.B.4.1).
F i b r a t i o n s : We now formulate a general theorem about preservation of fibration by Lf. Later on we will find special cases of its usefulness. In what follows we denote by L f and P w the homotopy fibre of the coaugmentation maps.
74
3. Commutation rules
D.2 THEOREM: Let F , E P, X be a/ibration with connected F , E , X E S.. Assume that L ~ / X ~ L f X and that L2 f L~ f X is homotopically discrete and further that L f L / E ~_ .. Then L:F. L:E
L:p, L : X
is also a/ibration sequence. The following are particularly useful special cases: D.3 COROLLARY: Consider the case: L f = P w :or any space W, (1) I : P 2 w X = P w X then P w preserves the ~bration sequence, i.e. P w F PwE , P w X is a ~bration sequence. (2) If X is Wnull then P w preserves the ~bration sequence. Proos
Case (2) is a special case of (1).
In case (1) one uses the fact that
P w P w X ~ * for any A , X . (See (1.H.1) and (1.H.2).) Therefore the assumptions of Theorem D.2 are satisfied and the fibration is preserved by P w . Proof os D.2: Consider the following commutative diagram of fibrations built by taking homotopy fibres from the lower right square: ~:X
1 I
f~X
~L~fX
> T
~
1 I
~ F
)
G
~
~
~:E
~
Z:X
E
l I
~
X
LfE
~
LfX
= E~/X
1 I = LEfX
Looking at the diagram we see that it suffices to show that F , G induces an equivalence L f F ~, L f G since G, being the homotopy fibre of a map between two flocal spaces, is flocal (1.A.8(e.3)). We first show that G is connected. Since by elementary fact (1.A.8) (e.ll) G is the homotopy fibre of a map between two connected spaces, it is sufficient to show that Lf(p) induces a surjective map on the fundamental group. Since E f is a map of simply connected spaces, the map induced by the coaugmentation ~hX ~ ~ h L ~ f X is surjective. Since the map p itself also induces a surjective map on fundamental groups, because the fibre F is connected, we are done by commutativity. Since G is connected we can apply (1.H.1) to the fibration F ~
F ~ G,
to show that L / F ~ L f G ~ G since G is flocal by (e.3). For this we need to prove LfF ~ .. Our assumptions are designed to achieve just this. Turning to the fibration sequence over L / E we notice that this space is connected by (e.ll), since L I L L E ~ 9 by assumption. Therefore by (1.H.1) again it is sufficient to show
3. Commutation rules
75
~ , , but by commutation (A.1) above the latter localization is equivalent to ~ L E f L ~ . f X , which we assume by homotopy discreteness to be contractible.
Lf~L~fX
D.4 REMARK In view of (3.A.1), the condition L ~ B = ~2LB is equivalent for a connected space B to L ~ f B ~ L f B . Notice that this last condition is hardly ever true unless B is already flocal. There is one exception, however. If f : S 1 * S 1 is a selfmap of the circle, then for a simply connected space X it is clear L ~ f X . In that case L f is the localization at the subring E l l ] if deg f = the classical result that the SullivanQuillen rationalization functor X preserves any fibration of simply connected spaces is a special case of (D.2) above.
that L f X = r. Therefore + X  7/,[1] the (D.3) or
E. C W A a n d f i b r a t i o n s The commutation C W A ~ X '~ ~ C W I E A X given in (A.3) allows one to prove theorems about preservation of fibration by C W A under strong assumptions. Here too we will see in Chapter 5 that by allowing 'error terms' or 'near preservation' one can prove much stronger results with only mild assumptions on the given fibration and A. We will need the following: E.1 LEMMA: In any fibration sequence F , E , X in S , , i f X and E are in C'(EA) then F E C'(A). I f F and X are in C'(EA) then E E C'(A). Proof." This is immediate from (2.D.12) and (A.2) above by backing up the fibration. E.2 THEOREM : Given any fibration o f a pointed space F m a p the following fibration into it:
, E
, B one can
m
CWAF
* E
* CW~.AB
F
,E
,B
where E is Acellular and g a ~Aequivalence.
E.3 REMARK: Thus although E is Acellular it is slightly removed from being
CWAE, since it is only ~Aequivalent to E not Aequivalent to it. We shall see in Chapter 5 that the fibre of a canonical map E * CWAE fibration is a GEM for A = ~ A I, a suspension space.
associated to such a
Proo~ The proof is easy for principal fibrations with the fibre over the base point being the group F = G. One first pulls back the fibration o v e r CW~AB to get a principal fibration with the same fibre G = F over the needed base space. This
76
3. Commutation rules
last fibration is classified by a map C W E A B + W G to the classifying space of G. But this last classifying map can be factored uniquely up to homotopy through C W E A W G * W G by the universal properties of C W ~ A . The domain of this last map is by (A.2), the classifying space of C W A G . Therefore we have obtained a fibration over C W E A B with the fibre C W A G that maps naturally up to homotopy to the given fibration. The general case is similar, but we need to use the usual technique of expressing the base of a principal fibration as a realization of a bar construction associated with it [B4],[B3, 5.5]. We give only the outline. Here we also use both the equivalence C W A ~ t X ~ ~ t C W ~ A X and the natural equivalence (2.E.8):
CWA(X x
Y ) ,'., C W A X X C W AY~
We consider first the associated principal fibration:
~tB
*F
,E
where we consider E as the 'quotient space' up to homotopy of F under the action ~tB • F . F. This action gives rise to a map
CWA(~B) • CWAF
* CWAF,
, C W A F . We now use the equivalence as usual to show or ~ t C W E A B • C W A F that the original action gtB • F * F gives rise to an action of ~ C W E A B on CWA_F. Therefore we get a principal fibration
~CWEAB
* CWAF
,, E
associated to that action. Classifying this fibration gives us the desired sequence. Since in that last fibration both fibre and total spaces are Acellular, so is E by (2.D.11). Further, since in the map of fibrations ~ h is an Aequivalence and f is an Aequivalence, we get by the usual long exact sequence for Ahomotopy groups [ E e A ,  ] . that g is an EAequivalence. This completes the proof.
3. Commutation rules
77
E.4 CGROLLARY: Let F
, E
P, B be any fibration sequence in S. and let E2A
be any double suspension in S.. If B is E2Acel]ular and if p induces the trivial map on homotopy classes from EA, namely lEA, p] ~ *, then
CWEAF
*
CWEAE
* B = CWEAB = CWE2AB
is also a fibration sequence. E.5 EXAMPLE: Take A = S 1. Then (E.4) states the easily checked fact that, over a 2connected space B, one can define a fibrewise universal covering space. Proof orE.4:
We consider the diagram: y
CWEAF
=
, g
,
,
C W E 2 A B  C W E A B
1 F
,E
,B
This diagram is constructed using the theorem above. We claim that there is a natural equivalence: ~: E , C W E A E . First notice that since B is E2Acellular we get from Lemma E.1 that E is EAcellular. Therefore there is a unique natural factorization ~. To complete the proof one must check that ~ is a homotopy equivalence. To this end we use the AWhitehead theorem and check that the map
map.(EA,~): map.(EA, E)
.
map.(EA, C W A E )
is a homotopy equivalence. Notice that map.(EA, Y)  * since Y is the fibre of f and map. (EA, f) is a homotopy equivalence. We know from the universal property that f is a EAequivalence. Thus map.(EA, Y) ~ * since Y is the fibre of a EAequivalence f. Now this shows that the homotopy fibre of the map map.(EA,~) is contractible. This is not enough since the base space map.(EA, C W A E ) is not connected. To complete the proof one must show that the induced map of function complexes on the level of components, i.e. on zc0, is an isomorphism. Since all the components of function complexes out of suspensions are equivalent to each other. Our assumption guarantees that it is surjective on components since it implies by a
78
3. C o m m u t a t i o n rules
simple d i a g r a m chase t h a t any m a p from ~ A to E lifts up to h o m o t o p y to a m a p ~A
* E. Injectivity follows from the abovementioned p r o p e r t y of Y.
4. D O L D  T H O M S Y M M E T R I C P R O D U C T S AND OTHER COLIMITS Introduction
One of the main results of this chapter says that if X is an infinite loop space that is equivalent (as such) to a possibly infinite (weak) product of K(II, n)'s with H an abelian group and n > 0 (here called GEM), then so are both L / X and CWAX for any f: A * B and A E S.. Moreover, these functorial constructions inherit the abelian group structure that X possesses as a GEM and the coaugmentation and augmentation maps are essentially GEM maps, namely up to homotopy they can be realized as homomorphisms in the category of simplicial abelian groups. This implies immediately that when applying L / o r C W A to an EilenbergMac Lane space K(G, n) with G an abelian group, the resulting GEM has at most two nonvanishing homotopy groups and they are in adjacent dimensions. These results would follow immediately if we would have a version of functors L / a n d C W A that strictly commutes with products since then the functors would turn (simplicial) abelian groups into such and the (co)augmentation map would be a group map. So part of the following development is there because the present version of these functors commutes with finite products only up to homotopy. This property of idempotent functors is a step towards a better understanding of the behaviour of fibration sequences under L / , C W A and, in particular, it will lead to an understanding of the behaviour of fibrations under Bousfield's homological localizations for any generalized homology theory. The above result is proved by presenting the symmetric product on X as a homotopy colimit of a certain diagram consisting of finite powers of X even though they are defined as strict direct limits (quotient spaces) of some finite power of X. The question then arises about the general relation between the colimit and the homotopy colimit of diagrams of (pointed) spaces. This leads to the second concern in this chapter: We will 'estimate' the difference between the homotopy colimit (pointed) and the strict colimit of pointed diagrams in S.. We do that by estimating the cofibre of the natural map hocolim. X   § colim X. By an 'estimate' we mean a cellular inequality of the type X << (2.D.2.1) or ~,2A << cofibre(hocolim. X
SP~X
* colimX).
Namely, the symmetric products on X are Xcellular and the above cofibre is ~2Acellular.
80
4. Symmetric products
This inequality immediately implies, for example, that the said cofibre is always 1connected because double suspensions are always 1connected and cellular equalities preserve connectivity. A. D o l d  T h o m s y m m e t r i c p r o d u c t s as h o m o t o p y colimits Given an Acellular space X equipped with an action by a group G, we may ask under what condition the quotient space X / G = colimX is also Acellular. G Before entering the discussion let us recall the concept of cellular inequality, which is clearly a partial order on spaces. A.1 RECALL (2.D.2.1): We say that X is built from A and write A << X if X is Acellular. A more extensive discussion of << is given in Chapter 8 below. Our main result is not unexpected: We start with a topological or simplicial group G for our purpose it is sufficient to consider a discrete, finite group and a Gspace X that has at least one Gfixed point. We assume that the action preserves the cell structure of the space X. A.2 THEOREM: Let X be a pointed Gspace. Assume that for all H C G the subspace X H of Hfixed points in X is Acellular. Then so is the quotient space
X/G. Remark: Thus in the above notation the theorem says that if, for all H C G, A << X H then A << X / G . In particular we can apply this to symmetric products. In this partial order notation we now deduce that for all X one has the inequality X << S P k X . In the following we denote by Sk the symmetric group on k letters. A.2.1 COROLLARY: Let X be anypointed space. For all 0 < k < oo the kfold symmetric product S P k X = X k /Sk is Xcellular space. A.2.2 COROLLARY: If X is an E.acyclic space where E. is any homology theory then so is the kfold symmetric product on X for all 0 < k < co. Proof of Corollaries A.2.1 and A.2.2: The second follows immediately from examples (3.D.2.1) and (3.D.2.5). As for the first, by the theorem above one need only check that for any subgroup H _ ~k of the permutation group ~k, the space (Xk) H is Xcellular. But since H operates on the factors of X k, it either leaves a whole factor in its place or moves it wholly to another copy of X. Thus it is clear that all the fixed point subspaces are also product X x . . . x X (ltimes) for some l < k. But we saw in (2.D.16) above that X ~ is Xcellular for any X. The corollary follows. A.2.3 EXAMPLE: Thus as defined the space S P 2 X is a quotient of X x X under the action of W.2, the permutation group on two letters. In other words, it is the strict colimit of a diagram {X x X; r} with one space and one nonidentity selfmap. But the argument proving (A.2) given below presents the same symmetric
4. Symmetric products
81
product, up to homotopy equivalence, as a homotopy colimit of a diagram consisting of two spaces:
{x
,X•
and two nonidentity maps, one the diagonal 6 from X to X • X the other being the above selfmap T of X x X permuting the two factors. The essential difference between these two diagrams is that the second one is free as a diagram (Appendix HL) while the first one is not. These two diagrams have the same strict colimit, S P ~ X , but only in the latter is this colimit equivalent to the homotopy colimit of the same diagram. Proof of Theorem A.2: One shows that there is a pointed diagram of Acellular spaces whose pointed homotopy colimit is X / G . The diagram is assembled from the subspaces of fixed points of all the subgroups of G ([DF1], [DK2] and Appendix HL below), namely one first considers the category O = O(G) of G orbits. This is simply the small category of all the quotients G / H for H C_ G taken as Gsets with the natural left action of G. The morphism sets in that category O are the Gmaps of these Gsets. We get the usual diagram of fixed point s p a c e s { X H } H E O de_fx O . Since X has a Gfixed point, this point is common to all the subspaces X H, thus the diagram is pointed, i.e. it is a functor X ~ : 0 ~ , S. from the opposite category of orbits O ~ to pointed spaces. Notice the canonical equality X H = h o m a ( G / H , X ) . We now observe that colimX ~ the direct limit of the diagram of fixed points, is in Oop
fact isomorphic to X / G , since any two points identified by an action of a subgroup are certainly identified by the action of G itself. On the other hand, we claim that for the diagram X ~ the natural map
hocolim. X ~ Oop
~ colimX ~ O~
is a homotopy equivalence. This is the content of Lemma A.3 below. Given that lemma, our proof is finished since we assume that the diagram X ~ over the orbit category of G consists of pointed Acellular spaces, therefore by definition the pointed homotopy colimit hocolim. X ~ is also an Acellular space and, by the lemma below, this homotopy O(a)op colimit is homotopy equivalent to X / G . The following lemma compares pointed and unpointed hocolim with strict colim:
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4. Symmetric products
A.3 LEMMA: Let X be a pointed Gspace. Then the natural maps
colim X ~ 9 o(a)o,
hocolimX ~ o(a)o,
. hocolim. X ~ o(a)o,
are homotopy equivalences. Proo~
As for the map on the left from hoeolimX ~ to. colimX ~ it is a homotopy I equivalence since the diagram of fixed points is a free O(G)~ of spaces: In general, for any free diagram Y of (cofibrant) spaces over any small category I the canonical map from the homotopy colimit to the colimit itself is a weak homotopy
equivalence. Compare [DF1], [DK2], and Appendix HL. The map on the right is a homotopy equivalence because the nerve of O(G) is contractible, since O(G) has a terminal object G/G, the onepoint orbit. Therefore by (2.D.3) above and Appendix HL, the natural map from the unpointed hocolim to the pointed one is a homotopy equivalence. We now use the above argument to make a direct comparison between the pointed Borel construction E G Ha X and the quotient space X / G . A.4 T h e p o i n t e d B o r e l construction: One way to gain a better understanding of quotient spaces, such as symmetric products of X, is to consider the difference between the quotient space and the homotopy quotient space. The latter is just a special case of homotopy colimit: We just view the Gspace X as a pointed diagram over the usual small category with one object defined by G. Notice that if * is a base point in X, then the point (,, , , . . . , ,) is fixed by the action of any permutation group on the product X • X • ... x X. Therefore we are in a situation to take pointed homotopy colimits and we can apply it to spaces that have a Gfixed point. The pointed Borel construction (see Example 3.C.7 above) is just a special case of a pointed homotopy limit. We define E G Ha X , the pointed Borel construction on the pointed Gspace X, simply as the mapping cone of the natural cofibration: B G = E G x a {*} * E G x a X . Equivalently, it is (EG ~( X ) / G . A.4.1 EXAMPLE: For any group object G E S,, if one considers the left action of G on itself, it induces an action of G on the suspension EG, which makes the latter into a pointed Gspace. As we shall see below (9.D.3). the associated pointed Borel construction E G ~
* X/G
, C
4. Symmetric products
83
is E2 Acellular.
A central example for the above concerns the action of the symmetric group S~ on the power space X '~ = X • X • ... • X (ntimes). Here we consider X ~ as an S .  d i a g r a m and, if X has a base point (,), then (*, * , . . . , * ) is fixed under S~, thus X "~ is a pointed diagram. Since m a p s ~ ( e , X n) = X k for some k > 0, we have X << maps~(e , X ~) = (X'~) ~ for all the orbits e = S ~ / H with H C S~ of S~. Therefore, this special case says: A.5.1 COROLLARY : Let X be a connected space in S , . Then
2 2 X << cof(ES~ ~<s~ X ~
* X~/S~)
where X ~ / S ~ = S P ~ X is the D o l d  T h o m symmetric product of X . This result is just a cellular version of a similar result of Dwyer [Dw2]. It is not hard to see that under our assumption this natural map induces an isomorphism on the fundamental group, and surjection on Ir2. In fact the homotopy fibre is 1connected if X u is connected.
Remark:
Proof." The proof starts with Lemma A.3 above. We want to write both the homotopy colimit, i.e. the pointed Borel construction, and the actual orbit space X / G as the same kind of homotopy colimits. This allows one to write the cofibre also as a pointed homotopy colimit and it will be apparent from the latter presentation that we get a ~2Acellular space as needed. The kind of pointed homotopy colimits we need here is socalled coend, or rather homotopy coend (Appendix HC, [HV]). This is the basic construction that allows one to actually write down any Gspace as a homotopy colimit of a diagram that depends on the collection of fixedpoint subspaces as a diagram over the orbit category of G. In order to write the pointed Borel construction as well as X / G as a pointed homotopy colimit (more accurately as a homotopy coend), we start by writing the Gspace X itself as such as colimit. For this we recall a basic construction from the homotopy theory of Gspaces [DF1] (see also Appendix HC ). We can assume that X is a G  C W space in the usual sense and so X, being a Gspace, can be reconstructed from the diagram of fixed points { X H} = 0 ~ with 0 ~ = O(G) ~ the opposite category of Gorbits. Recall that X H = m a p c ( G / H , X), by definition, so X H is the 'space of all orbits in X of type G / H ' if we multiplyhalfsmashthis space by G / K , which we take now as Gspace (unpointed); we get the pointed Gspace, G / H ~4X H. Now whenever one has a diagram W over a category I and a pointed diagram V over I~ the opposite
84
4. Symmetric products
category, one can form the coend by gluing together half smashes W(i) ~< V(j) = W+(i) A V(j). (See Appendix HC.) NOTATION: We denote the pointed (homotopy) coend of W and V by
/ W ~
This double integral should remind the reader that there are two diagrams involved in its formation. Notice (Appendix HC) also that one can write a general homotopy colimit as a homotopy coend by
f V=//,~V. I
I~
In our case we take the coend of the diagram of all the orbits G/Httc_ G thought of now as a diagram of Gspaces over the orbit category O. We denote this diagram by 27o since it is, by inclusion, a functor from Gorbits to Gspaces, assigning to every orbit thought of as an object in O the orbit itseff thought of as a Gspace. Denote by X ~ the diagram over 0 ~ consisting of the fixedpoint spaces. We can sum up the above discussion by:
A.5.2 PROPOSITION: There is a natural Gequivalence
/ Io ~<X ~
~X
0
from the coend to X. Similarly one can express X/G as a coend as follows:
A.5.3 PROPOSITION: There is a natural Gequivalence
f * ~<X ~ ~'~, X/G o
from the coend to X/G. Proof'. This is clear from (A.3) above. Now we can use this to express the pointed Borel construction as follows:
4. Symmetric products
85
We need the diagram of classifying spaces {BH}Hc_G. This diagram can be viewed as an Odiagram, since there is a canonical homotopy equivalence B H ~
EG ~a G/H. We denote this diagram by {Bh}. A.6 CLAIM: There is a homotopy equivalence
EG
~
~ X H
o
where by {Bh} we mean the Odiagram of the classifying spaces { B H } as above. Proof" One simply substitutes the expression for X as a coend of the fixedpoint sets in the pointed Borel construction and use the commutation of homotopy colimits: ff EG ~
0
0
where in the third term of the equation the values of the diagrams on typical orbits G / K and G / H are displayed. We have used the usual expression for B K , the classifying space on K, as the (unpointed) Borel construction on G / K . We have also used the natural equivalence
A~<(B~<X)~(A•
for a pointed X and unpointed A, B. This is a special case of the commutation rule for pointed and unpointed hocolim (2.D.6):
A
To continue the proof:
86
4. Symmetric products
CLAIM: There is a natural cofibration (as follows) associated with every pointed Gspace given in terms of homotopy coends, see Appendix HC below:
(A.6.1)
//{Bh}
~<X ~
, //pt
0
~<X ~
, //{Bh}
o
*X ~
0
which is naturally equivalent term for term to the natural sequence in the theorem, so that the homotopy coend on the right is equivalent to C from (A.5). Proof of Claim: By claim A.6 above the first homotopy coend term in (A.6.1) on the left is E G ~
~T
, S*T
where S * T denotes here the reduced join, i.e., it is obtained from the join by collapsing a copy of the cone over S in the usual join to a point, and thus is equivalent to the usual join. P r o o f This follows directly from the simple observation that the mapping cone of the projection S • T , T is T * S V ES. This is clear from the following: If S is unpointed space and T a pointed one, one gets an immediate cofibration square built from the upper left corner square by taking cofibres: S
1
, *
'ES
1
1
SxT
, T
S~T
, T
, S,TVES
, S,T
where the space T 9T is best thought of as the reduced join, and is pointed. This completes the proof of the observation. Therefore one has a homotopy cofibration sequence of pointed spaces:
BH
~< X K
~ X K
, BH
9X K
4. Symmetric products
87
for every two orbits G / H , G / K E 0 = O(G). But taking pointed homotopy colimits  and pointed homotopy coends in particular  commutes with taking mapping cones, which are, themselves homotopy colimit. Thus we get the desired cofibration above by taking the homotopy coend over O. This concludes the identification of the coend on the right in the cofibration (A.6.1) with C in the theorem. This presents C as a pointed homotopy coend and thus as a pointed homotopy colimit of a diagram consisting of the joins. To conclude the proof of the theorem it is sufficient to show that for all H the join space B H * X H ~ S 1 A B H A X u is G2Acellular, since by definition any pointed homotopy colimit on E2Acellular spaces is E2Acellular. But S 2 = S 1 A S 1 << S 1 A B H since B H is connected. By assumption one has A << X H for all H G G. Therefore one can conclude by (2.D.8) that
E2A~ S 2 A A << S 2 A X H << S ~ A B H ^
as needed. This completes the proof.
B. Localization
and cellularization
X H ~ B H * X
H,

of GEMs
In this section we will start to consider the relations between generalized EilenbergMac Lane space localization and cellular approximations. To facilitate the discussion we consider here pointed, connected spaces since on them one has a better control on the infinite symmetric product. We prove that both L / a n d C W A turns a GEM into a GEM. In fact, the coaugmentation and augmentation maps can be thought of, up to homotopy, as group maps in the category of simplicial abelian groups. Furthermore we will see that, when applied to an EilenbergMac Lane space K(~r, n) with ~r abelian, both these functors produce a product of at most two EilenbergMac Lane spaces in adjacent dimensions. B.1 DEFINITION: Compare [SV, 2.2 (iii)]. B y a G E M space we mean an infinite loop space that is a generalized EilenbergMac Lane space as a loop space, i.e. it is equivalent as an infinite loop space to a (possibly infinite, weak) product o f K ( G , n) 's taken as abelian group spaces for discrete abelian groups G and for n > O. A map G1
, G2 between two G E M spaces is called a G E M map i f it is equivalent to a
group h o m o m o r p h i s m between the corresponding abelian group spaces.
B.I.1 REMARK: A homotopy characterization of these structures appears in ISV]. We see from the definition that the homotopy category of GEM spaces and GEM maps is equivalent to the homotopy category of simplicial abelian groups with group maps as morphisms. The latter is in turn equivalent to the homotopy category of
88
4. Symmetric products
chain complexes of abelian groups [Mayl]. Thus, for example, if there is a nonnull , K(G', j ) then j is either i or i + 1. GEM map K(G, i) Our aim first is to show that L I ( G E M )  GEM and C W A ( G E M ) = GEM, i.e. the functors L f, C W A turn GEM spaces into (generally different) GEM spaces. (Compare [BK, VI.2.2] and [B5].) Our main tool in analyzing the effect of L f and C W A on a GEM is DoldThorn symmetric products [DTh], [BK]. Recall that the infinite symmetric product S P ~ 1 7 6on a pointed connected space X is just the direct limit of S P k X = X k / S k . This direct limit is clearly also a pointed homotopy colimit. Thus
S P ~ 1 7 6= hocolim. S P k X .
It is well known that S P ~ 1 7 6is a GEM. In fact it is homotopy equivalent to the infinite product H K(H~(X, Z), i). Moreover, simplicially it can be built as the O
free abelian (simplicial) group on the simplicial set X which we denote here as ZX. Thus we have a coaugmented functor X ~ S P ~ 1 7 6 which is a stepping stone for many of the constructions and theorems of [BK]. This functor is in a weak sense the universal GEM to which X is mapped: It is easy to see that up to homotopy any map of any space X to a GEM space X ~ GEM factors (nonuniquely) up to homotopy through X ~ S P ~ 1 7 6 F~rther, it is well known that S P ~ X has the following universal property: B.2 THEOREM ( G E M s AS RETRACTS): I f X is a retract up to homotopy of a GEM space G then X is the underlying space of a GEM. Furthermore, any map X * F in $ of X to a GEM space F factors up to homotopy through the natural coaugmentation map X , SP~ of X to the canonical GEM space associated with X . Proof'.
The theorem reduces, as we shall see, to the following lemma.
B.2.1 LEMMA A space X is the underlying space of a G E M if and only if the natural map X , S P ~ 1 7 6has a left homotopy inverse. Proof os 1emma: (Arguing simplicially) If X is the underlying space of a GEM there exist a simplicial abelian group A that is equivalent to X in S,.q. But being an
abelian group there is an obvious (strict) left inverse to A , ZAL Hence we have a homotopy inverse for X. Now suppose the above map has a homotopy inverse. In that case the homotopy groups inject into the integral homology groups since [Mayl], [DTh], [BK] 7riZX ~ Hi(X, E) is a natural isomorphism. One deduces easily using the universal coefficient theorem for cohomology with coefficients in the homotopy groups of X that there is a weak equivalence X , II0<~
4. Symmetric products
89
by constructing maps to the individual factors. ( Compare [Mo, 3.29].) This follows from the fact that given an element u E H k ( X , G) represented by a map X * K(G, k) the induced map on the kth homotopy groups factors through the induced map on homology via the Hurewicz map and the image of the element u in the group H o m ( H k ( Z , Z), G).  Using this Lemma we now give:
Proof of B.2: If X is a retract of a GEM G we can construct the following ladder using (B.2.1): X
~G
ZX
, ZG
,X
, ZX.
Where the retraction in the middle arrow is guaranteed by the lemma and it gives the a section of X * ZX. Thus, again by the lemma above, X is an underlying space of a GEM. The second claim of the theorem is an immediate consequence. The two main theorems of this section are B.3 THEOREM: Let X , A E S.. Assume X is a GEM. Then C W A X is also a GEM. If, in addition, 7fiX ~ 0 for i > N, then ~riCW A X ~ 0 for i >_ N . Moreover, the augmentation map j : C W A X ~ X is, up to homotopy, a retract of SP~176 Furthermore, the same holds for the homotopy idempotent functor PA, i.e. the homotopy fibre of nullification (2.A.3.1) (2)above. B.4 THEOREM: Let f: A
, B, X E S.. Assume X is a GEM. Then L I X is also a GEM. If, in addition, lriX ~ 0 for i <_ N, then ~riLfX  0 for i <_ N . Moreover, the coaugmentation map j : X ~ L f X is, up to homotopy, a retract of SP~176 B.4.1 COROLLARY: For any K ( r , n ) with zr abelian group, the homotopy groups of CWAK(Tr,n) vanish in all dimensions except possibly n  1 and n; the homotopy groups of L f K ( zr, n )) vanish in all dimensions except possibly n and n + 1.
Proof of corollary: (Compare [BK, VI.2.2] This follows at once from Remark B.1.1 above and (1.C.4) since the projection of the (co)augmentation map to any other EilenbergMac Lane factors must be null homotopic.  Both proofs rely heavily on (4.A.2.1) above, namely on X << S P k X for all 0 < k < oc. In both (B.3) and (B.4) the vanishing of ~r~ follows easily from universality.
90
4. Symmetric products
Proof o[B.3: Let X , SP~X ~, X be the inclusion followed by the homotopy retraction guaranteed by the above (B.2) characterization of the GEM space X. We now show that the natural inclusion C W A X ~ SP~176 admits a homotopy left inverse S P ~ C W A X * C W A X . This left inverse will be given here as a composition
SP~CWAX
~ ~ CWASPOoX
CWAr, CWAX
'
where the map on the right is C W A applied to the given retraction r, and the commutation map r ~ is given as follows: B.5 CLAIM: For any 0 < k < oc there is a natural transformation in ho$,,
Tk: s p k C W A
. C W A S P k.
These transformations are compatible in k.
