HOMOLOGY THEORY ss An Introduction to Algebraic Topology
This is Volume 53 in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUEL EILENBERG A complete list of titles in this series appears at the end of this volume
homology theory AN INTRODUCTION TO ALGEBRAIC TOPOLOGY
JAMES W . VICK University of Texas
ACADEMIC PRESS
New York and London
COPYRIGHT 0 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York,New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI
LIBRARY OF CONQRESS CATALOG C m NUMBER: 72-84375 AMS (MOS) 1970 Subject Classifications:5541,55BlO,57-10 PRINTED IN THE UNITED STATES OF AMERICA
to Professor Ed Floyd
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contents
Preface
ix
Acknowledgments
xi
1 SINGULAR HOMOLOGY THEORY
I
2 ATTACHING SPACES WITH MAPS
40
3 THE EILENBERG-STEENROD AXIOMS
4 PRODUCTS
74
98
5 MANIFOLDS AND POINCARE DUALITY
6 FIXED-POINT THEORY
173
vii
124
APPENDIX I
201
APPENDIX II References Bibliography
209
219 221
Books and Notes 221 Survey and Expository Articles Index
233
227
preface
During the last twenty-five years the field of algebraic topology has experienced a period of phenomenal growth and development. Along with the increasing number of students and researchers in the field and the expanding areas of knowledge have come new applications of the techniques and results of algebraic topology in other branches of mathematics. As a result there has been a growing demand for an introductory course in algebraic topology for students in algebra, geometry, and analysis, as well as for those planning further work in topology. This book is designed as a text for such a course as well as a source for individual reading and study. Its purpose is to present as clearly and concisely as possible the basic techniques and applications of homology theory. The subject matter includes singular homology theory, attaching spaces and finite CW complexes, cellular homology, the Eilenberg-Steenrod axioms, cohomology, products, and duality and fixed-point theory for topological manifolds. The treatment is highly intuitive with many figures to increase the geometric understanding. Generalities have been avoided whenever it was felt that they might obscure the essential concepts. Although the prerequisites are limited to basic algebra (abelian groups) and general topology (compact Hausdorff spaces), a number of the classical applications of algebraic topology are given in the first chapter. Rather
x
PREFACE
than devoting an initial chapter to homological algebra, these concepts have been integrated into the text so that the motivation for the constructions is more apparent. Similarly the exercises have been spread throughout in order to exploit techniques or reinforce concepts. At the close of the book there are three bibliographical lists. The first includes all works referenced in the text. The second is an extensive list of books and notes in algebraic topology and related fields, and the third is a similar list of survey and expository articles. It was felt that these would best serve the student, teacher, and reader in offering accessible sources for further reading and study.
acknowledgments
The original manuscript for this book was a set of lecture notes from Math 401402 taught at Princeton University in 1969-1970. However, much of the technique and organization of the first four chapters may be traced to courses in algebraic topology taught by Professor E. E. Floyd at the University of Virginia in 1964-1965 and 1966-1967. The author was one of the fortunate students who have been introduced to the subject by such a masterful teacher. Any compliments that this book may merit should justifiably be directed first to Professor Floyd. The author wishes to express his appreciation to the students and faculty of Princeton University and the University of Texas who have taken an interest in these notes, contributed to their improvement, and encouraged their publication. The typing of the manuscript by the secretarial staff of the Mathematics Department at the University of Texas was excellent, and particular thanks are due to Diane Schade who typed the majority of it. Many helpful improvements and corrections in the original manuscript were suggested by Professor Peter Landweber. Finally the author expresses a deep sense of gratitude to his wife and family for their boundless patience and understanding over years during which this book has evolved.
xi
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HOMOLOGY THEORY ss An Introduction to Algebraic Topology
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chapter 1 SINGULAR HOMOLOGY THEORY
The purpose of this chapter is to introduce the singular homology theory of an arbitrary topological space. Following the definitions and a proof of homotopy invariance, the essential computational tool (Theorem 1.14) is stated. Its proof is deferred to Appendix I so that the exposition need not be interrupted by its involved constructions. The Mayer-Vietoris sequence is noted as an immediate corollary of this theorem, and then applied to compute the homology groups of spheres. These results are applied to prove a number of classical theorems: the nonretractibility of a disk onto its boundary, the Brouwer fixed-point theorem, the nonexistence of vector fields on even-dimensional spheres, the Jordan-Brouwer separation theorem and the Brouwer theorem on the invariance of domain. If x and y are points in Rn, define the segment from x to y to be {(I - t ) x ty I 0 5 t I l}. A subset C z Rn is convex if, given x and y in C, the segment from x to y lies entirely in C. Note that an arbitrary intersection of convex sets is convex. If A c R",the convex ku// of A is the intersection of all convex sets in Rn which contain A . A p-simplex s in Rn is the convex hull of a collection of (p 1) points { x o , . . . , x P } in W n in which x, - xo,. . . , x p - xo form a linearly independent set. Note that this is independent of the designation of which point is x o .
+
+
I
2
1 SINGULAR HOMOLOGY THEORY
1.1 PROPOSITION Let { x o , .. . , x p } G R". Then the following are
equivalent :
-
(a) x1 - x,, . . . ,x p xo are linearly independent; (b) if sixi = C tixi and C si = C t i , then si = ti for i = 0,. . . ,p. PROOF
(a)
-
=
(b): If
C sixi
=
C tixi and C si
= C ti, then
P
(Si
- ?()(Xi - xo).
i-1
By the linear independence of x1 - x o , . . . ,x p - x0 it follows that si = ti for i = 1 , . . . ,p. Finally, this implies so = to since C si = t i . (b) (a): If CE, ( t i ) ( x i - xo) = 0, then tixi = (C:=, t i ) x o and by (b) the coefficients t , ,. . . , t, must all be zero. This proves linear independence. 0
-
Let s be a p-simplex in Rn and consider the set of all points of the form foxo t,x, . . t p x p ,where C ti = 1 and ti 2 0 for each i. Note that this is the convex hull of the set { x o , . . . ,x p } and hence from Proposition 1.1 we have the following:
+
+- +
1.2 PROPOSITION If the p-simplex s is the convex hull of {xo, . . . ,x p } ,
then every point of s has a distinct unique representation in the form C tixi, where ti 2 0 for all i and C ti = 1. 0 The points xi are the vertices of s. This proposition allows us to associate the points of s with (p 1)-tuples ( t o , t , , . . . , t p ) with a suitable choice of the coordinates t i .
+
Exercise I Let y be a point in s. Then y is a vertex of s if and only if y is not an interior point of any segment lying in s. If the vertices of s have been given a specific order, then s is an ordered simplex. So let s be an ordered simplex with vertices xo , xl,. . . , x p . Define up to be the set of all points ( t o ,t , ,. . . , rp) E RP+' with ti = I and ti 2 0 for each i. If a function f: u p - s is given by f(to,. . . ,rp) = C tixi, then f is continuous. Moreover, from the uniqueness of representations and the fact that up and s are compact
SINGULAR HOMOLOGY THEORY
3
Hausdorff spaces it follows that f is a homeomorphism. Thus, each ordered p-simplex is a natural homeomorphic image of u p . Note that up is a p-simplex with vertices xo’ = (1, 0,. . . , 0), x l ‘ = (0, 1,. . . ,0), . . . , xp’ = (0, . . . ,0, 1). up is called the standard p-simplex with natural ordering. Let X be a topological space. A singular p-simplex in X is a continuous function 4: u p + x . Note that the singular 0-simplices may be identified with the points of X , the singular I-simplices with the paths in X , and so forth. If 4 is a singular p-simplex and i is an integer with 0 5 i 5 p, define a,($), a singular ( p - 1)-simplex in A’, by &#(to,.
. . ,tp-1)
= +(to,
4 , . . . ,L , O ,
ti,. ..
9
tp-1).
of 4. For example, let 4 be a singular 2-simplex in X (Figure 1.1). Then, a14 is given by the composition shown in Figure 1.2. That is, to compute ai# we embed up-, into up opposite the ith vertex, using the usual ordering of vertices, and then go into X via 4.
ai4 is the ith face
XO
Figure 1.1
Figure 1.2
I
1
SINGULAR HOMOLOGY THEORY
-
Iff: X Y is a continuous function and $ is a singular p-simplex in X , define a singular p-simplex f,($) in Y by f,(#) f o $. Note that if g: Y W is continuous and id: X + X is the identity map,
-
= I
An abelian group G isfree if there exists a subset A E G such that every element g in G has a unique representation g=Cnz.x, red
where n, is an integer and equal to zero for all but finitely many x in A . The set A is a basis for G. Given an arbitrary set A we may construct a free abelian group with basis A in the following manner. Let F ( A ) be the set of all functionsffrom A into the integers such thatf(x) # 0 for only a finite number of elements of A . Define an operation in F ( A ) by (f g ) ( x ) = f ( x ) g(x). Then F(A) is an abelian group. For any a E A define a function& in F ( A ) by
+
1
=
{0
+
if x = a otherwise.
Then { & I a E A } is a basis for F ( A ) as a free abelian group. Identifying a with fa completes the construction. For example, let G = { ( n l , n 2 , . . .) I ni is an integer, eventually O}. Then G is an abelian group under coordinatewise addition, and furthermore it is free with basis
( I , O ,... ), (0,l , O , . . .), (O,O, l , O , . ..), . ... For convenience we say that if G = 0, then G is a free abelian group with empty basis. Note that if G is free abelian with basis A and H is an abelian group, then every functionf: A H can be uniquely extended to a homomorphism f:G H . If X is a topological space define S , ( X ) to be the free abelian group whose basis is the set of all singular n-simplices of X . An element of S , ( X ) is called a singular n-chain of X and has the form
-
-
c 4
n4
$9
where n4 is an integer, equal to zero for all but a finite number of
4.
SINGULAR HOMOLOGY THEORY
5
Since the ith face operator d i is a function from the set of singular n-simplices to the set of singular (n - 1)-simplices, there is a unique extension to a homomorphism
given by d,(C ng homomorphism
. 4) = C n4 . air$. Define the boundary operator to be the
a:
S,(X) + S,-,(X)
given by
a=
a0
-
d,
+ a, +
* * *
+ (-l)"d,
=
f (-l)%+.
i-0
1.3 PROPOSITION The composition d o d in
is zero.
Exercise 2 Prove Proposition 1.3.
0
Geometrically this statement merely says that the boundary of any n-chain is an (n - 1)-chain having no boundary. It is this basic property which leads to the definition of the homology groups. An element C E S,(X) is an n-cycle if d(c) = 0. An element d E S,,(X) is an n-boundary if d = a ( e ) for some e E S , + , ( X ) .Since is a homomorphism, its kernel, the set of all n-cycles, is a subgroup of S , ( X ) denoted by Z,(X). Similarly the image of d in S , ( X ) is the subgroup &(A') of all n-boundaries. Note that Proposition 1.3 implies that B , ( X ) C Z,(X) is a subgroup. The quotient group H,(X)
=
Z,(X)/4(X)
is the nth singular homology group of X . The geometric motivation for this algebraic construction is evident; the objects we wish to study are cycles in topological spaces. However, in using singular cycles, the collection of all such is too vast to be effectively studied. The natural approach is then to restrict our attention to equivalence classes of cycles under the relation that two cycles are equivalent if their difference forms a boundary of a chain of one dimension higher. This algebraic technique is a standard construction in homological algebra. A graded (abelian) group G is a collection of abelian groups {Gi}
6
1 SINGULAR HOMOLOGY THEORY
indexed by the integers with componentwise operation. If G and G‘ are graded groups, a homomorphism
f: G-+G‘ is a collection of homomorphisms
{fi},
where
for some fixed integer r . r is then called the degree off. A subgroup H G G of a graded group is a graded group {Hi} where Hiis a subgroup of G.i. The quotient group G / H is the graded group { G i / H i } . A chain complex is a sequence of abelian groups and homomorphisms Jn+I ... -+cn--+cn-,-+ an
an-1
*
*
a
in which the composition o d, = 0 for each n. Equivalently a chain complex is a graded group C = {Ci} together with a homomorphism a: C + C of degree - 1 such that 3 o a = 0. If C and C‘ are chain complexes with boundary operators a and a’, a chain map from C to C’ is a homomorphism
0: C + C ’ of degree zero such that a‘ o = @,- 0 d for each n. (Note that the requirement that @ have degree zero is unnecessary. It is stated here only as a convenience since all chain maps we will consider have this property.) Denoting by Z , ( C ) and B,(C) the kernel and image of d, respectively, the homology of C is the graded group H*(C) = Z*(C)/B*(C). Note that if @ is a chain map, @ ( Z , ( C ) ) c Z,(C’)
and
@(B,(C)) E B,(C’).
Therefore, @ induces a homomorphism on homology groups @*:
H*(C)
-+
H*(C’).
In this sense the graded group S , ( X ) = {Si(X)} becomes a chain complex under the boundary operator a, so that the homology group of X is the homology of this chain complex. Iff: X Y is a continuous function and q5 is a singular n-simplex in X, there is the singular n-simplex f,(q5) = S o q5 -+
SINGULAR HOMOLOGY THEORY
7
in Y. This extends uniquely to a homorphism
f#: S , ( X ) - S , ( Y )
for each n.
To show that f# is a chain map from S , ( X ) to S , ( Y ) it must be checked that the following rectangle commutes: Sn(X)
f#
Sn( Y )
First note that it is sufficient to check that this is true on singular n-simplices 4, and second, observe that it is sufficient to show a&(#> = f,ai(q5). Now
Thus, f#:S , ( X ) + S , ( Y ) is a chain map and there is induced a homomorphism of degree zero
f * : H*(X)
-
H*(Y).
-
Note that this is suitably functorial in the sense that for g: Y W a continuous function and id: X - X the identity, (g of)* = g, o.f* and id* is the identity homomorphism. As a first example take X = point, Then for each p 2 0 there exists a unique singular p-simplex q5p: ap --t X. Note further that for p > 0, di4p = $ p - l . So consider the chain complex * * *
- - - S,(pt)
S,(pt)
So(pt)
0.
Each S,(pt) is an infinite cyclic group generated by operator is given by
Thus,
a&,.-l
=
0 and
bn. The
boundary
a#,,, = q32n-1 for n > 0. Applying this to the chain
8
1 SINGULAR HOMOLOGY THEORY
complex it is evident that Z,(pt)
=
B,(pt)
for n > 0.
However, Zo(pt) = So(pt) is infinite cyclic, whereas Bo(pt) = 0. Therefore, we conclude that the homology groups of a point are given by
A space X is pathwise connected if given x, y function y : [O, 11-X
E
X , there is a continuous
such that y ( 0 ) = x and y (1 ) = y. Note that instead of [0, I] we could have used 0,. Suppose that X is a pathwise connected space, and consider the portion of the singular chain complex of X given by
S , ( X ) .5S,(X)
-
0.
Now S o ( X ) = Z o ( X ) , which may be viewed as the free abelian group generated by the points of X. That is Z o ( X ) = F ( X ) . Hence, an element y of Zo(X) has the form
where the n, are integers, all but finitely many equal to zero. On the other hand, Sl(X) may be viewed as the free abelian group generated by the set of all paths in X . If the vertices of c1are uo and u1 and q5 is a singular 1-simplex in X , then
aq5 = 4(u,) - q5(uo> E ZO(X). Define a homomorphism a : So(X)+ Z by a(C n, x ) = C n,. Note that if X is nonempty, then a is an epimorphism. Since for any singular 1-simplex q5 in X , .(a$) = a(q5(u1) - q5(uo)) = 0, it follows that B o ( X ) is contained in the kernel of a. Conversely, suppose that nlxl . . nkxk G Zo(X)with C ni = 0. Pick any point x E X and note that for each i there is a singular 1-simplex &: u1+ X with ao(q5i)= xi and = x. Taking the singular 1-chain in S , ( X ) we have a(C ni4i) = C nixi - (Z ni)x = C nixi. Therefore, the kernel of GL is contained in Bo(X). This proves that the kernel of a equals Bo(X) and we conclude the following:
+ -+
SINGULAR HOMOLOGY THEORY
9
1.4 PROPOSITION If X is a nonempty pathwise-connected space, then
Let A be a set and suppose that for each ( X E A there is given an abelian group G,. Define an abelian group ~ a c AG, as follows: the elements are all functions f: A - U G, UEA
such that f(a)E G, for each a, and f(a) = 0 for all but finitely many elements IX E A ; the operation is defined by (f g)(a) = f(a) g(a). Setting g, = f(u) E G, we write f = (g,: a E A ) and call the g, the components off. The group G, is the weak direct sum of the Ga's. If the requirement that f ( a ) =: 0 for all but finitely many a is omitted, then the resulting group is the strong direct sum or direct product of the Ga's, denoted naaA G,. Note that if G is an abelian group and {G,}aEA is a family of subgroups of G such that g E G has a unique representation
+
+
c
g =
C g,
with g,
E
G,
=, :I
and g, = 0 for all but finitely many a, then G is isomorphic to Now for each a E A suppose we have a chain complex Ca
Define a chain complex d(c,: a E A )
=
caEEs C" by taking (C C"),
C CPa and setting
(dac,: a E A ) .
1.5 LEMMA Hk(C C") PROOF
=
xbEA G,.
C,
Hk(cI1).
Note that by the definition of the chain complex C Ca we have
1 SINGULAR HOMOLOGY THEORY
10
Let X be a topological space and for x, y a path in X from x to y. It is evident that that is, (1)
E
X, set x
-
y if there exists
is an equivalence relation,
x-x,
(2) x - y (3) x - y
and y - z implies ximplies y - x ,
z,
for all points x, y, and z in X. Such a relation decomposes X into a collection of subsets, the equivalence classes, where x and y are in the same equivalence class if and only if x y. For this specific relation on X the equivalence classes are called the path components of X . Note that if X E X the path component of X containing x is the maximal pathwise-connected subset of X containing x.