For any given space X the map T k : S P k C W A X * C W A S p k x is given as follows: If k is finite we notice that, since S P k Y is Ycellular for all Y and since C W A X is Acellular, we get that S P k C W A X is Acellular. Therefore by universality of C W A a map ~_k is given, uniquely up to homotopy, by the factorization of the map S P k C W A X * S P k X , which we take a s s p k ( C W A X * X ) . Since all maps we took are natural or uniquely determined by such up to homotopy, and are compatible with the inclusions S P k X ~~ S P k + I X , the claim follows. Proof
Now the claim of (B.3) is a special (k = co) case of the following CLAIM: For any 0 < k < oc, if X is a homotopy retract of S P k X then so is C W A X . P r o o f The map follows:
Tk
just defined fits into a homotopy commutative diagram as
Z
J
, SPkX
r
~ X
f
(B.5.1)
SPkCW AX
CWAX
cwa(j)
CWASPkX
CWA(r), CWAX
4. Symmetric products
91
Notice, e.g., that the two compositions CWA(r)o'rkoi and CWA(r)oCWA(j) = C W A ( i d ) = id are homotopic, since they are homotopic after composition into X and factorization of (2.E.8) is unique up to homotopy. For a similar reason T k o i  C W A (j). Thus i has a left inverse as needed, since C W A (r) o C W A (j) ~ 1. To continue with the proof, since we assume the vanishing of the homotopy groups above dimension N,the vanishing of higher homotopy groups of C W A X is now a simple consequence of universality or idempotency: Since the augmentation map to X is universal, if it is null on any retract W of CWAX, then W ~ *. Compare (1.C.4) above. Since we now know that C W A X is a GEM, every homotopy group of this space is carried by an appropriate EilenbergMac Lane space that is a retract of C W A X itself. But all maps from higher EilenbergMac Lane spaces to our X are null, since by assumption this GEM is concentrated in lower dimensions. Hence these higher groups must vanish as needed. Dually, it follows that the lower homotopy groups of L f X vanish, as needed. Now the proof for the augmented functor PA is similar using universality and the consequence of (A.2.1), that if PA kills a space X it also kills any symmetric product of X. Since the above proof constructs a natural retraction, it presents by the same token the augmentation map as a retract of the symmetric product on itself.
Proof orB.4: The proof here is similar to the proof of (B.3) above. We will only , L I X for any 0 < k < oc, given a retraction give the retraction R : SPkLyX X
9s p k x
~* X.
Again R is given as a composition
SP~LyX ~ , L f S P k X
t(~), LyX.
Here T k is defined as a composition
SPkLyX ~, LySPkLfX ~_ LfSPkX. We now explain the homotopy equivalence. This is not immediate since as it stands SP k is a direct limit construction, not a homotopy direct limit.
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4. Symmetric products
B.6 PROPOSITION: For any connected space X there is an equivalence which is natural up to homotopy:
L f S P k L f X ~_ L f S P k X . Proof," This would be a direct consequence of (1.D.3) above once we present S P k Y as a homotopy colimit of a canonical diagram associated with Y and the integer k. But this presentation is given in the proof of (A.2) and (A.2.1) above: Thus S P k Y = Y k / S k is given as a hocolim, of a diagram consisting of finite products Y~ for 0 < t < k. Since L f commutes with products up to homotopy (1.A.8) (e.4), we can write S P k X and s p k ( L f X ) as homotopy colimits of similarly shaped diagrams of partial products X t and ( L f X ) e, as needed in (1.D.3). Compare (A.2.3) above.
For the case of the nullification functor PA, this result is just a formal consequence of the long exact sequence of the homotopy groups of fibration sequence P X ~ X ~ P X using the vanishing of the higher homotopy groups of the fibre and the vanishing of the lower homotopy groups of the base as given in (B.3) and (B.4). In the general case it follows directly from the fact that not only the localization and cellularization C W A of a GEM are GEMs, i.e. homotopy retracts of their own infinite symmetric power S P ~, but the augmentation map is a GEM map as given by the following diagram:
Proof of B.4.1:
X
, SP~176
, X
(B.6.1) LfX
. SP~176
, LfX
C. Relation between colimits and pointed homotopy colimits In this section we outline a ~2Alower estimate for the cofibre of the natural comparison map f X , colimi X, in terms of the 'fixed points' spaces in X, i.e. mapi(e, X) for the orbits e that occur in X. This is carried along the lines of (A.5) above. Recall [DF1] that an orbit e over a small category I is a n /  d i a g r a m e with colimx(e)  *, i.e. whose strict colimit is a point. For I = G a group, this is the usual definition. In [DF1] we associated in the usual way, with any small category O of/orbits and a n y /  d i a g r a m X, an O~ X ~ with X ~ = map/(e, X).
4. Symmetric products
93
If 0 is the set for all orbits X, namely all the pullbacks pt
~, colim ~" , I
> )( for
all u, then the associated Odiagram X ~ is free over 0 ~ C.1 EXAMPLE Here is a generic example: Let I be the 'pushout' category . . . . and consider the following /space: X = {{1} , {0, 1} , {1}}. Then the comparison map from the homotopy colin~t to the strict colimit is
s 1
, (pt)
and the cofibre is S 2. Since X itself is an orbit m a p i ( X , X) is a discrete space with few points. Hence in this case the statement of Theorem (C.8) below is S 2 V S 2 V S 2 << S 2, which is clearly correct. C.2 POINTED, UNPOINTED DIAGRAMS.
In general, the diagram pt : I
9 S . with
pt(a) = 9 a single point for all a E obj(I) is not a free diagram   it has no free generator. Think of a group acting trivially on onepoint space; see Appendix HL below. In the pointed category, however, every (pointed) diagram has pt as an orbit, and a free diagram in S.x is one that is free away from that orbit. In other words, all the orbits except the orbit of base points are free. Now if X is any pointed Idiagram, then of course {map/(e, X ) } e E o will also be pointed since, for each e C O, there is a single map e
* pt and so the map e
, pt
, X gives us a base point
in X(e) which turns X ~ into a free pointed diagram over O. In the discussion below we restrict our attention to diagrams X in S I, the pointed category. Thus 'free' means free in the pointed sense. Recall ~ e definition of pointed homotopy limit: We start with an arbitrary p o i n t e d /  s p a c e (/diagram) and resolve it by a pointedfree/space E X , X , with E X ~ , X~ a pointed homotopy equivalence for all a E I. Then the stric~ colimit o f E Z is t~he homotopy colimit of X: colim E X  / X. In [BK], we get f X as a functor by taking a canonical pointed free resolution of X.
I
C . 3 PRESENTATION OF STRICT COLIMIT AS A HOMOTOPY COLIMIT: L e t e~
be the free orbit generated at a. Then map/(e~, X )  X . Now it is immediate from the definition of orbit e that
94
4. S y m m e t r i c p r o d u c t s
C.3.1
PROPOSITION: / f O
:
O(X), t h e n
colim X ~ ~_ colim X 0 ~
~
I
is a natural equivalence. Proof:
T h e free orbits e~ are in O.
The m a i n advantage of the presentation of coliml as colimoop is t h a t , for a free O~ X ~ the l a t t e r is also the h o m o t o p y limit:
C.4 PROPOSITION: If O ( X ) = O, then the maps
c o l i m X ~ c o l i m X ~ ~_ hocolim. X 0 I
~
O~
~
Oov
are natural homotopy equivalences.
Proof." Under the a s s u m p t i o n X ~ is a pointed free O~
so its colimo X
and hocolim. X ~ f X are equivalent by definition. O
~
O
~
Cofibre of the comparison map. The e s t i m a t e for the cofibre of the comparison m a p follows from a formula for it as a pointed h o m o t o p y coequalizer, and thus pointed h o m o t o p y colimit. e: I
Let I be a small index category and let O be a set o f /  o r b i t 0  {e} with 9 S . , an orbit, and colimi e = (pt). An/space X: I
~ S . is called of o r b i t  t y p e O if it is built only from orbits
i n O . We assume t h a t for a l l a E I , the free orbit e~ generated at a E I , is in In addition, we consider W O an O. We denote by X ~ the associated O~ O  d i a g r a m defined as follows. C.5 DEFINITION: For all e E O, W O ( e ) = h o c o l i m e (free hocolim). Since hocolim I
is a functor we get a functor WO: 0
, S. m
Notice t h a t we used free hocolim and thus W O is an unpointed diagram. For example, if I = G a group and O is a category of orbits O = { G / H } H c G , then
W O ( G / H ) = h o c o l i m G / H _~ E G x v G / H = B H . G
W i t h this definition, we can formulate the m a i n technical result. This result expresses the cofibre of the comparison m a p between a strict direct limit and the
4. Symmetric products
95
corresponding homotopy direct limit in terms of a certain homotopy coend where we use the double integral notation from (A.5.1) and Appendix HC. C.6 THEOREM: For any pointed Idiagram X of type 0 = O ( X ) , one has a natural homotopy equivalence:
cof(hocolim. X
, col.im X) ~ / / W O
9X ~
o
where 9 denotes the reduced joint between unpointed and pointed spaces. C.7 REMARK: If A is unpointed and B is pointed, one can consider A * B as the pointed reduced join which is homotopy equivalent to the usual join and is obtained by collapsing the subjoin A 9 {b}, with b E B being the base point (A * {b} Cone(A)). We denoted by f f the homotopy coend over O, which is a special case of homotopy colimit.
O
C.8 THEOREM: Under the conditions of (C.6) above, if all the spaces in X ~ are Acellular then the cofibre cof(hocohm. X
, colim. X) is ~2Acellular.
Proo~ Since the coend in the theorem is also a pointed homotopy colimit, in order to show that it is constructed out of Z2A it is sufficient to show that for each (e, e') E O ~ x O, the space Be 9X e' >> Z2A. But since colims e is a single point Be, the nerve of e is connected so S 1 << Be and, by assumption, A << X e' for all e' C 0. Therefore ~,2A = S 1 9A << Be 9X e', as needed. Proofo[ Theorem C.6:
The proof is analogous to the above computation (A.6) and
we omit it here. D. A p p l i c a t i o n : Cellular v e r s i o n o f B o u s f i e l d ' s key l e m m a One formulation of the lemma is as follows: D . 1 PROPOSITION: Let X be a connected space. For any integer k > 0 the suspension of the DoldThom map E X , E S P k X induces a homotopy equivalence
pr~2x~ X
~, P~.2x~,,SPkX
D.2 REMARK: This innocentlooking lemma which is due to Bousfield [B4] is really a key technical result on which much of the present development rests. The statement is equivalent in less technical terms to the following statement originally given by Bousfield: If map.(E2X, Y) ~ 9 for Y a 1connected space, then the DoldThom map induces an equivalence on function complexes map, (X, ~tY) "~
96
4. Symmetric products
map,(SPkX, ~ Y ) for all k, or equivalently map(EX, Y) ~ map(ESPkX, Y). To see the equivalence note that the condition on Y is that it is E2Xnull, so map, (T, Y) "~ m a p ( P r 2 z T , Y) for all T 6 S,, hence if we assume (D.1) we get the equivalence of the function complexes into Y. On the other hand the statement given by Bousfield guarantees that the suspension of the DoldThom map on X is E2Xnull equivalence (1.C.5) which is precisely what (D.1) claims again by (1.C.5).
Proof"
This is based on [Dw2]. The key element behind the present proof is Theorem A.5 above. One considers S P k X as the quotient space colims~ X k and we notice that X itself is equal to the quotient space: colims~ V x . The map X k
,
S P k X is induced by V X ~ 1IX by taking colimsk, where Sk is the symmetric k
k
group on kletters. So the proposition is translated via the Sk equivariant map:
Eak: E c o l i m V X Sk
k
* E colimIIkX S~
to the statement PE2x(Eak) is a homotopy equivalence. The idea of the proof is to notice that had these colimits been (pointed) homotopy colimits, then the result would have been immediate. Since the cofibre of the map of suspensions before taking these hocolimits is clearly E2Xcellularsee belowit would have remained so after taking pointed hocolims. The body of the proof uses (A.5) to circumvent the difficulty by comparing the colimits with the homotopy colimits of the same spaces over Sk which are pointed with respect to the action. Let us begin with the
Observation:
Er/k: E Vk X
~ E I I X induces a homotopy equivalence P~2xE~k k
upon taking the E2Xnullification. Let us prove this for the case k = 2. We get E(X • X) = E X v E X v X 9X. Since X is connected, we have S 1 << X, so S 1 * X << X * X or E2X << X * X. Therefore we can compute using (1.D.2), (1.D.5)(4) above:
P ~ 2 x ~ ( X x X ) ~ P ~ 2 x ( ~ X V ~ X V X * X ) '~ ~ P ~ 2 x ( P ~ 2 x ( ~ X V ~ X ) V P ~ 2 x X * X ) ~ P~z2x(ZX V ~X). For k > 2 the argument is identical involving the decomposition of suspension of the higher powers of X as a wedge sum. We now compare in the following diagram wherein all hocolims are over Sk. We claim that upon application of P~2x to diagram D.2 it becomes a diagram of weak equivalences so at the bottom row we get the claim of (D.1) as needed. The
4. Symmetric products
97
vertical maps Ecl, Ec2 are the suspensions of the natural comparison maps from the pointed Borel constructions to the orbit spaces as in (A.5) above. The pointed homotopy colimits appearing below and denoted here by f are all pointed Borel constructions (with respect to the symmetric group on k letters) or suspensions thereof:
f v.v X
f~,7~ ~ f EIIkX
+
+
k
(D.3)
k
P~colims~ Vk X
~:a~,~ N colimsk IIkX
:1EX
+ Ec
, ESPkX
By the observation above the map P~2x E~k is a homotopy equivalence. We will now show using standard properties of pointed homotopy colimit ( f ) that this implies: CLAIM: The top horizontal map in the ladder (D.3) namely, f Er/k, induces an equivalence upon applying Pr~2x to it. Since E~k is an equivariant map we consider it as a map of diagrams and apply to it Proposition 1.D.2 to show that f Er/k also induces a homotopy equivalence on P~2x: P r o o f o f claim:
f
f
where we have applied the comparison (1.D.2) to the map E~k. In other words, we have the following two special cases of (1.D.2):
k
and
k
98
4. Symmetric products
Now the observation above shows the the the two spaces on the right of these equivalences are equivalent since it says that before applying to both P~2x /
they
,J
were already equivalent via an equivariant map P~2x E~ksince nullification is functorial it turns an equivariant map into such. But taking homotopy colimit always turns an equivariant map that is an equivalence into an equivalence. Therefore f Pr2xEnk and thus also P z 2 x f P~,~xEnk are equivalences. We conclude from this that f E~/induces an equivalence on P22x as claimed. By commutation of pointed homotopy with suspension E f 71k ,,~ f ~ / i n d u c e s homotopy equivalence upon applying PE2x. This takes care of the top two arrows in figure (D.3) above. Notice that g1,92 are homotopy equivalences, again since suspension commutes with pointed hocolim,. The last point of the proof is now to show that the two vertical comparison maps comparing the pointed Borel constructions with the strict direct limits again induce equivalence upon applying PE2x. To show that P~.2 x applied to Eei is a weak equivalence we use (1.D.4) and (1.D.5) (5) above: As we shall immediately see it is enough to show that Pz2xCof(c~ ) ~ * (where Cof() denotes the cofibrei.e. mapping coneof a map) and then use the usual Puppe sequence. For these we claim that both cofibres are E2Xcellular. In fact we claim that Cl, c2 satisfy the conditions of (A.5) and thus by (A.5) Pz2xCof(ci)  * which implies that PE2x(Ecl) and P~2x(Ec2 ) are homotopy equivalences. The map c2 is exactly the map appearing in (A.5.1) so the inequality E2X << Cof(c~) holds. The same reasoning as in (A.5.1) holds for the map cl since again the fixed points spaces of the action of the permutation group are Xcellular: Notice that the orbit category of the symmetric group is {Sk/H}Hc_S. and the spaces (IIkX) &'/H are partial products Xil x X~2 x .. x Xi, while the spaces (Vk X) &/H are always of the form: X~I v X~2 v ... V X~,~ because of the nature of the action. Therefore, all these fixedpoint spaces with respect to any subgroup are in fact Xcellular and connected. Therefore, in the cofibration Puppe sequence
f HkX el SPkX
C1
b
EfHX
~c2 ESPkX P
d
S j,
S~
we have E2X << C1, so P~2xC1 ~ * and thus the induced map Pp.2x(EC2) is a homotopy equivalence. A similar argument works for C2, where we replace H X by the wedge V X and SPkX by X; we get that the map P ~ 2 x ( E c l ) is a homotopy equivalence. To conclude, we saw, using standard PAcalculus (1.A.8) and Theorem A.5, that all the maps in the above comparison ladder induce equivalence on P~2x, therefore so do the maps Eak and Ec as claimed. 
4. Symmetric products
99
The following proposition formulates a weak 'geometric' aspect of the key lemma above. It show that to construct the suspension of the symmetric product of X out of the suspension of X one needs precisely one copy of E X and then one can proceed by attaching copies of higher suspensions E2+'~X of X: D.4 PROPOSITION: Let X be a connected space. For any integer k >_ 0 the suspension of the DoldThom map ~ X ~~ E S P k X gives cellular inequality:
~ 2 x << ~ ( S P k X / X ) . Proo~ This follows easily from diagram D.3 above by taking cofibres of the maps involved.
5. G E N E R A L
THEORY
OF FIBRATIONS,
GEM ERROR TERMS
Introduction In the present chapter we consider the behaviour of fibration sequences under L/, PA and C W A as well as some cofibration sequences. In particular, in continuation of the work in Chapter 3, more precise information about the way these functors commute with the loop space functor and other mapping spaces lies at the center of our interest. We also consider briefly commutation with suspensions. We would also like to relate, as closely as possible, Lymap(V, X) and map(V, L : X ) . The main steps in this direction were taken in [B4] and were further developed in [DFS]. It turns out that if A or f are suspensions or double suspensions, then these functors come very close to preserving general fibration sequences over connected spaces. In practice, we must first consider the functor PA. One can show that, short of a 'small abelian error term', the nullification functor X ~ P A X with respect to any suspension A  ~A ~ preserves homotopy fibration sequences. Explicitly, for any map g, the homotopy fibre of the natural map from the nullification of the homotopy fibre of g to the fibre of its nullification is always a GEM (see 4.B.1) for such an A. More generally, the flocalization functor X ~ L / X with respect to double suspensions f = ~2f~ preserves homotopy fibration sequences up to a 'generalized Postnikov stage'polyGEM. This implies as a special case that Bousfield's homological localization functor LE with respect to any general homology theory E, preserves fihrations of the form ~ 2 F ~ ~2E * ~2B up to polyGEM. Moreover, for ptorsion theories such as modp Ktheory and higher Morava Ktheories, the error term is rather small and lives in three dimensions typical of the given theory. Let us quote here two results from Section B below that are typical. These are dual results, one dealing with fibratious the other with cofibrations. MAIN THEOREM B . I : Let f : A ~ D be a m a p o f connected spaces. I f F * E * B is any fibration with L ~ f E ~_ L ~ f B ~_ , then L ~ / F has a natural structure o f a GEM.
Dually for cofibration one has:
COROLLARY B.4.1: Let E A ~ E X * E ( X / A ) be a suspension o f a cofibration sequence. I f L I E X ~ L I T ( X / A ) ~_ , then the localization L I E A has a natural structure o f a GEM.
This in turn has an interesting implication on the partial order (<) defined in (1.A.5):
5. GEM error terms
101
COROLLARY B.6: For any spaces X , Y and integer k >_ 1 one has: ~ X < ~ Y iff
EkX < SkY. Proofs: The proof of (B.1) occupies most of Section B below. The two corollaries (B.4.1) and (B.6) are treated at the end of Section B.
Remark:
We see that if L z f kills both base and total space of a fibration sequence, then it turns the fibre in a natural way into an infinite loop space that is equivalent as such to a product of (abelian) EilenbergMac Lane spaces equipped with their standard infinite loop space structure. Thus L~f 'abelianizes' the fibre, killing in the process all its kinvariants (and presumably much of its homotopy groups, too). In fact, in many important cases under the above assumption L I F is just a K(G, n) for some abelian group G and integer n. Notice that by (1.H.1) above and in contrast to (B.1), if Lf kills any other pair of spaces in F * E * B then it kills the third too, and here f need not be a suspension. In a suspension of any cofibration sequence similar behaviour occurs.
Example:
A typical example of the situation depicted above occurs when one applies the Postnikov section functor P~  Ps~+I to a fibration of the form fiX * 9 ~ X. If X is nconnected, then P,,X ~ 9but Pn~2X is equivalent to an EilenbergMac Lane space K(H, n) with II the first nontrivial homotopy group of X. Similarly, if a Postnikov section of a double suspension of any space is contractible then the same section of the single suspension must be an EilenbergMac Lane space. It is surprising that, despite its simplicity, this is the generic situation. It motivated Bousfield in the first place to look and find his early, most important, version of (B.1.1) below. The control gained on the behaviour of fibration sequences under localization will allow us to prove Theorem 6.A.1 (in the next chapter) which gives information about homological localizations of doubleloop fibre sequences. A. G E M a n d p o l y G E M e r r o r t e r m s  m a i n r e s u l t s In this section we state most of the main results about preservation of fibrations by homotopy localization functors. Since some of the proofs are a bit lengthy we will give these proofs together with several interesting additional propositions in sections B, C and D below.
A.1 DEFINITION: We define the class of (twisted) polyGEMs (respectively, oriented polyGEMs) as follows: It is the smallest full subcategory that includes all the underlying spaces of GEM spaces as members and is closed under taking the total space (respectively, homotopy fibre) of any fibration sequence for which the other two members are polyGEMs. Thus an oriented polyGEM is built from GEMs by a finite number of oriented, principal fibratious while in a general twisted polyGEM arbitrary fibrations are allowed. In particular, oriented polyGEMs are all nilpotent spaces whereas in general
102
5. GEM error terms
(twisted) polyGEMs the action of the fundamental group on the higher homotopy groups is not restricted, while the fundamental group itself can be any solvable group. The class of oriented polyGEMs is filtered naturally as follows: A 1polyGEM is the underlying space of a GEM. If W1 * W2 is a map of kpolyGEMs, then the homotopy fibre is a (k + 1)polyGEM. A . I . 1 EXAMPLE: Any finite Postnikov section of a space with abelian fundamental group is a polyGEM and if it is nilpotent as a space [BK] then it is an oriented polyGEM. In the present context compare (F.7) below.
A.2 DEFINITION: We say that the coaugmented functor X * L X preserves the fibre sequence F * E ~ B over the pointed space B up to a (poly)GEM error term i f one can construct a m a p e into the h o m o t o p y fibre o f L E + L B ,
c : L F * fibre(LE * LB),
whose h o m o t o p y fibre J = Fib(c) is a ( p o l y ) G E M (see A.1).
A.3 EXAMPLE: For L = L f, the homotopy flocalization functor, one can associate with any fibration F * E + B over a pointed space B a map L F * fibre ( L E * L B ) that is natural up to homotopy: This comparison map is given by universality of L  L f and the fact (1.A.8)(e.3) above, that the space fibre ( L E * L B ) is flocal. For the standard loop space fibration f~X ~ 9 * X the above map is the 'commutation map' Lf~X ~ f~LX. Let us now formulate the three main consequences of Theorem B.1 quoted above when coupled with results of Chapter 3: The following is an interesting special case of (A.5) given below when applied to the path fibration over X : A.4 THEOREM: Let A = E A ' be any suspension space and let X be any simply connected space. Then the fibre J o f the natural m a p a,
a : P A ~ X + ~ P A X ,
is a GEM. In other words, P A commutes with f~ up to a GEM. Moreover, J is Anull and P A, J ~ {*}.
5. GEM error terms
103
Remark: For a given space A ~ of finite type it is a simple matter to determine the local E X  n u l l G E M ' s which localize to (*} under PA,.
A.5 THEOREM: L e t A = E A ~ be any suspension. Then P A preserves any fibration sequence up to a G E M error term J. In fact, P A' preserves up to a G E M any loop fibration ~ F * ~2E ~ YIB for any space Aq This G E M error term J is Anull and P A' kills it: P A' J ~ *. Proo~
The proof is given in Section C.3 below.
Remark: A further interesting property of J is explained in (A.10) below. And finally,
A.6 THEOREM: For any m a p g : A ~ D the functor Lg preserves the double loops o f any fibration sequence: ~ 2 F +~ 2 E   ~ ~~2j~, up tO a p o l y G E M error term ~/2j which is glocal and satisfies L g J "~ . . Proofi
The proof is given in Section D below, see (D.6)(1),(2),(3).
If one applies (A.6) to the standard loop space fibration one gets:
A.7 COROLLARY: W h e n localizing a connected pointed space X with respect to any double suspension map, the h o m o t o p y difference between the localization o f the loop o f X and the loop o f the localization o f X is always a p o l y G E M .

A.8 Homotopically discrete mapping spaces In Theorem A.5 we saw that the error term is intuitively 'small' since it is E X  n u l l but it is killed by PA'. such spaces arise naturally here and they have a very interesting and useful property noticed in certain cases in [B4]: A.9 PROPOSITION: i f Y is a E W  n u l l space that is killed by P w i.e. with P w Y ~ * then the space o f pointed self m a p s y V = map.(Y, Y) has a contractible nullcomponent. Proo~ The proof is easy: All we need to show is that the loop Space of the mapping space is contractible. But we have
~ m a p , ( Y , Y) _~ ~ m a p , (Y, P r w Y ) ~ map , ( Y , ~2P~w Y ) ~ m a p , ( Y , P w ~ 2 Y ) ~ *
The first equivalence follows from Y being EWnull,the third is standard commutation from Chapter 3, the last one follows from the assumption that P w kills Y i.e. Y , 9 is a Wlocal equivalence.
104
5. GEM error terms
In the circumstances of (A.5) we see that the error term is a GEM that satisfies the conditions of (A.9) with W  A p. But since it is a GEM it is a loop space and all the components of map.(V, J) for any V are equivalent to each other and hence they are all contractible. Therefore we conclude from (A.5) and (A.9): A.10 PROPOSITION: The error term J in (A.5) is a G E M whose space o f selfmaps is homotopically discrete.

B. T h e m a i n t h e o r e m o n G E M e r r o r t e r m s In the present section we are concerned with the following: B.1 THEOREM: I f F * E ~ B is any fibration sequence and f: A ~ D any m a p o f connected spaces with L ~ f B ~_ Lr~f E ~ *, then L~ I F has a natural structure o f a GEM.
Moreover, it is probable that if the fibration is principal, F being a group, then the induced group structure on L f F coming from (3.A.1) is homotopically equivalent to its group structure as a GEM. See diagram (B.1.4). We will not need to use this extra structure in the present notes. Remark:
L~fX
B . I . 1 COROLLARY: For any m a p f o f connected spaces, i f L ~ f X has a natural structure of a GEM.
~ * then
It might as well be assumed that the spaces in the above E, B, X are all 1connected since this follows directly form the assumptions.
Remark:
The proof of (B.1) and (B.I.1) will occupy most of the rest of this section after the following
Proo~
1. For the case f : A *., i.e. L f = PA, and under special assumptions on A, this key result was found and proven in [B4]. As will be clear from the Remarks:
following Theorem B.1 can be considered as a relative version of (B.I.1). 2. Notice the nonexample for Theorem A.4 for nonsuspensions: If A is an acyclic space, then P ~ A A ~ A even if P A A ~ *. As a simplyconnected counterexample, when A is not a suspension one can take A = X = B S 3. By a theorem of Zabrodsky the null component of m a p . ( B S 3, B S 3) is contractible. Therefore B S 3 is ~BS3null. So in this case P E A X ~ X , but P A X ~ * and B S 3 is not a GEM. We conclude that one cannot dispense with some assumption on A or X in order to allow the deduction in Corollary B.I.1 below:
P A X ~ * ~ P ~ A X ~ GEM.
3. It is not hard to construct examples in which P~2A(~A) has nontrivial homotopy in infinitely many dimensions. Take A to be the wedge of all Moore spaces
5. GEM error terms
105
MP(Z/pZ). p > 2; then P~2AE A is the product of all K ( Z / p Z , p  1), p > 2 when p is a prime. 4. Notice the immediate interesting implication: For any connected space X the space G = P ~ 2 x E X has a natural structure of a GEM. This last implication does not hold for the threepoint space since we get a wedge of two circles for the space G. 5. It was shown by Casacuberta and Peschke [CP] that the localization with respect to the degree p map between circles does not behave as nicely as in (A.4). That map, of course, is not a suspension. Still, it can be partly understood using homology with local coefficients. 6. We were unable to determine whether the polyGEM in (A.6) is in fact a GEM. Proof of Corollary B.I.I: Since the proof here is an easier version of the proof of (B.1), we give of a direct, independent, proof of it below before that of (B.1) itself. However, using the loop structure on localization of loop spaces it can be deduce from the remark following (B.1) as follows: We can assume that X is a connected space by looking at each component. Recall L~2X ~_ ~ L ~ I X , so our assumption implies that Lf~2X ~_ .; in particular it follows that ~tX is connected and X is simply connected (1.A.3)(e.ll). The deduction of the corollary is immediate from the commutation formula (3.A.1) above once the theorem is applied to the principal fibration ~2X ~ 9 ~ X. The assumption L ~ f X ~ * gives, by Theorem B.1, L E f f l X ~ GEM. Therefore by (3.A.1) there is an equivalence ~IL~2fX ~ GEM. Since these two are equivalent as abelian infinite loop spaces one can take classifying spaces using (3.A.1) to get: L n 2 f X is also a GEM, as needed. We now turn to an independent proof of (B.1.1). The first main step in the proof is: B . 1 . 2 LEMMA: I f L f ~ X ~ *, then:
(i) The natural axis map LE2f(X V X V ' " V X ) ""* L E 2 f ( X x X x "." x X ) in an equivalence. (ii) Condition (i) implies that LE2f X has a natural infinite loop space struct ure. Proof of B.1.2 (i): For simplicity of exposition we consider only the case involving two copies of X; the general case proceeds verbatim with the obvious changes. We start our proof by recalling [G] (see examples in Appendix HL) that the homotopy fibre of the natural inclusion i : X V X ~ X x X is homotopy equivalent to the join ~tX 9 fiX ~ E ( f t X A ~tX) for any pointed connected CWcomplex X. Therefore, by (1.H.1), in order to show that Ln2f(i) is a homotopy equivalence it is sufficient to show that LE2 f ( ~ X * ~tX) ~_ ..
106
5. GEM error terms
But by (1.A.8)(e.10) we always have the implication L f A ~ 9 ~ L ~ f E A ~ *. Since X * Y ~ E ( X A Y) it is sufficient to show that L ~ . f ( f l X A f~X) ~ *. For this we use(1.A.8) (e.10) above: Since the assumption with (3.A.1) gives L I f ~ X ~ *, 12X is connected since (e.ll) the map f is between connected spaces. Thus P s l Y / X ~ *. Using (e.10) we get L s l ^ f ( f l X A ~2Y) ~ . , where S 1 is the circle, as needed. Proo[ o f B.1.2 (ii):
We proceed to show that the natural equivalence L ~ 2 f ( X v
X . .. v X ) ~ L E 2 f ( X • X . . . • X ) implies that L ~ 2 f X is an ooloop space. For this we use [A], [S], [BF], see also (3.C.3) above. According to Segal an
infinite loop space structure on a space X is specified completely once the space is embedded in a diagram of spaces over the small category of pointed finite sets F + = {n + : n > 0} (called very special Fspace) as the space corresponding to (1+). A special Fspace is just a diagram over the category of pointed finite sets F + such that it assigns a single point to (0 +) and its value on n + is homotopy equivalent to the nth power of its value on 1 + via the obvious map which is the product of the maps di : n + ) 1 + sending every element to 0 except i that goes to 1 C 1 +. It is very special if the induced monoid structure on the path components of F(1 +) is in fact an abelian group. In our case all spaces involved in the diagram are connected, in fact 1connected, so the condition on 7r0 is automatically satisfied. Therefore, to exhibit the infinite loop space structure on the space L ~ 2 f ( X ) as needed, we construct a (very) special Fspace X . with X0 "~ *, X1 "~ X. First V
V
we construct a 'nonspecial' F space X . by setting X,~ = V X, the pointsum of n
V
ncopies of X. If S is a pointed finite set, we really take the halfsmash X s = X D<S which we write, with a slight abuse of notation, as V X. Clearly for any map of S V
finite pointed sets S ~ T we have a corresponding map V x ~ V x , so x . is a functor from the category of finite sets to spaces with
S T V V X(empty)+= X 0 =
pt. The v only condition of a 'special' Fspace that is not satisfied by X ~ is that the map V
V
V
X,~ ~ X1 • " " • X1 (ntimes) is not a homotopy equivalence. But now we define 9
V
X ~ = L ~ 2 f X ~ ; since L~.2! is functorial we still get a Fspace. It is special because we have the equivalence:
L,.,f( V x)_ S
][ 1.,.,f x. S
5. GEM error terms
107
This follows by the natural equivalence proven above (B.1.2 (i)), together with the natural equivalence that gives the multiplicative property of L f (1.A.8)(e.4) L f ( W • W I) = L f W • L I W ~ for any f and W, W ~. This concludes the proof of Lemma B.1.2.
We now conclude the proof of (B.1.1): By Lemma B.1.2 our space L ~ 2 f X is an infinite loop space and therefore we can write L ~ 2 f X ~_ ~ Y . We saw above that ~tX is connected and X is simply connected. Therefore so is L ~ 2 f X ; we conclude that Y is simply connected. Consider map.(E 2 X, Y). We claim that it is contractible. This is true since any map E X ~ ~tY = L ~ 2 f X factors through L~2fE X. But since L ~ f X "~ * by assumption, the latter is equivalent to a point by (1.A.8) (e.10). Similarly, any map EkX ~ ~ Y must be null for all k > 1. Alternatively use (1.A.8) (e.9). Therefore the condition of Bousfield's key lemma (4.D.1) above and (B.2) below are satisfied, Y being simply connected. Thus any map X * ~lY = L E 2 f X factors uniquely through SPkX for all k _> 1. Because of the uniqueness of the factorization we can conclude that the factorizations are compatible for various k.
Therefore we get a factorization through the infinite symmetric power of X for the localization map on X:
SPo~ X
(B.I.3)
1
/ X
~ LE2fX
If we now apply LE2$ to this triangle, using LE2fLE2f ~ LE2f we get that L ~ 2 I X is a homotopy retract of L n 2 ( S P ~ 1 7 6 But (4.B.4) asserts that L f turns any GEM into a GEM. Since a retract of a GEM is a GEM we get the desired result that L ~ 2 f X  being up to homotopy a retract of L ~ 2 f S P ~ X  is a GEM. We now address the relations between the GEM structure and the infinite loop space structure. Notice that the infinite loop structures in S P ~ X and L n 2 f X are given by definition in the same manner by the concatenations of words in the underlying spaces, so that they are naturally compatible. Explicitly, the infinite loop space structure is given in terms of special Fspace via (B.1.2)(ii) since by the DoldThom theorem the axis map induces an equivalence
SPc~(X V X V... V X)
~, S P ~ X
x SP~X
x ... x SP~176
108
5. GEM error terms
Such an equivalence yields as above an infinite loop space structure on S P ~ 1 7 6 Therefore the homotopy uniqueness of the vertical factorization arrow in the triangle (B.1.2) implies that we have a homotopy commutative diagram in which the horizontal arrows give the homotopy product structures on the spaces involved in view of the equivalences above: SP~176 V X)
, SP~176
11
(B.1.4) L~:(X
v X)
, Lr.~:X.
Therefore the GEM structure and the infinite loop space structure on Lm2:X are compatible. This proves (B.I.1). 
We now turn to the proof of the main theorem (B.1). We use the same line of argument as in the proof of (B.I.1) so details are omitted. P r o o f of B.1 Referring to the proof of (B.1.2) above we first show that L E / F is an coloop space in a natural way using [S], [A]. We define a (nonspecial) Fspace as follows: V
Fn=fibreof(EV...vE~Bv.VB)
(ncopies o f E , B).
The homotopy fibre being a functor in S, and { V E }
+ { V B } n>O
being n_>O
V
a map of Fspaces, we conclude that F~ above is a Fspace. B.1.5 Claim: The natural map (see diagram B.I.4 below): V
f~: F . + F x ... x F
V
induces an equivalence on L~/(f,~) and therefore the space L ~ / F 1 = L ~ I F is an coloop space.
5. GEM error terms
109 k/
Proof of Claim: Since by definition F1  F the equivalence in the claim implies that: V
Lr.fF,~ ~ L ~ . f ( F
x
...
x
F ) ~ ( L ~ f F ) '~ V
~_ ( L ~ f F 1 ) n. V
V
Thus L r . f ( F . ) is a special Fspace and therefore L E f F 1 = L ~ f F is an ooloop space. Let us prove now the first part of the claim: Consider the diagram that depicts the above constructions for n = 2. This diagram is built from the lower right square by taking homotopy fibres: fl(~B * ~B)
(B.1.6)
X
~E * ~E
~B * ~B
l
l
1
V
F2
l
FxF
EVE
l
ExE
BVB
t
BxB
To facilitate the exposition we prove the equivalence in the the case n  2 the general case follows along the same path. By (1.H.1), in order to prove the claim it is sufficient to show that L ~ f X ~ , . In order to show that we use (1.H.1) again and show that both the fibre and the base in the fibration sequence with X as a total space in (B.1.6) above are killed by L~I. To this end it is sufficient to show L ~ I ~ ( ~ B * ~ B ) ~ L ~ f ~ 2 ( ~ E 9 f i E ) ~_ , . Because then Lr~f certainly kills ( ~ B 9~ B ) and (~2E 9~ E ) , again by (1.H.1). Now consider e.g. L ~ . f ~ ( ~ B 9 ~'lB) ~_ , . By the usual identities as above:
L ~ I ~ E ( f I B A ~ B ) '~ F/L~2fE(~B A 12B).
Therefore it is sufficient to show that L ~ 2 I E ( ~ B A ~ B )  *. In order to do that we use (1.A.8)(e.10) with W = ~ B and n = 0: We know that B is a 1connected space since it is killed by assumption by E f a map of 1connected spaces. Therefore ~ B is connected. L~f(~2B A ~ B ) ~ 9 since L f l 2 B ~_ 9 and P s ~ B ~ *. thus: LE2fE(g/B A ~2B) _~ ,. By the same argument we get Lr~2fE(~E A
~ E ) ~ *. This proves our claim since it implies:
L ~ I X ~_ L ~ l ( g l E 9 12E) ~_ , .
110
5. GEM error terms
Therefore L E f F is an ocloop space and, in particular, we can write: L r . f F = f l Y . We now prepare the ground to the use of the key lemma (B.2) below: Claim:
map,(E2F, Y) _~ map,(EF, flY) ~ *.
This follows from universality (1.A) as follows: We have a factorization: EF
,~ f l Y = L ~ f F
1/ L~f2F
in which: L z f E F "0 *. Moreover: L f F ~_ 9 (1.A.8,e.10). This is clear from (1.H.1) for the fibration:
Claim:
flB~F~E
and LI~2B ~_ f l L r 4 B ~ f~, ~_ ,. All the more so L ~ I E k F ~ *, and thus any amp EkF * f l y is null which is equivalent to the claim. The claim being proven we can conclude the proof of (B.1) just as we did in the proof of (b.l.1) below, from Bousfield's key lemma (B.2) below that F ~ L~.IF factors through the universal GEM associated with F, namely the infinite symmetric product: S P ~ F . SP~176
/1 F
, flY = Lr.IF
But S P ~ F , the DoldThom functor on F, is a GEM. Applying Lr. f to the factorization we get that L r . I F is a retract of a GEM, since L~I = L~ILr.I. But a retract of a GEM is a GEM. The naturality of this GEM structure and its compatibility with ooloop structure on L ~ I F constructed above follows as in (B.1.6) above from the naturality of construction. This concludes the proof of Theorem B.1. 
5. GEM error terms
111
B . 2 BOUSFIELD'S K E Y LEMMA: L e t X be a connected space, and let Y be a simplyo
connected
space.
Assume
map,(E2X, y )
_~ *.
map,(X,~Y)
Then
~
m a p , ( S P k X , ~ 2 Y ) for any k > 1 [B4, 6.9]. B . 3 REMARK: A proof of (B.2) was given earlier in Chapter 4 using only some basic material about homotopy colimits. A way to understand (2.2) is to interpret it as saying that the space E S P k X can be built by successively gluing together copies of ~ e X for / > 1 with precisely one copy for g = 1. Since the higher suspension ~ 2 + J X (j _> 0) will not contribute anything to m a p , ( Z S P ~ X , Y ) , we are left with map. (EX, Y).
G E M ERROR TERMS IN COFIBRATIONS: We conclude this section by presenting two symmetrically looking results. The first is just (B.I.1) above and the second is in fact a dual theorem for the localization of cofibration as its corollary B.4.1 shows. COROLLARY B . I . I : For any X and f : A * B in S, i l L , i X has a natural structure o f a GEM.
~ 9 then L ~ 2 f X
The proof was given above. B.4 THEOREM: For any space X and any m a p in 8 , i f L f E 2 X
~ * then L f E X has
a natural structure o f a GEM.
This has a corollary that is a sort of dual to Theorem B.I: B.4.1 COROLLARY: Let E A * E X * E ( X / A ) be a suspension o f a cofibration sequence. I f L f E X " L I E ( X / A ) ~_ 9 then the localization L f E A has a natural structure o f a GEM. Proof'. It follows from (3.D) that the assumption implies L y E 2 A ~ * using the Puppe sequence. Thus from (B.4) we get the desired result.
This follows formally from (B.1.1). First notice that L y n X ~This follows immediately from the assumption L f E 2 X ~ 9 by (1.D.7). But from (B.1.1) applied to PE2x we get that P E 2 x E X is a G E M   since P E x G X ~9. Therefore by (4.B.4) it follows that L / P E 2 ~ X is also a GEM, hence the conclusion follows. P r o o f o f B.4: LyPE2xEX.
The above result has an interesting corollary that will be used later on. This corollary is a partial inverse to elementary fact 1.A.8,(e.10). B.5 PROPOSITION: For any m a p f and space X , i f L ~ . f y , 2 X "" * then L f E X
~ *.
112
5. G E M error terms
B.6 COROLLARY: For any spaces X, Y and integer k > 1 one has: E X < E Y iff EkX < EkY. Remark:
Of course this is not true if we omit one suspension from the a s s u m p t i o n
and conclusion, since X m a y be an acyclic space whose suspension is equivalent to a point. Proof orB.5: By (B.4) we know t h a t the m a p E X + L f E X is a m a p of E X to a loop space, in fact to an infinite loop space. Therefore it factors, up to homotopy, through the canonical m a p to the James functor on E X : E X * g t E E X . Thus it is sufficient to check t h a t localization of the l a t t e r L f F t E 2 X is contractible. But using (3.A.1) this is exactly our assumption. Proo[ orB.6: E X ~ , .
This follows i m m e d i a t e l y from (B.5) by t a k i n g the m a p f to be say
C. T h e n u l l i f i c a t i o n a p p l i e d t o f i b r a t i o n s a n d f u n c t i o n
complexes
In this section we consider the action of nullification functor P A on fibration sequences and on function complexes. The s t a t e m e n t s of the m a i n theorems a p p e a r in Section A above. T h e n in Section D below we consider general localizations. In all cases one has to assume t h a t the m a p s with respect to which one localizes are at least one suspension maps, or equivalently, the spaces and m a p s have a loop structure. We employ r e p e a t e d l y the following diagram:
(C.1)
i I ~LB ~tB
I 1 X
~ F
,
,
,
I I LE E
, ~
I I LB B
where L is L f, a localization functor for some m a p f , and L (  ) denotes the homot o p y fibre of the coaugmentation. Let X , X , Y be the a p p r o p r i a t e h o m o t o p y fibres t h a t render every sequence of two collinear arrows a fibre sequence. The n a t u r a l m a p F ~ X is the m a p F ~ fibre ( L E ~ L B ) . We get a well defined h o m o t o p y class L F * X or L F * fibre ( L E * L B ) , as in (A.3) above. Now let us restrict to the case L = P A . In t h a t case the fibration sequence X * F ~ X is preserved by L = P A by (3.D.3)(2) above, since X is Anull by (1.A.8) (e.3) above. Therefore, upon t a k i n g P A of t h a t fibration sequence, we get a fibration
P A X + P A F + P A X = X
5. GEM error terms
113
or
m
P A X ~ P A F + fibre ( P A E + P A B ) .
Therefore P A X is the 'errorterm' of PA, and whenever P A X is a (poly)GEM,
PA preserves the fibration F + E   B up to a (poly)GEM (A.1 above). C.2 OBSERVATION: I f a localization functor L = P A commutes up to h o m o t o p y with fIB, i.e. i f L ~ B ~
~ L B , then it will preserve any fibration over B . Moreover,
for a general localization functor L = L f , if we assume in addition L I ~ t E ~ ~ L f E and L I ~ F ~_ ~ L f F , then we can conclude that L I preserves the fibration F +E + B. Proof'.
We use diagram (C.1). Under the assumption the map ~ B + ~tLB is
a PAlocalization map and thus, by (1.H.2) above, implies L Y ~ 9 for L = Therefore L X ~ L L E by (1.H.1) above. Since L ( L E ) ~ * (1.H.2) we get L X Since the fibration X ~ F + X is preserved in such a case, L F = L X ~ X; since X is local, being the fibre of local spaces we are done: L F ~ L X  X. proof in the general case is similar.
PA. ~ .. and The
PROOF OF (A.5) We argue with diagram (C.1) using Theorem 3.A.1. We now read (C.1) with L = P A , L PA. By (1.H.1) above we get P A P A B ~ P A P A E ~ *. Therefore (A = EA') we can use Theorem B.1 to deduce P A X ~ GEM. Notice that X ~ F ~ X in diagram (C.1) is a fibration with an Anull base space X, so by (3.D.3)(2) it is preserved under PA. Thus the fibre of P A F + X = PA x is P A X , an Anull GEM. Finally, P A , X ~ * since PA' kills both Y and P A E . The following is a generalization of Theorem B.1 and a weak inverse to (1.H.1) that follows easily by chasing the diagram (C.1) above. C.3 COROLLARY: In any fibration F + E ~
B that induces an equivalence
P n A E % P ~ A B , the nullification P y . A F is a GEM.
C.4 Localization of function complexes We now show that the above theory gives rather good control of the localization of function complexes when we consider nullification with respect to finitedimensional suspension spaces. The main result here is the following strong version of [B4, 8.3].
C.5 THEOREM: If V, W E S , are connected spaces with W finite dimensional, then the h o m o t o p y f b r e F = F~, over any vertex qa : V + P n w X
P ~ w ( X v) ~ ( P g w X ) V
o f the natural m a p
114
5. GEM error terms
has vanishing homotopy groups ~riF ~ 0 for i > dim W 4 1 = dim ~ W . The proof of this theorem is based on several useful lemmas and is given after (C.8) below. C.6 LEMMA: It for some integer n > d i m W , the abelian group G satisfies P w K ( G , n ) ~ K ( G , n ) and P w K ( G , n + 1) ~ *, then G ~ O. Another formulation of the same lemma is C.7 LEMMA: I f P w K ( G , n +
1) ~ * with n > d i m W , then P w K ( G , n ) ~ *.
C.7.1 REMARK: Clearly the opposite direction is true without any assumption on dim W or n. Intuitively we think about an equation P w K ( G , n + 1) ~ * or
W < K(G, n + 1) as saying that Z W ~
dim W
II K(H~W, i) has groups in dimensions
i=1
not above n + 1 that 'support' the group G, i.e. from which this group can be built, so that killing these H~(W) kills the group G too. In that case, lowering n by one without coming below the dimension of W will leave the inequality W < K(G, n + 1) true in a stronger form: W < K(G, n). In other words, the inequality:
i=l
K(C,, i) <
is equivalent to: K(~C~, m) < K ( G , m).
Proof of Lemma C.6: We are given that K ( G , n ) is Wnull. This can be written as ~tmap.(W, K(G, n + 1)) ~ *. Since we assume n > d i m W , this function complex is connected before taking the loop and so its loop, being contractible, implies that the function complex itself is ~ ,. Thus K(G, n § 1) is also Wnull. But P w K ( G , n + 1) "~ * by assumption, so G ~ 0.
Proof of Lemma C.7: Let K ( G , i ) be a factor in the GEM space P w K ( G , n) and we wish to show G  0. To this end we use (C.6) that we have just proven. In order to satisfy the conditions of (C.6) we need to verify three things: (1) The integer ~ > d i m W. This is clear since by assumption n > dim W and we know from (4.B.2) that t > n. (2) P w K ( G , i ) = K ( G , / ) . This follows since by definition the latter is a retract of a Wnull space. (3) P w K ( G , ~1) ~ .. This is true since the map K(G, n) = FtK(G, n + 1) P w ~ K ( G , n + 1) is a loop map by (3.A) above and therefore, by induction on the connectivity of range, we may assume that K ( G , n ) * K(G,g) is also a loop map. Taking the classifying space functor on the latter map we get K(G, n + 1)
5. G E M error terms
115
K(G, g + l ) . But by L e m m a (C.8) below K(G, g + l ) is a factor in P w K ( G , n + l )  ~ . , thus G  0 as needed. C.8 LEMMA: ( C o m p a r e [B4, 4.7]) Let Y, Y' be connected spaces. If the map t2g : ~ Y ~ t2Y' is a loop map with the induced map Pwi2 9 being a weak equivalence, then so is the map P w g : P w Y ~ P w Y ~.
Proo~ above.
This follows at once from the formula for localization of loop spaces (3.A.1)
Before proving the theorem (C.5) above we deduce the following neat consequence from the l e m m a above. C.9 THEOREM: / f ~ Z
< Y and dimEX
< connY, then (i) E 2 X < Y a n d (ii)
E X <
Proof: Since P ~ x Y ~ *, we get from G E M theory (B.I.1) above t h a t J = P z 2 x Y is a GEM. The space J is ~ 2 X  n u l l and P ~ x J ~ *. Let K(G, n) be a factor in J. We first show t h a t n < connY. Assume n > connY _> d i m ~ X . Since P ~ x K ( G , n) ~ * one gets P E ~ x K ( G , n + 1) ~ , . But K(G, n) is ~ 2 X  n u l l and n _> dim E2X, so we are under the a s s u m p t i o n of L e m m a C.6 with W = ~ 2 X . Therefore G ~ 0. Hence each factor K(G, n) with G r 0 in J satisfies n < connY. But then [Y, K(G, n)] ~ *, thus the m a p Y ~ J = P ~ 2 x Y is null homotopic and, by i d e m p o t e n c y PA, we get P~.2xY ~ * as claimed. F r o m (3.B.3) above we get (ii). We now t u r n to Proof of Theorem C.5:
We proceed by induction on the skeleton of V. For V = S 1
we must consider the h o m o t o p y fibre F of the m a p
P~.wI2X * i 2 P z w X
over a component of the base. We have shown in (A.5) t h a t this fibre is a E W  n u l l G E M with P w F
~ *. Consider a typical factor K(G, n) in the G E M F . This factor
is again E W  n u l l and, by L e m m a C.7 above, P ~ w K ( G , n + 1) ~ . . Therefore by L e m m a C.6 unless n < d i m E W one has G ~ 0. It follows t h a t 7riF ~ 0 for all i > dim E W , as claimed. To continue the proof of the t h e o r e m we first notice t h a t , by induction, the claim of the t h e o r e m is true for V  S " by examining the h o m o t o p y fibres of the m a p s in a square of the form (for n = 2):
PzwfI2X
~
flPzwflX
ci ~2p~wX
=
~2P~wX
116
5. GEM error terms
It follows, by considering corresponding exact sequences of fibrations in the square, that the fibre of the map c has vanishing homotopy groups above dimension dim W + 1 for this particular V = S ". Now for a general V we first use a similar squareargument for the fibration induced by the cofibration S k ~ V 1 ~ V, where we assume that S k and V 1, playing the role of V as exponents in (C.5), already satisfy the conclusion of Theorem C.5. This gives by finite induction the claim for any finite V. But since maps from an infinite complex V are just the homotopy (inverse) limit of the maps from the finitedimensional skeleta, and since taking homotopy fibres commutes with taking the linear direct limit (Appendix HL), we get the vanishing of the higher homotopy groups as needed for any V. This completes the proof. 
D. L o c a l i z a t i o n w i t h r e s p e c t t o a d o u b l e s u s p e n s i o n m a p In this section we apply the above material to a discussion of the fibre of L f Y t Y ~ Y t L f Y and prove Theorem A.6 for a general map f : A ~ D and a double
loop space Y = ~2X. The main observation is that the difference between Lf and Lv(f) (where C ( f ) = D U C A = the mapping cone of f) is an flocal GEM. First we make the following simple observation about any cofibration: f h D.1 PROPOSITION: I f A  * D ~ C is a co~bration with co~bre C = D u C A , then any flocal space is Cnull and any Anull space is hlocal.
This is immediate from the definition and the fact that, for any space X, the sequence: Proof"
map. (C, X)
~h map. (D, X) _.7 map. (A, X)
is a fibre sequence with map.(C, X), the homotopy fibre over the null component. So if m a p . ( A , X ) _~ . then h is an equivalence, while if 7 is an equivalence then map. (C, X) ~ *. D.2 REMARK: So when we consider a BarrattPuppe sequence:
A~DAC
~ EA~ED
~ EC ~ E2A ~ E2D ~ . . .
as we pick maps and spaces more and more to the right: A, h, EA, E h . . . , being local or periodic or null with respect to these maps and spaces, becomes a strictly weaker condition. D.3 COROLLARY: A s a direct result of (D.1) we get P A L h = P A and L f P c = L f . We are now ready to prove the very useful
5. GEM error terms
117
D.4 LEMMA: For any map f : A * D the fibre o f L n 2 f X ~ P n c X is a E2fnull G E M that localizes to a point under EC, where C is C ( f ) , the mapping cone o f f . Proof" The cofibration sequence depicted above together with (D.1) show that any EC periodic space is E2flocal. Therefore (D.3):
P E c L E 2 1 ~ PEG.
Hence, the following map is the EClocalization:
L~2fX ~ P~cX, and (1.H.1) its fibre F satisfies P ~ c F "" *. Moreover, F is E2C null, being E2f local. But by Corollary B.I.1 above we get that P~.2cF is a GEM. So F = Pp.2cF is a GEM which is E2fnull. This can be rewritten in a different form: D.5 COROLLARY: There is a natural map for any f:
a : LfFt2X * flPc(f)f~X,
whose fibre is an flocal GEM. Proof"
Consider the map Lg.2IX * P ~ c X
given in the proof of the lemma above. Since (3.A.1) Lf~tY ~ ~tL~fY, we can rewrite the map as W 2 L f ~ 2 X ~ W P C f ~ X with the fibre being G2Clocal GEM. Looping it down twice we get the desired map Lfgt2X * f~Pcf~X with C C ( f ) , the mapping cone of f . But now the fibre is a flocal GEM, being the double loop of a G2fnull space. Remark: The usefulness of (D.4) and (D.5) follows from the fact that P w , the nullification functor, behaves much more nicely with respect to fibrations as in Section C and (A.5) than L f for a general map f. We now turn to the Main Theorem about preservation of fibration under
L:E2g:
118
5. GEM error terms
D.6 THEOREM: Let g : A ~ D be a map of connected spaces, and F ~ E & B be a fibration sequence with B connected. Then the homotopy fibre A of the natural map L F ~ fibre(LE , LB), where L = Ln2g, satisfies (1) A is a polyGEM and flA is an oriented polyGEM, (2) A is E2glocal, (3) LgA ~ *. Proo~
The proof is presented after the next corollary.
Remark: Conditions (1), (2) and (3) force the error term A to be 'small' at each prime p, see Chapter 6 below. Here is one useful corollary which may hold under a weaker assumption.
D.6.1 COROLLARYIll/" iS a polyGEM, then for any g : A ~ B the localization of the double loop space Lg~2Y is again a polyGEM. Proof." By (4.B.4) Lg (GEM) is again a GEM. We now proceed by induction: Let W + X ~ Y be a fibration with Lg~t2X, L g ~ 2 y polyGEMs. Now by the above theorem (D.6) using adjunction ~2L~2g = LgO 2, we get a sequence of fibrations where the classification map to CtA is obtained using (3.A.1) above:
~2A ~ Lg~22W + Fibre (Lg~2X ~ Lg~2Y) ~ ~]A,
where A is as in (D.6). By (D.6)(1), CtA is a polyGEM and so, by the inductive definition (A.2), L g ~ 2 W is a polyGEM as needed. Proof of (D.6):
By (1.A.8) (e.3), (2) is immediate. We first deal with (3) by: D.7 LEMMA: Lg(A) '~ *.
Proo~ In diagram (C.1) with L = L22g we get, in view of (3.A.1) above, that Y is in fact the homotopy fibre of the localization map fibre of (~2B * L 2 g ~ B )
and therefore, by (1.H.3) above, L g Y ~_ .. So by (1.H.1) and (1.H.2) above we get LgX ~ Lg(LE) ~ .. Now consider the diagram of fibrations below which is derived from the relevant column in (C.1), turned into the middle row here, and A is defined
5. GEM error terms
119
as the homotopy fibre in the bottom row. Notice that X is E2glocal, being a fibre of a map between such spaces. LE~gF
LE2gF
i 1
l 1
X
~
A
~ L~2gF 
P
~
9
i 1
~ X ~ X
= L~gX

M
By (3.A) again one gets LgL~2gF ~_., and therefore, as usual, LgA _~ L g X ~ 9 as needed. D.8 REMARK: Therefore by the Main Theorem B.1, we get L~.g(A) is a GEM. However, A itself is not an Eglocal space, so we cannot conclude in general for Lg that A is a GEM even for L = L~39,. To circumvent this difficulty we use (D.4) above and must relax GEM to polyGEM. P r o o f of D.6(1):
Proceeding with the proof of D.6 we compare P~c(g) with L~2g, as follows using Lemma D.4 above. The diagram is obtained in the usual way by backing up from the lower right square in which the vertical arrows are Wlocalization maps in light of (D.3) above. Fz
(D.9)
l l F1
F2
,
~ ,~
(GEM)I
l 1 P~c(g)E L~:29E
,
, ~
(GEM)2
t 1 Pr~c(g)B L~2gB
As a result of (D.4) above, the homotopy fibres denoted by (GEM)l, (GEM)2 are in fact GEM spaces. It follows by definition that F3 is a polyGEM, being tb.e homotopy fibre of maps between GEMs. On the other hand, using Theorem A.5 to compare P w (fibre) and fibre P w ( E ~ B), we get the following diagram in which the central vertical sequence measures the difference between the fibre of the localization and the localization of the fibre. The spaces F~ are from (D.9) above.
120
5. GEM error terms
(D.IO) (PolyGEM)2
,
1 l (PolyGEM) = F3 (GEM)3
~
A
l 1 F2
LfF
,
, ,
~
(GEM)4
1 1 F1 = Fib(P~c(g)(E ~ P~C(g)F
B))
Notice again that diagram (D.10) is obtained by backing up from the lower right square taking homotopy fibres. The vertical arrows in that square are this time comparison arrows from the localization of the fibre to the fibre of localizations constructed just as in the proof of Theorem A.4 above according to the demands of Definition A.1 above. The above diagram shows that F3 is a polyGEM by Theorem A.5 as we saw in (D.9), and (GEM)4 is a GEM since F1 is given as a fibre of P w ( E * B). By (D.4) we get that (GEM)3 is also a GEM. Hence by definition A is a polyGEM. Backing up the fibration in the top row we also conclude therefore that ~ A is an oriented polyGEM by (A.2), as needed in (D.6)(1). This completes the proof of (D.6). 