-
1.6 PROPOSITION If X is a space and { X a : GI E A } are the path components of X, then H d X ) C Hk(Xa)* J
=A
PROOF There is a natural homomorphism
YJ:
C sk(Xa)
--t
aeA
sk(X)
given by
Since the groups involved are free abelian, Y must be a monomorphism. To observe that !P is also an epimorphism, note first that if
6:
0;G-X
is a singular k-simplex, then +(Q) is contained in some X, because 0, is pathwise connected. Hence, to any such 4 there is associated a unique +a E Sk(Xa) with !P(+a) = 9. Therefore, Y is an isomorphism for each k. Moreover, Y is a chain map between chain complexes so that
Finally, it follows from Lemma 1.5 that
which completes the proof.
0
HOMOTOPY
I2
This proposition establishes the intrinsic “additive” property of singular homology theory. Since the homological properties of a space are completely determined by those of its path components, and the homological properties of any path component are independent of the properties of any other path component, we may restrict our attention to the study of pathwise-connected spaces. Note that it follows from Propositions 1.6 and 1.4 that H,,(X) is a free abelian group whose basis is in a one-to-one correspondence with the path components of X . 1.7 THEOREM Iff: X+ Y is a homeomorphism, then
is an isomorphism for each p. Exercise 3 Prove Theorem 1.7.
0
The fact that this theorem, the topological invariance of the singular homology groups, is quite easy to prove is one of the major advantages of using singular homology theory. 1.8 THEOREM If X is a convex subset of R”,then
H,(X)=O PROOF Assume X
# 0 and let
for p > 0. x E X and
simplex, p 2 0. Then define a singular ( p follows :
+
X Figure 1.3
4:
ap + X be a singular p 1)-simplex 0: crp+l + X as
22
1 SINGULAR HOMOLOGY THEORY
That is, we are setting e(0, t l , . .
. , t,+d
= 4(tl,.
. . , t,+,)
and
O(1, 0,
. . , ,0) = x
and then taking line segments from to to the face opposite to linearly into the corresponding line segment in X (Figure 1.3). This construction is possible since X is convex. From its definition 8 is continuous except possibly a t (1, 0,. . . , 0). To check continuity there we must show that
k)) - (1 - t0)x 1 - to
I/
1 - to
+I
Since +(a,) is compact, (11 qWll(1 - t o ) , . . . 9 fP+l/(l - t o ) ) I x 11) is bounded. Thus, the final limit is zero because limto+l(l - to) = 0, and it follows that O is continuous. It is evident from the construction that do(@ = 4. Since this procedure may be applied to any singular k-simplex, k L 0, there is a unique extension to a homomorphism
T: &(X)
=
(1 - to)(#(',
On the other hand,
1 - to
...,
-
4-1
1-to
3
&+l(X)
ti 0,1-to 9
. . . I
1 - to
HOMOTOPY
Now let
4
23
be any singular k-simplex
So we have constructed a homomorphism T : S,(X) 4Sk+,(X) with the property that dT Ta is the identity homomorphism on Sk(X),whenever k 2 1. Now let z be an element of Z p ( X ) . From the above, for p > 0, (dT Td)z = z. Now since z is a cycle, Taz = 0. Thus, z = a(Tz) and z is in B p ( X ) . This implies that H,(X) = 0 for all p > 0. 0
+
+
The construction used in proving Theorem 1.8 is a special case of a chain homotopy between chain complexes. Suppose C = {Ci ,a} and C’ = { C i , a’} are chain complexes and T : C - C’ is a homomorphism of graded groups of degree one (but not necessarily a chain map). Then consider the homomorphism
d’T+ Ta: C+C’ of degree zero. This will be a chain map because
This chain map (a’T (d’T Now if z
E
+ Ta) induces a homomorphism on homology
+ Ta),:
Zp(C),
Hp(C)+ H,(C’)
for each p .
+ T ~ ) ( z=) ~ ’ T ( z ) (8’T + Ta), is the zero
(a’T which is in Bp(C’). Thus, each p .
homomorphism for
14
1 SINGULAR HOMOLOGY THEORY
Given chain maps f and g: C 4C‘, f and g are chain homotopic if there exists a homomorphism T: C + C’ of degree one with d‘T Td = f - g.
+
1.9 PROPOSITION Iff and g : C + C’ are chain homotopic chain maps, then f* = g , as homomorphisms from H,(C) to H,(C’). PROOF This follows immediately since
if T: C+ C’ is a chain homotopy
between f and g, then
As a special case, suppose that f and g: X induced chain maps
f, and
g,:
SdX)
-
-
Y are maps for which the
W Y )
are chain homotopic. If T is a chain homotopy between f#and g, , then T may be interpreted geometrically in the following way. Let q5 be a singular n-simplex in X . Then T(q5) may be viewed as a continuous deformation of &($) into g#(q5). From Figure 1.4, T(q5) appears as a prism with endsf,($) and g,(q5) and sides T(aq5).Thus, it is reasonable that Wq5) =f#(q5) - g,(q5) - T(d$>, which is the algebraic requirement for T to be a chain homotopy.
Figure 1.4
HOMOTOPY
15
If the chain c = mi+i is an n-cycle in X , then f#(c) and g , ( c ) are n-cycles in Y . T(c) is a collection of integral multiples of such prisms and the algebraic sum of the sides must be zero since ac = 0. Thus, the boundary of T ( c )is the algebraic sum of the ends of the prisms, which isf,(c) - g,(c), so that f#(c) and g , ( c ) are homologous cycles in Y. Given spaces X and Y, two maps fo,f,:X + Y are homotopic if there exists a map F : X x I + Y, I = [0, I], with F(x, 0) = f o ( x ) and F(x, 1) =fl(x), for all x in X . The map F is a homotopy between foand fi. Equivalently a homotopy is a family of maps {A}o5tsl from X to Y varying continuously with t. It is evident that the homotopy relation is an equivalence relation on the set of all maps from X to Y. It is customary to denote by [ X , Y ] the set of homotopy classes of maps. 1.10 THEOREM Iffo,fl: X + Y are homotopic maps, thenf,, as homomorphisms from H , ( X ) to H,( Y ) .
=A,
PROOF The idea of the proof is quite simple: if z is a cycle in X , then the images of z underf, andf, will be cycles in Y. Sincef, may be continuously deformed intof,, the image of z underf, should admit a similar continuous deformation into the image of z under f,.This should imply that the two images are homologous cycles. We now proceed to put these geometric ideas into the current algebraic framework. In view of Proposition 1.9 it will be sufficient to show that the chain maps fo# ,fi#: S , ( X ) + S,( Y ) are chain homotopic. Let
F: X X I + Y be a homotopy between fo and fl.Define maps g ,,g ,:
X
X+XxI
16
1
SINGULAR HOMOLOGY THEORY
Then in the diagram each triangle is commutative, that is, fo = F 0 go and
fi = F o g 1 . Now suppose that go, and g , , are chain homotopic as chain maps from S , ( X ) to S , ( X x I ) . This would mean that there exists a homomorphism T : S,(X)
of degree one with dT
-
S,(XX I )
+ Td = go, - g l , .
Applying F# to both sides gives
Then F,T is a homomorphism from S , ( X ) to S , ( Y ) of degree one and is a chain homotopy between fo, and &#. Therefore, it is sufficient to show that go, and g , , are chain homotopic. the element repreFor a, the standard n-simplex denote by t, E S,(a,) sented by the identity map. Note that if 9: a, -+ Xis any singular n-simplex in X , then the induced homomorphism
4,:
Sda,)
-
S,(x)
has #,(T,) = 4. It is evident that every singular n-simplex in X can be exhibited as the image of z, in this manner. Our technique of proof then will be to first give a construction involving t, and then extend it to all of S,(X) by the above approach. We construct a chain homotopy T between go, and g , , inductively on the dimension of the chain group. To do the inductive step first, suppose that n > 0 and for all spaces X and integers i < n there is a homomorphism
T : S,(X) -+ Si+l(Xx I )
+
such that dT Ti3 = go, - g l , . Assume further that this is natural in the sense that given any map h : X + W of spaces, commutativity holds in the diagram
lh# I
Si(X)
Tx
&+I
( X x I) ( h X Id)#
Si(W )-% Sj+1( wx I ) for all i < n. To define T on the n-chains of X , it is sufficient to define T on the singular n-simplices. So let #: a, .-+ X be a singular n-simplex and recall that r&(t,) = 4. Thus, by defining To,: S,(a,) -+ Sn+l(an x I ) , the naturality of
HOMOTOPY
27
the construction will require that
So to define T, it is sufficient to define To, on S,(u,). and consider the chain in S,(a, Let d be a singular n-simplex in c , ~ given by c = go,(d) - g 1 m - Tun(ad),
x I)
which is defined by the induction hypothesis since ad is in SnPl(an).Note that from the preceding discussion, c corresponds to the boundary of a certain prism in 0,. Then
Thus, c is a cycle of dimension n in the convex set a, x I. From Theorem 1.8 it follows that c is also a boundary. So let b E S,+,(o, x I ) with i3b = c. Geometrically b is the solid prism of which c is the boundary. Then define
Now for any singular n-simplex
4:
on
-
X define, as before,
So defined on the generators there is a unique extension to a homomorphism
This inductive construction indicates the proper definition for T on 0-chains. Recall that uo is a point and consider the chain c in So(uoxI ) given by
Take a singular I-simplex b in uoxI with boundary go,(to) - g,,(t,) and = b. This defines T on 0-chains by the same technique. define TUo(to)
1 SINGULAR HOMOLOGY THEORY
28
Finally it must be noted that in the definition given for Tx on n-chains of
x, aTx
+ Txa
= go,
- g1,
and that the construction is suitably natural with respect to maps h: X Note that if 4 is a singular n-simplex in X ,
-
W.
go,<4> = go###(rn) = (4 x id)#go#(tn)
and similarly g1,(4) = gl#4#(Tn>= (4 x id)#gl#(%)* Now consider
The naturality follows similarly. Therefore, Tx gives a chain homotopy between go, and g,, , and we have completed the proof thatf,, = fi, . 0 Note that this generalizes the approach in Theorem 1.8. There we used the fact that, since X was convex, the identity map was homotopic to the map sending all of X into the point x . Thus in positive dimensions the identity homomorphism and the trivial homomorphism agree, and the positive dimensional homology of X is trivial. Let f: X + Y and g: Y -+X be maps of topological spaces. If the compositions f o g and g of are each homotopic to the respective identity map, then and g are homotopy inverses of each other. A map f : X -,Y is a homotopy equivalence iff has a homotopy inverse; in this case X and Y are said to have the same homotopy type. 1.11
PROPOSITION If f: X+ Y is a homotopy equivalence, then
f,: H,(X)
+Hn(Y)
is an isomorphism for each n.
PROOF If g is a homotopy inverse for f, then by Theorem 1.10 f, o g, = (fog), = identity and g, of* = (g of ), = identity so that g, = f;'
and f, is an isomorphism.
0
EXACT SEQUENCES
19
Suppose that i : A X is the inclusion map of a subspace A of X . A map g : X + A such that g o i is the identity on A is a retraction of X onto A . If furthermore the composition i o g : X - . X is homotopic to the identity, then g is a deformation retraction and A is a deformation retract of X . Note --f
that in this case the inclusion i is a homotopy equivalence. 1.12 COROLLARY If i: A + X is the inclusion of a retract A of X , then i,: H , ( A ) + H,(X) is a monomorphism onto a direct summand. If A is a deformation retract of X , then i , is an isomorphism. PROOF The second statement follows immediately from Proposition 1.11. To prove the first, let g : X + A be a retraction. Then
g,
o
i,
=
(g o i ) , = (id), = identity on
H,(A).
Hence, i, is a monomorphism. Define subgroups of H,(X) by GI = image i, and G, = kernel g,. Let a E G, n G,, so that a = i,(B) for some B E H,(A) and g,(a) = 0. However 0 = g*(u> = g*i*(B) = B
so that a
=
i,(B)
must be zero. On the other hand, let y y = i*g*(y)
E
H , ( X ) . Then
+ (Y - i*g*(y))
-
expresses y as the sum of an element in G, and an element in G,.Therefore, H,(X) G, @ G, and the proof is complete. 0 A triple C A D A E of abelian groups and homomorphisms is exact if image f = kernel g. A sequence of abelian groups and homomorphisms
is exact if each triple is exact. An exact sequence f
g
O+C+D+E+O
is called short exact. This is a generalization of the concept of isomorphism in the sense that I t : GI + G, is an isomorphism if and only if 0-
is exact.
h
GI--* G, + 0
20
1 SINGULAR HOMOLOGY THEORY
Note that in a short exact sequence as above, f is a monomorphism and identifies C with a subgroup C’ G D. Also g is an epimorphism with kernel C‘. Thus up to isomorphism the sequence is just
0
-
C ’ A D A D/C‘
--+
0.
Suppose now that C = {Cn}, D = {Dn}and E = { E n } are chain complexes and 0-C-
f
D - Er - + O
is a short exact sequence where f and g are chain maps of degree zero. Hence, for each p there is an associated triple of homology groups, Hp(C)
- f*
H,(D)
ge
Hp(E).
We now want to examine precisely how this deviates from being short exact. So we are assuming that we have an infinite diagram in which the rows are short exact sequences and each square is commutative.
Let z E Zn(E), that is, z E En and dz = 0. Since g is an epimorphism, there’ exists an element d E D, with g ( d ) = z . From the fact that g is a chain map we have g(dd) = d(g(d)) = a2 = 0.
The exactness implies that dd is in the image o f f , so let c E Cnp1 with f(c) = ad. Note that
f(dc) = df(C)
=
Qdd)
= 0,
and since f is a monomorphism, dc must be zero, and c E Zn-,(C). The correspondence z + c of Zn(E) into Z,-,(C) is not a well-defined function from cycles to cycles due to the number of possible choices in
EXACT SEQUENCES
22
the construction. However, we now show that the associated correspondence on the homology groups is a well-defined homomorphism. Let z, z’ E Z,(E) be homologous cycles. So there exists an element e E En+, with d ( e ) = z - 2’. Let d, d’ E D, with g ( d ) = z, g(d’) = z’, and c, c’ E CnPl with f(c) = ad, f(c’) = dd‘. We must show that c and c’ are homologous cycles. There exists an element a E D,+l with g ( a ) = e. By the commutativity g(da) = dg(a) = de
= z - z’,
so we observe that (d - d ’ ) - da is in the kernel of g , hence also in the image off. Let b E C, withf(b) = (d - d ’ ) - da. Now we have f ( d b ) = df(b)
=
d(d - d’ - da)
=
dd - ad‘
= f(c) - f(c’) = f(c - c’).
Since f is one to one, it follows that c - c’ = db and c and c’ are homologous cycles. Therefore, the correspondence induced on the homology groups is well defined and obviously must be a homomorphism. This homomorphism is denoted by d : H,(E) + H,-,(C) and called the connecting homomorphism for the short exact sequence
0 + C t D+ E + 0. f
1.13 THEOREM If 0 -+ C
f
a
g D+ E + 0 is a short exact sequence of
chain complexes and degree zero chain maps, then the long exact sequence
-
- - J*
* *
H,(D)
Q*
A
H,(E)
H,-l(C)
- -f.
H,-,(D)
1 .
* *
is exact. Exercise 4 Prove Theorem 1.13.
0
It is important to note that the construction of the connecting homomorphism is suitably natural. That is, if O+C-
f
9
D-E+O
is a diagram of chain complexes and degree zero chain maps in which the
22
1 SINGULAR HOMOLOGY THEORY
rows are exact and the rectangles are commutative, then commutativity holds in each rectangle of the associated diagram
- - - la. - I - la* I
* * * +
H,(D)
.B
H,,(E)
g*'
*
..
H,(D')
y*
A
A,
H,,(E')
H,-,(C)
f*
H,,-,(D)
--j
'
* *
f*'
H,-l(c') -'H,,-,(D')
+'''
Let X be a topological space and A G X a subspace. The interior of A (Int A ) is the union of all open subsets of X which are contained in A , or equivalently the maximal subset of A which is open in X . A collection '% of subsets of X is a covering of X if X c U . Given a collection '%, let int '% be the collection of interiors of elements of '%.We will be interested in those '% for which int '% is a covering of X . For '% any covering of X , denote by SnQ(X)the subgroup of S , ( X ) generated by the singular n-simplices 4:u, + X for which #(a,) is contained in some U E '%.Then for each i
uusq
image
ai4
E
image
4
so that the total boundary
So associated with any covering '% of X there is a chain complex S:(X) and the natural inclusion
i: Q ( X )
-
S,(X)
is a chain map. Note that if "c3 is a covering of a space Y andf: X - Y is a map such that for each U E '%, f(U)is contained in some V of 93, then there is a chain map f#: S X X ) S%Y)
-
and f# o ix = iy o f#. We are now ready for the theorem which will serve as the essential computational tool in studying the homology groups of spaces. 1.14 THEOREM If '% is a family of subsets of X such that Int '% is a covering of X , then i,: f W " H X ) ) % ( X )
-
is an isomorphism for each n.
MAYER-VIETORIS SEQUENCE PROOF
See Appendix I.
23
0
The proof is deferred to an appendix to avoid a lengthy interruption of the exposition. It should not be assumed that this implies the proof is either irrelevant or uninteresting. Indeed this argument characterizes the basic difference between homology theory and homotopy theory. Intuitively the approach to proving this theorem is evident. Given a chain c in X we must construct a chain c’ in X such that c‘ is in the image of i and dc = dc’. Moreover, if c is a cycle we will want c’ to be homologous to c. This is done by “subdividing” the chain c repeatedly until the resulting chain is the desired c’. The technique of subdivision is possible in homology theory because an n-simplex may be subdivided into a collection of smaller n-simplices. However, the subdivision of a sphere does not result in a collection of smaller spheres. It is the absence of such a construction, that makes the computation of homotopy groups extremely difficult for spaces as simple as a sphere. To see the requirement that I n t GLI covers X is essential, let X = S’, xo E S’, and ‘h = ( { s o }S’ , - {xo}}.Then any chain c in S,’”(S’) may be uniquely written as the sum of a chain c1 in (xo} and a chain cz in S’ - { x o } . Moreover, since the image of c2 is contained in a compact subset of S’- {x,,}, c will be a cycle if and only if each of c1 and cz are cycles. Now both c1 and cg must then also be boundaries; hence, H , ( S $ ( S ’ ) ) = 0. However, it will soon be shown that H,(S’) Z. The first application of Theorem 1.14 will be the development of a technique for studying the homology of a space X in terms of the homology of the components of a covering GLI of X . In the simplest nontrivial case the covering ‘4 consists of two subsets U and V for which Int U u Int V = X . For convenience let A‘ be the set of all singular n-simplices in U and A” be the set of all singular n-simplices in V. Then
-
S,,(U) = W’), S,JU n V ) = F(A‘ n A ” ) ,
S J V ) = F(A”), SnQ(X)= F(A’
u
A”).