E. T h e f u n c t o r C W A a n d f i b r a t i o n s
In this section we consider the behaviour of fibration sequences under CWA. The results of Chapter 3 above suggest that we can expect here GEM error terms just as in applying PA to such sequences for a suspension space A. Unfortunately we get somewhat weaker results. For example, the analogue to the Main Theorem above is that, for a connected A, C W z A X ~ * implies that C W ~ A X is a GEM. However, we can only show that the fibre of C W ~ A X
~ CW~AX
is a polyGEM.
The slogan is: 'Whenever the function complex map.(~.A, X)  S is homotopically discrete, the space C W 2 A X is a GEM thus the above function se_t_tS has a natural abelian group structure' (compare [B4]). Recall that if P E A X ~ *, then P~.2AX is a GEM. Here we have a similar result about C W A X . E.1
THEOREM: Assume C W E 2 A X ~ *. Then C W ~ A X
hD,s a natural structure
of a GEM. Proof:
Consider the natural square of maps associated to any. space X: j: C W 2 A X
Pr~2Aj: Pr~2ACWr, A X
~
X ~ Pr~2AX
5. G E M error terms
121
Our a s s u m p t i o n is equivalent to m a p . ( ~ 2 A , X ) ~ *, i.e. X is ~2Anull and the right vertical m a p is an equivalence. Now since P ~ A C W ~ A X ~" * for any X, A above, we get from (B.I.1) t h a t P ~ 2 A C W 2 A X is a GEM. Therefore we r t h a t the canonical m a p CWr~AX + X factors up to h o m o t o p y t h r o u g h a GEM. Since by L e m m a 4.B.3 C W A (GEM) is always a GEM, we conclude t h a t C W n A X is a r e t r a c t of a GEM, thus a GEM. E.2 PROBLEM: IS it true t h a t for any A and a p o l y G E M X the space C W A X
is
also a p o l y G E M ? The following m a y not be the best possible result: E.3
THEOREM: For any A, X E S. the homotopy fibre F of
j: CWr~2AX + C W ~ A X
is a (twisted) polyGEM.
Proof This follows from (B.1). Notice t h a t P 2 A kills b o t h the d o m a i n and range of j above. Therefore P ~ A F is a GEM. But m a p . ( ~ 2 A , j ) is a h o m o t o p y equivalence. Therefore m a p . ( ~ 2 A , F ) ~ * and F is ~2Anull. Therefore by (B.1.2) the fibre of the m a p from F to P 2 A F is also a GEM, thus F itself sitting in a fibration between two G E M s is a p o l y G E M and we are done.

E . 4 COMMUTING C W A WITH TAKING HOMOTOPY FIBRES: W e will now address the question of the preservation of fibration by C W A . Looking at A = S ~ we see i m m e d i a t e l y t h a t X ( n ) = C W s ~ + I is the nconnected cover of X a n d so it does not preserve fibration in general. But again in this example fibrations are nearly preserved up to a single E i l e n b e r g  M a c Lane space. In general we shall see t h a t when A is a suspension the functor C W A ' a l m o s t ' preserves fibrations, the error t e r m being a G E M or a p o l y G E M . In order to measure the extent to which C W A preserves fibrations we will now compare the fibre of the C W  a p p r o x i m a t i o n w i t h the C W  a p p r o x i m a t i o n of the fibre via the following n a t u r a l map: (E.5)
A: C W A F ~ F i b ( C W A E + C W A B )
This is associated to any fibration sequence F * E b B
over a connected B.
For E ___ 9 we get, as a special case, a m a p C W A ~ B + 1 2 C W A X for any space B. In order to construct A one notices t h a t the fibre of the m a p C W A ( p ) , denoted here by F i b C W A ( p ) , m a p s n a t u r a l l y to F . This m a p induces an equivalence
122
5. GEM error terms
on function complexes map. (A, F i b ( C W A p ) ) ~ map. (A, F) since map. (A,  ) commutes with taking homotopy fibres. Therefore, by the universal property (2.E.8) there is a factorization C W A F + fib(CWA(p)) unique up to homotopy. Now in general one shows: E.6 PROPOSITION: W h e n e v e r A = E2A ' is a double suspension, the h o m o t o p y fibre A o f the above natural A is an extension o f two G E M spaces, i.e. it is a (twisted) 2polyGEM:
(GEM),) ~ A .~ (GEM)I. Moreover, A is an Anu//, A'cellular space. Proo~ First we notice that by a straightforward argument one shows that m a p , ( A , A ) _~ ,, i.e. A is Anull. This is because the map map,(A,A) is a homotopy equivalence, since the fibre J of A = pr2A, A ~ P~A,A is a GEM (by 5.A.5). By (3.A.2) above both the domain and range of A are EWcellular and thus both are killed by P~A'. Therefore the condition of (B.1) is satisfied and P ~ A , A is a GEM. This completes the proof. 
E.7 COROLLARY: For A = E2A ' the fibre o f C W A ~ X
~
~'~CWAX is a 2
polyGEM. Proof"
Apply the above to ~ X ~
9 ~ X.
F. A p p l i c a t i o n s : A g e n e r a l i z e d S e r r e t h e o r e m , N e i s e n d o r f e r t h e o r e m It is well known that finite, 1connected and noncontractible CWcomplexes have nontrivial homotopy groups in infinitely many dimensions. This has been generalized in many directions, relaxing the assumption of finiteness. In this section we consider a different direction of generalizing. Instead of considering [S ~, X] we will consider [E~A, X] for an arbitrary space A: Instead of assuming X is a finite simply connected CWcomplex we assume X is a finite EAcellular space f o r any connected A. Namely X (2.E.7) a space obtained by a finite number of steps starting with a finite wedge of copies of ZA and adding cones along maps from EA to the earlier step:
X "' (VEA) UCEe'AUCE~2AU ...UCE4A F.1
(~ _> 1).
THEOREM: L e t A be any pointed, connected space o f finite type. L e t X be any finite EAcellular space, with H * ( X , Z / p Z ) r 0 for some p. Then ~ri(X, A) = [EiA, X] r 0 for infinitely m a n y dimensions i >_ O. One immediate corollary is for X = EA.
5. GEM error terms
123
F.2 COROLLARY: Let A be any pointed, connected space of finite type with H.(A, Z / p Z ) ~ 0 for somep. There are infinitely many e's for which [EeA, EA] ~ *. Proof: First we note that, since we consider spaces built from EA, by a finite number of cofibration steps we get a space which is conic in the sense of [HFLT], namely it is derived from a single point by a finite number of steps taking mapping cones.
Now since /~* (X, Yv) ~ 0 for some p, X satisfies the hypothesis of [HFLT]. Their argument now shows that X cannot have a finite generalized Postnikov decomposition, i.e. in the present terminology X cannot be an oriented polyGEM, since X is of finite type. On the other hand, suppose lEnA, X] ~ 0 for i ~ N. Then map.(ENA, X) ~ 9 since all the homotopy groups of this space vanish. In other words, X is ENAnull o r P E N A X ~ X . We claim that P E A X "~ *. This is true since, by assumption, X is a EAcellular space (Example 2.D.2 above). But by induction from (B.I.1) we show below (F.6), (F.7) that the homotopy fibre FN of P E N A X * P E A X is an oriented polyGEM for any connected A, X. However, we just saw that the homotopy fibre of that map is X itself. But this space by [HFLT] cannot be an oriented polyGEM (A.1). This contradiction implies [EiA, X] ~ 0 for infinitely many i's, as needed. Remark: Notice that in order to prove Corollary F.2 above we do not have to use the heavy result of [HFLT], since H . ~ E Y is a tensor algebra and is not nilpotent. Therefore, already by MooreSmith [MS] EA cannot be an oriented polyGEM (A.1)
if H.(EA, Z / p Z ) ~ 0. Hence there must be infinitely many maps E~A ~ EA for any such A. F.3 THE MAP O'N: P E N w X Let X, W map
be any connected spaces. We now apply (A.8) to analyze the fibre of the
aN: PENwX
P~,wPENwX
" PEwX
~ PEwX
map: aN: PENwX
, PEwX
Since it follows directly form the definitions that
the following map , PEwX
is up to homotopy
a EWnullification
for any spaces W, X. Consider this map for N = 2.
F.4 PROPOSITION: The homotopy fibre F of the map a2 is a GEM. Proof: We know from (1.H.2) that P E w F ~ * since F is the fibre of EWnullification. Therefore by (B.1.1) we get that P E ~ w F is a GEM. However being the homotopy fibre of a map of E2Wnull spaces F is also E2Wnull. Therefore IF is equivalent to its E2Wnullification and thus it is a GEM.
The GEM appearing in (F.4) has several special properties that we have already seen in (A.8) and (A.9) above. Therefore one can deduce directly from the argument of (F.4) above:
124
5. GEM error terms
F.4.1 COROLLARY: The homotopy fibre ~ of the map a2 is a GEM whose space of pointed self maps map. (IF,IF) is discrete. Proo~
See (A.8)(A.10) above.
It is killed by P~.w and it is also ~2Wnull. We now proceed to consider the fibre of aN. We use the usual comparison diagram: F2
' F3
1
(F.4.2)
F2
*
' F1
1
1
, Pz~wX
, P~.~wX
, P~wX
, P~wX
Since by Proposition F.4 just proven the fibre F1 and F2 a2e GEMs then by definition IF3 is a (twisted) polyGEM. This argument carries over by induction to show: F.5 PROPOSITION: The homotopy fibre ]FN of the map aN is a polyGEM. But we now use (A.5) to show F.6 PROPOSITION: The homotopy fibre FN of the map aN is an oriented polyGEM. Proof." We show that the inductive construction of the polyGEM uses principal fibrations. This follows from the fact that the fibres G~ in the tower GN
(F.6.1)
1 P~.~wX
GN 1
G2
1 " P~~wX
1 . . . . .
P~.2wX
, PEwX
(where G~ = F i b ( P ~ x X , P ~ ,  l x X ) ) are all connected GEM spaces that have homotopically discrete spaces of pointed selfmaps as was shown in (F.4.1) above, and thus of course discrete spaces of pointed selfhomotopy equivalences. Their connectivity follows e.g. from (1.H.2) since each of the horizontal maps is a nullification with respect to a connected space. Since the base spaces of all the fibrations in the tower are 1connected the classifying map can be lifted to the classifying space WGi = BGi, which exactly means that these fibrations are principal. Stated otherwise for each Gi denoted here generically by G the fibration from which the universal fibration with a connected fibre G arise [D=Z1], see (1.F.1):
5. GEM error terms
125
aut'G
* autG
~ G
is a covering space fibration, since the fibre is discrete. But the base is 1connected, so the total space is a disjoint union of copies of the base space G and as a group it is a twisted product, thus we have a fibration:
BG
* BautG
, K(Tr0aut~
1)
Recall [Mayl] that fibration with fibre G are classified by homotopy classes of maps to BautG. Since the spaces in the tower above are simply connected the classifying maps to BautG can be lifted to BG, rendering the fibrations in the tower above principal as needed.  The following is an immediate F.7 COROLLARY: For any connected space X the nullification P Z N x E X is an oriented polyGEM. 
Neisendorfer's Theorem on connected
covers
of finite complexes
Using the nullification functor and Miller's theorem, Neisendorfer has shown [N] that one can recover the pcompletion of, say, a 2connected finite complex from any high connected cover of this complex. Since the nullification of a point is always to a point, we get that, if any high covering is contractible, then the original space must have a trivial pcompletion. In particular, if one takes a sphere we detect in this way the nontriviality of ptorsion elements in infinitely high dimensions. Another result in the same spirit was discovered by McGibbon [McG] says that under mild conditions a finite space X can be recovered up to homotopy from the homotopy fibre of its canonical map to its stabilization Q X . Here is one formulation of this remarkable theorem [N]. F.8 THEOREM: If X is a simply connected finite complex with 7r2(X) finite and W = K ( Z / p Z , 1), then for any integer n > 0 the nullification P w ( X p < n >), of the pcompletion of the nth connected cover of X , has Xp itself as its pcompletion. Proof:
The proof is similar to that of (F.1). One interprets the Sullivan conjecture
as saying that finite complexes are Wnull and therefore do not change under nullification. On the other hand, the nullification of the fibre of the map from a high cover of X to X is killed by P w , assuming everything is pcomplete, since these fibres are polyGEMs and so one can use (1.H.1) inductively on the finite construction of this polyGEM out of EilenbergMac Lane spaces. The following interesting variant of (F.8) is due to McGibbon [McG]:
126
5. GEM error terms
F.9 THEOREM: I[ X is as in (F.8) above then the m a p
PwFib(X
, f~~176
, PwX
~ X
induces an equivalence on the pcompletions [BK], where W = K ( • / p Z ,
1) and p
is any p r i m e number.
We will not reproduce the proof here but remark that the theorem follows from another result in [McG] that shows that the nullification P w kills up to pcompletion any connected infinite loop space whose fundamental group is a torsion group. 
6. H O M O L O G I C A L L O C A L I Z A T I O N NEARLY PRESERVES FIBRATIONS
A. I n t r o d u c t i o n , m a i n r e s u l t We now turn to homological localization LEZ/p for a generalized (nonconnected) homology theory and, in particular, to modp complex Ktheory and higher Morava Ktheories. This section is largely taken from [DFS]. Homological localization as defined by Bousfield [Bl] is a special case of homotopy localization with respect to a map f : V * W. Our concern in this chapter is to prove that in certain nottoorestrictive cases, Bousfield homological localization 'nearly' preserves fibration sequences. Notice the well known counterexample: Consider the fibre sequence
Since the base space in known to be Kacyclic JAnH] and the fibre CP ~ = BS 1, being a retract of BU, is Klocal, the fibration is not preserved as such under Klocalization. It turns out that in a certain precise sense this fibration plus a few others (see (C.2) below) exhaust the possible 'nonexactness' of Ktheory localization for loop fibrations. This is because the general fibration theorem (5.D.6) above specializes for homological localization to yield a rather small 'error term' when localizing double loop of a fibration sequence. It is very probable that the same theorem holds even for single loop spaces. This near preservation of fibration is of course very useful by itself. It will be used in the next chapter to show that under not too restrictive assumptions there is an accessible description of the Bousfield localization with respect to modp K theory in terms of a certain telescope. It will of course render the modp homotopy groups of that localization more accessible. A.1 THEOREM: Let LK be the homotogical localization with respect to complex modp Ktheory. Let F ~ E * B be a fibration over a 2connected pointed space B. The homological localization LK nearly preserves the fibration ~t~F ~ ~2 E ~ ~22B up to an error term J = K ( H , 2) x K(G, 1) x K ( S , 0), where H is torsion free and G, S are abelian. Let F * E ~ B be a fibration and Lg(n) the homological localization functor with respect to Morava Ktheory K ( n ) , where n > O. Up to an error term with at most three homotopy groups in dimensions n  1, n, n + 1 the functor Lg(n) preserves the double loop of the given fibration. More generally, for any ptorsion homology theory E Z / p Z , there exists an integer d(E), 1 < d < ~ , such that LEZ/p preserves
128
6. Homological localization
any doubleloop fibration up to an error term with possibly nontrivial h o m o t o p y groups only in dimension d  1, d, d + 1.
A. 1.1 REMARK ~ NOTATION: The proof is given in section C below. Standard examples show that these three 'homotopy groups of the error term' do in fact arise nontrivially in homological localization. See (C.2) below. We may assume that all the spaces involved are HZ/pcomplete with respect to usual homology. Since we work with a given prime p and HRlocalization preserves any fibration which is a doubleloop fibration by the fibre lemma of [BK], since it is principal, therefore one may take the pcompletion before applying LE. Condition (5.D.6) (3) leads us to consider E,acyclic spaces, i.e. spaces X with L E X ~ *. It will be convenient to denote by PE the 'nullification or localization functor with respect to Eaeyclic spaces', namely PE = Pc(g) where C(g) is the mapping cone of g appropriate to E,. In other words P E = VAcy(E), where Acy(E) is the wedge of all pointed E,acyclic spaces with cardinality not bigger t h a n / ~ , S ~ For example, if E is integral homology, then P E is the plus construction of Quillen. Notice that V E X ~ * ~=~ L E X ~ * and this distinguishes L E from a general L f . B. L o c a l i z a t i o n o f p o l y G E M s The main tool we need for homological localization is Theorem 5.D.6 above. In the present case we will need to analyze conditions (1), (2), (3) of that Theorem and, in particular, to gain some knowledge of the localization of polyGEMs. In general one would hope for a generalization of (4.B.3) and (4.B.4) for polyGEMs, namely that when the functor L f is applied to a polyGEM the result is still a polyGEM. This however is not known. Luckily we can do with a statement concerning the loop space of a polyGEM. B.1 LEMMA: Let W be a space such that for some integer n and for any abelian group G one has P w K ( G , n) ~ , . I f Y E S is an oriented p o l y C E M we have vanishing h o m o t o p y groups as follows: 7riPwf~Y ~ O, for a11 i >_ n. Remark: It is likely that also 7r~PwY ~ 0 for i _> n. But in our proof we need to loop down once.
B.2 COROLLARY: L e t E , be any homology theory for which there exists an integer n >_ 1 such that for any abelian group G one has P E K ( G , n) ~ * (i.e. E , K ( G , n) = 0). L e t T be a p o l y G E M with f~T being E,local. Then 7rif~T ~_ 0 for all i >_ n.
By (A.1.1), for E , we have a space W that satisfies the conditions of (B.1). But then P w f ~ T ~ f~T since F/T is E,local, thus the conclusion follows from (B.1.) Proof orB.2:
P r o o f o f B.I:
First assume that Y is a GEM. Then ~ Y and, by (4.B.4) above, is Wnull GEM, so is every retract of
P w ~ 2 Y are also GEM spaces. Since P w ~ Y
6. Homological localization
129
it. Now K ( ~ q P w ~ Y , i) for every i is a retract of Pw~2Y, thus this EilenbergMac Lane space is Wnull. By assumption on W we get ~riPw~Y ~ O, as needed. We proceed by induction on the construction of the oriented p o l y G E M Y. We use the inductive filtration of the class of oriented polyGEMs given in (5.A.2). We assume by induction that the conclusion of (B.1) holds for all kpolyGEMs. Let Y = Wk+l be a (k + 1)polyGEM with Wk+l * X * X 1 a fibration in which X, X 1 are kpolyGEMs. Consider the (nonfibration) sequence:
P w ~ W k + I ~ P w g t X ~ P w ~ t X 1.
By Theorem 5.A.5 above this is a fibration up to a K(G, n)null GEM, say A. In other words it induces a fibration:
A +P w ~ W k + I + Fibre ( P w Y I X *P w ~ X 1 ) .
Now A is K(G, n)null GEM and therefore, by the above argument, its homotopy groups vanish above dimension n  1. By the induction assumption the same holds for the base space of this last fibration sequence. Therefore, it is also true for the total space, as needed. B.3 COROLLARY: I f L f K ( G , n ) ~ 9 for some f and all G, then 7 r i L f ~ 3 r "~ 0 for i _> n for any oriented polyGEM Y.
Remark:
Again this is likely to be true without the looping.
Proos By (5.D.6)(1) L/~22Y is again a polyGEM. However, since L / K ( G , n ) ~ * we have L f = PK(a,n)Lf, or Lf~22Y is a K(G, n)null double loop space, and the result follows by (5.A.5) above. In the following lemma we show how (1), (2) and (3) of (5.D.6) combine to render ~ A a small oriented polyGEM. The main idea is that, if a space X is both E,local and E,acyclic, then it must be contractible. Since in (5.D.6) there is a 'shift', A can still have several (at most three) nonzero homotopy groups. B.4 LEMMA: For any p torsion homology theory ( E Z / p ) . there exists a typical number 1 < d < co such that if A is any 1connected space satisfying: (1) ~ A is an oriented polyGEM, (2) A is EZ/pacyclic, (3) ~22A is EZ/plocal, then ~riA ~_ O, except possibly for i 6 {d + 1, d + 2, d + 3}. Further, the groups lriA satisfy: (4) K(~rd+lA, d + 1) is EZ/pacyclic,
130
6. Homological localization
(5) K(Trd+3A, d + 1) is PEz/plocal. In [B6, 6.16.4] the homological localization of K(G, n)s with respect to any homology theory with Z/pZcoefficients EZ/p is given.
We say t h a t an abelian
group G is H Z / p  l o c a l if K(G, n) is H Z / p  l o c a l for some (and thus for all [BK]) integers n > 1. We need the following result of Bousfield: B.5 PROPOSITION [B6,6]: Let G be any abelian HZ/plocal group. Then there
exist 1 < m < oo such that:
LEZ/pK(G
, i) "~
K(G, i)
LEz/pK(G , i) ~ *
for 1 < i < m,
/'or m + 2 < i < oo.
B.5.1 REMARK: In the extreme case m = (x~ the space K(G, i) is E Z / p  l o c a l for all 0 < i < oo. It is clear from (3.A.1) above t h a t if LEz/pK(G,i) ~ * for some i = d, then the same holds for all i > d. Thus every homology theory E . has a transitional dimension (as explained in [B6, 8.1]); from this dimension and higher all E i l e n b e r g  M a c Lane spaces are E.acyclic. B.5.2 EXAMPLE: For complex or real K  t h e o r y we have m = 1. B.6 PROOF OF B.4: We use the framework of P r o p o s i t i o n B.5 above. The proof is then divided into three parts. We first deal with the easy cases m  ~ and m = 1. Then we proceed to 1 < m < oo. (m = ce). If for our homology theory EZ/p one has m = ~ , then we are in (3.5), (6.2) of [B6] and the E Z / p  l o c a l i z a t i o n for any space is the same as HZ/plocalization, since being E Z / p  l o c a l is equivalent to being H Z / p  l o c a l . But since we assume (3), t h a t Yt2A is H Z / p  l o c a l , we get from s t a n d a r d properties of HZ/plocalization t h a t A itself is HZ/p local and thus A is E Z / p  l o c a l . But by (2), A i s also E Z / p  a c y c l i c . Thus A  .. Therefore in t h a t case the t h e o r e m is proved. (m = 1) This means t h a t K(Z/pZ, 1) is also E Z / p  a c y c l i c a n d therefore [B6, 4.1] EZ/p is the trivial homology theory EZ/p ~ *, hence, s being E Z / p  l o c a l , this also means s _~ .. Since A is 1connected we get A has at most one nontrivial h o m o t o p y in dimension 1, so the conclusion holds. (1 < m < co). We now get to the nontrivial case where the integer m in B.5 is neither 1 nor oo. In this case the argument is more involved and occupies the rest of the present section B: Denote this integer specific to E by d = d(E) = m. We now use Corollary B.2 above, to show t h a t higher h o m o t o p y groups of A vanish. We can a p p l y (B.2) with T = s
for the theory EZ/p and n = d + 2, since we
have LEz/pK(G, d + 2) ~ 9 _~ PEz/pK(G, d + 2). Since ~ T = s
is E Z / p  l o c a l
6. Homological localization
131
and T = ftA is a polyGEM we conclude that 7riftT Therefore
(B.6.1)
7hA ~ 0

71i~ 2 A

0 for i _> d + 2.
for all i > d + 4.
On the other hand, we have chosen d such that any H Z / p  l o c a l K(G, i) for i < d is E Z / p  l o c a l [B6, 6.3, 6.4]. Therefore the dPostnikov section of A, namely PdA, is PEz/plocal since it is a finitelyrepeated extension of PEz/plocal spaces by PEz/plocal EilenbergMac Lane spaces (1.A.8) (e.6). Therefore, in the fibration
+ A + p d A.
The base is PEz/plocal and the fibre, being dconnected, has homotopy groups in at most three dimensions d + 1, d + 2 and d + 3, as we saw in (B.6.1) above. Now using (3.D.3)(2) to localize the fibration we apply PEZ/p to get (B.6.2)
PEz/p(/~) + PEZ/p A ~ PEZ/p(Pd A) PdA;
the base in the original sequence being PE.local, this is still a fibration. We will now show that all the three spaces in the fibration (B.6.2) are contractible: First, since by our assumption LEZ/pA _~., we get, using R e m a r k A.I.1 above, P E z / p A ~ *. We claim t h a t PEZ/p~ ~ * and therefore also Pd A '~ *. Second, notice that by the definition of d we have PEZ/BK(G, d + i) ~ * for all i >_ 2. Therefore, taking the (d + 1)connected cover fibration over /~ and using (1.H.I) we get PEZ/pA ~ PEz/pK(G, d + 1) with G = ~rd+l/X = rd+l A. Now by (4.B.4) above PEz/pK(G, d + 1) is a dconnected GEM. Now, from the above PEz/plocalized fibration (B.6.2) and from PEZ/pA ~_ *, we get that this dconnected GEM is the loop space over a dPostnikov section. This is possible only if both Pd A and its loop space PEZ/p~ ~ PEz/pK(G, d + 1) are contractible. In sum all the spaces in (B.6.2) are contractible. In other words, A has no homotopy groups below dimension d + 1 and, in conjunction with the first part of the proof, has nonvanishing homotopy groups at most in the three dimensions d + 1, d + 2, d + 3, as claimed. Now we turn to the proof of (4) and (5) in (B.4). To prove (4) consider the fibration sequence A(d + 2) ~ A +
Pd+iA
132
6. Homological localization
Pd+l denotes the ( d + 1)P0stnikov section of A. By our assumption on d any K(G, d + i) is EZ/pZ acyclic for i > 2. Since we know that the (n + 1)connected
where
cover A(d + 2) has homotopy groups only in dimensions d + 2 and d + 3, it must be EZ/pZ acyclic. Now in the above fibration both total space and fibre are EZ/pZacyclic, so the base is EZ/pacyclic too ((1.H.1) and (A.I.1) above). But the base space Pd+IA is K ( T r d + l A , d + 1), since we know that the lower homotopy groups of A vanish. This proves (4). Finally, to prove (5), we use the fibration sequence K(~'d+3A, d + 1) ~ ~2A * pd~2A. Again by our assumption on d every K(G, d  i) is EZ/pZlocal for i > 0. Therefore Pdgt2A is the total space of a fibration with PEz/plocal base and fibre, hence it is PEZ/pIocal by (1.A.8)(e.6) above. Since we assume (3), that ~t2A is EZ/plocal, we can conclude from the above fibration sequence that the fibre K(Trd+3A, d + 1) is PEz/plocal too, as needed. This completes the proof of (B.4).  C. L o c a l i z a t i o n w i t h r e s p e c t to M o r a v a K  t h e o r i e s We now prove Theorem A.1. Let K , be modp Ktheory also denoted here by KZ/p,. As far as localization goes, K , is the first in the series of Morava Ktheories K(n). We consider these theories here at an odd prime in line with [RW], which considers the value of the theories K(n) on EilenbergMac Lane spaces for odd primes.
Let LK be Bousfield's homological localization at modp Ktheory KZ/p,. The results above specialize to the effect of LK on a double loop of a fibration sequence ~ t 2 F ~ ~ 2 E ~ ~ 2 B for a 2connected space B. In that case we consider the 'error term', namely the homotopy fibre J of the map:
LK~2f ~fibre(LK~t2E , LK~2B). Since L ~ 2 Y ~ ~2LE2gY we first consider LE2g. We claim that the 'error term' fibre J is of the form J = ~2A where A satisfies conditions (1), (2), (3) of B.4 above. First notice that since every KZ/plocal space is also HZ/plocal, all our spaces here are HZ/pZlocal. Second, J is clearly a double loop space since, for any map f: V ~ W and any fibration F ~ E ~ B over a 2connected B, the canonical comparison map Lf~t2F ~ fibre(Lf~2E + Ly~2B) is the double loop of
L ~ 2 f F ~ fibre(L~2/E *L~2fB);
6. Homological localization
133
so the fibre of that canonical comparison m a p is the double loop of the fibre of that latter m a p (3.A.1). It now remains to recall that by JAnHI the space K(G, n) is KZ/pZacyclic for all n > 3 and all G. On the other hand, by [B6] K(G, 1) is KZ/pZlocal if it is H Z / p  l o c a l and G is abelian. Therefore for modp K  t h e o r y one has d ( K Z / p ) = 1, where d is as in (B.4). Therefore, besides lr0 the possible homotopy groups of the error term J above are in dimensions one and two. Thus J is a disjoint union of 2stage Postnikov sections J = IIJ0 sitting in a fibration sequence:
K(~2, 2) + Jo + K(~h, 1). Now by (B.4) (5) we know that K(~r2, 2) is KZ/plocal. Therefore 7r2 is complete without Z/pZtorsion, i.e. a free module over Zp. Since J0 is a loop space, this fibration splits and Jo ,2_ K(~r2, 2) x K(~rl, 1); see L e m m a C.1 This concludes the proof for the case of Ktheory. We now turn to the higher Morava Ktheories at a given odd prime p. calculations in JRW, 12.1], for odd primes,
Z/pZdouble below. By the
[((n)K(G, n + 2) = 0 and K ( n ) K ( Z / p Z , n + 1) = 0. We can apply (B.4) above to conclude: If Y is the homotopy fibre of
LK(n)I'I2F . fibre(LK(n)fl2E . LK(n)I'12B), it fits into a fibration:
K(F, n + 1) x K(G, n) ~ Y * K(S, n  1) where F is a torsion free group. Again, Y being a double loop space and F torsion free, we can conclude that the possible Kinvariant connecting the two higher groups must vanish, since there are no unstable elements in this range. This concludes the proof of (B.4) for Morava Ktheories at odd primes. 
134
6. Homological localization
C.1 LEMMA: Let Y "" 122Y 1 be a Klocal space with 12Y 1 a polyGEM. Then Iio, the null component of Y, fits into a fibration K(F, 2) ~ ]Io * K ( C , 1) where F is a torsion free group. Proof'. We use (B.2) below for T = ~2Y1. Since by [AnH] reduced Ktheory vanishes on K ( Z , 3), we get L K K ( Z , 3) ~ .. Applying (B.4) above we get that Y0, our Klocal polyGEM, fits into a fibration:
K(G', 2) ~ Yo * K(G, 1),
since it only has two nonvanishing homotopy groups. We claim'that G ~ is a torsion free group. To see this we observe that both base and total space Y are K(Tor, 2) null with respect to any abelian torsion group Tor: For the base observe directly that m a p . ( K ( T o r , 2), K(G, 1)) ~ .. And the total space Y is, by assumption, Klocal, so it is null with respect to any Kacyclic space, such as K(Tor, 2) [AnHI. Therefore the fibre K ( G ' , 2) must also be Klocal and, in particular, K(Tor, 2)null, so it can admit no maps from K(Tor, 2). Hence Tor G ~ = 0 and G ~ is torsion free. C.2 Standard examples: We now give a short list of examples that show how the three possibly nontrivial homotopy groups in the error term in (A.1) and (B.4) actually arise in standard fibration sequences. We consider the effect of L g on fibrations: 1. In the fibration
SUZ/p ~
9 ~ BSUZ/p
the error term is K ( Z / p Z , 0). This is true since the Klocalization of the infinite loop space BSUZ/p is homotopy equivalent to B U Z / p using, say, [B3] while SUZ/p is Klocal. 2. In the fibration K(G, 1) + * + K(G, 2)
for any abelian G, the error term is K(Tor G, 1). 3. In the fibration K(Z, 2) ~ 9 ~ g ( z , 3) the error term is K ( Z , 2). Thus the three possible dimensions actually arise. Similar examples can be given for the higher Morava Ktheories.
7. C L A S S I F I C A T I O N O F N U L L I T Y A N D C E L L U L A R T Y P E S O F FINITE pTORSION SUSPENSION SPACES
Introduction Let P** be the full subcategory of S, consisting of finite ptorsion spaces, namely W is finite and H,(W,Z) is a finite ptorsion group for W E 9~,. We will consider nullity classes (  ) and cellular classes C ' (  ) . It turns out that after a single suspension both classes in P** can be understood in terms of Morava Ktheories and, in particular, HopkinsSmith theory of types. These two classifications are closely related in view of (3.B.3) above. Recall that (W) = (V) or V and W have the same nullity type if V < W and W < V or P v W ~ * and P w V ~ *, alternatively, if for all X E S, one has the double implication.
map,(V, X) ~ 9 ~
map.(W, X) ~ *.
WARNING: Here the partial order < is used in the opposite sense to that of [B4]. The present notation is consistent with connectivity i.e. X < Y implies connX < connY. Also S ~ < S T M < S ~+2... etc. It is also consistent, as we will see here, with the HopkinsSmith type.
Also recall that C'(A) = C'(B) or A and B have the same cellular type if A << B and B << A o r i f C W A B ~ B and C W s A ~ A. Both < and << define partial orders on S, and .~,. Our aim is to determine these partially ordered sets of types up to isomorphism. This can be done in F~.T~~ the suspension spaces in 9~,. It turns out that in ~P** the above partial order can be reduced to Ktheoretical and homological properties.
A. Stable nullity classes and HopkinsSmith types Associated with nullity and cellular types are the corresponding stable notions. It turns out that these agree with each other and with the notion of HopkinsSmith type [HS]. With each W E S, we have a (generally strictly) increasing series
(*)
W < ~ W < ~2W < ... < ~kW < . . . ,
(*)
W ~ ~,,W <~ ~2W << ... << ~kW <~ . . . .
136
7. Classification of nullity types
Therefore we can define 'stable' versions of < and <<: namely A <s B (or A <<~ B) if for some k A < ~ k B (or A << EkB). In view of (3.B.3) above the partial orders <~ and <<~ are the same: A <<~ B if and only if A <~ B. We denote in line with Bousfield the stable equivalence classes by {A}, namely {A} = {B} if A <8 B and B<sA.
First let us recall (5.B.6) above. In the present context it says that on suspension spaces the partial order (<) is inherently stable: COROLLARY 5.8.6: For any spaces X , Y and integer k > 1 one has: ~ X < P~Y iff EkX < ~Y.
But the main observation about stable types is the following relation between stable classes {A} and HopkinsSmith types: A.1 THEOREM: For any prime p and spaces W, V E P . one has {A} <s {B} i f and only i f type A <_ type B . Recall: For each space X C 9~, with H . ( X , Z ) r 0 there exists an integer n > 0 for which ~ [ ( m ) . X = 0 for all m < n and K . ( m ) Z r 0 for all m _> n, where / ( ( m ) . is the ruth reduced Morava Ktheory at the prime p. This integer n is called the type of X at the prime p. For a clear survey see [H], [R]. Further, the type of X is related to classification by Hopkins and Smith of socalled thick subcategories. A full subcategory ~ ~ U C )v,p is called thick if it is closed under the third term in a cofibration sequence A ~ X ~ X / A and under homotopy retracts. Their main theorem classifies all thick subcategories:
A.2 THEOREM (Thick subcategory): I f L t is a thick subcategory in FV,, then ld is equal to the thick subcategory o f all spaces o f t y p e >_ n for some 0 < n < oc. Using <8 we can define for each space W E F,v a thick subcategory as follows: A.3 DEFINITION: The thick subcategory associated with W , denoted by U ( W ) , consists o f all V c FP, for which W < s V.
We must show t h a t / J ( W ) is closed under retracts and third terms in cofibrations. Since for any retract V I of V one has V < W, and thus V <s W, the first is clear. Using the Puppe sequence A ~* X * X / A ~ E A and (3.D.1) above, the second closure property follows immediately: P w A ~ * and P w X ~ * implies P w ( X / A ) ~ * etc. On the other hand, if ~ [ ( r n ) , A ~ 0 and P A E k B ~ *, then c l e a r l y / f ( m ) , E k B _~ 0, thus B is also/((m),acyclic. So we have type A _< type B if A <~ B. To conclude, one can reformulate the above discussion as follows: Proof of A.l:
7. Classification of nullity types
137
A . 4 THEOREM: The following classifications of ~'~. are one and the same as partially ordered sets of equivalent classes: (1) stable classes of nullity types, (2) stable classes of cellular types, (3) thick subcategories of J~..
B. U n s t a b l e nullity t y p e s Without stabilization as above there is a difference, 'at the bottom dimension', between nullity type and cellular type of spaces in S. or in 3~.. The canonical example comes from (3.C.9) and (2.D.11), namely K(Z/p2Z,n) and K ( Z / p Z , n) have the same nullity type while they do not have the same cellular type. Rather we have K(Z/p2Z, n) << K ( Z / p Z , n), but the opposite cellular inequality does not hold.(Compare [B12, 3.1].) B.1 EXAMPLE: Since we have a fibration sequence for each n > 1,
K(Z/p2Z, n) x_,pK(Z/p2Z, n) * K ( Z / p , n + 1) • K ( Z / p , n), it follows from (3.C.9) and (2.D.11) above that the product at the base of the fibration and each of its factors is a K(Z/p2Z, n)cellular space. Thus K(Z/p2Z, n) << K ( Z / p Z , n). Similarly, one has for Moore spaces Md(G) the cellular inequalities Md(Z/p2Z) << Md(Z/pZ) but not in the opposite sense. Using the technique of Chapter 5 the classification of nullity type can be reduced to the stable classification for ~)rp, i.e. suspension spaces in .~.. NOTATION Let connX denote the connectivity of X. B.2 THEOREM [B4]: For a given prime p let W, V be two spaces in :~.. Then the following two conditions are equivalent:
(i) (ii)
~ w < r~y, ~ W <s ZV
and
conn~W < conn~V.
Proof: The implication (i) =~ (ii) is the easier one. Assuming (i), then ~ W <s ~ V is immediate while connZW < conn~V follows from: B.3 PROPOSITION: For any A , B E S., irA < B then connA _ connB.
Proof: Assume, by negation, that connA > connB = n. Then for dimensional reasons it follows that for every group G we have map.(A, K(G, n + 1))  *. But the canonical map B + K(~rn+lB, n § 1) is not null if connB = n (where corm
138
7. Classification of nullity types
denotes the connectivity), so m a p . ( B , K ( z c n + l B , n + 1)) ~ ..
This contradicts
P A B ~ *. The proof of the implication (i) =a (ii) is now complete.
To prove the opposite implication we follow the approach of Bousfield in identifying the nullity type of a finite space E X E S.p with that of a certain infinite space E k X V K ( Z / p Z , d) for any k _> 0 and d = connEX + 1. Assuming (B.4) below the proof is easily completed: We must show that for some integers k, l there is an inequality:
EkWvK(Z/p,n+2)
< EtVvK(Z/p,m+2)
where n = c o n n E W , m = c o n n E V . But this is clear since by assumption n < m so K ( Z / p , n + 2) < K ( Z / p , m + 2) and also by choosing l large enough in relation to k one gets EkW < EIV. This completes the proof of (B.2).  In the proof of (B.2) we have used the following basic result is due to Boasfield. We give here a somewhat different proof based on Chapter 4 and the useful (B.4,B.5) below. For simplicity we consider here only finite ptorsion spaces; a similar statement holds for all ptorsion spaces. B.4 THEOREM: Let X E ~
be a finite, ptorsion space and let n = connX. Then
for each k >_ 1 the spaces E X and Y = E k X V K ( Z / p , n + 2) have the same nullity type.
The proof runs over the next two pages using several propositions and lemmas of independent interest. We must show P z x Y ~ * and P y E X ~ *. Consider the first and easier part. Since k > 1 we have P z x Y ~ P ~ x K ( Z / p Z , n + 2 ) by ( * ) i n (A) above. By Proposition B.5 below one has P r o o f o f (t3.4):
P ~ x K ( Z / p Z , n + 2) ~ P z z x K ( Z / p Z , n + 2).
If G = H,~+I(X, Z), then K ( G , n + 2) is, up to homotopy, the bottom factor in the GEM product Z E X . Thus P z x Y ~ * will follow from PK(C,,~+2)K(Z/p, Z, n + 2 ) 9. Since by assumption G is a nontrivial ptorsion group, the last equivalence follows from our example (3.C.9) above, proving P z x Y ~ * 9 Now consider P y E X . By (5.F.7), P z ~ x E X is an oriented polyGEM with ptorsion homotopy groups. In fact by (C.7) below it is a finite Postnikov stage with a finite number of finite ptorsion homotopy groups. This polyGEM is (n + 1)connected and thus the desired P y E X ~ * follows from the following lemma (B.4.1).
7. Classification of nullity types
139
B.4.1 LEMMA: If W E .~P is an mconnected finite ptorsion polyGEM
then PK(Z/pZ,,~+I)W ~ *. Proo~
This is true if W is an mconnected ptorsion GEM. Now use (1.H.1) to
proceed by induction on the finite construction of the p o l y G E M .

B.5 PROPOSITION: For any GEM space X and any W 6 $., there is a natural
homotopy equivalence PzwX ~ PwX. We can reformulate this proposition as follows: B.5.1 PROPOSITION: For any GEM space X and any W E S., we have the
double implication:
m a p . ( W , X ) ~ 9 4=~ m a p , ( Z W , X ) ~ *.
These two propositions are one and the same, since we know a priori t h a t if X is a G E M then the spaces P z w X and P w X
are b o t h G E M spaces, and we use
the following l e m m a in conjunction with W < Z W (since, in fact, W << Z W ) ; see C h a p t e r 4. In fact all one needs to show in order to get the h o m o t o p y equivalence is t h a t P z w X is W  n u l l if X is a GEM.
B.6 LEMMA: I f W < V, then we have natural equivalences of functors
P w ~ P w P v ~ P v P w . Proof: Given the first equivalence, the second follows since P w X is W  n u l l and thus Vnull for all X . To get the first equivalence we s t a r t with the m a p P w X * P w P v X , which is P w applied to the coaugmentation. To get a h o m o t o p y inverse P w P v X .   * P w X we s t a r t w i t h X   * P w X . Since W << V this i s a m a p t o a Vnull space, and it factors uniquely up to h o m o t o p y t h r o u g h X * P v X * P w X . But now the m a p on the right is a m a p of P v X into a W  n u l l space, so it factors t h r o u g h P v X + P w P v X * P w X . H o m o t o p y uniqueness now gives t h a t the m a p P w P v X ~ P w X we have obtained is the desired h o m o t o p y inverse. Proof of Proposition B.5.h
One implication follows i m m e d i a t e l y from W << Z W
as in (4.A.2.1) above. To get the other direction we use the basic l e m m a (4.B.2) above: A n y m a p W * G E M from W to a G E M space factors up to h o m o t o p y t h r o u g h W * Z W * GEM. Since we assume m a p . ( Z W , X ) ~ *, we can conclude
[W,X] ~ *. Therefore, to get m a p . ( W , X ) _~ 9 it is sufficient to show [EkW, X] ~ * for every integer k > 0. Since X is a GEM, it is sufficient to show t h a t [T7.EkW, X] ~
140
7. Classification of nullity types
9. Since we are given map(ZW, X) ~ *, we know that [EkZW, X] ~ ,. Therefore our claim follows from BI7 LEMMA: For any k >_ 0 and W C S, we have EkZW << ZEkW. Proof." In fact ZEkW is obtained from ZW by taking ktimes the classifying space functor B k Z W ~ ZEkW. The last equivalence follows directly from the basic property of Z ~ S P ~ , the infinite symmetric product functor as investigated by Dold and Thorn [DTh]; namely if A * X ~ X / A is any cofibration, then ZA Z X * Z ( X / A ) is a fibration sequence. Therefore (letting B denote the classifying space functor) Z X ~ 9 + Z E X is a fibration sequence; in other words Z E X = BZX. But we will show in Chapter 9 below (9.D.3) that BG >> EG for any group object in S, andthus, by induction, BkG >> EkG for any abelian group object in S,. This completes the proof of the B.7 and thus of Theorem B.4. II An immediate deduction is B.8 COROLLARY: The following two conditions are equivalent in i~, : (i) E X and E Y have the same nullity, (ii) type E X  type E Y and connEX = connEY. C. U n s t a b l e cellular t y p e s The classification of cellular types is very closely related to that of nullity types. We saw above that, while the two Moore spaces M d ( Z / p ) and Md(Z/p2Z) have the same nullity, they are not of the same cellular type so that the inequality M d ( Z / p Z ) << M d ( Z / p Z 2) is strict. The classification is related to the ptorsion in the bottom dimension of E X [B12]. C. 1 DEFINITION: Let G be a finite abe~an ptorsion group. We denote by t(G) the exponent of G i.e. the maximal integer ~ for which Z/p~Z is a direct summand of G or the minimal ~ with p~. G = O.
With this integer t(G) we can formulate the classification of cellular types in EF,p as follows: C.2 THEOREM: Let X e J:P, and let n = connX and ~ = t(Hn+I(X,Z)). Then for each k > 0 the spaces E X and E k X V K ( Z / p e Z , n + 2) have the same cellular types. Proof'. The proof is very similar to that of Theorem (B.4) above, (Compare [B1, 3.1] except that we use the following version of Proposition (B.5).
7. Classification of nullity types
141
C.3 PROPOSITION: I [ X is a GEM and A is any space in S., then there is a natural homotopy equivalence e : CWzAX ~ CWAX. Proof." Since A << ZA by (4.A.2.1) above we conclude C W z A ( C W A X ) ~ C W z A X , hence the natural map. To get an inverse we use (C.4) below: Since by (4.B.3) C W A X is a GEM and also Acellular, it is a ZAcellular space by Proposition C.4 below. Therefore the natural augmentation C W A X * X factors up to homotopy through C W z A X * X , which is the universal map of ZAcellular spaces to X. Hence we get a map f : C W A X ~ C W z A X . Universality and uniqueness up to homotopy now imply as usual that f is a homotopy inverse to e. We have used C.4 PROPOSITION: I[ X is an Acellular GEM, then X is also ZAcellular. Proof:
This follows from Proposition (C.5) below about symmetric products. Since
A << X by assumption, w e g e t ZA<< Z X b y (C.5). But X i s a G E M , s o X i s a retract, up to homotopy, of ZX; thus Z X << X by (2.D.1 (5)) above. It follows that ZA << X, as required. C.5 PROPOSITION: For any A , X E S., if A << X then Z A << Z X . Proof: (Arguing simplicially.) We proceed by induction on the construction of X from A, namely assume X is given as a pointed homotopy colimit by f A~ and / Ai 7> A, where we assume by induction that ZA~ >7 ZA and prove Z X 77 ZA. C.6 LEMMA: I f Y is a free Idiagram in S., then
Z/ Y ~ dirlim/ZY I
where the direct limit is taken in the category of abelian groups. Proof: This is immediate, since for a free diagram Y in S, the homotopy colimit colim fY is just colimiY and the free abelian group functor Z [BK], being left I
adjoint, commutes with colimits (= direct limits). To proceed with the proof we write Z X as Z f A~. I
Now by cellular induction we may assume that the category I is either discrete, so that colimiA~ = V A~, or a coequalizer diagram, so that I  9~ .. By our lemma Z f A~ is isomorphic to lim ZA~ in the category of abelian groups, where the direct limit is taken over the appropriate small category. To proceed with the proof of
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7. Classification of nullity types
(C.5) we examine each of the two small categories I. In both cases we assume by cellular induction that A~ >> A have the desired property, namely that ZA{ >> ZA. Let us start with the second, coequalizer case: We may assume that (Ai) ]1 (A1 ==~ A2) is free, i.e. fl and f2 are cofibrations that agree only on the base point.
12
In that case colimi ZAi in the category of abelian groups is given by a short exact sequence of simplicial abelian groups
ZA1 f~f2 ZA2 ~ Z X = colirnZA~.
Thus we have a principal fibration sequence with the base Z X and, by induction, both fibre and total space are in C'(A). It follows by (2.D.11) above that the base Z X E C'(A), as needed. Turning now to the first case of a wedge the direct limit in the category of abelian groups is given by Z X = @ZAi. Again we assume ZAi >> A and must show ZAi)~_ where (a) represents a well that G{ZA{ )> A. But OielZAi = colim(~)( ~ ordering of I. Thus since C' (A) is closed under linear direct limits which now are taken in $ and finite products, we get the desired result.
Special cases: nsupported W, ptorsion W The functor P E w is best behaved when we assume a certain relation between its first nontrivial and higher homology groups. We would like to know when the error terms have at most one nontrivial homotopy group. It turns out that this can be guaranteed if we assume that K(H,~, W, n) satisfies
K ( H , W , n) < K ( H , + i W , n) where H n W is the first nontrivial integral homotopy and i _> 0. Alternatively, we say that W is nsupported (compare [B4, 7.1]) if P p ~ z w Z W ~ *, where n c o n n H . W + 1. In other words, W is nsupported if P n Z W < Z W . C.7 PROPOSITION: Let W be an nsupported space. Assume that in the fibration
F + E ~ X we have P r . w E ~ P ~ w X ~ *. Then P ~ w F h ~ a single homotopy group in dimension n = c o n n H , W + 1. C.7.1 EXAMPLE: If the nhomotopy group of W has the group of integers Z as a retract, then W is nsupported. The reason is, in that case, K(H~W, n) < K ( Z , n) and for any G we have K ( Z , n) < K(G, n + i) for all i _> 0. C.7.2 EXAMPLE: [B4]: I f / ~ i ( W , Z ) ~ 0 for i < n a n d / t , ( W , Z ) is Jtorsion for some set of primes (resp. Jlocal) and H ' ( W , Z/p) ~ Hom(Hn(W, Z), Z/p) is
7. Classification of nullity types
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not trivial for each p E J (resp. Hn(W, Z(j)) ~ 0), then W is nsupported. This is true, since the existence of a nontrivial, and thus surjective, m a p H,~(W, Z) * Z / p guarantees that K(H,~(W,Z),n) < K ( Z / p , n ) , for each p E ,7. Since Hn+iW are J  t o r s i o n groups by assumption, we get
K ( Z / p , n) < K ( Z / p Z , n + i) < K(Hn+i(W, Z), n + i), for all i _> 0 as needed (resp. for Jlocal). C.7.3 EXAMPLE: If Hk(W, Z) are allptorsion for a fixed prime p and H,~(W, Z) is finitely generated, then W is nsupported. In this case H,~(W, Z) contains a finite ptorsion group Z/p*Z as a direct summand. Therefore
K(H,~W, n) < K(Z/peZ) < K ( Z / p Z , n) < K ( H i W , n + i) for each i _> 0, since H i W is a ptorsion group.
Proof of Proposition C. 7: As we have seen in Chapter 4 above, in such a fibration P w F ~ *, F is EWnull and F is a GEM. Therefore by (B.5), P z w F ~ *. But we assume P ~ Z W < Z W . Therefore P p ~ z w F "" *. On the other hand F is EWnull, and since E W < Z E W = W Z W (where W is the classifying space functor) we get that F is WZWnull; in particular it is K(H~W, n + 1)null, the latter space being a retract of W Z W . But since K(H~W, n) < F we get K(H~W, n + 1) < F(n) (the nconnected cover of F). Now we obtain that F(n) is both K(H~W, n + 1)null (since F is a G E M and F(n) is a retract of F ) and K(H,~W, n + 1)supported. Therefore F(n) ~ 9 and F has a single homotopy group in dimension n. C.8 COROLLARY: If W is nsupported, and F + E + X a fibration sequence over a connected space X and W is an nsupported space, then the error term F i b ( P n w F + F i b ( P ~ w E ~ P ~ w X ) ) has a single homotopy group in dimension c o n n H . W + 1.
8. v l  P E R I O D I C S P A C E S A N D K  T H E O R Y Introduction In this chapter we concentrate on the relationship between two localization functors: homological localization with respect to Ktheory and homotopy localization with respect to the well known Adams map from a certain suspension of a modp Moore space to the Moore space. This map was constructed by Adams as a Kisomorphism from some suspension of a Moore space to the Moore space in the stable range. Using [CN], [Oka], [MR] we know that one may choose the minimal dimensions of the Moore space at the range of the Adams map to be 3 for p odd and , EeM3(Z/pZ), 5 for p = 2. Thus the Adams maps are EeVl : Eq+eM3(Z/pZ) where s _> 0, q = 2p  2 for p odd, while for p = 2 the lowest dimensional map is vl : M13(Z/pZ) , M5(Z/pZ), i.e. the number of suspensions is q = 8 for p = 2. It turns out that, as envisaged by Mahowald, Miller, Ravenel and others and proved in [MT], [T], [B4] under mild assumptions, the two are closely related and, moreover, the modp homotopy groups of these localizations 7r.(LKX, Z/pZ) and ~r. (P1 X, Z/pZ) (where Pvl as usual is the nullification with respect to the mapping cone of vl) are related to 7r.(X, Z/pZ) by a simple algebraic localization of the latter with respect to firstorder operation of vl by composition. As we saw above homological localization is always a special case of homotopical localization with respect to a certain map f between 'large' spaces. In general, the map f is quite inaccessible since we must take the wedge of all possible homology isomorphisms among two spaces with cardinality not bigger than a wellchosen cardinality A. This can be taken as the first infinite cardinal with IE.(pt)l <_~. However, in practice we now know that under quite mild assumptions the above map f can be taken to be a well known map, such as the Adams map for modp Ktheory, or a map between a wedge of circles for integral homology [DF4]. Thus, loosely speaking, the homotopy groups of L K X should be, under mild assumptions, isomorphic to the algebraic localization of ~. (X, Z/pZ) with respect to vl. This allows one to prove several results that are in close analogy with the classical results about rationalization. In this chapter we give an exposition of this analogy. Still, some severe limitations remain. Firstly we know that vl localization renders spaces periodic and, in particular, their modp homotopy groups become periodic after this localization process. But this also means that 'periodic families' which in the original space started in an arbitrarily high dimension are presumably 'pulled down' to the lower dimension within the first range of periodicity. This means, in particular, that even if one starts with a highly connected space of finite type, its vllocalization will have, in general, low connectivity and will have an
8. vlPeriodic spaces
145
infinite number of vlperiodic families starting at the first few dimensions. Hence this periodic space is expected to be 'thick' and not of finite type. Nonetheless its Ktheory is the same as that of the original space, since we construct it by gluing on Kisomorphisms. We shall not treat here localizations with respect to higher v,~maps, but Bousfield shows that the initial steps are similar although the deeper properties seem much harder [B4]. A major obstruction to the natural generalization is the failure of the telescope conjecture of Ravenel. This failure has prevented up to now the characterization, in the stable category, of K(n)local infinite loop spaces in algebraic terms of the homotopy groups, where K(n) denotes as usual the Morava Ktheory. In order to demonstrate the structures that we seek, we first consider the classical case of localization with respect to (a subring of) the ring of rational numbers
Q. Rational homology and rational homotopy: Let us summarize some wellknown results, due mostly to Serre and Quillen, about the relationship between rational homotopy and homology. This will serve us mostly as a paradigm for a similar relationship between 'periodic Vlhomotopy' and Ktheory. For each prime p > 2 we have an operation on 7~.X: multiplication by p,  x p : ~rnX ~ ~nX. Tensoring with rationals is the universal manner of turning all these maps into isomorphisms. For 1connected (or even nilpotent) space one can associate with this operation of tensoring a space level rational localization X ~ XQ that relates nicely to the algebraic operation of tensoring with the rationals Q, compare e.g. (1.E.23). Let X, Y be simply connected spaces: (1) A map f : X * Y induces an isomorphism on ~.  Q if and only if it induces an isomorphism on rational homology. (2) The homotopy groups of XQ, the rationalization of X, are canonically isomorphic to ~ . X  Q. (3) The map X ~ XQ induces an isomorphism on the rational homology groups. (4) The Quillen rationalization functor X ~ XQ is naturally equivalent to the homotopy localization LVp at the map V p : V $2 ~ V $2 of the infinite p
P
wedge of 2spheres into itself. (5) The class of nilpotent, rationally acyclic spaces, i.e. { W I / t . ( w , Q) _~ 0}, is precisely the class of spaces supported by VM3(Z/pZ). Further, if we consider p only simply connected spaces that are Qacyclic, then all such spaces are VM2(Z/pZ)cellular. Thus the basic building block of Qacyclic 1connected P
spaces are Moore spaces M2(Z/pZ) = S 1 Up e2.
Proo?: Of the above (1)(3) are wellknown facts and (4) is explained in (1.E.2) above. So we turn to (5). Compare [B12,3.1]. Since fI.(M~(Z/pZ), Q) ~ 0, every
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8. vlPeriodic spaces
space supported by Mn(Z/pZ) for any prime p and any n > 1 is rationally acyclic. Given a space W w i t h / ~ . (W, Q) ~ 0 and ~rlW ~ 0, we claim it is supported by T = VM3(Z/pZ) = VM(Z/pZ, 2) (i.e. it has nontrivial reduces integral homology only P
P
in dimension 2). To see this, notice that WQ ~ * by (3). We claim that PTW "" *. This is so, since PTW being Tnull has no ptorsion for any p and its rationalization ( P T W ) Q ~ WQ ~ *. The latter follows using e.g. (1.D.3) to the actual construction of P T as a homotopy colimit, at each step the Q localization is contractible so it is also contractible when applied to the homotopy colimit. Thus our space PTW has neither torsion nor rational homotopy and so it is contractible as claimed: P T W " ' * or T < W, as claimed. Further, by (3.B.3), since T = E(VM2(Z/pZ)), we deduce P
VM2(Z/pZ) << W, as claimed. P
A. T h e v l  p e r i o d i z a t i o n of spaces Given the Adams map vl : EqMa(Z/pZ) ~ M3(Z/pZ), the most direct way to proceed in inverting vl is to consider the localization functor L.~ as in Chapter 1 above. This functor will turn (in a functorial fashion) any space X into a vlperiodic space. In particular, the modp homotopy groups: Tr,~(L~,~X,Z/pZ) = [M'~(Z/pZ),L.~X], are periodic with period q. This periodicity isomorphism is obtained by composing a given element a : Mn(Z/pg) ~ X of ~r~(X, Z / p Z ) with the Adams map which, as Adams proved, is a Kisomorphism vl : M'~+q(Z/pZ) ' Mn(Z/pZ) to get Vl 9a C ~rn+q(X,Z/pZ). We can consider the graded polynomial ring Z/pg[vl] with degvl = q as operating on 7f.(X, Z,pZ). By inverting v~ we can localize the vlmodule 7r.(X, Z / p Z ) to get a vllocal module v~l~.(X, Z/pZ). Explicitly, v~lTr.(X, Z / p Z ) is the direct limit of
..(x, z/pz)
z/pz)
. . + : q ( x , Z/pZ) . . . .
.
However, we saw above that localization L I behaves much better for the case f : E W ~ *. Therefore, we try to consider localization (nullification) with respect to the mapping cone of vl, or rather its suspension (see Chapter 3). So let E(M3(Z/pZ) uvl CM3+q(Z/pZ))= EV(1). Here, as usual, V(1) denotes the mapping cone of the Adams map vl. Since vl induces an isomorphism on Ktheory, the space V(1) is Kacyclic. It is clear from the discussion in section 5.D above that localization with respect to EV(1), i.e. the functor P•v0), is 'weaker' than L,1; in other words L,1P~.~I ~ P~v~Lv~ "" L ~ . But for any X the localization P ~ X has periodic homotopy groups above the first dimension, since it is always local with respect to y]2Vl : E2EqM3(Z/pZ)~ E2M3(Z/pZ).