Note that there is a natural homomorphism
h : F(A’) @ F(A”) + F(A’ given by h(Ui‘, Q!‘) I
a,’
u A”)
+ g!1‘.
It is not difficult to see that h is an epimorphism. On the other hand, €here
24
1 SINGULAR HOMOLOGY THEORY
is the homomorphism g : F(A’ n A”) + F(A’) @ F(A”)
given by d b i ) = (bi,
It follows immediately that g is a monomorphism and h 0 g = 0. Now suppose h(C nisi', C mjaj’) = 0. That is
C niq‘ + C mjaj‘ = 0.
Since these are free abelian groups, the only way this can happen is for each nonzero n i , air = uj’ for some j and furthermore mj = -ni. All nonzero coefficients mj must appear in this manner. This implies that all air are in A‘ n A” and if x = C nisi', then C mjaj’ = - x . Hence, x E F(A’
n A”)
and
g(x) =
(C niui’, C mp,!‘).
This proves that the kernel of h is contained in the image of g, and interpreting these facts in terms of the chain groups gives for each n a short exact sequence O
-
S,( U n V)
8%
S,( U)@ S,( V )
- h#
S,‘“‘(X)
0.
Define a chain complex S,(U) @ S,( V) by setting (S,(U) @ S,( V ) ) , = S,(U) @ S,(V) and letting the boundary operator be the usual boundary on each component. Then the above sequence becomes a short exact sequence of chain complexes and degree zero chain maps. By Theorem 1.13 there is associated a long exact sequence of homology groups,
..
-
-- - ..
A
H,(U n V )
A
H,-,(U n V) +
g*
H,(S*(U)
-
0S*(v)) h* Hn(S$(X))
From the definition of the chain complex it is evident that H,(S,(U) @ S,(V)) H , ( U ) @ H,(V), and by Theorem 1.14 we have Hn(S’(X))
Hn(x>*
Incorporating these isomorphisms into the long exact sequence, we have
HOMOLOGY OF SPHERES
established the Mayer- Vietoris sequence
-
...
--
.I
H,(U n V )
A
H,-,(U n V ) --+
--+
0.
H,(U) @ H,(V)
25
h,
H,(X)
a .
Note that if we define by
U
uv
U n V
V=X
V
the respective inclusion maps, then g*(x) = ( i * ( x ) , - j * ( x ) ) and /I&, z ) = k,(y) /*(z). The connecting homomorphism d may be interpreted geometrically as follows: any homology class w in H J X ) may be represented by a cycle c d where c is a chain in U and d is a chain in V. (This follows from Theorem 1.14.) Then d(w) is represented by the cycle ac in U n V. The construction of the Mayer-Vietoris sequence is natural in the sense that if X’ is a space, U’ and V’ are subsets with Int (I’u Int V’ = X‘, and f: X-. X’ is a map for which f(V) E U‘andf(V) c V’, then commutativity holds in each rectangle of the diagram
+
+
...-
..-
-
A
H,(U n V )
1’ H
,
(
(
I
J
~
lr* v
’
)
~
a.
H,(u)
0H J V ) 5 H,(x) 5H,-,((I
1
r, r, Q
h,‘
If*
n v)+.
s.
If*
H,(u’)OH~(V”-H,(X’)~H,-,((I’nV’)-.
Example Let X = S1 and denote by z and z’ the north and south poles, respectively, and by x and y the points on the equator (Figure 1.5). Let U S’ - { z ’ } and V = S1 - { z } . Then in the Mayer-Vietoris sequence associated with this covering we have :
H,(u)
0H,(v) -L H,(P)
A
H,(U n V) 2H,,(u)
0H,(v).
The first term is zero since U and V are contractible. Thus, d is a monomorphism and Hl(S1)will be isomorphic to the image of d = the kernel of g , . An element of H,(U n V ) = 2 @ 2 may be written in the form ax by, where a and b are integers.
+
26
1 SINGULAR HOMOLOGY THEORY
+
Since U and V are pathwise connected, i,(ux by) = 0 if and only if a = -b and similarly for j,. Thus the kernel of g , is the subgroup of H,(U n V ) consisting of all elements of the form ax - uy. This is an infinite cyclic subgroup generated by x - y . Therefore, we conclude that
To give geometrically a generator LC) for this group, we must represent o by the sum of two chains, c d, where c is in U and d is in V, for which d ( c ) = x - y = -ad. The chains c and d may be chosen as shown in Figure 1.6. For any integer n > 1 the portion of the Mayer-Vietoris sequence
+
Hn(W 0Hn(V
- h.
Hn(S)
A
Hn-l(U n V )
has the two end terms equal to zero; hence, Hn(S') = 0. This completes the determination of the homology of S'. We now proceed inductively to compute the homology of S" for each n. Recall that S"
=
{(x,,.
. . ,Xn,.1) I
xi E
w,c xi' = l } G Pf'.
In the usual fashion consider W n c Rn+l as all points of the form (xl,. . . , x,, 0). Under this inclusion Sn-' c Sn as the "equator." Denote by z = (0, . . . , 0, 1) and z1 = (0,. . . , 0, - 1 ) the north and south poles of Sn. Then by stereographic projection Sn - { z } is homeomorphic to Rn, and similarly for Sn - { z ' } . Furthermore, Sn - { z u z'} is homeomorphic to Rn - {origin}.
Figure 1.5
Figure 1.6
APPLICATIONS
27
Exercise 5 Show that Sn-' is a deformation retract of Rn - {origin}. NowletU-Sn-{z},V=Sn{ z ' } s o t h a t U n V = P - {zuz'}. Then by the observations and the exercise above, the Mayer-Vietoris sequence for this covering becomes Hm(R")0H,(Rfi)
- h*
A
Hm(S")
Hnl-*(S"-')B1' H,-,(R,)
0H,-,(R,).
For m > I , the end terms are zero so that d is an isomorphism. For m = 1 and n > 1, g , and A must both be monomorphisms so that H l ( S n ) = 0. This furnishes the inductive step in the proof of the following: 1.15 THEOREM For any integer n 2 0, H , ( S n ) is a free abelian group with two generators, one in dimension zero and one in dimension n. 0 1.16 COROLLARY For n # m,Snand Sm do not have the same homot0PY type. 0
Exercise 6 Using only the tools that we have developed, compute the homology of a two-sphere with two handles (Figure 1.7). Define the n-disk in Rn to be Dn = {(x',.
. . , x,)
E
RnI C X: 5 l }
and note that S"-' G Dn is its boundary. 1.17 COROLLARY There is no retraction of Dn onto Sn-'. For n = 1 this is obvious since D' is connected and So is not. Suppose n > 1 and f:D n+ Sn-' is a map such that f o i = identity, where i is the inclusion of Sn-' in D". This implies that the following diagram of homology groups and induced
PROOF
Figure 1.7
28
1 SINGULAR HOMOLOGY. THEORY
homomorphisms is commutative:
However, this gives a factorization of the identity on an infinite cyclic group through zero which is impossible. Therefore, no such retraction f exists. 0
1.18 COROLLARY
(Brouwer $xed-point
theorem)
Given a map
f: D" -+ Dn,there exists an x in D nwith f ( x ) = x .
-
PROOF Suppose f: Dn D" without fixed points. Define a function g : Dn Sn-l as follows: for x .E Dn there is a well-defined ray starting at f ( x ) and passing through x . Define g ( x ) to be the point at which this ray intersects Sn-' (Figure 1.8). Then g: D n.+ Sn-l is continuous and g ( x ) = x for all x in Sn-'. But the existence of such a map g contradicts Corollary 1.17. Therefore, f must have a fixed point. 0 --f
Exercise 7 Show that Corollary 1.18 implies Corollary 1.17.
-
Let n 2 1 and suppose that f:Sn+ Sn is a map. Choose a generator a of H,,(Sn) Z and note that the homomorphism induced by f on Hn(Sn) hasf,(a) = m a for some integer m. This integer is independent of the choice of the generator since f*(-a) = -f*(a) = -m * a = m . (-a). The integer m is the degree off, denoted d ( f ) . This is often referred to as the Brouwer degree as a result of his efforts in developing this idea. The degree of a map is a direct generalization of the "winding number" associated with a map from the circle into the nonzero complex numbers. +
Figure 1.8
APPLICATIONS
29
The following basic properties of the degree of a map are immediate consequences of our previous results : (a) (b) (c) (d) (e)
-
d(identity) = 1; i f f and g : Sn Sn are maps, d ( f o g) = d ( f ) d(g); d(constant map) = 0; iff and g are homotopic, then d ( f ) = d(g); i f f i s a homotopy equivalence then d ( f ) = f l .
-
A slightly less obvious property (a future exercise) is that there exist maps of any integral degree on Snwhenever n > 0. All these properties are results of homology theory, and as such are easily obtained. A much more sophisticated property is the homotopy theoretic result of Hopf, which is the converse of property (d),if d ( f ) = d(g) thenfand g are homotopic. Thus, the degree is a complete algebraic invariant for studying homotopy classes of maps from Sn to S". 1.19 PROPOSITION Let n
> 0 and define f:Sn
--f
Sn by
Then d ( f ) = -1. First consider the case n = 1 (Figure 1.9). As before let z = (0, I), z'=(O,-l) and x = ( - l , O ) , y = ( l , O ) . Thecovering U = S ' - { z ' } and V = S' - { z } has the property that f(U)G U and f(V ) E V . Thus, by the naturality of the Mayer-Vietoris sequence the diagram PROOF
Figure 1.9
30
1 SINGULAR HOMOLOGY THEORY
has exact rows, and the rectangle commutes wheref, is the restriction off. Recall that a generator a of H , ( S ' ) was represented by the cycle c d where dc = x - y = -ad, and d ( u ) is represented by x - y . Now
+
df*(a)= f 3 * d ( x ) =fa*(x - y ) = y
-x =
-O(a)
=
d(-a).
Since d is a monomorphism, d ( f ) = - 1. Now suppose the conclusion is true in dimension 11 - 1 2 I and consider S"-' 5 S" as before. Taking U and V to be the complements of the south pole and the north pole, respectively, in S", the inclusion i : Sn-'+ U n V
is a homotopy equivalence. Since n 2 2, the connecting homomorphism in the Mayer-Vietoris sequence is an isomorphism. Thus, in the diagram J
H , , ( s ~y )
If*
H,-,(un V )
+& - H,_,(S~-I)
I
f*
each rectangle commutes and the horizontal homomorphisms are isomorphisms. If a is a generator of Hn(Sn), f*(a) = d-'f,,d(a) = d-'i*f*i;'d(a) = - P i *i-,'O(a) = -a.
This gives the inductive step and the proof is complete.
0
For a given map f: Sn + S", n 2 0, there is associated a map + Sn+' called the suspension o f f and denoted by Cf.Intuitively, the idea is that the restriction to the equator (Sn)in Sn+'should be f and each slice in S"+' parallel to the equator should be mapped into the cor-
g : S"+'
c 3
Q _------
/------
_----
Xl Figure 1.10
APPLICATIONS
32
responding slice in the manner prescribed by f (Figure 1.10). Specifically consider S"" s R1I2 R J t i l x R 1so that the points of SJ2+Iare of the form (.Y, i), where .Y E R'*f', r E R' and 1 .Y 1 t l2 = 1. Then define ~~
+
It is not difficult to see that If is continuous and has the desired characteristics. The technique used in proving Proposition 1.19 may be applied to establish the following: 1.20 PROPOSITION Iff: S"+ S", ?I
2 1 is a map, then d ( C f ) d ( f ) . 1
0 Note that if f ( x l ,. . . ,. Y " + ~ ) -- (-s1 ,. . . , x,+J and g(.ul,.. . , x,,+~) (--sl,. . . , s,,,,), then g = C f a n d Proposition 1.19 is a special case of
-~ ~
Proposition 1.20.
1.21 COROLLARY I f f : S"
-
S" is given by
then d( f ) = -1. Let h : S" + S" be the map that exchanges the first coordinate and the ith coordinate. Then h is a homeomorphism (h-l = h), so d(h) = * l . Let g ( x l , . . . ,.Y,+,) = (-&,. . . , x,+~) so that d ( g ) = -1. Then
PROOF
d ( f ) = d ( h o g 0 h ) = d(h)"(g)
=
(*l)*(-1)
=
0
-1.
1.22 COROLLARY The antipodal map A : Sn + S" A ( x l , .. . , x,) = (-xl,. . . , - x , ~ ) has d ( A ) = (-lp+I.
From Corollary 1.21 A is the composition of (n having degree -1. 0
PROOF
defined by
+ 1)-maps, all
-
Exercise 8 Show that for n > 0 and m any integer, there exists a map Sn of degree m.
f: Sn
1.23 PROPOSITION If J g : Sn + Sn are maps with f(x) #g(x) for all x in S", then g is homotopic to A of.
32
1 SINGULAR HOMOLOGY THEORY
Graphically the idea is as follows: since g(x) f f(x), the segment in Rn+l from Af(x) to g(x) does not pass through the origin. Thus, projecting out from the origin onto the sphere yields a path in s" between Af(x) and g(x) (Figure 1.1 1). These are the paths which produce the desired homotopy. In particular we define a function PROOF
which gives the homotopy explicitly.
-
0
1.24 COROLLARY If fi P" Pnis a map, then there exists an x in S2"with f(x) = x or there exists a y in Sm with f(y) = -y.
Iff(x) # x for all x, then by Proposition 1.23fis homotopic to A. On the other hand, iff(x)# -x = A(x) for all x, then f i s homotopic to A o A = identity. When both of these conditions hold, we have
PROOF
d ( A ) = d ( f ) = d(identity). However, d ( A ) = (-l)zn+l = -1 and d(identity) ditions cannot hold simultaneously. 0
=
1, and the two con-
1.25 COROLLARY There is no continuous map f:Pn+ Pnsuch that x andf(x) are orthogonal for all x. 0 Although these ideas have not been defined, Sn is a manifold of dimension n. That is, it is locally homeomorphic to R".As such it has a tangent x ) at each point x in Sn. With Sn identified with the unit sphere space T(Sn,
Figure 1.11
APPLICATIONS
33
in Wn+l, T(Sn,x) is the n-dimensional hyperplane in an+'which is tangent to S" at x (Figure 1.12). We may translate this hyperplane to the origin where it becomes the n-dimensional subspace orthogonal to the vector x. Of course, as x varies over Sn,these subspaces will vary accordingly. A vectorfield on S n is a continuous function assigning to each x in S" a vector in the corresponding linear subspace. A vector field # is nonzero if #(x) # 0 for each x in S". 1.26
COROLLARY There exists no nonzero vector field on S2n.
-
If # is a nonzero vector field on SZn,then y ( x ) = #(x)/ll #(x) 1 is a vector field on S2" of unit length. Thus, y : SZn SZnis a map for which y ( x ) is orthogonal to x for each x. But this is impossible by Corollary 1.25. Hence, no such vector field exists. 0
PROOF
Nonzero vector fields always exist on odd-dimensional spheres. A collection of vector fields 4, ,. . . , #k on S" is linearly independent if for each x in S" the vectors #,(x), . . . ,#,(x) are linearly independent. A famous problem in mathematics is the determination of the maximum number of linearly independent vector fields which exist on SZn+lfor each value of n. The work of Hurwitz and Radon [see Eckmann, 19421 gives a strong positive result; that is, a specific number of linearly independent vector fields (varying with the dimension of the sphere) is shown to exist. The solution of the problem was completed by Adams [ 19621 who showed that these positive results were the best possible. Before proceeding with further applications, we digress in order to introduce some necessary algebraic ideas. A directed set A is a set with a partial order relation 5 such that given elements a and b in A there exists an element c in A with a 5 c and b 5 c. A direct system of sets is a family of sets where A is a directed set, and functions S,b:
Figure 1.12
X,
+x
b
whenever a 5 6,
34
1 SINGULAR HOMOLOGY THEORY
satisfying the following requirements : (i) 2 = identity on Xu for each a in A ; (ii) if a 5 b 5 c, then f," =fbe o f a b . The particular case of interest to us is where the Xu are abelian groups be a direct system of and the f,b are homomorphisms. So let {Xa,fab} abelian groups and homomorphisms. Define a subgroup R of Xu as follows :
xa
xalI there exists a c E A , c 2 ai for all i, and f&:(xu,) i=l
= 0). I
Then the direct limit of the system {Xa,fab} is the group
-
lim Xu =
Xa/R. a
a
Note that if xa is in Xa and xb is in X , , then they will be equal in the direct limit if for some c in A, c 2 a and c 2 b andQ(xa) = fbc(xb). 1.27 LEMMA Let X be a space and denote by { X u } the family of all compact subsets of X, partially ordered by inclusion. Then the family of groups { H * ( X a ) ) forms a direct system where the homomorphisms are induced by the inclusion maps. Then
lim H,(Xa)
+ a
PROOF
-
H,(X).
For each X,, let the homomorphism
be induced by the inclusion map. Then set
Now suppose that
xr=lxai is in R ; that is, there exists a compact subset
Xb C X such that Xat C X b for each i and n
C &*(XaJ
i-1
=0
in H*(x,)-
APPLICATIONS
35
Then from the commutativity of the diagram
it follows that g(C:=l xaf) = 0 and R is contained in the kernel of g. Thus, g induces a homomorphism
g:
c H,(Xa)/R a
=
lim H,(Xa)
------c
+
H,(X).
a
For any homology class x in H,(X), represent x by a cycle C njq5j. Since is compact, q5j(a,) is compact in X for each j . Then the chain C njq5j is q5j(qJ,which is compact since the sum is finite. “supported” on the set Thus (Jq5j(crn) = Xu for some a, gn
uj
j
x
and njdj must represent some homology class xa in H,,(Xa). Moreover, it is evident that gul(xa)= x ; hence, x is in the image of g and g is an epimorphism. Now suppose that C!=l xa, is in H,,(Xu) with g(x.i”=l xaf) = 0. Each xaf may be represented by a cycle Cj in XoI. Then g(zTE1ai) is represented in X by the cycle nij&. Since we have assumed that this cycle bounds, there exists an (n 1)-chain mkvk in X with mkvk) = .&nijq5ij. Once again define a subset of X by
xu
zi,j +
zk
and note that Xb is compact. Since C mkykis an (n ni&, it follows that d ( x mkyk)=
f g:,, ( c nij&)
1=1
is a boundary in
a(x
+ 1)-chain in Xb with s,(xb)
3
and
Thus,
Zy=lxa, is in
R, R
=
kernel of g, and g is an isomorphism.
0
36
1 SINGULAR HOMOLOGY THEORY
1.28 LEMMA If A 5 Sn is a subset with A homeomorphic to I k , 0 5 k 5 n, then
Proceeding by induction on k, if k = 0 then A is a point and Sn - A is homeomorphic to Rnfrom which the conclusion follows. Assume then that the result is true for k < m and let
PROOF
h: A
-
I"'
be a homeomorphism. Split the m-cube Zm into its upper and lower halves by setting
I+ = {(x, ,.. . ,x,)
E
Im(x1 2 0) and I-
=
{(xl ,. . . ,x,) E Im( x1 5 0)
so that If n I- is homeomorphic to P - l . For the corresponding decomposition of A denote by A+ = h-'(Z+) and A- = /+(I-). The set Sn - (A+ n A - ) may be written as the union of two sets (Sn- A + ) u (Sn- A - ) satisfying the requirements of the Mayer-Vietoris sequence. So there is an exact sequence
-
-
Hj+l(Sn - (A+ n A - ) ) -+Hj(P - A ) H ~ ( s ~A+) 0H , . ( s ~- A - ) Hj(Sn- (A+ n A - ) ) . By the inductive hypothesis, for j yields an isomorphism Hj(S" - A )
> 0 the end terms are both zero. This
=
ate6
H j p - A + ) @ Hj(P - A-).
So if X E H j ( S n - A ) and x f 0 , then either i t ( x ) # O or i;(x)f 0. Suppose i:(x) # 0. Now repeat the procedure by splitting Af into two pieces whose intersection is homeomorphic to In this manner a sequence of subsets of S" may be constructed A = Al 2 A , 2 A , z having the property that the inclusion e
Sn - A E S,, - Ak induces a homomorphism on homology taking x into a nonzero element of Hj(Sn- A k ) , and furthermore that Ai is homeomorphic to P-'. Now every compact subset of (S"A i ) will be contained in some (Sn- Ak).Thus the isomorphism of Lemma 1.27 factors through the direct
ni
ni
APPLICATIONS
37
limit lim H j ( S n- A P ) , 4
k
ni ni
so that this direct limit must also be isomorphic to Hj(P Ai).By the construction, the element of this direct limit represented by x is nonzero; however, by the inductive hypothesis the group H j ( S n Ai)= 0. This contradiction implies that no such element x exists and H j ( S n - A ) = 0. For the case j = 0, the Mayer-Vietoris sequence yields a monomorphism rather than an isomorphism. If x and y are points in Sn- A with ( x - y ) # 0 in Ho(Sn- A ) , then the above argument may be duplicated to imply that (x - y ) must be nonzero in H,(S" Ai),a contradiction. 0
ni
1.29 COROLLARY If B C_ P is a subset homeomorphic to Sk for 0 5 k 5 n - 1, then H*(Sn - B ) is a free abelian group with two generators, one in dimension zero and one in dimension n - k - 1. PROOF Once again inducting on k, note that for k = 0, Sk is two points and Sn - B has the homotopy type of Sn-'. Since H,(Sn-') satisfies the description, the result is true for k = 0. Suppose the result is true for k - 1 and write B = B+ u B-, where B+ and B- are homeomorphic to closed hemispheres in Sk and B+ n B- is homeomorphic to Sk-l. The MayerVietoris sequence of the covering
Sn - (B+ n B-)
=
(P- B+) u (P- B-)
has the form
Hj+l(Sn - B + ) @ Hj+l(Sn - B-) 4Hj,,(P - (B+ n B - ) ) 4 Hj(Sn- B ) +
Hj(Sn - B+) @ Hj(Sn - B-).
For j > 0, both of the end terms are zero by Lemma 1.28. The resulting isomorphism furnishes the inductive step necessary to complete the proof. 0 This result may now be applied to prove the following famous theorem. 1.30 THEOREM (Jordan-Brouwer separation theorem) An (n - 1)sphere imbedded in Sn separates Sn into two components and it is the boundary of each component.
1 SINGULAR HOMOLOGY THEORY
38
Let B s S” be the imbedded copy of 9 - l . Then by Corollary 1.29, H,(Sn - B ) is free abelian with two basis elements, both of dimension zero. So S” - B has two path components. B is closed, so S” - B is open and hence locally pathwise connected. This implies that the path components are components. Let C, and C, be the components of S” - B. Since C, u B is closed, the boundary of C, is contained in B. (Here we mean by the boundary of C , , the set dC, = - C,”). The proof will be complete when we show that B c d C , . Let x E B and U be a neighborhood of x in S”. Since B is an imbedded copy of Sn-l, there is a subset K of U n B with x E K and B - K homeomorphic to D”-’(Figure 1.13). Now by Lemma 1.28 H*(Sn - ( B - K ) ) = 2 with generator in dimension zero. Thus, S” - ( B - K ) has one path component. Let p 1 E C,, p , E C, and y a path in S” - ( B - K ) between p 1 and p , . Since C , and C , are distinct path components in S” - B, the path y must intersect K . As a result, K contains points of and c2. We have shown that an arbitrary neighborhood of x contains points of both and hence x is in the boundary of C, and the proof is complete. 0 PROOF
el
el
c,
c,,
One final application is the Brouwer theorem on the invariance of domain.
-
1.31 THEOREM Suppose that U, and U, are subsets of Sn and that h : U , U2is a homeomorphism. Then if U , is open, U2is also open. It should be observed that this is a nontrivial fact. Of course, it is obviously true if “open” is replaced by “closed,” or if the homeomorphism is assumed to be defined over all of S”. This need not be true in spaces in general. For example, let W , = (4,11 and W, = (0, 41 be subsets of [0, 11. If h : W , -+ W , is given by h(x) = x - B, then h is a homeomorphism,
NOTE
Figure 1.13
APPLICATIONS
39
W , is open, but W , is not. It should be evident that there is no extension of h to a homeomorphism of [0, I ] onto itself. PROOF Suppose x, = h ( x , ) is some point in U 2 .Let Vl be a neighborhood of s1 in U , with V , homeomorphic to D nand dV, homeomorphic to Sn-l. Set V2 /z(V,) and denote by aV, = h(dV,), so that d V , is a subset of Sn homeomorphic to S"-' (Figure 1.14). Then by Lemma 1.28 S" - V, is connected, while by Theorem 1.30 S" - d V2has two components. So S" - d V, is the disjoint union of S" - V, and V2 - d V 2 , both of which are connected. Hence, they are the components of S" - d V 2 . This implies that V , - d V, is open, contained in lJ2, and s2E V, - d V 2 . Hence, U , is open. 0 7
Figure 1.14
chapter 2 ATTACHING SPACES WITH MAPS
The purpose of this chapter is to develop the basic theory of CW complexes and their homology groups. An equivalence relation on a topological space is seen to produce a new space whose points are the equivalence classes. This gives a means of attaching one space to another via a mapping from a subspace of the first to the second. The case of particular interest is that of attaching a cell to a space via a map defined on the boundary. This leads naturally to the definition of CW complexes. To serve as tools in the study of these spaces, relative homology groups are introduced and the excision theorem is proved. It is shown that the relative groups of adjacent skeletons produce a finitely generated chain complex whose homology is the homology of the space, and this is applied to compute the homology of real projective spaces. Recall that a relation on a set A is an equivalence relation if the following are satisfied: ‘v
(i) a- a, (ii) a - b - b - a , (iii) a-b, b - c = - u - c , for all a, b, and c in A. Such a relation on A gives a decomposition of A into equivalence classes. On the other hand, a decomposition of A into 40
EQUIVALENCE RELATIONS
42
disjoint subsets defines an equivalence relation on A ( a - b o a and b are in the same subset) under which these subsets are the equivalence classes. Denote by A / - the set of equivalence classes under -. By the quotient function x:A + A / - we mean the function which assigns to a E A the equivalence class containing a. More generally, iff: A -+ B is a function of sets, there is naturally associated an equivalence relation on A . Specifically, a, a, if and only if f(a,) = f ( a , ) . In particular if B = A / - , for some equivalence relation -, and f = x, then we recover the original relation in this way. is an equivalence relation on a topological space X . Now suppose The quotient space XI- may be topologized by defining a subset U 5 X / to be open if and only if n-l(U) is open in X . Note that under this topology, x becomes a continuous function. Since our main interest is in Hausdorff spaces, we will want to restrict our attention to those equivalence relations on a Hausdorff space X for which the quotient space X/- is Hausdorff. For example define an equivalence relation on [- 1, 11 by a -a if I a I < 1 and a a for all a. Then the images of 1 and -1 in the quotient space cannot be separated by mutually disjoint open sets. If X is a topological space define
-
-
-
-
-
D = {(x,x)~ X E X }C X X X
-
the diagonal in X x X . Recall that X is Hausdorff if and only if the diagonal is a closed subset of X x X . Now let be an equivalence relation on X and denote by L3 the diagonal in ( X / - ) x (X/-). Note that the continuous function x x n: X x X + (X/-) x (X/-) has (nx n>-'(d) = ((x, Y ) 1 x Y } .
-
This subset of X x X is the graph of rhe relation. closed if and only if its graph is a closed subset of the above that if X / - is a Hausdorff space, then We now show that the converse is true whenever
-
-
The relation on X is X x X . It is evident from is a closed relation on X . X is compact.
PROPOSITION If is a closed relation on a compact Hausdorff space, then X / - is Hausdorff.
2.1
Recall that a subset of a compact Hausdorff space is closed if and only if it is compact. Denote by p 1 and pz the projection maps of X x X
PROOF
42
2 ATTACHING SPACES WITH MAPS
onto the first and second factors, respectively. Let C be a closed subset of X and G G X x X the graph of -. Then p,(p;'(C) n G ) = { y E XI
y
-
x
for some x E C}
= n-l(n(C)).
Now p;'(C) n G is closed, hence compact, and so p,(p;'(C) n G ) is compact, hence closed. Thus, for any closed C E X , n-'(n(C)) is closed in X ; hence, n(C)is closed in X / - . If R and j j E X / - are distinct points, then they are closed in X / - since they are images of single points in X . Thus, n-'(Z) and n-l(y) are disjoint closed subsets of X . Since X is compact Hausdorff, it is normal, and there exist open sets U,V , in X containing n-'(R) and n-I(jj), respectively, with U n V = 0. Let U' and V' be the complements of U and V, so that n ( U ' ) and n( V ' )are closed subsets of A'/--. Then their complements X / - - n ( U ' ) and X / - - n ( V ' ) are open, disjoint and contain X and j j , respectively. Thus, X / - is Hausdorff. 0
-
Exercise Z (a) Give an example of a closed relation on a Hausdorff space X such that n: A'-+ X / - is not a closed mapping. (b) Give an example of a closed relation on a Hausdorff space X such that X / - is not Hausdorff.
-
If a partial relation -' is given on a space X , it is possible to associate with -' a specific equivalence relation on X . Define an equivalence relation on X by x y if there exists a sequence x,,, . . . ,x,, in X with xo = x , x, = y, and
-
-
(i) xi+l = xi or (ii) xi+'-' xi or (iii) xi -' xi+l
-
for each i. Then is the equivalence relation generated by -'. It is the least equivalence relation that preserves all of the relations from -'. For example, let X = Sn,n 2 1, and define to be the least equivalence relation on S" for which x - x for all x. The graph of in S" x Sn is the union of the diagonal D and the antidiagonal D' = { ( x , - x ) I x E S"}. This is obviously closed; hence, s"/- is a compact Hausdorff space called real projective n-space, RP(n). Suppose A , X, and Y are spaces with A c X and X n Y = 0. Let f: A Y be a continuous function. We consider X v Y as a topological space in which X and Y are both open and closed, carrying their original
-
--f
-
-
ATTACHING CELLS
43
-
topologies. Let be the least equivalence relation on X u Y such that x - f ( x ) for all x E A . The identification space X u Y/- is the space obtained by attaching X to Y via f: A + Y. It is customary to denote X u Y/- by X L J ~ Y.
-
Exercise 2 Suppose in the above that X and Y are Hausdorff spaces and A is closed in X . Then show that is a closed relation. 2.2 COROLLARY If X and Y are compact Hausdorff spaces, A is closed in X and f:A -+ Y is continuous, then X uf Y is a compact Hausdorff space. 0
-
It is not difficult to see that there is a homeomorphic copy of Y sitting in X u, Y . We denote by i: Y X u, Y the homeomorphism onto this subspace; i may be thought of as the composition of the inclusion of Y in X u Y followed by the quotient map n: X u Y + X uf Y. A case of particular importance is when X = Dn and A = Sn-' = dD". The space D" u, Y is called the space obtained by attaching an n-cell to Y viaJ When it may be done without causing confusion, we will denote D" Y by Yj. ~f
-
Example Let X = D2,A = S' = dD2and Y be a copy of S'disjoint from X . Let f: A Y be the standard map of degree two given in complex coordinates by f(eio)= eZi".The identification space X u, Y is then the real projective plane, RP(2). The homology groups of this space may be computed by applying the Mayer-Vietoris sequence. In the interior of D2 pick an open cell U and a
D2 Figure 2.1
44
2 ATTACHING SPACES WITH MAPS
point p contained in U (see Figure 2.1). Setting V = RP(2) - {p}, consider the Mayer-Vietoris sequence of the covering { U, V}. U n V and V both have the homotopy type of S’,whereas U is contractible. In the portion of the sequence given by
~ ( n uV )A H,W)o K ( V ) -L ~ 11
( ~ p ( 2 ) )
1
z
Z
it is easy to check that t9 is an epimorphism. A generating one-cycle in U n V , when retracted out onto the boundary, is wrapped twice around S1 sincefhas degree two. Thus, 01 is a monomorphism onto 22, and H1(RP(2)) 2 / 2 2 = 2,. Moreover, the connecting homomorphism
-
H,(RP(~)) -LH,(U n V ) is a monomorphism whose image is the kernel of 01, so H,(RP(2)) = 0. All higher-dimensional homology groups are easily seen to be zero, and RP(2) is pathwise connected, so its homology is completely determined. The technique used in this example may easily be adapted to prove the following proposition :
2.3 PROPOSITION Iff: Sn-’ then there is an exact sequence
---
-+
Hm(sn-’)
+
Ho(P-’)
-
Y is continuous where Y is Hausdorff,
f*
H,( Y )
i*
Hm( Y,)
Ho( Y ) @ z -+ Ho( Y,).
A
H,-,(sn-’)
-+ *
* *
0
-
This exact sequence shows how closely related are the homology groups i, H,( Y,) of Y and Y,. If an n-cell has been attached to Y, then Hn( Y) is a monomorphism with cokernel either zero or infinite cyclic. In this sense we may have created a new n-dimensional “hole.” On the other
i,
hand, H,-l( Y) If,-,( Y,) is an epimorphism with kernel either zero or cyclic, so the effect of this new w e l l may have been to fill an existing (n - 1)-dimensional “hole” in Y. Away from these dimensions, the addition of an n-cell does not effect the homology. Let (X, A) be a pair of spaces and Y = point. Then there is only one The space X u, Y is then denoted by X / A because map A A Y for A # 0. it can be pictured as the spaced formed from X by collapsing A to a point.
ATTACHING CELLS
45
Note that if X is compact Hausdorff and A is closed in X, then X/A is compact Hausdorff. 2.4 PROPOSITION If X and W are compact Hausdorff spaces and g : X-. W is a continuous function onto W such that for some w,, E W, g-'(w,) is a closed set A c X , and for w # w,,g-'(w) is a single point of X , then W is homeomorphic to X/A. This follows immediately from the following more general fact.
-
2.5 PROPOSITION Suppose X , Y, and Ware compact Hausdorff spaces and A is a closed subset of X.Letf: A Y be continuous and g : X u Y+ W continuous and onto. If for each w E W, g-'(w) is either a single point of X - A or the union of a single point y E Y together with f - l ( y ) in A, then W is homeomorphic to X u,Y. If n: X u Y-. X u, Y is the identification map, g may be factored through n to give a commutative triangle
PROOF
xu, Y where k is induced by g. Then k is one to one and onto by the properties of g. To see that k is continuous, let C be closed in W. Then k-'(C) is closed if and only if n-'k-'(C) is closed. But n-'k-'(C) = g-'(C), which is closed since g is continuous. Since X u, Y and W are compact Hausdorff spaces, k is a homeomorphism. 0 Example Consider Sn-' as the boundary of Dn and let h,: Dn- P-l+ Rn be a homeomorphism. Let z E Sn and set h2: Sn - { z } -+ R" to be the homeomorphism given by stereographic projection. Now define a function
Then checking that g satisfies the hypothesis of Proposition 2.4 with A = Sn-' we conclude that Dn/Sn-l is homeomorphic to S". Thus, SR may be viewed as the space given by attaching an n-cell to a point.
46
2 ATTACHING SPACES WITH MAPS
Many of the spaces which concern algebraic topologists may be constructed in a similar fashion, that is, by repeatedly attaching cells of varying dimensions to a finite set of points. Before giving a formal definition, we consider a number of important examples.