8. vlPeriodic spaces
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This will force us to restrict ourselves to a double loop space ~~2X in order to get good control. According to (5.D.12) any EV(1)null space is also EVllocal space. In general, we do not know the relation between Lvl and PP.v(I) since vl is not a suspension map. Notice also that the general localization functor Lp with respect to the degreep map p : S 1 * S 1 on the circle is much more complicated than the localization with respect to its suspension LEp with Ep : S 2 * S 2, when applied to nonnilpotent space [Ca]. However, the relations between LEkvl and Pr.~v(1) for various k, ~?are not difficult to consider in view of Chapter 5. Without being the 'full' Vllocalization P~v(1)X still has vlperiodic homotopy groups above the bottom dimension: A.1 PROPOSITION: The algebraic localization m a p ~r,P~.v0)X * v~l~riP~v(1)X is an epimorphism for i = 3 and an isomorphism for i > 3. Proof." This is immediate from the usual functionspace fibration sequence associated with the cofibration that defines V(1). A.2 Realizing v~lTr,(X, Z/pZ) as a m o d  p h o m o t o p y of a m a p p i n g t e l e s c o p e One way to invert the action of vl on the modp homotopy of a given space is to look for a direct way to build a space whose modp homotopy groups are precisely the algebraic vllocalization of the homotopy groups of the given space. This cannot be done in general by an idempotent coaugmented functor, but there is a nice and useful telescope that realizes these groups in a canonical fashion. Consider, after [MT], the following analogue of the direct limit construction of a rationalization functor. We shall soon see (A.7) that this infinite telescope realizes as a space the vlperiodic homotopy groups v~lTr,(X, Z/pZ) above.
A.3 DEFINITION: Let ca : ~,qw ~ W be any 'selfmap' from some suspension of W to W . Define T ~ X to be the h o m o t o p y direct limit (i.e. infinite mapping telescope) of the tower of function complexes:
map, (W, X) K~ map, (Eqw, X)
(~~)*
map, (E2qW, X) + . . . .
An important property of T ~ X is given by: A.4 PROPOSITION: For a map of finite spaces w : E k W * W , the telescope T ~ X is naturally an infinite loop space. Proof'. Mapping a finite space to a direct limit of a telescope commutes with taking factors through the tower. Thus clearly we have a homotopy equivalence Y~kT~X "" T ~ X , making T ~ X into a periodic infinite loop space. This periodicity equivalence is induced on T ~ X by the map w* : ( X ~ ' " w ) w * ( x ~ W ) z ~ w ""
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8. vlPeriodic spaces
( ~ k x Z ~ W ) W ~ (XE'+~w) W. The composition ( x ~ ' W ) w ~ ( X ~ + ~ w ) W is simply
(~.)w. Further, one can conclude from the above argument: A.5 PROPOSITION: For any map of finite complexes w : ~ q W ~ W , the space T ~ X is wlocal. Proo~ We have map.(W, T~X) ~ map.(W, f~qT~X) ~ m a p . ( Z q w , T ~ X ) and this composition is induced by w as above. In more detail: The map in question, w* : map.(W, T ~ X ) ~ map(EqW, T~X), is the direct limit of a tower of maps since W is a finite complex: w* : map. (W, X ~tqw) ~ map. (~qw, X ~lqw)
which is the same as s w where s is
s : m a p . ( ~ q w , X) * map.(~(t+l)qW, X).
Since the shift map of towers induces a homotopy equivalence on their common limit T~X, it is still an equivalence after taking map. (W, s). A.6 PROPOSITION: If F ~ E ~ X is a fibration sequence in S., then so is T~F
T~E ~ T ~ X , where the homotopy fibre is taken over the null component of the telescope. Proo~ Both function complex functor map. ( W ,  ) and telescopes preserve fibrations see Appendix HL.
THE MAHOWALDTHOMPSON TELESCOPE OF THE ADAMS MAP Vl Let us consider the modp homotopy groups of this infinite telescope. For any finite complex V there is a natural homotopy equivalence, which is the map induced on the homotopy colimit: hocolim map.(V, map. ( Ekqw, X ) ) ~_ map.(V, T ~ X ). k In particular, for the path component level we have a direct limit of sets of components:
dirlim[Y, Z z~'W] ~_ [V, T~X]. We denote by rk(X, V) the homotopy classes [EkV, X]. Thus ~rk(X, M 3 ( Z / p Z ) ) ~[M k+3, X]  rk+3(X, Z/pZ). Notice the shift in dimension with respect to the usual notation of homotopy groups with coefficient in Z / p Z .
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It will be more convenient from the point of view of indexing to use ~r. (X, M) with M  M 3 ( Z / p Z ) rather than 7r.(X, Z/pZ). We now turn to special properties of T~, where w = Vl is the Adams map that induces an isomorphism on Ktheory. In this case we denote the telescope by T, 1. In particular, we have from the above considerations that the telescope T,1 realizes the vlperiodic homotopy: We have defined algebraically v~llr.(X, M) as the localization of the groups 7r.(X, M) with respect to the composition operation by Vl: ~qM + M. A.7 PROPOSITION: There is a canonical isomorphism v~ l ~r, (X, M) ~ ~r,T~ X . Proof:
We have 7r,T,1X = colimTr,map,(ZqeM, X ) .

In light of this canonical identification we will use T~IX instead of v ~ l r , X . The former has the advantage of being a space. Not only the homotopy groups of the telescope are periodic, the space itself is local with respect to Ktheory [T]: A.8 PROPOSITION: Tv~X is always Klocal in the sense of Bousfield. In [B3] Bousfield proves that an infinite loop space is KloCal if and only if its modp homotopy groups are vlperiodic. Proof:
RELATIONS BETWEEN KHOMOLOGICAL LOCALIZATION, THE NULLIFICATIONPvl AND THE TELESCOPE. W e n o w use the above general construction to show that a small modification of both the nullification functor ('periodization' might be a better name here) and the Khomological localization are closely related to the much more accessible telescope T,I X. Start by specializing the above to W E ,q.v, which is also finite dimensional and nsupported (7.C): A.9 PROPOSITION [B4, 11.5]: Let o~: ~ k W + W be a selfmap of a finite space W with C = cof(w). For any X C S. the map j : X ~ P ~ c X induces an equivalence T~(j) : T ~ X * T ~ ( P ~ c X ) . Proof: By (5.C.5) above, since C is a finite space the homotopy fibre of is a finite Postnikov stagenamely all its higher homotopy groups vanish above some fixed dimension d=dim C + 1. Therefore, taking telescopes we get the following infinite ladder of pointed function complexes and their nullifications:
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8. VlPeriodic spaces
P ~ c ( X W)
'~'. P~,c(X ~qw)
(~'~)'. P z c ( X ~'2"w) . . . .
11 (P~cX) w
P~cT~X
11
"*. (P~,cX) ~qw
(~q')', (P~cX) ~2'w . . . .
T,,PEcX
using [HL], the fact that P A commutes with telescopes for a finite space A (1.D.6), and that taking homotopy fibre commutes with taking telescopes we deduce that the homotopy fibre of T,,X "~ P~cT,~X ~ T , , P z c X is also a finite Postnikov stage. But by periodicity of T~Y, that fibre is a fibre of maps between two periodic infinite loop spaces (A.4) so the homotopy fibre must be contractible as needed. The first equivalence in the last map comes from C = cof(w). 
REMARK: We now come to the main formula of this chapter. Notice that T~X and P ~ c whose C = cof(w) are closely related: Both associate a E2wlocal space to any space X, both 'come close' to preserving fibrations. Notice also that (P~cX) ~2W is an infinite loop space, since we have a cofibration EC ~ E2EqW ~ E2W, hence map.(E2W, P ~ c X) ~map.(EqE2W, P~.cX) or ( P ~ c X ) ~2W ~ ~q(P~cX) ~2W. The next proposition shows a relationship between ( P c X ) W and T,,X. We formulate the main reduction of the nullification functor with respect to the cone of a selfmap to the telescope of this map. This result, when applied to the only nonnilpotent selfmaps on finite ptorsion space, yields a better understanding of the Klocalization and periodization of a space. The proof relies heavily on the main results of this and previous chapters. The proposition says that, while the localization itself might still be mysterious, if one looks at the modp information or the function complex of maps from the relevant Moore space to the localization, it is as simple as one can expectafter double looping: We first formulate a general result about selfmaps of finite complexes that, inter alia, shows how to modify P c so that it preserves fibration sequences unconditionally. A.IO PROPOSITION: If C = cof(~]qw ~ W) is the mapping cone of a selfmap of a finite space W, then there is a homotopy equivalence
~T,,~X ~ ~ ( P c ~ X ) w or equivalently, f~2T,,X ~ gt2(PzcX) w.
8. vlPeriodic spaces
Proof"
151
By the definition of C as a cofibre, we have a cofibration
E C * E 2 E q W * E2W.
This shows t h a t ( P E c X ) ~'2W "~ ( P ~ c X ) Eq+~w is a h o m o t o p y equivalence, since this induced m a p is a p a r t of a fibration with contractible base. Furthermore, this 9 can be continued to a telescope of eqmvalences {(P~cX)
E2EqtW
}l>_o (where we have
suppressed the m a p s from the notation) t h a t converges by the above definition of T~ as a functor, to ~ 2 T ~ P E c X . Therefore we get an equivalence
( P z c X ) ~'~w ~ f t 2 T ~ P ~ c X . Now using A.9 above one can rewrite the righthand side:
~2T~P~.cX ~_ ~t2T~X. Use (3.A.1) to o b t a i n an equivalence:
f ~ 2 ( p z c X ) W = f~(Pcf~X) w ~ f ~ T ~ X ~ ~t2T~X,
as needed.

A . 1 1 COROLLARY: I f C = cof(w) is a finite space then the (nonaugmented) functor
that takes X to ( P c ~ X ) Ew ~ f ~ 2 ( p ~ c x ) W preserves fibre sequences. Proofi
B o t h T~ and ~t preserve fibre sequences.
A.11.1 EXAMPLES: F i r s t a trivial example. If W = S n and our m a p w = E W ~ W is just the null m a p S n+l * S ~ with cofibre S ~ Y S '~+2, then P c = P ~ is P s i , i.e. the (n  1)Postnikov stage 9 PEG is P ~ , the n t h Postnikov stage, and f~2(pnx)S~ = ~+2p~x "~ *. This is the trivial functor, so it certainly preserves fibrations. This example underlines the radical modification t h a t is sometimes done here to a functor like the Postnikov section functor t h a t makes it preserve fibrations unconditionally 9 For a more interesting example consider in (A.10) the A d a m s selfmap w = vl.
In this case the function complex m a p . ( M , P ~ c X ) ,
where C = V(1), the
m a p p i n g cone of vl, contains in fact the most interesting information a b o u t the nullification functor, namely its m o d  p h o m o t o p y groups. So in this case P r o p o s i t i o n
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A.10 precisely identifies the relevant homotopy groups of the uptonow mysterious ~V(1)nullification functor. The last result shows that T ~ X is very closely related to the space of maps map. (W, P ~ X ) of nullification with respect to cof(w). Further, we now deduce the same relationship to Bousfield homological localization. This establishes a relation between vlperiodic homotopy and Ktheory. Proposition A.t0 has an immediate corollary relating the telescope, nullification and Khomological localization. It shows in essence that in the category of ptorsion spaces modulo low dimensions, and after concentrating on the modp information, it is sufficient to homotopically invert one Kisomorphism, the Adams map, in order to invert all Kisomorphisms. A.12 THEOREM: For any X E 8v. with p an odd prime, we have h o m o t o p y equivalences f l 2 T v l X ,,~ ~ ( p v l ~ X ) M ~ ~ ( L K ~ X ) M
where M ~ M 3 ( Z / p Z ) . Outline o f Proo~ This follows from (A.8) above when coupled with (A.10), since by (A.10) the function complex into the nullification is Klocal.
B. K  i s o m o r p h i s m s , K  a c y c l i c s p a c e s First we begin with a theorem of Thompson: B.1 THEOREM: L e t X be a 3connected space. I f K , ( ~ 3 X ) ~ 0 then T v l X ~ *. Proo~ Since the space M = S 2 tap e 3 is three dimensional we have a M * S 3 ~ S 3 cofibration that gives a principal fibration ~ 3 X * ~ 3 X * X M whose base is connected, since X E ,9,(3). Therefore by (1.H.2), [ ( , ( X M) ~_ O. This means that any map X M * Tvl X is null homotopic, since Tv~X is a Klocal space and the said
map factors through L K X ~ *. Now take the natural maps of the telescope that defines T. 1X to T,~ X itself, namely ao : m a p , ( M , X ) ~ T , I X , a~ : m a p , ( ~ . q ~ M , X ) , T v l X .
Since by definition ~ q a t = a~+l we see that o~1 = ~qo~0 ~ * and a t ~ *, and this immediately implies T v I X ~ , , since colim(Xi * X ~ ) is the identity on X ~ : T e t ( X i ) for any telescope X{ ~ X { + l   
SPACES WITH TRIVIAL NULLIFICATION P2V(1): Recall that M = M 3 ( Z / p Z ) is the 3dimensional Moore space and V(1)  col(v1 : 2 q M * M ) . The space V(1) is clearly 2supported, see (7.C.7). Although the relation between the nullification and the telescope goes in (A.10) through function complexes, in the case of a trivial telescope or nullification the relation is more direct.
8. VlPeriodic spaces
153
B.2 THEOREM: If X be an nsupported Mcellular space with n=connX +l then PEv(1)EX ~ * if and only if there exists an integer N > 0 with Tv~ENX ~ * . REMARK: The conditions are satisfied by any finite ptorsion 2connected space
[B12, 3.11.
Proof
Thus we must show that Pzy(1) kills the suspension of X iff the vltelescope kills some, possibly higher, suspension of X. We will prove one direction; the other direction follows similarly by reversing the arguments. Assume T, I E N x ~ *. First, since the integer q is at least 2, we have as in Theorem A.10 and (A.11.1) above the homotopy equivalence \EqM
P~v(I)X)
~ TvtP~.v(1)X.
From the assumption we get by Proposition A.9 above T v l P E v ( 1 ) E N x ~ *. But, as earlier, we get from q _> 2 that
(PEv(1)•N x ) EgM ~ TvlPEv(DEN x and so the lefthand side ~ .. Using the cofibration EEqM * E M * EV(1) that induces a fibration upon taking a function complex into Pzv(1)ENX, we conclude that map.(EM, P E v ( D Z N x ) ~_., since it is the total space in a fibration with contractible fibre and base ( P z v ( 1 ) E N X is by definition EV(1)null). But for N _> 1 the space PEv(1)ENX is EMcellular by the construction of PEv(D as a homotopy limit. Therefore, we can conclude from the Whitehead theorem (2.E.1) that PEV(1)ENx ~ *. If N E {0, 1} we are done; otherwise, since X is nsupported we have by (7.B.4) :
(EX} = (ENX} V K ( Z / p Z , c+ 1), where c  connEX + 1.
But by discussion in example 3.C.5 we know that
P~.v(1)K(Z/pZ, 3) ~ * and connX > 1, so we can conclude that P z v ( 1 ) E X ~ *, as needed. The other direction is obtained more easily by reversing the above arguments.

We now apply (B.2) to Ktheory. In the next few results we relate the vanishing of Ktheory on some iterated loop space of a given space X to the partial ordering (<) and (<<). The idea here is to have a Ktheoretic analog of item (5) in section A above concerning trivial rational
154
8. vlPeriodic spaces
homotopy. We see in (B.6) a reasonable analog to item (5) in the Introduction to the present section. B.3 COROLLARY: If~(.,~3X "~ 0 then P E w ( 1 ) E X ~ *. Proof
T~IX ~_ * by Proposition B.1 above, so by Theorem B.2, P E v ( 1 ) E X ~ *.
B.4 COROLLARY: IY K , X ~_ 0 then P z v ( 1 ) E 3 X  *. Proof." This is true since we h a v e / ~ , ( f l a E 3 x ) _~ 0, so P z v ( 1 ) E 3 t ~ 3 E a x ~ *. But E3123Y << Y. Take Y = E3X. B.5 COROLLARY: / f / ~ , ( X ) 0 and X is 1connected then P E v 0 ) E X ~ *. Proof <EX} = V K ( Z / p Z , c o n n E X + 1) and P E v o ) K ( Z / p Z , 3) ~ *, while c o n n E X + 1 > 3.
B.6 COROLLARY:
/ < . ( ~ X ) ~_ * ~ P ~ v o ) X ~ * ~ V(1) << X. Proof." In general EI2X << X (3.C.7). Now by (1.A.8) the assumption implies PEv(1)E~IX ~ *. Therefore P z v ( 1 ) X ~ *. But then the conclusion follows by (3.B.3). II
9. C E L L U L A R I N E Q U A L I T I E S A. I n t r o d u c t i o n a n d m a i n results
In previous chapters we have profitably used cellular inequalities of the form W << X several times. Thus to show that Lf(GEM) is again a GEM, we used the observation X << SP~X for all 0 < k < oo, namely that S P k X is Xcellular. Also, in proving a cellular form of Bousfield's key lemma (4.D) above we show that E2X << E(SPkX/X). This last inequality can be rephrased by saying that E X is the leading term in the construction of E S P k X out of E X and all higherorder terms are E2Xcellular. We observe that if W << X, namely if X can be built from W by a possibly infinite process of taking cofibration or iterated pointed homotopy colimits, then X retains several important properties of W. For example, connW _< connX. Another example is the vanishing of generalized homology or the vanishing of the singular homology with coefficients in G up to some fixed dimension d. If W << X, then any homology theory that vanishes on W also vanishes on X. Still another property is one associated with each given space T in S. That property is the 'vanishing' of the pointed mapping space map.(W, T). If X is Wcellular, then X also has the same property. In particular, any cohomology theory vanishing on W also vanishes on X. Notice that the above two properties are not preserved under unpointed homotopy colimits: Any space whatsoever is homotopy equivalent to an unpointed homotopy colimit of a diagram of contractible spaces (Appendix HL). Let us list several problems whose understanding relates to cellular inequalities: A.1 For a given homology theory E. such as Morava Ktheory, an important problem is to decide when an E.acyclic space is a direct limit of its finite E. acyclic subcomplexes. This is closely related to the question of whether or not the theory E. is 'smashing': namely whether or not Bousfield's E.localization is simply the smash product with the E.local sphere. We approach this by asking whether there exists a finite space AE which is a generator of C"(acyclics)= the closed (under pointed hocolims) class of all E.acyclic spaces, or a generator for a large subclass thereof: Namely we look for a finite AE with C(AE) = C (acyclics). For example, we saw in (8.B.6) that the mapping cone of the Adams map V(1) generates a large class of Kacyclic spaces. A.2 Another example is the general problem of relating (co)limits to homotopy (co)limits. We would like to show that in some way the 'difference' between colim and hocolim is 'smaller than expected'. For example, often, as we saw
156
9. Cellular inequalities
in Chapter 4, if X is a pointed diagram in g" (A) (i.e. consisting of Acellular spaces) then the fibre and cofibre of the natural map
hocolim, X ~ colimX
are in C'(EA) or even in g ( ~ 2 A ) . A.3 Both the fibre and the cofibre of a map g : X ~ Y can be considered as 'measures of the deviation' of g from homotopy equivalence. Yet few general relations between them are available. For example, assume that g is E,homology isomorphism with the special property that the fibre of g is E,acyclic. %Chat can we deduce on cof(g) = Y t3 C X beyond the immediate consequence that it is E,acyclic? It turns out that f~(Y U C X ) is also E,acyclic under this condition. For a proof see (D.2)(ii) below. A.4 Moreover, inequalities will help us in general to find relations between hocolim, 3holimiX~j and holimi hocolim, X~j for a given doubly indexed diagram. In general, inverse limit and direct limit functors do not commute, but often they are related by cellular inequalities such as EI2X << f/EX. A.5 RECALL (Compare [B4] [DF5]: We say that A 'supports' X and denotes it by A < X if P A X ~ * or, equivalently, if for all Y one has the implication map.(A, Y) ~ * ~ m a p . ( X , Y). A.6 PROPOSITION: For all A, X E $ . one has
(i)
A << X =~ A < X ,
(ii)
EA < X =~ A << X.
(i) We know that X is built from A by a process of taking hocolim,. One needs to check that if in a diagram A one has A < A~ for all a, then
Proof'.
A < hocolim, A. This follows directly from (1.D), explicitly: Let Y be any Anull space so that m a p , ( A , Y )  *. we must show that map,(hocolim, A,Y) ~. Notice map, (hocolim, A, Y) is equivalent to holim map,(A~, Y) which is contractible because holimV~ with V~ ~ 9 is contractible. (ii) This is Proposition 3.B.3 . Therefore, in some cases, we get interesting relations of the form A < X by first showing A << X and vice versa.
9. Cellular inequalities
157
A.7 NOTATIONS: Recall that for any pointed diagram X and any diagram Y, we denote the pointed homotopy colimits by f X and the unpointed one by j~ X, namely: I
I
f X  hocolim, X I
I
and J Y  hoeolimY I
I
for X: I ~ S. and Y: I ~ S are diagrams. This notation is supposed to remind the reader that in going from f to f , we must collapse to a point the classifying space of the small category [/[which in the simplest connected case is a circle. It should also remind the reader that homotopy colimits, both pointed and unpointed, formally share some of the axiomatic properties of the definite integrals of real valued or other functions. Thus they are homotopy additive: hocolim of a homotopy pushout is weakly homotopy equivalent to the pushout of the homotopy colimits. The pointed homotopy colimit of 'the point' (i.e. the diagram that assigns a single point to each object in I) is a point, while in the unpointed homotopy colimit the point serves as a 'unit' and not as a 'zero' and the unpointed, free homotopy colimit of a diagram of points is the nerve or classifying space of the underlying small category that serves here as the 'domain of integration'. It is also additive in the obvious sense with respect to the 'domain of integration'. This notation also serves to emphasize the distinction between homotopy colimits and homotopy limits. Although they are formally dual from a homotopical algebra point of view, they share only few properties and it might be better not to create a false analogy between them. We also find this notation shorter, more convenient and visually appealing. Main Results We consider two main directions: one concerns the homotopy fibres of a map between colimits; the other relates strict colimits to homotopy colimits. A.8 THEOREM: Let I be a small category and A a pointed space. Let {E~ ~ B~} be an Idiagram of maps in S over connected spaces Ba with pointed homotopy fibres F~. Assume that bl is a dosed class such that F~ E/g for a11 a. In that case also:
<wos/
ib<J I
J Bo/ u I
158
9. Cellular inequalities
In particular, i r A << F~ for all a then A << F i b ( d E s * f B ~ ) . A similar statement holds for diagrams in S, and pointed hocolim,. Proos
See (C.1) below.
A.9 Remarks and Examples: (i) In several important cases a much stronger version of the above is true, namely
iosl
i,o..,i, iI,.i,.,.
This is true if, for example, the situation falls under the assumptions of V. Puppe's theorem (Appendix HL): e.g. all the maps in (B~),~eI are identity maps and [I[ ~ * so that I is a contractible category. For a concrete example consider the diagram:
T *
X
1
1
1
The stronger inequality (CS) gives E ~ X << ~tEX, which is true for any connected X (see 3.A). The last case is a special case of the inequality: For any map f: X + Y the homotopy fibre of the suspension of f can be built from the suspension of the homotopy fibre of f. Compare [Ch2],[DF5]:
EFib(X * Y) << F i b ( E X + EY). (ii) We refer to the inequality in the theorem as weak CauchySchwartz (WCS) and to the above (CS) as CauchySehwartz, since they are reminiscent of the inequality x 2 + y2 _< (x + y)2 for nonnegative real numbers, the analogue of the product being limit and that of the sum being colimit. (iii) Nontrivial examples of the (WCS): (1) The inequality E F << B / E for any fibre sequence F ~ E ~ B over a connected B and (2) the inequality F << F i b ( E / F ~ B) for such a fibration.
9. Cellular inequalities
159
Proof: T h e y arise in the maps of diagrams:
l
E
and
~ E
1
E
F
1
,
*
l
B
i
E
B
where the vertical arrows are maps inside diagrams a n d the horizontal ones are components of m a p s between d i a g r a m s and the m a p E + E is the identity. So this is a special case of the above with I being * *   *
~ ..
To derive the first inequality from (WCS) one uses the fact t h a t for connected X, Y one has ~ X << Y r
X <~ f/Y. See C h a p t e r 3.
Consider the first diagram: *
1 *
~
E
<
E
l
,
E
,
B
l
The h o m o t o p y colimit of the t o p row is ( . ) and of the b o t t o m row is B / E . Then the m a p hocolim. F~ ~ hocolim. B~ is the m a p * ~ B / E . Thus by T h e o r e m A.8 above, F << fI(B/E); by (3.A), E F << B I B . In fact, as we will see in (D.1) below one can express the cofibre B / E directly as a pointed h o m o t o p y colimit as follows: Recall (1.F), (HL) t h a t the t o t a l space itself is the u n p o i n t e d h o m o t o p y colimit of the d i a g r a m of 'fibres' indexed by F B .
If we suspend this d i a g r a m of
fibres at each o b j e c t we get a p o i n t e d d i a g r a m   a c t u a l l y with two possible base points. We can then take the pointed h o m o t o p y colimit with respect to one of these two base points. One gets: B/E ~ /
EF.
FB
A striking e x a m p l e is: A . 1 0 COROLLARY: The homotopy fibre of X * X / A is A cellular for any cofibra
tion sequence A + X  , X / A in S,. The inverse image of a simplex in X / A is either a point or A itself, so by (A.11) below the h o m o t o p y fibre is Acellular.
Proof:
This corollary is clearly a special case of the more general theorem:
160
9o Cellular inequalities
A. 11 THEOREM: Let f: X ~ Y be any map of connected pointed simplicial spaces. Assume that the inverse image f  l ( a ) of each simplex a in Y is connected and Acellular. Then the homotopy fibre of the map f is also Acellular. Similarly, if the map is a simplicial map between two simplicial complexes then one can consider the point inverses of a11 the barycentres in the base space; if they are a11 Acellular then so is the homotopy fibre of the map. Proof: The proof is lengthy and will occupy the next section. We will first demonstrate how to present the homotopy fibre as an unpointed homotopy colimit of a special diagram. B. T h e h o m o t o p y fibre as h o m o t o p y colimit We shall see below that it is often easier to present a space as an unpointed homotopy colimit of a specific kind than as a pointed one. This is clear in view of the fact that, given a n /  d i a g r a m X, we cannot take its pointed hocolim unless the diagram is pointed. But there is no reason to assume that it has a consistent choice of base points, namely a map pt ~ X of/diagrams. The latter is equivalent to lim X ~ 0. But we will be interested in cases when not only the limit of our I
diagram is empty, but also the homotopy inverse limit might be empty. The feature that renders an unpointed homotopy colimit nontrivial from the present point of view is the restriction one puts on the homotopy type of the classifying space of I. As we saw above it is otherwise pointless from the present point of view to consider the construction of spaces via unrestricted use of unpointed homotopy colimits. The restriction that occurs most naturally is that of restricting the possible shapes of the small categories over which one takes these unpointed homotopy colimits. This was investigated elsewhere in some detail [Chol,2]. For our purpose here we will consider in particular contractible small categories, namely categories I with III  .. We will see immediately that unpointed homotopy colimits of a diagram of nonempty spaces X over a contractible small category I can always be reconstructed up to a weak equivalence as a pointed homotopy colimit out of the same spaces with a suitable choice of base points in these nonempty spaces X~. In applications, we often wish to obtain inequalities of the type W << Fib(X ~ Y), since the fibre as opposed to cofibre is harder and often more useful to estimate. It turns out that one way to arrive at such inequalities is to express the homotopy fibre Fib(X ~ Y) of a map g : X ~ Y as a homotopy colimit of the inverse images of points g  l ( y ) for y E Y. This is easier to consider when X, Y are simplicial sets, or in fact simplicial complexes. Let FY be the small category of all simplices of a simplicial complex Y with IFYI ~ Y (see (1.F) above).
9. Cellular inequalities
161
If Y is a simplicial complex, then FY is just the category of simplices of Y: It has one object for each simplex and one map for each face inclusion, so that Hom(a, a I) in that category FY is either empty or a singleton. We associate with the map g a diagram (Appendix HL) denoted here fancifully by ~x = Dx(g) (the 'decomposition of X according to the map g') over the small category FY as follows:
Dx(g)(a)  g  l ( o ' ) C X . Thus g  l ( a ) is the (strict) pullback in: gl(a)
'
X
As we saw above in 1.F, X is weakly equivalent to the homotopy colimit over FY of Dx(g). Symbolically X ~ ~ Dx(g). The map g itself can be recovered from FY
Dx(g) by taking the unpointed homotopy colimit of its map into a diagram (*) of points in the shape of FY. But our point here is that Dz(g) is the diagram of actual fibres or pointinverses of g. (Working with simplicial complexes one can actually take the point inverses of all the barycentres of linear simplexes in the base.) The diagram D x (g) is unpointed; a choice of base point in each g  l ( a ) amounts to a choice of a homotopy section of g, a section that may not exist. We now show that one can rearrange the spaces g  l ( a ) = Dx(g)(a) over a contractible small category denoted by AY (it depends only on Y) in such a way that the (unpointed) homotopy colimit will be homotopy equivalent to Fib(X ~ Y). B.1 PROPOSITION: For every map g : X ~ Y there exists a diagram A(g) : AY ~ ,.q
of unpointed spaces such that: (a) For each i E AY the space A(g)(i) that appears in the diagram is equal to gX(a) for some a : A[n] ~ Y. (b) The small category AY is contractible. (c) The unpointed homotopy colimit of A(g) is weakly equivalent to Fib(g), namely:
f A(g)~Fib(X AY
a,Y).