-
-+
Example Recall that RP(n) = S/-, whele is the least equivalence relation on S" having x - -x for all x. Denote by z: Sn RP(n) the quotient map. What space is produced by attaching an (n 1)-cell to RP(n) via n? Regard Sn E Sn+lby identifying ( x l , .. . ,x,+,) E S with (xl ,. . . ,x , + ~,0 ) E Sn+l.This induces an inclusion map i: RP(n) RP(n l), of a closed subset. Write Sn+l as the union of two subsets E,"+' u E!+' which correspond to the upper and lower closed hemispheres, that is, E,"+' n En+' - = Sn. There is a homeomorphism g : Dn+l E,"+'. Denote by fi: Dn+l + RP(n 1) the composition of the maps
-
+
-
+
5 E T + ~E sn+l-.
~ n + l
h
RP(n
+ 11,
where h is the quotient map on S"+'. Thus, we have a mapping of the union Dn+l u RP(n) 3RP(n
+ 1).
It is not difficult to check thatf, u i is onto; in fact,fl is onto. Note that l), f,-'(z) is either a single point of Dn+l - Sn or a pair for z E RP(n {x, -x} in Sn,the latter being true if and only if z lies in the subspace RP(n). Thus, the hypothesis of Proposition 2.5 are satisfied and we conclude that RP(n 1) is homeomorphic to Dn+l u nRP(n), the space given by attaching an (n 1)-cell to RP(n) via z.
+
+ +
Suppose X and Y are topological spaces and xo E X, yo E Y are base points. In X x Y there are the subsets { x o }x Y and X x { y o } . Define X v Y, the wedge of X and Y, to be the union of these two subsets, X x {Yo}
ubO)X
y.
Example Denote by Z the unit interval in R', dZ = (0,1 }. The n-cube
In E Rn
has
So Zm x In = Zm+, and
d P = {(x, ,. . . ,x,)
I
some xi
=0
or 1 }.
47
ATTACHING CELLS
Let z,,, E S'n and z,, E Sn be base points. There exists a map of pairs f : ( I m , dI"') 4 (SnL,z,~),which is a relative homeomorphism; similarly there is a g : (PI, d P ) (Sn,z J . Taking Cartesian products gives a map ---f
f
x
g:
P X
I"-+ S"XS"
To see what happens on d ( I m x I")note that
From the properties off and g , f x g maps this one to one onto (S~~-Zln)X(s"-Zn)=
sn,xsn- ( S m x { z n }u
=s m x s n -sm
v
{Zm}XS")
sn.
Furthermore, f x g maps C ~ ( Ponto + ~ Sm ) v Sn, so by Proposition 2.5, Smx Sn is homeomorphic to the space obtained by attaching an (m n)-
+
cell to Sm v Sn via the map a(p+n)
sm+n-l+
s1n
v
sn.
Smx S" is called a generalized torus.
Example For each integer n identify WZnwith C" and denote the points by ( z l , . . . ,z,,). Then S2J1-1 G CJa is given by
by (zl ,. . . , z I , ) - (zl',. . . ,z , ~ ' )if Define an equivalence relation on P-' and only if there exists a complex number I with I ilI = 1 such that zl' = Az, ,. . . , zJl' = Az,,. This is a closed relation, and the space S2"-'/is denoted CP(n - I), (n - 1)-din?ensionaZ complex projectioe space. [This because its complex dimension is (n - l ) , real dimension (2n - 2).] Recall that in the case of real projective space, a point on the sphere determined a unique real line through the origin and the point was set equivalent to all other points on that line, that is, the antipodal point. In the complex case, a point on the sphere determines a unique complex line through the origin and the point is identified with all other points on that line.
Exercise 3 Let) S.rn-l SBn--l /- = CP(n - 1) be the identification map. Show that the space formed by attaching a 2n-cell to CP(n - 1) via f is homeomorphic to cCP(n). -+
2 ATTACHING SPACES WITH MAPS
48
For the case n = 1, any two points in S1 are equivalent; hence, CP(0) is a point. Now CP(1) is formed by attaching D2to @P(O), which must yield S2.Thus, @P(1) is homeomorphic to S2.A matter of particular interest is the identification map
9- P/-= aY(1)
= 9.
This map
h : SY-+S2 is called the Hopf map and is of particular importance in homotopy theory. Example In the same manner as for the complex number field, we may identify R4 with the division ring of quaternions by ( x l , x2, x 3 , x 4 )-+ x1 ix, j x , kx,. This identifies with Hn,and the sphere
+ +
+
s4n-1
= {(a,,.
-
. . ,a,)
E
HnI C I a11, = I}.
On S4"-' set ( a l , .. . ,a,) (a1',.. . , a,') if there exists a y E H with I y = 1 such that al' = y a l , . . . , a: = ya,. Then S4.-l/- is wP(n - l), (n - 1)-dimensional quaternionic projective space. As before we find that m(0) = pt, HP(1) S4 and WP(n) is the space given by attaching a 4n-cell to HP(n - 1) via the identification map S4,-' -+ HP(n - 1). The identification map h : s7 + E P ( 1 ) = S4 is once again called the Hopf map.
I
-
We now want to compute the homology groups of some of these examples. Leaving the real projective spaces for later in this chapter, first consider the generalized torus Smx Snand assume m,n 2 2. Recall that Smx Sn is given by attaching an (m n)-cell to Sm v 9. Denote by -zm and -z,, the antipodes of the base points Z,E Sm and z,, 6 S" (see Figure 2.2). Define
+
u = S m v s n - - {-z,}
Figure 2.2
and
V = P v S n - {-zm}.
ATTACHING CELLS
49
Then { U , V } gives an open covering of Smv Sn, U admits a deformation retraction onto Sm,and V admits a deformation retraction onto Sn.Finally, note that II n V has the homotopy type of a point. Thus, in the MayerVietoris sequence for this covering we have
Therefore, H*(Sm v Sn) is a free abelian group of rank three having one basis element of dimension zero one of dimension m and one of dimension n. Now by Proposition 2.3 there is an exact sequence * * *
+ Hi(Sm+n-l)
f,
v sn)
__*
--f
H i ( P x Sn) + H&l(Sm+n-l)+
+
+
*
-.
Since m, n 2 2, m n - 1 > m and rn n - 1 > n. It follows that f* is the zero map in positive dimensions. On the other hand, if i = m n, the connecting homomorphism
+
must be an isomorphism. This information may be combined with special arguments for dimensions zero and one to prove the following: 2.6 PROPOSITION H,(Smx Sn), m,n 2 0, is a free abelian group of n. 0 rank four having one basis element of each dimension 0, m,n, and m
+
NOTE
This is our first encounter with a nonspherical homology class. Let
a E Hk(Sk) = Z be a generator. An homology class j3 E H k ( X ) is spherical if there exists a map f: Sk+ X such that f*(a) = 8. Specifically, if
H,(S1 x Sl) is a generator, then j3 is not spherical. Although we are not equipped to prove this at the present time, the basic reason is that j3 is a product of two one-dimensional homology classes, while u E H,(S2) is not. E
-
Next consider complex projective space @P(n). For n H,(@P(O)) H*(pt) and H,(@P(l)) = H,(S2).
= 0,
1 we know
2.7 PROPOSITION
=
{ 0"
for i = 0, 2, 4,. . . , 2 n otherwise.
We proceed by induction on n. From the remarks above, the result is true for n = 0 or 1. So suppose it is true for n - 1 2 1 and recall that CP(n) may be constructed by attaching a 2n-cell to CP(n - 1) via the
PROOF
50
2 ATTACHING SPACES WITH MAPS
identification mapf: exact sequence
. . . -,~
~
(
p
)-
Pn-l+ CP(n
f* 1
- 1). By Proposition 2.3 this yields an
-
j
Hi(cP(n - 1)) -A Hi(eP(n))
A
H&l(P-l)
-
* * *
for i > 0. For strictly algebraic reasons the homomorphism f* must be zero in positive dimensions. So for i > 1, this gives a collection of short exact sequences 0 + H,(CP(n - 1))
-
7Hi(eP(n)) 7Hi&l(S2n-1)0.
These, together with the induction hypothesis, the fact that j , is an epimorphism in dimension one and the fact that CP(n) is pathwise connected, complete the inductive step and the result follows. 0 2.8 PROPOSITION Hi('dP(n))
{ 0"
for i = 0, 4, 8,.. . ,4n otherwise.
PROOF The proof of this is entirely analogous to that for Proposition
2.7.
0
With these examples in mind we now develop some of'the basic properties of spaces constructed in this way. To do so it is necessary to introduce the relative homology groups, a useful generalization initiated by Lefschetz in the 1920s. The concept is entirely analogous to that of the quotient of a group by a subgroup. If A is a subspace of X then we set two chains of X equal modulo A if their difference is a chain in A . In particular a chain in X is a cycle modulo A if its boundary is contained in A . This reflects the structure of X - A and the way that it is attached to A . In a sense, changes in the interior of A , away from its boundary with X - A , should not alter these homology groups. To introduce the necessary homological algebra, let C = { C , l ,a } be a chain complex. D = {D,,, a } is a subcomplex of C if D, c C,, for each n and the boundary operator for D is the restriction of the boundary operator for C. Define the quotient chain complex CID = {CnlDn
9
ar},
where 8 { c } = { a c } for { c } the coset containing c. For convenience the prime will be omitted and all boundary operators will continue to be denoted by a.
RELATIVE HOMOLOGY GROUPS
52
There is a natural short exact sequence of chain complexes and chain maps i
O-tD-CAC/D+O,
where i is the inclusion and n is the projection. From Theorem 1.13 this leads to a long exact sequence of homology groups * *.
+
H,(D)
.2
n*
H,(C)
H,(C/D)
- A
H,-,(D)
i*
* * *.
For clarity denote by { } the equivalence relation in C / D and by ( ) the equivalence relation in homology. To see how the connecting homomorphism A is defined let {c} be a cycle in Z,2(C/D).To determine d(({c})), represent {c} by an element c E C,, having ac E Dn-l. Of course, ac E Z,-,(D) and hence represents a class in H,-,(D). Thus, we have d(({c}))
=
.
More generally, if E c D c C are chain complexes and subcomplexes, there is a short exact sequence of chain complexes and chain maps O+D/E+C/E-+C/D+O.
In the corresponding long exact homology sequence
-
-+
-
H,,(D/E)-t Hn(C/E)
H,-,(D/E)
H,(C/D)
the connecting homomorphism is given by d'(({c))) be viewed as the composition H,(C/D)
A
H,-,(D)
= ({ac)),
---
*
which may
H,-l(D/E).
This is natural in the sense that if E' E D' E C' are chain complexes and subcomplexes and f:C-. C' is a chain map for which f ( 0 ) E D' and f ( E ) c E', then the induced homomorphisms on homology groups give a transformation between the long exact homology sequences in which each rectangle commutes. By a pair of spaces ( X , A ) we mean a space X together with a subspace A c X . If ( X , A ) is a pair of spaces, S , ( A ) may be viewed as a subcomplex of S , ( X ) . The singular chain complex of X mod A is defined by S*(X, A )
=
S*(X)/S*(A).
52
2 ATTACHING SPACES WITH MAPS
The homology of this chain complex, the relative singular homology of X mod A, is thus given by
From the previous observations any pair (X, A) has an exact homology sequence
In this sense H , ( X , A ) is a measure of how far i,: H , ( A ) - + H , ( X ) is from being an isomorphism. That is, i, is an isomorphism of graded groups if and only if H,(X, A) = 0. Thus, we have immediately the following: 2.9 PROPOSITION If (X, A) is a pair for which A is a deformation retract of X, then H,(X, A) = 0. 0
More generally if (X,A, B) is a triple of spaces, that is, B there results a short exact sequence of chain complexes
cA
C X,
0 -+ S*(A, B ) --t S,(X, B ) -+ S,(X, A ) -. 0 which yields the corresponding long exact sequence of relative homology groups. It is conventional to define S,(@) = 0 so that H,(X, 0) = H,(X) and all homology groups may be viewed as groups of pairs. Given pairs (X, A) and (Y, B ) a map of pairs f: ( X , A ) -+ (Y, B) is a continuous function f: X - . Y for which f ( A ) c B. Note that for such a map f,(S,(A)) c S,(B), so that there is associated a homomorphism
which is a chain map, hence also a homomorphism on the relative homology groups. Note that the homomorphisms of degree zero in the exact sequence of a triple are induced by the inclusion maps of pairs. Two maps of pairs f, g : ( X , A ) + ( Y , B) are homotopic as maps of pairs if there exists a map of pairs F: ( X x Z , A x Z ) - + ( Y , B)
such that F(x, 0) = f ( x ) and F(x, 1) = g(x). Note that this says that in continuously deformingf into g , it is required that at each stage we map A into B.
RELATIVE HOMOLOGY GROUPS
53
2.10 THEOREM If f , g : ( X , A ) + ( Y , B ) are homotopic as maps of pairs, then f* = g, as homomorphisms from H,(X, A ) to H,(Y, B).
-
As before define io, i, : ( X , A ) ( X X I, A x I ) by io(x) = (x, 0) and i l ( x ) = ( x , 1) and note that it is sufficient to show that io# and i l , are chain homotopic. Using the same technique as for the absolute case of Theorem 1.10 we construct a natural homomorphism
PROOF
T : S,(X) having
dT
-
S,+,(Xx I )
+ Td = io, - it,
and observe that the restriction of T has T(S,(A)) G S,+,(AxZ). Thus, there is induced the desired chain homotopy
T : S,(X, A )
--+
S,+,(XX I, A X 0.
0
Example To illustrate the difference between maps being absolutely homotopic and homotopic as maps of pairs, consider the following example. Let X = [0, 11, A = (0,l}, and Y = S', B = {l}. Define
by f(x)= e2nizand g ( x ) = 1. Then fand g are maps of pairs ( X , A ) (Y,B) and f and g are absolutely homotopic as maps from X to Y but they are not homotopic as maps of pairs. --+
Exercise 4
(The five lemma) Suppose that
is a diagram of abelian groups and homomorphisms in which the rows are exact and each square is commutative. Then show (i) if fi ,f4are epimorphisms and f 5 is a monomorphism, then f3is an epimorphism ; (ii) if fi ,f4 are monomorphisms and f,is an epimorphism, then f3is a monomorphism.
54
2 ATTACHING SPACES WITH MAPS
Note that, as a special case of this exercise, if fi ,fi,&, and f, are isomorphisms, then f3 is an isomorphism. As pointed out before it seems that those points of A which are not close to the complement of A in X (see Figure 2.3) make no contribution to the relative homology group of the pair ( X , A ) . This property is formally set forth in the following excision theorem.
2.11 THEOREM If ( X , A ) is a pair of spaces and U is a subset of A with 0 contained in the interior of A , then the inclusion map
i : (X - U,A - U)---+ ( X , A )
induces an isomorphism on relative homology groups
i,:
H , ( X - U, A - U)--t H , ( X , A ) .
That is, such a set U may be excised without altering the relative homology groups.
Gu the covering of X given by the two sets X - U and Int A . By assumption their interiors cover X ; thus, their interiors also cover A and we set Gu' to be the covering of A given by { A - U , Int A } . Then by Theorem 1.14 the inclusion homomorphisms of chains PROOF Denote by
i: e ( X ) - + S , ( X )
and
both induce isomorphisms on homology.
Figure 2.3
i': c ' ( A ) + S , ( A )
EXCISION
55
Considering s:'(A) as a subcomplex of e ( X ) there is a chain mapping of chain complexes j:
XW>/Q'(A)
-
S,(X)/S,(A)= S,(X,
4.
The chain mappings i, i', and j give rise to the following diagram of homology groups
Since i*' and i* are isomorphisms, it follows from the five lemma (see Exercise 4) that j , is an isomorphism. Now we can write S$(X) as the sum of two subgroups Q(X)
=
S,(X - U )
+ S,(rnt
A),
but not necessarily as a direct sum. Similarly
Then by elementary group theory
Composing this isomorphism with the chain map j there is induced on homology the desired isomorphism H * ( X - U,A - U )
+
H*(X, A ) .
0
A short exact sequence of abelian groups and homomorphisms
O+A-
f
g
B--+ C-0
is split exact if f ( A ) is a direct summand of B. Exercise 5 Suppose 0 ing are equivalent:
-
A -+ f B A C +0 is short exact. Then the follow-
(i) the sequence is split exact; (ii) there exists a homomorphism B + A with f o f = identity; (iii) there exists a homomorphism 1: C + B with g 0 g = identity.
fi
56
2 ATTACHING SPACES WITH MAPS
Let X be a space and y a single point. Denote by a: X - + y the map of X into y. Then there is the induced homomorphism on homology
a*: H * ( X )
-
H*(y).
Denote the kernel of a* by A,(X). This subgroup of H , ( X ) is the reduced homology group of X . Note that since Hi(y) = 0 for i f 0, & ( X ) = Hi(X) for if 0. Furthermore, if X # 0, then a* is an epimorphism so that Bo(X) is free abelian with one fewer basis element than Ho(X). Note that if f:X-+ Y is a map, then f,: A,(X) + A,(Y). For example, A,(Sn)is free abelian with one basis element in dimension n. 2.12 PROPOSITION If xo E X, then H,(X, xo) = A&'). PROOF In the exact homology sequence of the pair ( X , x , ) the homo-
morphism Hi(xo) H i ( X ) is a monomorphism for each i. Thus, the long sequence breaks up into a collection of short exact sequences: -+
0
--f
Hi(X0)
- i.
H&Y)
j.
-
&(X, xo)
0.
The map a: X + xo induces a,: Hi(X) + Hi(xo)which splits the sequence. Thus there is a homomorphism ,8: H i ( X , xo)+ H , ( X ) with j*,8 = identity. This is then an isomorphism onto the subgroup Ai(X). 0 A subspace A of a space X is a strong deformation retract of X if there exists a map F: X x Z X such that --f
(i) F(x, 0) = x for all x E X ; (ii) F(x, 1 ) E A for all x E X ; (iii) F(a, t ) = a for all a E A and t E I. Let X be the space given by the unit interval together with a family of segments approaching it as pictured in Figure 2.4. If A is the unit Exercise 6
A
Figure 2.4
REDUCED HOMOLOGY GROUPS
57
interval, show that A is a deformation retract of X but not a strong deformation retract. PROPOSITION Let ( X , A ) be a pair in which Xis compact Hausdorff, A is closed in X and A is a strong deformation retract of X . Let n:X + X / A be the identification map and denote by y the point n ( A ) in X / A . Then { y } is a strong deformation retract of X / A .