162
9. Cellular inequalities
Proof'. Notice t h a t g is not necessarily a fibre m a p so the h o m o t o p y t y p e of g  l ( a ) varies as with a. We first give a few examples and draw i m p o r t a n t consequences, and then give the simple proof below. C o m p a r e section E.4 below. B. 1.1 EXAMPLE: Here is an illuminating example of how the h o m o t o p y fibre of a m a p g arises as a h o m o t o p y direct limit of a d i a g r a m consisting of the a c t u a l fibres, i.e. inverse images of g. Notice t h a t in this example there is no difference whether one takes a pointed or unpointed h o m o t o p y colimit since the small category we will consider is linear and contractible. Consider any selfmap w : W ~ W of an a r b i t r a r y space W. F o r m the m a p p i n g torus T ( w )  W • I / ( x , O) ~ (w(x), 1). The m a p W ~ 9 induces a m a p g : T ( w ) ~ S 1 on m a p p i n g toruses since S 1 is the m a p p i n g torus of the identity: 9 ~ .. Consider the strict pullback diagram, which is also a h o m o t o p y pullback since the universal covering m a p over the circle is a fibre map: Fib(g)
l
T(w)
,
]~
,
S1
l
This d i a g r a m gives the usual n a t u r a l construction of the h o m o t o p y fibre of g, namely F i b ( g ) as a pullback from ~ ~ S 1. But a direct e x a m i n a t i o n of the strict pullback F i b ( g ) in this square reveals t h a t Fib(g) is in fact isomorphic (homeomorphic) to the infinite m a p p i n g telescope Tel(w) of the original m a p w : W ~ W . This telescope is clearly equivalent to the h o m o t o p y colimit of the tower ( W w, W w W * . . . ) . Now notice t h a t W is equal to g  l ( y ) for any y E S 1. So the h o m o t o p y fibre is presented here as an infinite telescope in terms of the strict inverse images of our m a p g. Notice that, although the infinite telescope is not given as a pointed d i a g r a m or a pointed h o m o t o p y colimit, one can choose a consistent base point system, and pointed and unpointed h o m o t o p y colimits are equivalent here. We now prove (B.1).
P r o o f of B.l:
We consider the construction of Fib(g) as a strict pullba.ck in:
Fib(g)
l
X
~,
AY
g ~
1,Y
The space AY is the simplicial p a t h space over the pointed space Y [Mayl]. The space AY is a contractible space and we take the category AY to be P(AY). We
9. Cellular inequalities
163
define the diagram A(g) to be the functor A(g): AY ~ S with A(g)(~) = 91(~). In other words, A(g) is the diagram of inverse images of the map g. Since AY ~  . , (B.1)(b) is satisfied. We now consider the map 9; by the decomposition of the domain of a map as a homotopy colimit of the inverse images, as discussed above, the domain of ~ is given as: Fib(g) = hocolim{gl(a)} so A(g) satisfies also (B.1)(c). aEFAY
To prove (B.1)(a) notice that, since the diagram defining Fib(g) is a strict pullback for any simplex ~ E A Y , we have an isomorphism:
(~)l(~)_~gl(e(~)). In fact g and ~ have the same point inverses over corresponding simplices. This completes the proof of (B.1). I
Presenting the homotopy fibre as a homotopy direct limit construction over a contractible category out of the actual inverse images of simplices (or points) in the range, over a contractible small category, was our first step. The second step is a theorem which guarantees that, if space X is homotopy equivalent to J~xY with I ~ *, and Y(i) r 0, then X can also be obtained up to homotopy by repeated construction of a pointed homotopy colimit over a sequence of small categories J~ of pointed diagram Y. 5, which F R O M UNPOINTED TO POINTED HOMOTOPY COLIMITS:
consists of the same spaces Y(i) or spaces built from them by pointed hocolims: B.2 THEOREM: Let Y: I * S be a diagram of nonempty space Y(i) ~ 0 with I ~ *. There exists a choice of base points * E Y(i) with respect to which
hocolimY E I
e( V r(i)). i
An outline of a proof is given after the proof of (B.3) below. The proposition says, loosely speaking, that any unpointed homotopy colimit over a contractible category can be obtained as a repeated pointed homotopy colimit using the same spaces with a proper choice of base points. An immediate consequence of (B.1) and (B.2) is B.3 COROLLARY: Let g: X ~ Y be any surjective map of pointed simplicial sets (or complexes). Iflg is any closed class such that, for each a E Y , one has g  l ( ~ ) Elg
164
9. Cellular inequalities
with respect to some choice of base point * E g  l ( a ) , then the homotopy fibre Fib(g) is also in the class lg.
SLOGAN: Thus: 'The homotopy fibre Fib(g) can be built as a pointed homotopy colimit out of the 'real fibres' the point inverses {gl(y)}., In particular, as a special case we get the following result of Quillen [Q2]: B.3.1 COROLLARY: If g: X +r is a simplicial map with contractible point inverse g  l ( y ) ~ ,, then g is a weak equivalence, namely its homotopy fibre is also contractible. Moreover if the point inverses are all nconnected spaces then so is the homotopy fibre. Nonexample: The exponential map for topological spaces exp : [0,1) * S1 has single points as all its point inverses but it is not an equivalence. As it stands it is not a simplicial map though. Similarly one has:
B.3.2 COROLLARY: If g: X + Y is a simplicial map with g  l ( y ) and an E,acyclic space E, is any homology theory, then the homotopy fibre is also E,acyclic. Proof orB.3: Apply (B.1) to the map g: X ~ Y. We get that Fib(g) is a homotopy colimit as in (B.1)(c). Now apply (B.2) to (B.1)(a), by using surjectivity which gives us a diagram of nonempty spaces g  l ( a ) , where (r E FY. By (B.2) we get Fib(g) as a pointed homotopy colimit, as needed. Proof of B.3.1 and B.3.2: Since if Y ~ * every space in C ' ( Y ) is contractible, we get from (B.3) that Fib(g) ~ ,. Similarly, if ~:,(Y) ~ 0 then E , ( Y ' ) ~ 0 for any Y' E C" (Y). So again we get (B.3.2) f~om (B.3) Outline of proof of (B.2): [Am][Ch2] We must show how to choose pointed versions of Y(i) from which we construct hocolim Y by repeatedly taking pointed homotopy
colimits. We give an outline of a proof based on the work of Amit [Am]. (An independent proof was given in [Ch2] that actually has a stronger version and consequences.) Amit reduces everything to taking the colimit over finite collapsible categories that are the underlying categories of simplicial complexes. The proof rests on two examples of unpointed direct limits over contractible small categories that can be easily made pointed. B.4 LEMMA: Theorem B.2 is true if I is either the category associated to a partially ordered set satisfying (with x * y being x < y) VxVy3z(x < z)&(y < z) (filtering indexing category) or the 'elementary pushout' category given by the diagram * + 9 * ,. Proof of Lemma B.4: If X0 + X1 + X2 is a diagram of nonempty spaces, we c a n always choose a base point consistently by starting with one in X1. If X: I + S and
9. Cellular inequalities
165
i is any fixed object in the filtering category I as above, any direct limit of X: ~ S is isomorphic to the direct limit of X>~: I >~ * S, where I >i is the subcategory of all elements greater than or equal to i. But in X >~ we again can choose a base point by first taking an arbitrary point in X(i) which we assume to be nonempty. Since the homotopy direct limit over I can always be written as a strict direct limit over the associated free diagram (see Appendix HL), we are done. Next, given any contractible category I ~ 9 Amit has shown how to write any direct limit over I as a series of direct limits over categories which are special, i.e. are of the forms in L e m m a B.4 above. This he does in four steps: Step 1: We can always reduce the situation to diagrams over the category of simplices of a contractible simplicial complex. Step 2: We can assume that our category is the category of a finite contractible simplicial complex. Step 3 : B . 2 is true for the category of simplices of a collapsible simplicial complex
[Co]. Step 4: We can assume that our category is the category of a finite collapsible simplicial complex. B.5 OUTLINE OF THE PROOFS OF 14 Step 1: This rests on the following two claims:
1.a Claim:
If X: I ~ S is a diagram over a small category I, then the X induces
as in [DK3, sec. 5] a diagram sdX, over the opposite of the subdivision of I denoted by sdI. The subdivision is a certain quotient category of finite chains of composeable arrows in I. See [DK3]. The colimit of s d X over s d I is naturally isomorphic to the colimit of X over I. The following interesting combinatorial property of subdivision is crucial for the present approach:
1.b Claim: If I is any small category then the third subdivision of I, namely (~~)3i, is the category associated with a simplicial complex. Both claims are proved by a straightforward detailed computation. Step 2: Any contractible simplicial complex is the direct limit of the partially ordered collection of all finite contractible simplicial subcomplexes. Therefore any direct limit over a given infinite complex can be written as a composition of direct limits over finite complexes followed by a direct limit over the partially ordered indexing category of all finite subcomplexes. Now we can use L e m m a B.4 above, since this partially ordered set is 'filtered' as in (B.4).
166
9. Cellular inequalities
Step 3: Again by direct computation one shows that, if a simplicial complex K collapses to L, then any diagram over K induces one over L and vice versa 9 Moreover, these diagrams will have naturally isomorphic direct limits 9 This is done inductively by collapsing one simplex at a time. Given a diagram X: K ~ S, where K is a category of simplices of a complex and g: K ~ K1 is an elementary collapse, then with each diagram X: K1 * S over K1 one can associate a diagram g*(X) over K whose spaces over the extra (two) simplices are obtained from X ( i ) ' s via an elementary pushout over the category ~ 9* .. Moreover, there is a natural isomorphism between the two colimits: 9
Step 4: This follows from Step 3 when we recall that any contraetible finite complex is the collapse of a finite collapsible complex. This completes the proof of (B.2). 
C. T h e w e a k C a u c h y  S c h w a r t z i n e q u a l i t y Theorem B.3 above has a direct consequence that also gives us a generalization. Given a map o f /  d i a g r a m s in 8 (or $ . ) g: X * Y,~ one would like to estimate the homotopy fibre of the induced maps between the homotopycolimits, namely to estimate Fib(hocolim g). Since we are in an unpointed context we put ourselves in a connected context: We denote by cS or c~. the subcategories of connected spaces. C.1 THEOREM: Let g: X ~ Y be a m a p of Idiagrams in cS (corresponding o5,) and 142 a closed class. A s s u m e that for each i E o b j I the h o m o t o p y fibre Fib(G(i)) is connected and belongs to 14;. Then the homotopy fibre ofhocolim g is connected I ~ and belongs to 14;:
(Correspondingly, the homotopy fibre of the pointed hocolim F i b ( f I g) also belongs I ~ to
W.)
Proo~
We may assume that for each i E o b j I the map g(i): X(i) ~ Y(i) is a fibre
map, otherwise we functorially turn all these maps into fibre maps. Since Y(i) is connected g(i) has a welldefined (up to homotopy) fibre and .we assume that this fibre is connected and, with respect to any choice of base point, is a member of the closed class )4;. Now we observe that for every simplex a E hocolimY, there is a I simplex ai E Y(i) such that
(g(i))1((7i) _~ g  l ( ~ )
9. Cellular inequalities
167
where g = hocolim g is the induced map on the homotopy colimits. This follows I
~
directly from the universal construction of hocolim in [BK]. Therefore, by assumption for each a E hocolimY, we have g  l ( a ) E 14~. Hence by Theorem B.3, Fib(g) I
is connected and is a member of 14;, as needed. Once we have proved the unpointed version, the pointed version follows from it as a special case or can be proven by an analogous pointed version. C . I . 1 REMARK: Theorem B.3 is also a special case of (C.1). Consider any map g: X ~ Y in $. The map g can be presented, up to homotopy, as a homotopy colimit of a map of diagrams as follows: we write X as the homotopy colimit of the FYdiagram D x (g), which is our notation for the diagram whose value at a simplex E FY is gl(a). Now hocolim(pt) _~ Y and hocolimDx(g) = Y. FY
FY
Therefore, on taking the homotopy colimit, the map of diagrams D x (g) * (pt) into the constant pointdiagram induces the given map g. Further, the inverse image of a simplex a E Y under g is precisely the homotopy fibre of D x ( g ) ( a ) ~ pt. Therefore, if one applies (C.1) to the map Dx(g) * (pt), one gets (B.3).
D. Examples D.1 THE FIBRE AND COFIBRE OF A MAP. For any map g: E * B to a connected space B with homotopy fibre F, we have E F << B / E ~ B O CE. Stated otherwise, the cofibre of g is cellular with respect to the suspension of the fibre. This follows directly from WCS, as we saw earlier in B.2 (iii). In fact, one can write B / E as hocolim, of a FBdiagram consisting entirely of spaces homotopy equivalent to E F : Let F = F(p) be the FBdiagram of the fibres of p; namely we take F B to be a small category whose classifying space is homotopy equivalent to B, e.g. F B can be taken to be the small category of (nondegenerate) simplices of B. Then F is a functor F : FB ~
S with hocolim F ~ E and the natural map hocolim F ~ hocolim {pt} FB
~
FB
~
FB
is the map p : E ~ B under the natural equivalences. The FBdiagram F is not pointed. However, take the free suspension objectwise in F to get
E F : F B ~ $.
The diagrams F and E F consist of spaces which are homotopy equivalent to F or E F and all The m a p s i n the diagram are homotopy equivalences. Now E F is pointed by (any) one of the suspension points. So consider E F : F B * S. as a pointed diagram.
168
9. Cellular inequalities
CLAIM: There is a natural h o m o t o p y equivalence
f
EE
~ B / E
~
FB
that realizes the inequality E F << B / E .
To prove the claim we recall the relation between pointed and unpointed homotopy colimits [BK]. For any pointed diagram X we have a cofibration (2.D.3)
III + hocolimX~ ~ f X
= hocolim,x X.~
I
So to compute f E F we first compute h o c o l i m E F . Since FB
~
FB
~
E F = hocolim(pt + F + pt), thus hocolim E F is, by commutation of hocotims: FB
hocolim(hocolimpt + hocolim F + hocolimpt) 9"~''*"
FB
FB
~
FB
__ Dcyl(B + E + B) (because hocolimptrB B). Therefore, since f EF~ = ( h o c ~ i m E F ) ~ / B , we get FB
/
E F ~_ Dcyl(B +E ~ B ) / B = colim(, ~ E ~ B)
FB
~ B Up C E
as needed. As a minor example of the use of the cellular inequality E F << B / E , here is a quick cellular proof of a well known theorem of Hopf: For any connected space X the map H 2 X ~ H 2 K ( I I , 1) is surjective. Consider the fibration )( + X + K(YI, 1). We have an exact sequence on integral homology of a pair: Example:
9. Cellular inequalities
169
H2X * H2K(II, 1) ~ H2(K(II, 1), X). Now H2(K(II, 1), X) = H2(K(H, 1)/X). First, notice that by the above inequality E)( << K(H, 1)IX. Second, E J( is 2connected as is any space built from it by a pointed homotopy colimit (2.D.2.5), so K(II, 1 ) / X is 2connected too, hence H2(K(H, 1), X) ~_ 0. Therefore, by exactness, we get the desired surjection.
D.2 D U R A B L E OR STRONG HOMOLOGY ISOMORPHISMS. G i v e n a m a p p : X ~ Y that induces an E.isomorphism with respect to some homology theory, it does not follow that its homotopy fibre is E.acyclic. If it is acyclic, then we have a special 'strong' homology isomorphism. Similarly, though the cofibre of p is acyclic its loop space ~2Y/X is not in general acyclic. If, however, the homotopy fibre is acyclic, then the loop on the cofibre is also an acyclic space with respect to E.. This occurs, for example, for any suspension of an E.isomorphism. Thus the suspension of a homology isomorphism is always a 'strong' homology isomorphism. (I) CLAIM: If p above is an E.isomorphism, then the homotopy fibre of ~p is E.acyclic. Proo~ We have a cofibration Y / X ~ ~ X * E Y with Y / X acyclic. Therefore, by (A.10) above the fibre of ~p is Y/Xcellular, and thus also aeyclic (2.D.2.5). (II) CLAIM: If X +B is an E.isomorphism with an E.aeyclic homotopy fibre, then the loop space of the cofibre ~ ( B / X ) is E.acyclic. Proo~ Let PE. = PA be the functor ( )+., namely the functor that kills precisely all E.acyclic spaces. We need to show P E . ~ 2 ( B / X ) ~ *.
P A ~ ( B / X ) ~ ~ P ~ A ( B / X ) ~ ~tP~A J ~ F , FB where F is the homotopy fibre of X ~ B. We continue (1.D):
~~P~.A J P~A~F. FB
Now since P A F ~~ * by assumption on F, we get by (1.A.8)(e.10) P ~ A ~ F ~ *. Thus f P~zAEF ~ *
B
and
P A ~ ( B / X ) ~ *
170
9. Cellular inequalities
as needed. D.3 POINTED BOREL CONSTRUCTION. We now give the classifying space W G of any topological group G, as a pointed homotopy colimit of the suspension of the group 2 G with respect to the natural action of G: We saw in (3.B.3) above that for a connected space X we always have 2f~X << X. In other words, there exists a process of repeatedly taking homotopy colimits that leads from EG to the classifying space W G . An explicit construction of X as pointed hocolim out of 2 f t X is given as follows. Consider the map 9 ~ X into a connected X. The homotopy fibre being f~X and cofibre being X itself, we get as in (D.2) above: X = /
~2X.
,J
UX
We claim that one can replace the domain FX by the topological category f~X. Pick a model for F X to be a topological (or simplicial) category with one object 9 and map(., .) = ~2X as a simplicial group. Then the classifying space of the (small) topological category ftX is [ftX[ _~ X. So ftX as a topological category is a good model for (FX): a diagram with hocolim(pt) _~ X. r(nx) Therefore using halfsmash:
x
f xax FX
f xax
EaX nx XaX
~X
In other words, if f/X = G is any topological group, then X = W G in the above considerations yields:
W G = E G ~
In particular, we get (see A.6 above): D.4
PROPOSITION: For
any
group
G
map,(EG, Y) ~ 9 ~ map(WG, Y ) ~_ *,
the
following
implication
in other words E G < W G .
REMARK: The last implication follows also from (3.A), since P r . a W G ~ *.
holds:
9. Cellular inequalities
171
D.5 COROLLARY: In any cofibration A * X
J ~ X / A the suspension o f the fibre
F o f j is nullequivalent to EA. Thus for all W
m a p . ( E F , W ) ~ 9 ~=~ m a p . ( E A , W ) ~ 9 or Proof
E A < E F < EA.
Since cof(j) = EA, we have, by (D.1) above, E F < EA. But we saw (A.10)
t h a t A << F , so by (A.6) A < F and thus by (1.A.S)(e.10) E A < E F . D.6 EXAMPLE:
For any fibration F ~
E ~
B over a connected B and any
connected X , the T h e o r e m C.1 asserts
F << F i b ( E v X , B v X ) .
To see why, we simply consider the d i a g r a m of fibrations: F
~
pt
~
pt
E
1 i
~
pt
~
B
~
1 I
pt
~
1 I: X
X
It follows t h a t the fibres of E * B and E V X ~ B V X are cellularequivalent to each other. Similarly F i b ( X ~ Y) << F i b ( X / V ~ Y / V ) for any V  * X  *
Y.
D.7 CELLULAR VERSION OF BLAKERSMASSEY THEOREM: On the level of hom o t o p y groups this classical theorem asserts t h a t in a certain range, depending * I h ( X / A ) is an isomorphism: If on the connectivity, the n a t u r a l m a p lr{(X, A) connA = s  1 and conn(X, A) = n  1 then the l a t t e r m a p is an n + s  1  i s o m o r p h i s m (see [G, 16.30]). Reformulating in terms of h o m o t o p y fibre the result can be viewed as a s t a t e m e n t a b o u t the connectivity of the m a p F . A from the fibre F of X , X / A to A itself. A recent result of Chacholsky gives a cellular version of B l a k e r s  M a s s e y t h e o r e m t h a t is much stronger t h a n the original:
172
9. Cellular inequalities
PROPOSITION D.7.1 fibre of the m a p X
[CH4]:
Let A
, X be a cofibration and F be the homotopy
, X / A . If F is connected then the following inequality holds:
A 9F i b ( A
* X ) << E ( F / A )
We will not reproduce the proof t h a t uses W C S above.

Since cellular inequality preserves connectivity the classical B l a k e r s  M a s s e y theorem follows without difficulty. However this inequality holds without higher connectivity assumptions and estimates, in general, from below, the 'difference' between the fibre and the cobase of X
. X / A in t e r m of A and Fib(A
~ X).
I n [Ch4] a more general theorem is proven comparing h o m o t o p y pushouts and h o m o t o p y pullback is general squares.
E. A v e r a g e or w e a k c o l i m i t o f a d i a g r a m In P r o p o s i t i o n B.1 above the h o m o t o p y fibre of an a r b i t r a r y simplicial m a p is constructed out of the strict inverse images of simplices by t a k i n g a certain h o m o t o p y colimit over a contractible category associated to the range of the map.
In this
section we want to p u t this construction in a wider perspective: Given any d i a g r a m of objects in any small category one m a y want to t u r n this diagram, in a universal way, into a d i a g r a m of isomorphisms or equivalences. Thus we might want to t u r n a d i a g r a m of groups into a d i a g r a m of group isomorphisms in a universal way, or a d i a g r a m of sets into a d i a g r a m of set isomorphisms. This reminds one of the process of taking a direct limit (i.e. colimit). Forming a colimit can be viewed as a process of turning a d i a g r a m X: I
, g given over a category g into a d i a g r a m of identities.
If we relax the d e m a n d and ask for isomorphisms r a t h e r t h a n identities we get a construction t h a t might be called a weak colimit or average. E. 1 EXAMPLES 1. Consider the small category D = * 3 * consisting of two objects and two nonidentity arrows between them. So the classifying space of D is the circle, and let X = { ~ x 0 }
be a d i a g r a m over D consisting of the e m p t y set m a p p i n g twice to
a single point Xo. The colimit of X is a single point, b u t the weak colimit or average a v X of X consists of a d i a g r a m of two infinite sets m a p p i n g via b o t h arrows by isomorphisms. Notice t h a t no d i a g r a m consisting of two isomorphisms between two finite sets can be universal among all such i.e. in the present case, has a m a p to all such. It is not accidental t h a t we get here the loop space of the circle as equivalent to the average. 2. Consider the category T with one object and an infinite number of nonidentity selfmaps {g'~ = g o g . . . o g : n > 0}, n a m e l y m o r I = N, the n a t u r a l
9. Cellular inequalities
173
numbers. Then if we take N itself as a set on which T acts by shifting n to n + 1 we see t h a t g acts by a nonisomorphism. Turning this action in a canonical way into isomorphisms we get t h a t the weak colimit is the d i a g r a m consisting of the integers Z with g acting as a shift m a p which is now a nonidentity isomorphism. In short avT
= Z.
E.2 DEFINITION: Given a diagram X: I
9 E the diagram avX: I
, E will be
called a weak limit or average o f X i f all the maps in a v X are isomorphisms and it comes with coaugmentation 2(
, a v X that is a universal m a p o f 2( into diagrams
o f isomorphisms over I. Remark:
Thus av is the left adjoint to the inclusion functor from d i a g r a m s of
isomorphisms over I to all diagrams over I. Such an adjoint exists whenever the category E is coclosed, i.e. has a r b i t r a r y direct limits (=colimits). AVERAGE AS A LEFT KAN EXTENSION. It might be illuminating to present the weak colimit as a left K a n extension [Mac]. Consider the (noncommutative) diagram, where the diagonal is the K a n extension of the vertical arrow. I
. I[I I]
E By I[11] we denote as usual the localization of I with respect to all its arrows, namely, the category obtained by formally adding inverses to all arrows in I. Assuming E is coclosed we always have a canonical extension of the functor X: I
, E
to the category I[I1]. Since all morphisms in the l a t t e r are isomorphisms the dia g r a m o b t a i n e d over the localization of I consists of isomorphisms in E. Now one can construct the average of the given X by precomposing into I.
E.3 DEFINITION: avX =
the compositon
I x.
i[/1]
IiX.
]~.
It now follows from the general properties of left K a n extensions t h a t a v X is in fact universal in the above sense. F r o m the point of view of the present section and (B.1)(b) above the following is a crucial p r o p e r t y of the weak colimit construction.
174
9. Cellular inequalities
PROPOSITION: For each i E I the object in E given by a v X ( i ) is obtained as a
colimit of a contractible diagram of objects in E of the form X ( a ) for a C I . Proof." See [Mac]. The value of the K a n extension is o b t a i n e d by t a k i n g colimit over certain undercategory which are always contractible, having an initial object. E.4
SIMPLICIAL
LOCALIZATION,
HOMOTOPY
AVERAGING
AND
FIBRATIONS.
We now extend the notion of weak colimit to its h o m o t o p y version. Given a dia g r a m of spaces we would like to t u r n it in a canonical way into a d i a g r a m of weak h o m o t o p y equivalences.
Thus if the d i a g r a m comes from a simplicial m a p as in
(B.1) above, once we t u r n it into a d i a g r a m of equivalences the typical h o m o t o p y type appearing in the h o m o t o p y average d i a g r a m will be t h a t of the h o m o t o p y fibre of the m a p we s t a r t e d with. Again, this typical value of the h o m o t o p y average will be o b t a i n e d by t a k i n g a h o m o t o p y colimit over a contractible category. Thus we see t h a t turning a m a p into a fibre m a p is the universal construction t h a t turns this m a p into a m a p with h o m o t o p y equivalent inverse images (fibres). The actual construction is analogous to the above except t h a t one uses the D w y e r  K a n simplicial localization [DKl] L(I) of the category I rather t h a n its localization I[I1]. Explicitly, we define the h o m o t o p y averaging as the composition I . L(I)
,
S in the triangle below t h a t gives the h o m o t o p y K a n extension L ( X ) of the given d i a g r a m of spaces A" :
I
, L(I)
$ We can therefore conclude t h a t the h o m o t o p y fibre of a simplicial m a p X
.
Y to a connected space Y is the h o m o t o p y average of the d i a g r a m of the actual fibres defined over the small category F Y associated to Y.
F. A list o f q u e s t i o n s Below we list a r a t h e r r a n d o m list of open problems relating to cellular constructions. P r o b l e m 23 seem hard and interesting. At present the only known case is where A is a circle or a higher sphere. It is curious t h a t P r o b l e m 7 does not seem to be easy. 1. Develop a r i t h m e t i c square techniques for PA, L/. 2. Give a recognition principle for a space Y to be equivalent to a function complex m a p . ( A , X ) for a given A. For A = S 1 this leads e.g. to the Segal loop machine from C h a p t e r 3.
9. Cellular inequalities
175
In particular, for a given X and A, when is there a space Y with PA(X B) ~ yB?
3.
IS it possible to recover C W A ( X )
map.(A,X)
from the function complex X A =
and A itself in a n a t u r a l way. Of course here the function complex
must be equipped with some e x t r a structure such as the action of A A on it. 4. For a nonconnected homology theory give a construction of all acyclics or a large subclass thereof.
5. Prove: if X is a pcomplete polyGEM then [BZ/pZ, X] is not trivial. 6. Discuss the effect of PA, Lf on the fundamental group. In particular the effect of/(theory localization should be the same as that of homology with the corresponding coefficients. In general the effect on the fundamental group of any homology theory localization might be the same as that of the corresponding connective theory localization. 7. Prove or disprove: For any map f and pointed A, If X is nilpotent then so are LfX and CWAX; the same question for X a PolyGEM or X a lconnected space. 8. For a given generalized homology discuss the associated plus construction that kills all the acyclics: Pacyctics. 9. Give necessary and sufficient conditions for a space to be cellularequivalent to the nsphere. i0. Prove or disprove: if X, Y are Acellular then X A V yA << (X V y)A. ii. Can one show that BZ/pZ ~ BGp for any connected compact Lie G? Dwyer [Dwl] shows by a JackowskyMcClure induction that PBz/pBG ~ BG[I/p], McGibbon has considered a similar question recently [McG, 2.1]. 12. Let Zs be the BousfieldKan tot~functor. Is X << ZsX or at least X < Z ~ X for s < oo?
13. Give conditions on a (coaugmented) h o m o t o p y functor t h a t will ensure t h a t it can be t u r n e d into a simplicial one by a universal arrow. 14. Is it true t h a t X << Y implies S P k X << S P k Y ? True for k = cx~.
UD[]
Appendices
A p p e n d i x H L : H o m o t o p y colimits a n d f i b r a t i o n s
We give here a brief review of homotopy colimits and, more importantly, we list some of their crucial properties relating them to fibrations. These properties came to light after the appearance of [BK]. The most important additional properties relate to the interaction between fibration and homotopy colimit and were first stated clearly by V. Puppe [Pu] in a paper dealing with Segal's characterization of loop spaces. In fact, much of what we need follows formally from the basic results of V. Puppe by induction on skeletons. Another issue we survey is the definition of homotopy colimits via free resolutions. The basic definition of homotopy colimits is given in [BK] and IS], where the property of homotopy invariance and their associated mapping property is given; here however we mostly use a different approach. This different approach from a 'homotopical algebra' point of view is given in [DF1] where an 'invariant' definition is given. We briefly recall that second (equivalent) definition via free resolution below. Compare also the brief review in the first three sections of [Dw1]. Recently, two extensive expositions combining and relating these two approaches appeared: a shorter one by Chacholsky and a more detailed one by Hirschhorn. Basic property, examples Homotopy colimits in general are functors that assign a space to a strictly commutative diagram of spaces. One starts with an arbitrary diagram, namely a functor X: I ~ (Spaces) where I is a small category that might be enriched over simplicial sets or topological spaces, namely in a simplicial category the morphism sets are equipped with the structure of a simplicial set and composition respects this structure. To such a diagram the functor hocolim (otherwise called homotopy direct limit and denoted holimi) assigns a space hocolimX c(Spaces). A basic property *
I
of this assignment is that it respects 'local weak equivalence of diagrams'.