2.13
Denote by F : X x I -+ X the map given by the fact that A is a strong deformation retract of X . We must exhibit a map P : ( X / A )x I- X / A having F(2,O) = 2, P(2, 1) = y for all 2 E X / A and P ( y , r) = y for all t E I. Thus, it would be sufficient to define a map so that the following diagram is commutative:
PROOF
F
Xx l -
x
So define P = 3t o F o (nxid)-l. To see that this is single valued, let (2,r ) E ( X / A )x I . Then (nx id)-I(Z, t ) is just (x, t ) if x 4 A and is A x { t } if x E A . So if x 4 A , this is obviously single valued. If x E A , note that F(A x { t } ) c A and n ( A ) = y. Hence, P is single valued. To show that P is continuous, let C c X / A be a closed set. Then F-' o n-'(C) is closed in X x I, hence compact. Thus, (nx id) o F-' o n-'(C) is compact in X / A x I, hence closed. Therefore, P is continuous.
THEOREM Let ( X , A ) be a pair with X compact Hausdorff and A closed in X , where A is a strong deformation retract of some closed neighborhood of A in X . Let n:( X , A ) ( X / A , y ) be the identification map. Then n*: H*(X, A ) H * ( X / A , Y ) 2.14
-
-
is an isomorphism. PROOF Let U be a compact neighborhood of A in X which admits a strong deformation retraction onto A (see Figure 2.5). Applying Proposition 2.13
58
2 ATTACHING SPACES WITH MAPS
to the pair ( U , A ) we observe that { y } is a strong deformation retract of n ( U ) . Thus, in the exact sequence of the triple ( X / A , n ( U ) , y),
it follows that H,(n(U), y ) = 0. Hence, the inclusion map of pairs induces an isomorphism
H * ( X / A , u) = H * ( X / A , n ( U ) ) . Recall that since X is compact Hausdorff, it is also normal. Now Int U is an open set containing the closed set A , so there exists an open set V with A E V and B E Int U . Thus, V may be excised from the pair ( X , U ) to induce an isomorphism H * ( X - V, u - V )
-
H*(X, U).
Since A is a strong deformation retract of U, it follows from the exact sequence that
H*(K A )
-
H*V,
w.
These two isomorphisms may be combined to give
H*(X, A ) = H*(X - V, U
- V).
In similar fashion the set n( V ) may be excised from the pair ( X / A , n( U ) ) to give an isomorphism
Figure 2.5
RELATIVE HOMEOMORPHISM THEOREM
59
Now note that since Vis a neighborhood of the set A which was collapsed, the restriction of the map 7c gives a homeomorphism of pairs 7c:
( X - v, I/ - V )
-
( X / A - n ( V ) ,n ( V ) - n(V)),
and so an isomorphism of their homology groups. All of these combine to give the desired isomorphism
COROLLARY If ( X , A ) is a compact Hausdorff pair for which A is a strong deformation retract of some compact neighborhood of A in X , then H*(X, A ) = f i * ( X / A ) . 0 2.15
Iff: (X,A ) 4 ( Y , B ) is a map of pairs such that f maps X - A one to one and onto Y - B, then f is a relative homeomorphism. Under certain conditions on the pairs a relative homeomorphism will induce an isomorphism of relative homology groups. 2.16 THEOREM (Relative homeomorphism theorem) If f: ( X , A ) 4 ( Y , B ) is a relative homeomorphism of compact Hausdorff pairs in which A is a strong deformation retract of some compact neighborhood in X and B is a strong deformation retract of some compact neighborhood in Y, then
f*:
H,(X, A ) + H,(Y, B ) is an isomorphism.
Consider the diagram of spaces and maps, where n and n' are the identification maps and f ' = n' of o 7c-l:
PROOF
X-Y
f
In In* I'
X/A-
Y/B
As in the proof of Proposition 2.13 it is easy to see thatf' is single valued and continuous. Since f is a relative homeomorphism, f ' is one to one and onto. But X / A and Y / E are compact Hausdorff spaces, so f ' is a homeomorphism.
60
2 ATTACHING SPACES WITH MAPS
Denoting xo = n(A) and yo = n’(B) there is the corresponding diagram of relative homology groups and induced homomorphisms : H*(X A )
I
f*
H*( y, B )
n.
I
n,’
By Theorem 2.14 the homomorphisms n* and nk are isomorphisms. Also f*’is an isomorphism since f’is a homeomorphism. Thus
f*: H,(X, A ) + H,( Y, B) is an isomorphism.
0
Exumples (1) There is a relative homeomorphism
f: (Dn, Sn-l)
-
(P, z),
where z is any point in Sn. Both pairs satisfy the hypotheses of Theorem 2.16, so there is an isomorphism
(2) To see that the hypotheses of the theorem are actually necessary, consider the following example. Using the curve sin(1 /x) construct a space as shown in Figure 2.6a, where X is the curve together with those points “inside,” and A is the boundary. Let Y = D2 and B = LID2 = S1(Figure 2.6b). Then (X,A) and (Y, B) are compact Hausdorff pairs. By flattening the pathological part of A it is possible to define a map of pairs f: (A’, A) (Y, B) which is a relative homeomorphism. However, it cannot induce an isomorphism on homology because H&f, A) = 0 and H2(Y,B ) = Z .
-+
Figure 2.6
RELATIVE HOMEOMORPHISM THEOREM
62
The result fails because A is not a strong deformation retract of some compact neighborhood of A in X. The fact that H,(X, A) = 0 is an easy consequence of the exact sequence of the pair (X, A), *
-
*
+
H,(A) + H 2 ( X )
-+
H,(X, A ) + Hl(A) +
-
f
*.
Now X is contractible, so H , ( X ) = 0. On the other hand, if C nidi is a 1-chain in A, the sum must be finite. Since the curve A is not locally connected, the union of the images of these singular simplices cannot bridge the gap in the sin(l/x) curve. Thus, the chain is supported by some contractible subset of A, so that if it is a cycle, it is also a boundary. Therefore, H , ( A ) = 0 and by exactness H2(X,A) = 0. 2.17 LEMMA Let f: Sn-' Y be a map, where Y is a compact Hausdorff space. If Y, is the space obtained by attaching an n-cell to Y via A then Y is a strong deformation retract of some compact neighborhood of Y in Y,. -+
Let U C Dn be the subset given by U = { x E DnI 11 x 1 2 i}, and observe that U is a compact neighborhood of Sn-'in Dn.Define a map F: ( U U Y ) x I + U U Y by PROOF
if X E Y F(x, t) =
(1 - t ) x
+t IxI X
*
if
XE
U.
Then F is continuous, F(x, 0) = x and F(x, 1) E Sn-l u Y for all x , and if x E Sn-' u Y, then F(x, t ) = x for all t. Thus, F is a strong deformation retraction of U u Y onto Sn-' u Y. Now let n : Dn u Y + Y, be the identification map and consider the diagram F
(UU Y)xI-Uu
Y
As before define
F'=noFo(nxid)-'. Then F' is well defined, continuous and gives a strong deformation retraction of the compact neighborhood n( U u Y) of n( Y)onto n( Y). 0
2 ATTACHING SPACES WITH MAPS
62
NOTE
Denote by h the composition incl
Dn-Dn
v Y Z Yf.
-
Then h gives a map of pairs h: (Dn, S"-') (Yf,Y) which is a relative homeomorphism. The hypotheses of Lemma 2.17 and Theorem 2.16 are satisfied, so we may conclude that h,:
H*(Dn, S"-')
-
H*( Yf, Y)
is an isomorphism. Therefore, H , ( Y f , Y ) is a free abelian group on one basis element of dimension n. Suppose that Din,. . . , Dkn is a finite number of disjoint n-cells with boundaries Sy-l,. . . , Sg-l. For each i = 1,. . . , k letfi: ST-l- Y be a map to be the least equivalence relation on into a fixed space Y. Define u Dkn u Y for which xi -fi(xi) whenever xi E Dln v Then Dln v v Dknu Y/- may be denoted Yf,,...,fk, the space obtained by attaching n-cells to Y via fi,. . . ,&. Conversely, if ( X , Y ) is a compact Hausdorff pair for which there exists a relative homeomorphism
-
-
then X i s homeomorphic to Y f l , . . . , fwhereJi k = F Is;-l. A Jinite C W complex is a compact Hausdorff space X and a sequence X o G XI C C X" = X of closed subspaces such that (i) X o is a finite set of points; (ii) Xk is homeomorphic to a space obtained by attaching a finite number of k-cells to Xk-l. Note that Xk- Xk--' is thus homeomorphic to a finite disjoint union of open k-cells, denoted Elk,.. . , EI",. These are the k-cells of X . Using the convention that DO = point and dDo = S-' = 0, the requirements (i) and (ii) may be replaced by the condition that for each k there exist a relative homeomorphism
f: (D,k v
* * *
v DFk, St-' u * * - u Sc)+ (Xk, Xk-').
It is easy to verify that the cells of X have the following properties: (a)
{@ I k = 0, 1,. . . ,n; i = 1,. . . ,rk} is a partition of X into disjoint sets;
FINITE CW COMPLEXES
63
(b) for each k and i the set I?? - E t is contained in the union of all cells of lower dimension; (c) X k = U p r k E,”’; (d) for each i and k there exists a relative homeomorphism 11:
(Dk
Sk-1)
--f
( E .1k 9
E t - E.kz ).
These properties characterize finite CW complexes and will be used as an alternate definition whenever it is convenient. The closed subset Xk is the k-skeleton of X . If X ” = X and X”-l# X , then X is n-dimensional. Example It should be evident that for a given space there may be many different decompositions into cells and skeletons (see Figure 2.7). For example, let X = s?. If z is a point in S2,then S2 may be described as the space obtained by attaching a 2-cell to z. This gives Sza cell structure in which there is one 0-cell and one 2-cell (Figure 2.7a). If z’ is another point in S’and u is a simple path from z to z’, we have a cell structure with two 0-cells, a 1-cell, and a 2-cell (Figure 2.7b). Why was it necessary to include the 1-cell (x when two vertices were used? Further cells may be included as shown in the third figure, in which there are two 0-cells, three I-cells, and three 2-cells (Figure 2.7~).While there is considerable freedom in assigning a cell structure to a finite CW complex, it is apparent that a n y change in the number of cells in a certain dimension dictates some corresponding change in the number of cells in other dimensions. Note that one apparent advantage CW complexes have over simplicia1 complexes is that considerably fewer cells are generally necessary in the decomposition of a complex. 2.18 PROPOSITION If X and Y are finite CW complexes, then X x Y is a finite CW complex in a natural way.
Figure 2.7
2 ATTACHING SPACES WITH MAPS
61
PROOF Suppose the cellular decompositions of X and Y are given by and {Ejl}. The obvious candidate for a cellular decomposition for X x Y is the collection { E t x Eil}. First note that this is a partition of X x Y into a finite number of sets homeomorphic to open cells. Also
{Et}
+
which is contained in the union of all cells of dimension less than k 1. To check the third requirement we may assume that there are relative homeomorphisms
fi
(Ik, dZk) + (&,
,!?t- E t )
and
g:
(Z1, aZl)
+
(,!?it, ,!?!I3
- Ejz).
Then f x g : (Ik+', aZktZ)-+ ( E t x Eil,Ek x Ejt - E t x Ejl) gives the desired relative homeomorphism. 0
Examples (1) Taking the decomposition of S ' into one 0-cell (z) and one 1-cell (a)as in Figure 2.8a, the torus S'xS' is naturally given the decomposition into one 0-cell ( z x z ) , two 1-cells ( z x a and a x z ) and one 2-cell (a x a ) (Figure 2.8b). (2) Recall that RP(0) = pt and RP(k) is obtained by attaching a k-cell to RP(k - 1). Thus, WP(n) is an n-dimensional finite CW complex with one cell in each dimension 0, . . . , n. Moreover, the k-skeleton of WP(n) under this structure is just RP(k). (3) Similarly @P(n)is a finite CW complex of dimension 2n with one cell in each even dimension, 0,2,4, . . . ,2n. Also CP(k) = the 2k-skeleton of @P(n)= the (2k 1)-skeleton of @P(n) for 0 5 k 5 n. An anologous structure may be given to quaternionic projective space.
+
(a)
Figure 2.8
FINITE CW COMPLEXES
65
If X is a finite C W complex with cells {E?}, then a subset A of X is a subcomplex of X if whenever A n E? # 0 then l?? E A . Note that if A is a subcomplex of X , then A is a closed subset of X and inherits a natural CW complex structure.
THEOREM If A is a subcomplex of a finite CW complex X , then A is a strong deformation retract of some compact neighborhood of A in X .
2.19
PROOF Denote by N the number of cells in X - A . We proceed by induction on N . If N 0 the result is trivial and if N = 1 we may adapt the proof of Lemma 2.17 to give the desired result. So suppose the result is true for any finite C W pair ( Y , B ) where the number of cells in Y - B is N - 1. Let Eimbe a cell of maximal dimension in X - A , and define X, = X - Eim.Note that X , must be a finite CW complex since any cell in X - Eimeither lies in A so that its boundary must also lie in A or has dimension less than or equal to m. In either case its boundary does not meet Eim, Moreover, A is a subcomplex of X I . Now the number of cells in X, - A is N - 1, so by the inductive hy. pothesis there exists a compact neighborhood U,of A in X I such that A is a strong deformation retract of U,. There is a relative homeomorphism :
Figure 2.9
66
2 ATTACHING SPACES WITH MAPS
given by the structure of X as a finite CW complex (Figure 2.9). Define the radial projection map r: Dm - (0)+ Sm-'
by d x ) = Xlll 11. Since V , is a compact subset of XI, #-'(V,) is a compact subset of Sm-'. Now define V=
{#WI
x E Drn, 11 x
11 L 4
and r(x) E
#-'(U1>>.
Then V is a compact subset of X which admits a strong deformation retraction onto V , n V. Thus, V u U,is a compact subset of X which admits a strong deformation retraction onto A . We must now make certain that the interior of V u U , contains A . Let y be in the interior of U, in XI. If y is not in V, y must also be in the interior of U , in X , hence also in the interior of V u U,. So suppose y is in V or, in other words, y is in (Eim- Eirn). Now 4-l of the interior of V , in X , is an open subset of Sm-' containing the compact set #-'(y). By the description of V it follows that #-'(y) is contained in the interior of #-I( V ) in Dm.Thus, y must be in the interior of V in Therefore, we have shown that any point in the interior of V , in XI lies in the interior of V u U,in X. So V u U,gives the desired compact neighborhood of A in X . 0
F.
Note that as an immediate consequence of this result, the conclusions in Theorem 2.14, Corollary 2. I5 and the relative homeomorphism theorem, Theorem 2.16, will hold whenever the spaces involved are finite CW pairs. These have some very useful applications. 2.20 PROPOSITION If Xis a finite CW complex and X kis the k-skeleton Xk-') is a free abelian of X , then H j ( X k ,Xk-') = 0 for j f k and Hk(Xk,
group with one basis element for each k-cell of X.
Xk-' is a subcomplex of Xk,so by Theorem 2.19 it is a strong deformation retract of a compact neighborhood in Xk.Since X is a finite CW complex, there is a relative homeomorphism
PROOF
#:
(D,k u
- - - u D,k, Sf-'u
* * *
u St-')+ (Xk, Xk-'),
Then applying Theorem 2.16 yields the desired result from the corresponding fact about H*(Dlk u * * * u D:, St-' u * * * u S!-'). 0
SKELETAL HOMOLOGY
67
For any finite CW complex X define
C,(X)
=
Hk(Xt, Xk--1).
Then C,(X) = C C,(X) is a graded group which is nonzero in only finitely many dimensions, moreover it is free abelian and finitely generated in each dimension. The connecting homomorphism of the triple (Xk, Xk-' X k - 2 defines an operator 3
-
)
a: c,(x) c,-,(x).
Recall that these connecting homomorphisms may be factored in the following way: Hkp2(Xk-3)
where d' and d" are boundary operators for the respective pairs and i and j are inclusions of pairs. So a o a = j , o d" o i, 0 8'. But d" o i* is the composition of two consecutive homomorphisms in the exact sequence of the pair (Xk-', P2) and hence must be zero. Therefore, d o i3 = 0 and {C,(X), a} is a chain complex. Of course, the obvious question is then to ask how the homology of this chain complex is related to the singular homology of X .
2.21 THEOREM If X is a finite CW complex, then H,(C,(X)) = H,(X)
for each k.
NOTE This is an extreme simplification. The chain complex used in defining H , ( X ) was, in general, a free abelian group with an uncountable basis. Here we have reduced the chain complex, not only to a finite basis, but these generators are in one-to-one correspondence with the cells of X .
PROOF
We must analyze the composition Hk+l(Xk+',X k ) 2Hk(Xk, X k - 1 )
and show that kernel d2/image a,
1 :
Ja __*
Hk(X).
H,-,(Xk--l,
Xk-2)
68
2 A’ITACHING SPACES WITH MAPS
First consider the diagram
in which i* and j , are induced by inclusion maps of pairs. The row and the column are exact sequences of triples in which the zeros appear by Proposition 2.20. The triangle commutes by the naturality of the boundary operators. Let x E kernel a,. Then a,i*(x) = 0 and i*(x) = j , ( y ) for some y E Hk(Xk+l, Note that since j , is a monomorphism, this y is uniquely determined. Thus, we define a homomorphism
4:
kernel
a, -+Hk(Xk+’,Xk-,)
by 4(x) = Y . If y’ E Hk(Xk+l, X k - 2 ) then j * ( y r ) is in the image of i* because i* is an epimorphism. So there exists an x r E Hk(Xk, Xk-’)with i*(x‘) = j*(y’). Then a,(x’) = L+,i*(x’) = a, j*(y’) = 0 so x r is in the kernel of a, and #(x’) = y’. We conclude that 4 is an epimorphism. Since i* o a, = 0, it is apparent that the image of a, is contained in the kernel of 4. On the other hand, let X E kernel 8, with # ( x ) = 0. But the fact that j , is a monomorphism implies that i*(x) = 0. Then by exactness x is in the image of 8,. Hence, we have shown that 4 is an epimorphism with kernel given by the image of dl and we conclude that
In the remainder of the proof we show that
Suppose that X is n-dimensional, so that
REAL PROJECTIVE SPACE
69
Consider the sequence of homomorphisms Hk(X) = Hk(Xn, X-1)
+Hk(Xn,
XO) 4
*
-
+
H k ( P , Xk-Z),
each induced by an inclusion of pairs. In general, a homomorphism in this sequence is a part of the exact sequence of a triple
where i 5 k - 2. But by Proposition 2.20 for this range of values of i the first and last group must be zero. Hence, each homomorphism in the sequence is an isomorphism and Hk(X)
-
Hk(Xn, Xk-2).
Similarly the homomorphisms
induced by inclusion maps are all isomorphism, so that
and the proof is complete.
0
A map f:X + Y between finite CW complexes is cellular if f ( X k ) G Y k for each integer k. Although we do not need this at the moment, it is true that any map g : X + Y is homotopic to a cellular map. For a proof of this result see Brown [1968, p. 2611. Iff: X + Y is cellular, then f defines a map of pairs f: ( X k , X k - 1 ) + ( Y k , Yk-1) for each k, and hence a chain mapping
One should check that the homomorphism induced by f * on the homology of the chain complex C , ( X ) corresponds with the homomorphism induced by f on H,(X) under the isomorphism of Theorem 2.21. We now want to compute the homology of RP(n). To do this, give Sn the structure of a finite C W complex so that the k-skeleton is Sk.That is
9 c S'C
... g Sk E ... g sn
70
2 ATTACHING SPACES WITH MAPS
so that there are two cells in each dimension, denoted by E+k and E-k. Similarly give RP(n) the structure of a finite CW complex so that RP(k) is the k-skeleton. Thus RP(0)
c RP(1)
E
* * *
c RP(k)
E
*
'* E
RP(n)
and there is one cell in each dimension. With these structures, the identification map n: Sn+ RP(n) is cellular. By Proposition 2.20 the group C,(RP(n)) is infinite cyclic for 0 5 k 5 n and we denote a generator by ek'. In order to compute the homology of RP(n) we need to know what the boundary operator d: Ck(RP(n)) .+ C,_,(RP(n)) does to the element ek'. To answer this question we first study the situation in S". Recall that the antipodal map of S", A : Sn Sn,is cellular and furthermore maps E+k homeomorphically onto E_k and vice versa for each k . Denote by Fk the composition of maps of pairs
-
Sk--'), then F*k(ik) = ek is a basis If we choose a generator ik of Hk(Dk, Sk-l) = Ck(Sn). We view ek as the basis element correelement in Hk(Sk, sponding to the cell E+k. Since the following diagram commutes (Dk,Sk-1) 5 ( E + k ,
Sk-1)
JA
( E - k , Sk-1)
L5 ( S k , Sk-1) (Sk,Sk-1)
we may take the element A,(e,) to be the basis element corresponding to the cell E-k. Thus, Ck(Sn) is the free abelian group with basis { e k ,A*(ek)}. To determine the boundary operator d: Ck(S") Ck-,(Sn) consider the following diagram:
-
J
REAL PROJECTIVE SPACE
72
in which each triangle and rectangle is commutative. The homomorphism A* in the center has been previously computed, specifically it is multiplication by Starting with e, E Hk(Sk, Sk-I) = C,(S") we have dA,(ek) = i*d'A,(e,)
i,A,d'(e,)
=
= (-l)ki*d'(e,) = ( - l ) k d ( e k ) .
+
Thus, e, (-l)k+1,4*(ek)is a cycle in Ck(S7L). In fact, the set of cycles in C,(Sn)is an infinite cyclic subgroup generated by ex-4-(-I)k+lA*(ek). Before proceeding with the proof, note that this algebraic fact is entirely reasonable from a geometric viewpoint. Since ek and A,(e,) correspond to the upper and lower halves of the sphere Sk, and they are being combined in such a way that the respective boundaries will cancel each other, geometrically we see this generating cycle as the sphere Sk itself. So suppose 0 = d(nlek
+ n,A,(ed)
+ n,dA*ek = n,de, + (- I)kn,dek = (nl + (-l)kn2)de,.
= nldek
Since C,-,(S")is free abelian, it must be true that either dek = 0 or (n, (-l)%,) = 0. Suppose de, = 0. Then also dA,ek = 0 and d: C,(S") -,Ck-,(Sfl)is identically zero. But we have observed that there are nontrivial cycles in C,-,(S"), so if k > 1, these cycles must bound because Hk-,(Sn) = 0 in this range. [It is also easy to see that d: Cl(S") C,(S") cannot be identically zero because every element of C,(Sn) is a cycle.] This contradiction implies that de, f 0 and we conclude that n, (--l),n, = 0 or n2 = (- I)k+lnl. Therefore
+
--f
+
nlek
+ n,A,ek
= nl(ek
+ (--l)k+lA,ek)
as desired. Since H,(Sn) = 0 for 0 < k < n, we must have a(e,+,) = *(ek
+ (-l)k+l&(ed).
Once again, this formula may be shown to hold as well for k = 0. We may as well suppose the sign is f. The identification map n: (Sk, Sk-')+ (RP(k),RP(k - 1)) is a relative homeomorphism on the closure of each k-cell. The generator e,' could have
72
2 ATTACHING SPACES WITH MAPS
been chosen so that e,'
= n,(ek). Then
Therefore, the boundary operator in the chain complex C,(RP(n)) is given by
&i+A
= nd(ek+J
= = n,(ek
=e i
+ (-l)k+lA,ek)
+ (-
={' :
I)k+*ei
for k odd for k even.
This completely determines the boundary operator in the chain complex C,(RP(n)) so that we may apply Theorem 2.21 to conclude the following:
2.22 PROPOSITION The homology groups of real projective space are given by for i = 0 for i odd, 0
2.23 PROPOSITION If (A', A) is a finite CW pair, then H,(X, A) is a finitely generated abelian group. By Corollary 2.15 we know that H , ( X , A) = A , ( X / A ) , and since H , ( X / A ) = l?,(X/A) @ Z, it is sufficient to show that H , ( X / A ) is finitely generated. X / A may be given the structure of a finite CW complex directly from the structure of X and A. The cells of X / A correspond to the cells of X which are not in A together with one 0-cell corresponding to A, thus
PROOF
EULER CHARACTERISTIC
73
dim ( X / A ) 5 dim X . It follows from Theorem 2.21 that H , ( X / A ) is the quotient of a finitely generated abelian group by a subgroup, and is nonzero for only finitely many values of k. Therefore, H , ( X / A ) is finitely generated and the result follows.
For a space X the ith Betti number of X , b,(X), is the rank of H,(X). From Proposition 2.23 we see that if X is a finite CW complex, b,(X) is finite for all i, and nonzero for only finitely many values of i. It was noted previously that b,(X) is the number of path components in X . In a corresponding sense the number b,(X) is a measure of a form of higher-dimensional connectivity of X . The Eufer characteristic of X is given by
x(X)=
c (-I)ib,(X). i
2.24 PROPOSITION If X is a finite CW complex with aicells in dimension i, then (- l>i ai = x ( X ) .
c
PROOF
Exercise 7.
0
Exercise 8 If X and Y are finite CW complexes, show that
chapter 3 THE EILENBERGSTEENROD AXIOMS
Following the necessary algebraic preliminaries, we introduce the homology of a space with coefficients in an arbitrary abelian group. Combined with the results of the previous chapters this establishes the existence of homology theories satisfying the Eilenberg-Steenrod axioms for arbitrary coefficient groups. The corresponding uniqueness theorem is proved in the category of finite CW complexes. Finally, the singular cohomology groups are introduced and shown to satisfy the contravariant analogs of the axioms. If A, B, and C are abelian groups, a mapping
is bilinear (or is a bihomomorphkm) if
Note that if A x B is given the usual product group structure, # will not be a homomorphism except in very special cases. 74
TENSOR PRODUCTS
75
Denote by F ( A x B ) the free abelian group generated by A x B. An element of F ( A x B ) has the form
where the sum is finite, u iE A , bi E B and ni is an integer. Let R ( A x B ) be the subgroup of F(A x B ) generated by elements of the form (a,
+ a 2 ,6 ) - (al,b ) - ( a 2 ,6)
or
(a, bl
+ b 2 ) - (a, 6 , )
-
(a, bd,
where a, a , , a2 E A and b, b,, b, E B. Then the tensor product of A and B is defined to be A @ B = F(A x B ) / R ( Ax B). Note that if 4: A x B -,C is any function, there exists a unique extension of 4 to a homomorphism
4':
F ( A x B ) + C.
Moreover, if 4 is a bihomomorphism, then 4' is zero on the subgroup R ( A x B), so that there is induced a homomorphism
4":
A @ B + C,
which is uniquely determined by 4. This universal property with respect to bilinear maps can be used to characterize the tensor product. There exists a bilinear map t:
AxB-+A@B
defined by taking (a, 6 ) into u @ b, the coset containing (a, b). Given a bihomomorphism 4: A x B C we have seen that there exists a unique homomorphism 4": A @ B + C such that commutativity holds in -+
AxB
dJ
C
A@B
On the other hand, if G is abelian and
t ' :A
x B + G is a bihomomorphism
76
3 THE EILENBERGSTEENROD AXIOMS
whose image generates G, such that any bihomomorphism 4: A x B + C can be lifted through G, then G is isomorphic to A @ B. Since the elements (a, b) generate F(A x B), it follows that the elements a @ b generate A @ B. Note that in A @ B we have
n(a @ b) = (nu)@ b = a @ (nb)
0@ b = 0 = a @ 0
+
(4 4)0(bl
for any integer n ;
for all a and b;
+ bz) = (a1 + a,) 0bl + (4+ ad 0b2 = a,
06,
+ a, @ b l + a, 0b2 + a2 Ob,.
3.1 PROPOSITION There is a unique isomorphism 8: A ~ B - B B A
such that 8(a @ 6 ) = (b @ a ) . Define p : A x B B 0A by p(a, b) = b @ a. This is a welldefined bihomomorphism. Thus, there exists a unique homomorphism 8: A @ B + B @ A with 8(a @ b) = b @ a. Similarly there exists a homomorphism 8’: B @ A -+ A @ B such that 8’(b @ a) = a @ 6. Then the compositions 8 o 8’ and 8’ o 8 are the identity on respective generating sets, and it follows that 8 is an isomorphism with inverse 8’. 0 PROOF
-+
3.2 PROPOSITION Given homomorphisms f:A +A’ and g : B-* B‘, there exists a unique homomorphism
with
Define a mapping p : A x B + A’ @ B‘ by p(a, 6) = f(a) @ g ( b ) , and observe that p is well defined and bilinear. Thus, there exists a unique homomorphism 8: A @ B A’ @ B‘ with 8(t(a, b)) = p ( a , b ) or 8(a @ b) = f ( a ) @ g(b). This 8 is the desired f@ g. 0
PROOF
---f
-
3.3 PROPOSITIONS (a) Iff: A g’ : B‘ B”, then
-+
A‘, f’: A’
-
A” and g : B
-
B‘,
TENSOR PRODUCTS
77
(b) i f A = C A j , then A @ B = C ( A j @ B ) ; (c) if for each j in some index set J there is a homomorphism fj: A ---L A‘ such that for any a E A , f i ( a ) is nonzero for only finitely many values of j , then we can define Cf;: A A’. For any homomorphism g : B B’ it follows that
-
-
(C&)@g =
c (hog);
-
(d) for any abelian group A , 2 @ A A; (e) if A is a free abelian group with basis { a i } and B is a free abelian group with basis { b j } , then A @ B is a free abelian group with basis {ai @ bj 1. We prove only Part (d). Note that Part (e) follows from Parts (d) and (b) and Proposition 3.1. Define p : Z x A + A by p(n, a) = nu. Then ,u is bilinear, so there exists a unique lifting 8 : Z @ A A with O ( n @ a ) = n a. Now define 8 ’ : A -+ Z @ A by 8’(a) = 1 @ a and observe that PROOF
-
-
eeya) = e(i 0 a) = u and
0a) =
=
1
0 na = 0a.
Thus, 8 and 8‘ behave as inverses on generating sets and 8 is an isomorphism. 0 Now suppose that A’ and B‘ are subgroups of A and B, respectively. We want to describe the tensor product AIA‘ @ BIB’. Denote by n1: A + A/A‘ and n,:B + BIB‘ the quotient homomorphisms. Then by Proposition 3.2 there is a homomorphism n1 @ n2: A @ B + A/A‘ @ BIB‘.
If a’ E A‘ and b E B, then n1@ n,(a’ @ 6) = nl(a’) @ n,(b) = 0. Similarly if a E A and b’ E B‘, nl @ nz(a 0b’) = 0. Thus, if we denote by i,:
A’+ A
and
iz: B ‘ - B,
the inclusion homomorphisms H = im (i, @ i d )
+ im (id @ i2) C ker n,@ n2.
This means that n, @ n2 induces a homomorphism @:
A @ BIH
+
A/A‘ @ BIB’.
78
3 THE EILENBERG-STEENROD AXIOMS
We now want to show that @ is an isomorphism. Define a function
!P: A/A' x BIB'
A @ BIH
-+
by Y ( { a } , { b } )= { a @ b } , where { } denotes the respective coset. It is evident that this is well defined, since if a' E A', then !P({a'}, { b } ) = {a' @ b } = 0,
and similarly for 6' E B'. Y is also bilinear, so there exists a unique homomorphism e : AIA' 0BIB' A 0BIH.
-
The homomorphisms @ and 0 are easily seen to be inverses of each other, so we have proved the following.
3.4 PROPOSITION If i,: A' groups, then AIA' @ BIB' =
+
A and i,: B'+ B are inclusions of sub-
A@B im(i, 0id) im(id @ i,)
+
'
0
Example Z , 0 Z, = Z(,,*,, where ( p , q) is the greatest common divisor of p and q. To see this, let ( p , q ) = r so that p = r s, q = r t with (s, t ) = 1. Denote by p Z G Z the subgroup divisible by p and identify Z, = ZlpZ and Zq = Z/qZ. Therefore
z, @ z, = z/pz 0ZlqZ zgz s im(i, 0id)
+ im(id 0i z )
Z
im il =
+ im i,
M
Z
rsZ
+ rtZ
3
ZlrZ
zr = Z(P,*)'
Specifically then Z , @ Z , = Z,,
Z , 0Z ,
= 0,
Z, @ Z,, = Z,,
and so forth.
3.5 PROPOSITION If A A B C + 0 is an exact sequence, then for any abelian group D A @ D . O l d B @ D - CB @ 0 idD + O is exact.
TENSOR PRODUCTS PROOF
79
Define a function
4:
C x D + B @ D/im(a @ i d )
as follows: for (c, d ) E C x D let b E B with @(b)= c. Then set 4(c, d ) = {b 0d } ,
where { } denotes the coset in the quotient group. If b’ is another element of B with p(b’) = c, then b - b‘ E kernel 9, = image a, so there exists an a E A with .(a) = b - b‘. Then note that { b @ d } - {b’ @ d } = {(b - b‘) @ d } =
M a ) 0dl
{ ( a 0id)(a 041 = 0. =
This implies that 4 is independent of the choice of b and so is well defined. Since 4 is also bilinear, there is associated a unique homomorphism
8: C @ D -
BOD im(a @ id)
’
On the other hand, (@@ id) is zero on the image of (a @ id) so we have a homomorphism @ @ i d : B@D/im(a@id)-,C@D. I t is evident that @ @ id and 8 are inverses. This isomorphism establishes the desired exactness. 0 NOTE If we had added a zero to the left of A in Proposition 3.5, the corresponding conclusion would not have been true. That is, in general, tensoring with D does not preserve monomorphisms. For example, let p : Z + 2 be given by p ( n ) = 2 4 so that p is a monomorphism. However
p@id: Z @ Z , + Z @ Z , is zero because ( p @ id)(l @ 1) = (2 @ 1 ) = 1 @ 2 = 0. For this reason we say that tensoring with D is a right exact functor. In trying to measure the extent to which this fails to be left exact, we introduce a useful idea which will be employed in later results.
3 THE EILENBERG-STEENROD AXIOMS
80
Recall that every abelian group A is the homomorphic image of a free abelian group and denote by F the free abelian group generated by the elements of A. Then let x: F + A be the natural epimorphism. If R c F is the kernel of x, then R must be a free abelian group since it is a subgroup of F. (The proof that any subgroup of a free abelian group is free is definitely nontrivial. See Spanier’s book for a proof of this fact.) Thus, there is a short exact sequence
o-+ R LF ~ * A + o . This is an example of afree resolution of the group A. In general, a free resolution of an abelian group A is a short exact sequence 5 o-+G ~--+ A +G O ~A
in which both GIand Goare free abelian groups. Given a free resolution of A and an abelian group D we know by Proposition 3.5 that exactness holds in G1@ D z Go@D %A @ D +O. Then define Tor(A, D) = kernel( j @ id). In a sense, this measures the extent to which j @ id fails to be a monomorphism. Exercise I
(a) Compute Tor(Z,, 2,) for any integers p and q. (b) Show that if A is free abelian, Tor(& A) = 0 for any abelian group B. (c) Show Tor@, B) is independent of the resolution chosen for A.
-
Exercise 2 Show that for any abelian groups A and B,
Tor(A, B)
Tor@, A).
Suppose that C == {Cn, a } is a free chain complex. That is, each C,, is a free abelian group. For any abelian group G define a new chain complex C O G by C @ G = {C, @ G,13 @ id}. It is evident that (a @ id) o (a @ id) = 0. Iff: C -+ C‘ is a chain map, the associated homomorphism faid:
C@G+C’@G
has
(f@ id) o (a @ id) =f o a @id = d’
of@
id = (a’ @ id) o (f@ id)
UNIVERSAL COEFFICIENT THEOREM
81
so that f@ id is also a chain map. Suppose T is a chain homotopy between chain maps fo andf, , that is, d’T
+ Ta = f i -f o .