Appendix HL: Homotopy colimits
177
PROPOSITION: Let f: X * Y be a map of Idiagram of spaces. Thus f is a natural transformation b~ween ~two functors X and Y. Assume that for each i C I the map X i ~ Yi is a weak equivalence. T'hen f reduces a weak equivalence of spaces hocolimf: hocolimX I
I
~
, hocolimY. I
In some sense the fnnctor hocolim is the 'closest one' to the actual direct limit or colimit that has this 'weak homotopy invariant' property. One can consider the functor hocolim in various categories {Spaces} of spaces that have an appropriate notion of weak equivalence or 'homotopy of maps' (for example, in chain complexes or in simplicial (abelian or not) groups). We will need to consider it only in two cases: pointed spaces ,9. and unpointed spaces S. It turns out that their values on these categories are different, as can be seen in the following examples:
Examples: (1) The most elementary examples of homotopy colimit are disjoint unions of (unpointed) spaces or wedges of pointed spaces. These are homotopy colimits over discrete categories, i.e. categories with only identity morphisms. A diagram over such a category is just a family of spaces. A pointed diagram is a family of pointed spaces. In this case the unpointed homotopy colimit is equal to the colimit itself. Already here the pointed and unpointed homotopy colimits are different. As always they are related by the cofibration B I ~ hocolimX ~ hocolim.X so that in the pointed case one gets the wedge / ~ / V Xi as the pointed homotopy colimit over categories with no nonidentity maps.. (2) If the category I = G is a (discrete or continuous) group, then a diagram over G is a space with a group action and a pointed diagram is a space with a specified fixed point (fixed under the action of G). The homotopy colimit in the unpointed case is weakly equivalent to the Borel construction E G x a X , and this is true whether X has fixed points or not. In fact X may well be a single point 9 and then hocolima(*) ~ BG, the classifying space of G. If G here is a simplicial or a topological category, we get B G as a topological or simplicial space by the definition of homotopy colimit. If X is a pointed Gspace, then hocolim, X, the pointed homotopy colimit, is the pointed Borel construction given as E G ~
178
Appendix HLi Homotopy colimits
A theorem of V. Puppe There are very useful relations between the concepts of 'fibre map', 'homotopy fibre' and that of homotopy colimits. These relations are useful since s o m e t i m e s they allow one to commute the operation of taking the homotopy fibre of a map with that of taking homotopy colimits of a system of maps. In general one does not expect a 'left adjoint', such as a homotopy colimit, to commute with a 'right adjoint', such as taking a homotopy fibre of a map. In fact, in general, these operations do not commute. The homotopy colimit of the twoarrow pushout diagram 9 ~ X   + * is, of course, the suspension E X of X. Now given a map g: X ~ Y with Y
Example:
connected, taking its homotopy fibre does not commute with suspension G(Fibg) Fib(Eg). But notice that taking homotopy groups commutes with linear homotopy colimits, namely in a tower of cofibrations of pointed spaces X :
X = (Xo ~
one
X1 ~
X2 ~
"")
has
PROPOSITION: For a linear tower as above there is a natural i s o m o r p h i s m
colim ~riX ~ Th h o c o l i m X . Proof."
This follows from the compactness of S ~ and the unit interval.

Now when we translate this to properties of homotopy fibres using long exact sequences and the fact that linear direct limits preserve exactness, we get PROPOSITION: L e t Eo Po~ Bo
+
E1
~
1 B1
+
E2
~
~ B2
"'"
En
"'"
P~ Bn
".
~
E~ P~
...
+
B~
be a tower of fibrations over c o n n e c t e d p o i n t e d spaces Bi. T h e r e is a w e a k equivalence relating h o m o t o p y fibres as follows
hocolimF~~Fib(hocolimpi) ~ Fib(poo: Eo~ ~
Boo).
Appendix HL: Homotopy colimits
179
More surprising perhaps is that under certain conditions homotopy pushout also commutes with taking homotopy fibre [Pu]:
THEOREM (V. PUPPE): Let E2 , A
Eo
A , El
B2 ~ f2
Bo
I1 , B1
be a commutative diagram and suppose that each square is a homotopy pullback, i.e. both (fl, fl) and (f2, f2) induce a homotopy equivalence as the homotopy Fib(po) ~ Fib(p1) ~ Fib(p2) via the obvious maps. Then the maps of Fib(pi) for i = 0, 1, 2 into the homotopy fibre of the maps ofhomotopy pushouts, namely
Fib(pi) ~, Fib(EIOEoE2 ~ BIOBoB2), are all weak equivalences, where 0 denotes the homotopy pushout, i.e. mapping cylinder.
double
General homotopy colimits of fibrations Since by induction one can build any homotopy colimit using direct towers, disjoint unions and pushouts, we can reach the following conclusion:
PROPOSITION[Pu]:
Let E + B be any map of Idiagrams with Bi connected for all i E I. Assume that for each ~ i ~ j in mor I the square: ~ Ej
Ei
1
l
is a pullback square up to homotopy, so that the induced map on the homotopy fibres:
Fi = Fib(E~
, Bi)
~
Fib(Ej
) Ej) = Fj
is a weak equivalence. Suppose in addition, for simplicity, that I is connected (i.e. BI is a connected space). In this situation the homotopy fibre of the induced map
180
Appendix HL: Homotopy colimits
on the homotopy colimits is equivalent to the common values of all the homotopy fibres, i.e. Fib (hoc/olimE * hoc~limB) is weakly equivalent to the 'common value' Fi. Diagrams over a fixed base space B The above theorem of V. Puppe has a very useful corollary, where instead of considering a diagram of fibrations with 'fixed homotopy type as a homotopy fibre' one considers diagrams of fibration sequences over a fixed base space B. See the examples following the next proposition. PROPOSITION: Let E * B be any map of an Idiagram to a fixed connected space B. Thus all we assume is that all the squares strictly commute:
,Ej
I B
l = ,B
Let F be the diagram of the homotopy fibres of the maps Ei , Bi. In th~s situation the homotopy fibre of the map hocolim E , hocolim B = B I
~
I
is equivalent to the homotopy colimit of the diagram of fibres: "hocolim F. I
Proof: It is not hard to reduce this to the theorem of V. Puppe above by backing up the fibration to get a diagram of fibrations with a fixed homotopy type fiB as the homotopy fibre. From the proposition above we get a fibration
fiB
, hocolim F
, hocolim E
I
I
~
that can be seen to be a principal one. Upon classifying this fibration sequence we get the desired fibration sequence over B. An easy direct proof can be obtained by decomposing each fibre map Ei , B into a commutative diagram of trivial fibrations Fi , (Ei)~ , a over the diagram of simplices { a } = F B as explained below. This gives a 'double diagram' of fibrations F~,o  , Ei,~ , a. The original diagram is obtained by 'integrating', i.e. taking homotopy colimit over the index a that varies over the simplices of B. But since we are allowed to change the order of taking homotopy colimits we first take the homotopy colimit over the index i E I.
A p p e n d i x HL: H o m o t o p y colimits
181
Namely, we take first the hocolim over each simplex in B. This gives us a d i a g r a m of maps:
(*)
( h o c o l i m Fi o I
'
~, hocolim Ei,o ~
a ) CerB"
I
Notice the weak equivalence between fibre and t o t a l space t h a t follows from the fact t h a t for each individual i the m a p is an equivalence. Now as we vary a over the d i a g r a m F B of simplices in B, for each fixed i a m a p a * a t induces an equivalence
(**)
F~,~ ~. Fr162
because for each i the m a p Ei * B is a fibration. Therefore, by the basic p r o p e r t y of h o m o t o p y colimit, u p o n taking h o m o t o p y colimit over i E I one still gets an equivalence (*), as claimed. Now (.), being a d i a g r a m of equivalences with respect to all m a p s in F B , it gives the desired fibration sequence over B upon t a k i n g hocolim with respect to ~ E F B . EXAMPLES: maps:
Let I be the small d i a g r a m 9 * 9 + 9and consider the d i a g r a m of
x
1
X
,
~=
(pt)
l
X
~
x
=1
X
1
Or in other words: x,
(pt)
, x
\1/ X T h e n the fibration of the hocolim is the fold m a p X V X * X .
In this
case we get G a n e a ' s theorem t h a t the fibre is E ~ X since the d i a g r a m of fibres is pt ~
f t X * pt, whose hocolim is E a X .
Similarly, by considering the analogous
d i a g r a m over B x B one gets t h a t the h o m o t o p y fibre of X v X join f~X 9 a X .
, X x X is the
182
A p p e n d i x HL: H o m o t o p y colimits
A n o t h e r special case is the fibre of the m a p E U C F * B for a fibration F
E
~ B. Here we use the d i a g r a m fiB
1 1 B *
~~
~
FxftB
1 l,B F
, *
~
F
1 1 B E
So the hocolim of the fibres is the join F * f t B . Notice t h a t a special case of the fibration E~tX ~ X V X ~ X is the wonderful fibration sequence:
S 2 * C P ~ v C P ~ ~ C P ~r
or
S 2 ~ K ( Z , 2) V K ( Z , 2) * K ( Z , 2). Decomposing general m a p s into free diagrams A p a r t i c u l a r case of V. P u p p e ' s theorem, from which in fact the general ease follows, is especially useful. This is the case when all the base spaces are contractible and thus all the spaces E~ are equivalent to each other and are just the fibres themselves. This case arises n a t u r a l l y when a fibration over, say, a simplicial complex is decomposed into a d i a g r a m of the simplices of the base spaces. C o m p a r e (1.F). We want to think a b o u t fibration as a m a p E ~ B in which all the fibres p  1 (b) are h o m o t o p y equivalent to each other; more correctly, a m a p in which each p a t h ~ : I ~ B, ~(0) ~ ~(1) in the base can be lifted to a h o m o t o p y equivalence of the fibres over these end points. Thus the t o t a l space of a fibre m a p is 'a family of equivalent fibres' glued together over B. A good example to consider is the m a p p i n g torus of a selfmap w : W ~ W discussed in (9.B.l.1), where other examples of this procedure are discussed. Now to make the above more formal we work, for simplicity, with simplicial complexes, b u t everything can be done similarly with simplicial sets [Ch3], [HH]. Consider
an a r b i t r a r y simplicial m a p of simplicial complexes: f : E * B.
For the m o m e n t we do not assume t h a t the m a p f is a fibre map. One can consider the small category defined by the simplices of the simplicial complex B. This category, denoted by F B , has one object a for each simplex in B and one m o r p h i s m a * r for each face inclusion of simplices in B. Thus Hom(x, y) in F B has at most one element and usually it is empty. Now define a d i a g r a m E of simplicial complexes indexed by F B as follows: E~ = f  l ( a ) ; this is a s u b c o m p l e x of E . Thus
Appendix HL: Homotopy colimits
183
E is simply the diagram of the inverse images of simplices in B. Notice that it is i~nmediate from the definition of E that its strict colimit over FB is equal to E itself: colimE = E. FB
The following is very useful (explanation below): PROPOSITION: In the situation above/'or any simplicial map f : E ~ B: (1) The diagram E is a free FBdiagram. (2) The homotopy colimit o r E is equivalent to the space E itsel/, hocolimE ~ E,
when the homotopy colimit is taken over the indexing diagram FB. In order to explain this proposition we recall the treatment Of homotopy colimit via free resolution of diagrams. Recall: Free Idiagrams By 'free diagram of spaces' we mean a 'cellular' diagram that 'can be built from free/cells of the form A[n] x F d by disjoint unions and elementary pushouts'. Let us explain [DFI]: (i) An orbit is an Idiagram e whose strict direct limit is a point. (ii) With each object d E I one associates an orbit denoted F d and called the free orbit at d. By definition Fd: I ~ S is given by Fd(d ') = Hom;(d, d'), where composition in I gives obvious maps F d ~ Fj d for each (i ~ j) E morl. Example:
Thus for the category I  G associated to a group there is only one object and only one type of free orbit F*  G  mapa(*, .), namely the group itself with the left action, since there is only one object in G. Notice that the
free orbit F d is a diagram of sets, or discrete spaces if I is a (nonenriched) small category, but if I is a simplicial category then F d is a diagram of simplicial sets. (iii) A free cellular/diagram is a diagram of cofibrant spaces built from free /cells: A[n] • F d by gluing them together along their boundaries ~[n] x F d.
Example: If we return to the simplicial map f : E ~ B in the above proposition it is not hard to see that E is a free FBdiagram. This follows from a basic property of simplicial complexes:~Each point in B belongs to a unique simplex as an interior point. Hence we can construct E as a diagram of simplicial complexes by induction over the skeletons of B. An illuminating example of the above is the diagram associated with the identity map f = id : B  L B of a simplicial complex. This diagram B: FB ~ S assigns to each simplex a E B the underlying space of that simplex, and to each face inclusion a C a ' the inclusion of spaces a C a'. We get a free diagram where every point b E B(cr) belongs to a unique free orbit: the orbit F ~ where ~ is the unique simplex 5 c a such that b E int ~.
184
Appendix HL: Homotopy colimits
Free diagram and homotopy colimits The basic properties of free diagrams are: PROPOSITION [DF1]:
For every diagram X : I ~ S there exist a free diagram N
X f~r
and natural m a p X f ~
+ X which is a weak equivalence, i.e. such that for
all i E I the induced m a p X [ ~
+ X i is a weak equivalence o f spaces.
(Compare [HH], [DK2]) The construction is analogous to that of the CWapproximation to a space and it is done again by induction on the dimension of the free ceils F d x A[n] in X f~r162Notice that in this way one g a s always a diagram of eofibrant spaces. Proof"
PROPOSITION: For any
free diagram X free the natural map:
hocolimX f~e~ ~, colim X f~er
from the h o m o t o p y colimit to the colimit itself is a weak equivalence. F R E E DIAGRAMS IN THE POINTED CATEGORY. I n the pointed category a freepointed diagram is not free, but it is free away from the base point orbit. Namely it is a diagram constructed from the point diagram over I denoted by (*) or (*I), with *i = *, by adding free cells as above. So it is free relative to *I.
Recognizing homotopy colimits Free resolution as above is often an effective method to compute hocolim X. For any space X with an action of a group G, the map E G x X * X is a free Gresolution since E G ~ * and the action of G on E G x X is free. Therefore we get the Borel construction Example:
hocolimGX ~ colim E G • X = E G x GX. G
Let " I be a point diagram. Since as a pointed diagram it is free, the pointed homotopy colimit hocolim ( ' I ) is equivalent to the colimit itself, which is
Example:
I
just a point (*) . On the other hand, the unpointed homotopy colimit hocolim *I is obtained by first constructing a free /diagram of constructible spaces, denoted here by E I , and since the only map E I ~ 9 is an equivalence on each space in the diagram, one now takes the colimit of E I . In fact one gets: colimi E I = B I , namely we get the classifying space of I.
A p p e n d i x HL: H o m o t o p y colimits
Example:
185
If B is any simplicial complex, then B is a free F B  d i a g r a m . Therefore
hocolim B = colim B = B I
~
I
since B is clearly o b t a i n e d by gluing its own simplices along all the simplicial face inclusions. F r o m d i a g r a m to m a p s Now for an a r b i t r a r y simplicial m a p p: E + B we have formed a d i a g r a m E. F r o m this d i a g r a m we can recover not only the space E , b u t the m a p p itself up 7o homotopy:
PROPOSITION:
A n y simplicial m a p p: E ~ B is equivalent up to h o m o t o p y to the
m a p hocolim E * W ( F B ) , where W denotes the classifying space o f a diagram. Proof:
Since b o t h E and B are free F B  d i a g r a m s we have the equivalences:
hocolimE
1 hocolimB
 , c o l i m E
1 ~ , colimB
= , E
1 ; , B
The d i a g r a m E comes with a n a t u r a l m a p to the point d i a g r a m *rB, so this m a p induces a m a p on h o m o t o p y colimits. Now the proof is concluded by using the proposition below for X  B and the unfortunate n o t a t i o n B is the classifying space functor W : PROPOSITION:
For any simplicial complex X there is an equivalence
B F X ~ X. Proof."
By definition B F X = h o c o l i m r x ( * ) , the h o m o t o p y colimit of the point
diagram. But X ~
9 is a free resolution of *, because it is free, as we saw above,
and X ( a ) = a ~ * is a contractible space, being a Simplex, so t h a t X ( a ) ~ * is a weak equivalence for each a E F X . Therefore h o c o l i m r x X ~ h o c o l i m r x (*) ~ B F , where B is now the classifying space functor. But we saw t h a t h o c o l i m r x X ~ X , so X ~ B F X , as needed.
186
Appendix HL: Homotopy colimits
Homotopy colimits and fibrations, Quillen's Theorem B We saw above how to write an arbitrary simplicial map f : E ~ X in the form hocolimrxE * hocolimrx*. Now we specialize to fibre maps. These are distinguished from other maps by the property that the associated diagram E over the small category FX is a diagram of spaces in which all the maps are weak equivalences. PROPOSITION: I f E ~ X is a simplicial fibre map, then for every map T + a in the small category F X the m a p E ( r ) ~ E ( a ) is an equivalence. Proof:
This is the homotopy lifting property that implies the equivalence of fibres
along paths in the base space. Diagrams of fibres computinK homotopv fibre For a simplex a E B in the base and a simplicial fibre map E * X, the inclusion of the barycenter of a to c~, namely b(a) * a, induces an equivalence pl(b(cr)) ~* p  l ( a ) = E(a). Therefore E ( a ) i s equivalent to the fibre over b(a). If X is connected, all these fibres are weakly equivalent. Conversely, given a diagram of weak equivalences, if one forms the associated map by taking homotopy colimits, one can identify the homotopy fibre of the resulting map as the (constant) homotopy type that appears in the given diagram: PROPOSITION: Suppose that I is a small category and let W : I * S be a diagram o f equivalences, so that for each i * j in mor I the map W i * W j is an equivalence. A s s u m e that the space F is equivalent to all the spaces W i for i E I. Then the h o m o t o p y fibre o f the natural map hocolim W * B I is again equivalent to F . I Proof: This follows by induction on the skeleton from V. Puppe's Theorem above, by decomposing the base. In fact it is also a formulation of Quillen's Theorem B
[Q2]. Relations to homotopy (inverse) limits Homotopy limits are 'homotopy approximations' to (inverse) limits of a diagram in an analogous way to the relation between colimit and homotopy colimit. Again, these can be defined in various categories. In general, in order to define holim as a functor one chooses once and for all a specific, free resolution of the onepoint diagram (.) = (*I), where *i(i)  *. Let us denote this resolution by E 1 * (*) so that E I is a free Idiagram with EI~ ~ (pt), a contractible space for each i C I. One then takes holimxX for any diagram to be the space of all maps of diagrams
holimlX = Hom(EI, X).
It has two crucial properties:
Appendix HL: Homotopy colimits
187
PROPOSITION: I f f : X + Y is a weak equivalence of an Idiagrams O.e. fi is a weak equivalence for all i) w i t h X i and Y~ fibrant, then the induced m a p holimlX , holimiY is a weak equivalence.
PROPOSITION: For any diagram X and any space W , there is a natural weak equiv
alence map(hocolimX, W) ~ holimioP map(X, W). Remark: A special case of the second property is the wellknown statement, that if A ~ X ~ X / A is a cofibration, then for any connected space W the sequence [S]:
m a p ( X / A , W ) * map(X, W) ~ map(A, W) is a fibre sequence, where map(X/A, W) is weakly equivalent to the homotopy fibre over the component of the null maps A ~ W in map(A, W). The same statement holds for pointed diagrams with homotopy colimit replaced by pointed homotopy colimit. Pushout and pullback squares These diagrams are especially useful examples of both homotopy colimits and homotopy limits. They also have special useful properties that we make explicit now. Given any strictly commutative square: X
"f , Y
V
g,W
(Q)
one can form the homotopy colimit of V ," X I . Y, called also the homotopy pushout, simply by taking the double mapping cylinder C ( f , v). This space comes with obvious maps Y , C ( f , v) and V 9 C ( f , v). Upon precomposing into X we get another square that is, however, commutative only up to homotopy. From the above square one can also form the strict pushout or identification space as a quotient of Y disjoint union with V. Denote this pushout by P O . As always there is a natural map from the homotopy colimit to the strict colimit. Now by the basic property of the strict direct limit the given square (Q) above induces a natural map P O ~, W . Therefore upon composition one gets a natural map C ( f , v) * W.
188
Appendix HL: Homotopy colimits
Similar maps exist for the homotopy pullback of the maps V , W , Y. This is the usual space of paths A(g, w). If we denote the strict pullback of the same pair of maps (g, w) by P B . Again there is a natural map P B * A(g, w). Therefore we get a natural map X * A(g, w).
Appendix HC: Pointed homotopy coends
Here we discuss briefly a certain kind of homotopy colimit called a homotopy coend; compare [DF1], [Dw2], [HV]. It can be used also to define the homotopy colimit itself. Given a small category I and two diagrams: an unpointed one over I denoted by U and a pointed one over I ~ denoted by P, one can form a homotopy version of their coend [Mac]. This version will depend only on the weak homotopy type of these diagrams and is equivalent to the coend itself if one of these diagrams is free as in Appendix HL above. We use the double integral notation to denote this type of pointed homotopy colimit (compare Appendix HL above); this notation is meant to remind the reader that the 'integrand' has two 'variables': One covariant the other contravariant.
DEFINITION: Let U : I ~ S be a functor to (unpointed) spaces and P : I ~ S. be a contravariant functor to pointed spaces. The pointed homotopy coend, denoted
h~176
//U+
AP=
f/U
I
K
I
is defined to be the realization (or diagonal) of the simplicial space B(U, I, P) which, in dimension k, consists of the one point wedge:
V co.Cl ......c
u+(co)^ P(ek), k
indexed by the ksimplices Co ~ Cl ~ ... ~ ck of the nerve of I. The face and degeneracy maps in the simplicial space B (U,I,P) are the usual ones as follows: Let f,~ o . . . o fl: a ~ ~ be composeable, and let p E P~,u E U~, one has:
(U; fl,"" ", fn1; f,~(P)), di(U;fl, f2 " ' ' , f n ; p ) :
i = 0
(U;fl,'",fi o fi+l"",fn;P)),
( f l ( U ) ; f 2 , " ", fn;P)),
1 <_ i <_ n  1
i : n.
The degeneracies are defined by insertion of identity maps. WARNING: Our notation regarding I and I ~ is opposite to the one used by
[HV].
190
Appendix HC: Homotopy coends
Free resolution approach Compare [HV, 3.2]. Just as in the case of homotopy colimit in appendix HL above, here too one can reduce the notion of homotopy coend to that of (strict) coend    by first resolving one of the two diagrams involved by free diagram. Recall (Appendix HL) that for a free diagram the notions of homotopy colimit and (strict) colimit are canonically equivalent. Analogously, in case the diagram P above is free, the strict coend is equivalent to the homotopy coend. To see that one shows by a direct computation that it is so for any free orbit over I ~ and then one proceeds by induction on the skeleta of P using the fact that both the coend and the homotopy coend commutes with the pushouts used to attach free cells A[n] x F i. To conclude, let us denote as usual the (strict) coend by the tensor product notation @i; we have: PROPOSITION: I f U : I
9S (Res. P : I ~
* S.) is free (res. pointed free) then
the natural map / U AP
@I P
. U+
I
is a weak equivalence.
Thus we have
f U hP
U + @I P Y ~ .
I
In this way we see that the homotopy coend is the total derived functor of coend or tensor product of two diagrams. A generalization: We also use a generalization of this concept when, instead of the smash product between two pointed spaces, one uses any binary homotopy functor, say W # V , such as the join W 9 V. In that case we denote by
hocoend,(U,i P; # ) = / / U # P I
the diagonal of the corresponding simplicial space B(U, I, P; # ) , which in dimension k consists of the one point wedge
V Co ,~,C1 ~, . .   * C k
U(co)#P(ck)
Appendix HC: Homotopy coends
191
indexed by the ksimplices Co * Cl ~ ... * ck of the nerve of I. The face and degeneracy maps in the simplicial space B(U, I, P; # ) are the usual ones. It follows from a direct inspection of the construction of a pointed homotopy colimit in [BK] that one can write the pointed homotopy colimit as a coend: PROPOSITION: There is a natural equivalence for any diagram of pointed spaces P
as above: h o c o l i m . P ~ hocoend.(*, P). I
I
EXAMPLE: GSPACE AS A (HOMOTOPY) COEND Given a pointed Gspace * E X, one can recover it from the diagram of the fixed points as a coend as explained in [DF1] and [DK2]. In this case the coend is equivalent to the homotopy coend since we get a free diagram as an input (see below). Namely we associate with X two diagrams over the categories O, O ~ of Gorbits: O is the category of Gorbits, i.e. the G spaces G/K, for K any subgroup of G where the maps are Gequivariant maps. First, the identity functor Z = Zg that assigns to each Gorbit of the form G/H, that orbit itself as a Gspace. This is a functor Z: O ~ S from O to unpointed spaces, since none of these orbits except the trivial one G/G has a Gfixed point. Second, the fixedpoint assignment is a contravariant functor denoted by X ~ 0 ~ S. that assigns the space X g = mapG(G/K , X ) to the orbit G / K and maps coming from composition on the domain in the mapping space presentation of the fixedpoint set. Notice that now the coend itself has a natural structure of a Gspace coming from the left action of G on the orbits that enter into the coend. There is an evaluation map coend(Z, X ~ . X which takes the s u m m a n d G / H • mapG(G/K , X ) that corresponds to a specific map w: G / H ~ G / K via the obvious composition and evaluation. We now quote from standard Gspace theory : PROPOSITION: The evaluation m a p gives a Gequivalence: coend(Z, X ~ form the coend to X .
* X
Outline of Proof: The main point is to notice that the functor mapv(G/L,   ) on G spaces where L is a subgroup of G commutes with homotopy colimits up to equivalence, because G/L is an 'orbit' so it is 'small'. So one shows that the evaluation map induces an equivalence on Lfixedpoint sets for any L C G. 
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INDEX
Acellular spaces Adams map Acyclic spaces Ahomotopy equivalence augmented functors AWhitehead theorem BlakersMassey Borel construction Borel construction, pointed BSUZ/p CauchySchwartz inequality cellular approximation cellular inequalities cellular spaces cellular types classifying spaces closed classes coaugmented coend coend, homotopy, pointed coequalizer colimits, homotopy, pointed comparison map connected covers continuity contractible small categories convergent functors CWA of products CWA and fibrations CWA, construction of CWapproximation decomposition divisibility durable homology isomorphisms E.acyclic EilenbergMac Lane spaces
2.D 8.A 2.D.2.5 2.A; 2.E 2.A 2.E 9.D.7 3.C.7; 9.D.3 4.A.4; 9.D.3 6.C.2 9.A.9(ii), 8.C 2.A.3 9 3.C 7.A, 7.C 3.C.6 2.D 1.A.2 3.C.4; Appendix HC Appendix HC 3.C.4 2.D.1; 9.B.l.1; HL 1.D.3 5.F.2 1.C 9.B 3.C.3 2.D.15 5.E 2.B 2.A.3.4 2.C; Appendix HL 1.C.3; 1.E.3 8.D.2 2.D.2.5; 3.C.13; 9.D.2 4
Index elementary facts error term factorization fequivalence fibrations & closed classes fibrewise universal covering flocal flocal equivalence free diagram free orbit Ganea's theorem GEM halfsmash hocolim homotopy coend homotopy coend, pointed homotopy colimit homotopy colimits, pointed homotopy colimits, unpointed homotopy direct limit homotopy flocal homotopy functors homotopy inverse limits homotopy limit homotopically discrete HopkinsSmith type idempotency infinite loop space James functor Kan complex localization, BousfieldKan localization, construction of localization, flocalization, fibre of localization, fibrewise (topological) localization, fibrewise localization, homological localization, Klocalization of a GEM localization of a polyGEM localization, QuillenSullivan
197 1.A.8 5.D.6 1.C.1; 2.C.2.1 1.C.5; 1.D 2.D.10 3.E.5 1.A.1 1.C.5; 1.D 4.C.2; Appendix HL Appendix HL 2.D.13; Appendix HL 4.B 2.D.4 Appendix HL 4.A.5.1; Appendix HC Appendix HC 4.A.5.1; Appendix HL 2.D.1; 9.B.l.1; HL 2.D.3; 9.B.1 Appendix HL 1.A.1.2 1.F.1 Appendix HL Appendix HL 5.E 7 Introduction, 7.A 1.A; 2 . B . 2 5.B.1.2; 8.A.8 3.C.1 1.B.1 1.E.3 1.B 1.A.3 1.H 1.F.6 1.F 1.E.4 8.A 4.B 6.B 1.E.2
198 localization, rigid fibrewise localization, Vlloop functor mapping torus Miller spaces model category Morava Ktheory nconnected null, Wnullification nullity classes nullity type ordinal number pdivisible periodic families periodicity functors plus construction polyGEM Postnikov section principal fibrations rationally acyclic realization retracts Segal's loop 'machine' Segal's infinite loop 'machine' simplicial diagrams simplicial functor singular functor small object argument special Fspace subdivision supported, nsupported W, supports suspension symmetric products thick subcategories universal Aequivalence universal cover universality properties unstable cellular type
Index 1.F.2 3.C.5; 8 3.A 9.B.1.1 2.D.2.4 2.C.2 3.C.5; 6.C 1.A.8 (e.11); 2.D.2.6 1.A.1 1.A.4; 5.C 1.A.4; 7.A 7 1.B.3 1.A.1.1 8
1.E.6, 8 1.E.5 5.A.1; 5.E.3 1.A.1.1; 1.E.1 3.E.3, 5.F &Introduction 1.F.6 2.D.1.5 3.A.3 5.B.1.2 1.D.1; Appendix HL 1.C.7 1.F.6 1.B.1 5.B.1.2 9.B.5.1.b 7.C.6 1.A.5 8.A.5 1.A.8 4 7.A.1 2.A.3.5 2.A.3 1.C; 2.E.8 7.C
Index unstable nullity type WCS
199 7.B 9.A.8 August 1995. The Hebrew University of Jerusalem. email address: [email protected] (emm)