Then
(8’@ id)(T @ id)
+ (T @ id)(a @ id) = d’T @ id + Td @ id = (d‘T + Td) @ id =
(fi - f o ) O i d
= A m i d -&@id. Hence, T 0 id is a chain homotopy between the chain maps fi @ id and
fo @ id. Now fix the abelian group G. For each pair of spaces ( X , A ) we may use the free chain complex S,(X, A ) to construct a new chain complex S,(X, A ) @ G. This chain complex, denoted S,(X, A ; G), is the singular chain complex of ( X , A ) with coefjcients in G. Since there is a natural isomorphism S*(X A ) 0z = S*V,
4,
we refer to S,(X, A ) as the singular chain complex with integral coefjcients. The homology of S,(X, A ; G) is denoted by H,(X, A ; G). Note that if f:( X , A ) -,(A”, A ‘ ) is a map of pairs, it follows from the preceding comments that there is an induced homomorphism
f*: H,(X, A ; G )
-
H,(X’, A ’ ; G).
In some applications it is desirable to have additional structures on these homology groups. For example suppose that R is an associative ring and G is a right R-module. Then S,(X, A ; G) may easily be given the structure of a right R-module in such a way that the boundary operators and induced homomorphisms are all homomorphisms of R-modules. In particular, if R is a field, then each H,(X, A ; G) is a vector space over R. Note that for any R we know that R is a free module over itself, so that S,(X, A ; R) is the free R-module generated by the singular n-simplices of X mod A . Suppose that ( X , A , B ) is a triple of spaces. We have observed previously that there is a short exact sequence of chain maps
0
-
S*(A, B)
-
S*(X, B )
-
S*(X, A )
--+
0.
82
3 THE EILENBERG-STEENROD AXIOMS
Since each chain complex is free, it follows from Exercises 1 and 2 that the exactness is preserved when we tensor throughout with G. Thus
0 + S*(A, B ; G ) + S,(X, B ; G )
-
S,(X, A ; G)+ 0
is a short exact sequence of chain complexes and chain maps. There results the long exact sequence of the triple (A', A, B) for homology with coefficients in G. As in the case of integral coefficients it is easy to show that
{ 0"
for n = 0 otherwise.
Returning to the general case of a free chain complex C, we now consider the problem of relating the homology of C @ G to the homology of C. For example, suppose that x E C, such that p x is a boundary for some integer p . Since C is free, this implies that x must be a cycle, so x represents a homology class of order p. Let b E C,+, with db = p x . Note that b is not a cycle unless x = 0. If we now tensor C with G = Z , , the element b @ 1 in C,,, @ Z, has (8 @ id)(b @ 1)
=
db @ 1 = p x @ 1 = x @ p
= 0.
Thus, b @ 1 is a cycle in C,,, @ 2,,where b had not been a cycle previously. In this way we see how torsion common to H,(C) and G produces new homology classes in H,+,(C @ G). Before proceeding we note the easily proved algebraic fact that if f: G G' and g: G' --+ G are homomorphisms of abelian groups with g of = identity, then --f
G'
= im
f @ ker g.
As usual we denote by Bn G Z , c_ C, the subgroups of boundaries and cycles, respectively. If C is a free chain complex, then each B, and 2, will be a free abelian group. Fix an abelian group G and consider the short exact sequence
0+Z ,
+
C,
a
Y
Bn-l
+
0.
First note that since B,-, is free, the sequence splits. That is, if { x i } is a basis for Bn-, , for each i there exists an element ci E C, with aci = xi. Define y ( x J = ci and note that y extends uniquely to a homomorphism which splits the sequence.
UNIVERSAL COEFFICIENT THEOREM
83
Now since BnPl is free, Tor(B,-,, G ) = 0 and the short exact sequence is preserved when tensored with G ,
0 --+ 2, @ G - +
c, @ G
a 6 id
m B,-, @ G
-+
y 6 id
0.
This sequence is also split by the homomorphism y @ id. On the other hand, the short exact sequence
0
-+
B,
A Z,
-+
H,(C)
-+
0
is a free resolution of H,(C); hence, it yields the exact sequence 0
-
Tor(H,(C), G ) 2 B, @ G
j6id
Z, @ G --t H,(C) @ G
-+
We want to compute the homology of C @ G, given by kernel image 8, @ id in the following diagram: B, @ G
0.
a @ id/
-
a,Bid
C,,, @ G
j 6 id
Z, @ G
-I k 6 id
ai @ i d
C, @ G
a6id
Cnd1@ G
Note that by the above remarks both i @ id and k @ id are monomorphisms and a, @ id is an epimorphism. Thus, kernel d @ id = kernel 8, @ id and image a, @ id may be identified with image j @ id. Now consider the groups and homomorphisms
where the horizontal row is exact. As we have observed, the cycle group in C, @ G is the kernel of ( j @ id) o (a, @id). Since 8, @ id is an epimorphism and kernel j @ id = image g, we have
84
3 THE EILENBERG-STEENROD AXIOMS
Thus, there are homomorphisms
(830id)-'g(Tor(Hn-,(C), GI)
a, 0 id y 0 Id
g(Tor(Hn-,(C), GI)
for which the composition (a, @ id) o (y @ id) is the identity. Combining these observations we have the cycle group expressed as a direct sum ker(j @ id) o (a, @ id) = ker(8, @ id) @ (y @ id)g(Tor(H,-,(C), G ) ) . Note that the first direct summand may be identified with Z , @ G, while in the second, both g and y @ id are monomorphisms. Thus, we may identify the groups of cycles in C, @ G with the direct sum Z, @ G 0Tor(H,-,(C), G).
Furthermore, the group of boundaries, which has been identified with the image of B, @ G-* Z , @ G , is contained entirely in the first summand. Finally, recalling the exactness of
B, @ G - + Z, @ G - + H,(C) @ G - 0,
-
we conclude that the homology of the chain complex C @ G is given by H,(C @ G) H,(C) @ G @ Tor(H,-,(C), G). This completes the proof of the universal coef$cient theorem :
3.6 THEOREM If C is a free chain complex and G is an abelian group, then 0 H,(C @ G) H J C ) 0G 0Tor(H,-,(C), G).
-
3.7 COROLLARY For any pair of spaces (X,A),
Example Recall that the integral homology groups of real projective space are given by Z for k = 0 or for k odd and = n 2, for k odd, 0 < k < n 0 otherwise.
EILENBERG-STEENROD AXIOMS
85
Applying Corollary 3.7 to compute H,(RP(n); Z,) we first note that Tor(&, Z,) =is Z, and Tor(Z, Z,) = 0 were results from an earlier exercise. We are thus able to conclude that
where H,(RP(n); Z,) results from H,(RP(n)) @ 2, for k = 0 or for k odd, 0 < k 5 n, and Hk(RP(n);Z,) results from two-torsion in Hk-,(RP(n)) for k even 0 < k 5 n. We are now in a position to characterize singular homology in terms of a set of axioms. Each of these axioms has been established previously as an intrinsic property of singular homology theory. Our main purpose here is to show that when restricted to a suitable category of spaces and maps, these axioms uniquely determine a homology theory. The formulation of the axioms and the proof of the uniqueness are due to Eilenberg and Steenrod [1952]. Suppose X is a function assigning to each pair of spaces ( X , A ) and integer n an abelian group X,(X, A ) , and to each map of pairs f:(X,A ) ( Y , B ) a homomorphism f*:SC,(X, A ) -+ X,(Y, B). Suppose further that for each n there is a homomorphism d : X,(X, A ) -+ X , - , ( A ) . This operation gives a homology theory if the following axioms are satisfied:
-
(1)
if id: (X,A ) -+ ( X , A ) is the identity map, then id,:
X,(X, A ) + %,(A’, A )
-.
is the identity homomorphism; (2) i f f : ( X , A ) + (Y, A r ) , g : ( X ’ , A ’ ) (X”, A ” ) are maps of pairs, then ( g of)* = g* of*; (3)
iff: (X,A ) -+ ( Y , B ) is a map of pairs, then 8 of*
(4)
=
( I J n ) * a; O
if i: ( A , 0) + ( X , 0) and j : ( X , 0) + (X, A ) are inclusion maps, then the following sequence is exact:
.
- - a
* *
X,(A)
-
i*
X,(X)
j*
X,(X, A )
a
X,-,(A)
--
( 5 ) iff, g : ( X , A ) ( Y , B ) are homotopic as maps of pairs, thenf, as homomorphisms;
*
*
;
=g ,
86
3 THE EILENBERG-STEENROD AXIOMS
-
-
if (X,A) is a pair and U E A has r7 E Int A, then the inclusion map i : ( X - U , A-U) ( X , A ) has i,: 3C,(X-U, A - U ) X,(X, A) an isomorphism; (7) X,(pt) = 0 for n # 0 [X,(pt) is called the coefJicient group].
(6)
3.8 THEOREM (Existence) Given any abelian group G, there exists a homology theory with coefficient group G. Let X , ( X , A ) = H,(X,A ; G), singular homology with coefficients in G. Then each of the axioms has been proved previously. 0
PROOF
3.9 THEOREM (Uniqueness) On the category of finite CW pairs and maps of pairs, homology theories are determined, up to isomorphism, by their coefficient groups. That is, if X, and X,‘ are homology theories and h: 3C, X*‘ is a natural transformation (that is, it commutes with induced homomorphisms and boundary operators) such that h: X,,(pt) + X,,’(pt) X,’(X, A) is an isomorphism for is an isomorphism, then h: X,(X, A) each integer n and each finite CW pair (X,A).
-
-
First note that the proofs of Theorems 2.14 and 2.16 (the relative homeomorphism theorem) only require that singular homology theory satisfy these axioms. So the analogs of these results will hold for any honiology theory. Denote the zero-sphere So as the union of two points SO = x u y , and consider the diagram
PROOF
%(Y) 4
Xk‘cv)
i,
Xk(X
i,
Xk/(X
uY,XI uY,x)
which commutes by the naturality of h. The horizontal maps are excision maps, so both horizontal homomorphisms are isomorphisms by Axiom 6. Since the first vertical homomorphism is an isomorphism by the hypothesis, we conclude that h : Xk(S0, x ) -+ X,’(S0, x ) is an isomorphism for each k. Now consider the diagram
EILENBERG-STEENROD AXIOMS
87
where the rows are exact by Axiom 4. By the five lemma (Exercise 4, Chapter 2), h : 3Ck(So)43Ck’(So)is an isomorphism. We now prove inductively that h is an isomorphism for spheres of all dimensions. Suppose It: 3Ck(SI1-l) 3Ck’(Sf’-1) is an isomorphism, n > 0. The n-disk Dn has the homotopy type of a point, so by using Axiom 5 we have an isomorphism h : 3Ck(Dn) 3Ck’(D”).In the commutative diagram
-
the horizontal homomorphisms are isomorphisms since they are induced by relative homeomorphisms, while the vertical homomorphism on the left is an isomorphism by the five lemma (Exercise 4, Chapter 2). So we conclude that h : 3Ck(S”,x) -+ 3Ck’(Sn,x) is an isomorphism, and again apply the five lemma to see that h : 3Ck(Sn)+ 3Ck’(Sn)is an isomorphism. This completes the inductive step. We are now ready to prove the theorem by inducting on the number of cells in the finite CW complex, X . Of course the conclusion is true if X has only one cell, so suppose that h is an isomorphism for all complexes having less than m cells. Let X be a finite CW complex containing m cells. If dim X = n, pick a specific n-cell of X and denote by A the complement of this top dimensional cell. Then A is a subcomplex of X having m - 1 cells and there is a relative homeomorphism 57:
(D, P-1)
-+
(X, A).
In the commutative diagram
the horizontal homomorphisms are induced by relative homeomorphisms, so they are isomorphisms. The first vertical homomorphism is an isomorphism by the inductive argument above. Hence h : %AX, A ) + SC,’(X, A )
88
3 THE EILENBERG-STEENROD AXIOMS
is an isomorphism for each k. Finally, the five lemma together with the inductive hypothesis imply that
h : XlC,(X)-+ X,’(X) is an isomorphism. This establishes the theorem for any finite CW complex, and the similar result for pairs follows by another application of the five lemma. 0 During recent years, many theories have been developed which satisfy all of the axioms except Axiom 7. These have been called “generalized homology theories” and include stable homotopy, various K-theories, and bordism theories. Some of these theories are able to detect invariants which cannot be detected by ordinary homology. As a result, problems have been solved by these techniques whose solutions in terms of singular homology were either extremely difficult or impossible. Certainly any thorough study of modern methods in algebraic topology should include a significant segment on generalized homology and cohomology theories.
NOTE
We now want to introduce singular cohomology theory. If A and G are abelian groups, denote by Hom(A, G ) the abelian group of homomorphisms from A to G, where (f+ g)(a) = f ( a ) g(a) for each a in A . If 4: A B is a homomorphism, there is an induced homomorphism
+
4”:
-
Hom(B, G) -+ Hom(A, G)
defined by qP(f)= f o 4. Note that if y : B + C is a homomorphism, then ( y 0 4)* = ## 0 y#. For a chain complex {Cn, a } and an abelian group G, define abelian groups Cn = Hom(C,, G). Then the boundary operator
-
a: C,,, C, has a#: C”+ c n + 1
and the composition d# o a# = (a o a)# = 0. So this resembles a chain complex except that the indices are increased rather than decreased. This ,leads us to define a cochain complex to be a collection of abelian groups and homomorphisms {C”, S} where S: C n+ Cnfl and 6 o S = 0. The homomorphism 6 is the coboundary operator. Note that if {C”,S} is a cochain complex and we define D,L= C-” and 8 = 6: D,, + D,-l, then { D , , a } becomes a chain complex. So the two
SINGULAR COHOMOLOGY
89
notions are precisely dual to each other and the use of cochain complexes is mainly a convenience. The basic definitions for chain complexes may be duplicated for cochain complexes. If {Pi, S} and {On,S'} are cochain complexes, a cochain map f of degree k is a collection of homomorphisms
such that . f o 6 = 6' of. Two cochain maps f and g of degree zero are cochain homotopic if there is a collection of homomorphisms
+
such that 6'T TS = f - g. T is a cochain homoropy. Note that if {C,,, a } and { O n ,a'} are chain complexes andf, g : {Cn,a } { D J Ja,' } are chain homotopic chain maps, then for any abelian group G the cochain maps
-
f #, g # : {Horn(&, G), a'#}
-
{Hom(C,, G ) , a#}
are cochain homotopic. Let C = {C", S} be a cochain complex and define Z"(C) = kernel 8: C n+ C',+l, the group of n-cocyc!es, and B"(C) = image 6: Cn-' C", the group of n-coboundaries. The nth cohomology group of C is then the quotient group H n ( C ) = Zn(C)/Bn(C).
-
If A
=
(A"}, B
=
( B n } , and C = { C n }are cochain complexes and
is a short exact sequence of cochain maps of degree zero, then there exists a long exact sequence of cohomology groups
. . . + H"(A) + H y B )
--t
A H y C )d H"+'(A)
--f
-
*
a ,
where the connecting homomorphism A is defined in a fashion analogous to the connecting homomorphism for homology. Now let ( X , A ) be a pair of spaces and G be an abelian group. Define Sn(X,A ; G ) = Hom(S,(X, A ) , G )
90
3 THE EILENBERG-STEENROD AXIOMS
as the n-dimensional cochain group of ( X , A ) with coefficients in G . Let
6: P(X, A ; G ) + P+l(X, A ; G ) be given by 6 = d#. This defines the singular cochain complex of (X, A) whose homolob is the graded group
H*(X, A ; G) as the singular cohomology group of ( X , A ) with coefficients in G . Each of the covariant properties of singular homology becomes a contravariant property of singular cohomology. In particular, iffi (A’, A ) + (Y, B) is a map of pairs, than there is induced a homomorphism
f *: H*(Y, B ; G ) + H*(X, A ; G ) . If g : ( Y , B) + (W, C) is another map of pairs, then ( g f ) * = f* o g*. Let o+FLH-I~,K+o be a short exact sequence of abelian groups and homomorphism which is split by a homomorphism y : H + F. If G is an abelian group and f is a nonzero element of Hom(K, G), then n # ( f )= f o n is a nonzero element of Hom(H, G) since n is an epimorphism. It is evident that i# o n # = (n o i)” = 0. On the other hand, let h: H + G be a homomorphism such that P ( h ) = h o i = 0. Since h is zero on the image of i = kernel of n, h may be factored through K. The resulting homomorphism h: K+ G will have n#(h)= h. Thus, the kernel of i# is equal to the image of n#. Finally, since y o i is the identity on F, ( y o i)# is the identity on Hom(F, G) But this implies that i# is an epimorphism. Therefore, we have completed the proof of the following. 3.10 PROPOSITION If 0 + FA H-3, K-+ 0 is a split exact sequence and G is an abelian group, then n#
0 -+ Hom(K, G) -+ Hom(H, G) is exact.
i#
Hom(F, G) + 0
0
For example, if (X,A) is a pair of spaces, the sequence
0 + S*(A) + S * ( X ) + S*(X, A ) + 0
SINGULAR COHOMOLOGY
92
is split exact since S,(X, A) is a free chain complex. Thus, by Proposition 3.10
0 + S*(X, A ; G) + S*(X; G) + S * ( A ; G) + 0 is a short exact sequence of cochain complexes and cochain maps. By the previous remarks, this produces a long exact sequence in singular cohomology,
It is important to note the necessity of the hypothesis in Proposition 3.10 that the original sequence be split exact. For example, if i : Z + Z is the monomorphism given by i(1) = 2, then i # : Hom(Z, 2,)
+
Hom(Z, 2,)
is zero and thus fails to be an epimorphism. The other conclusions of exactness will hold in general since they were established without using the fact that the sequence was split. As in the case of the tensor product, this failure to preserve short exact sequences may be measured. Let E be an abelian group and take a free resolution of E, '
n
O+R>F-E+O.
Then for any abelian group G the sequence n#
0 + Hom(E, G) --+
Hom(F, G)
i#
Hom(R, G)
is exact by the proof of Proposition 3.10. Define Ext(E, G)
= coker
i# = Hom(R, G)/im i#.
The basic properties of Ext are dual to those of Tor and may be established in the following exercises: Exercise 3 (a) If E is free abelian, Ext(E, C ) = 0. (b) Ext(E, G) is independent of the choice of the resolution for E. (c) Ext(E, G) is contravariant in E and covariant in G ; that is, given homomorphisms f:E + E' and h : G + G' there are induced homomorphisms f * : Ext(E', G) + Ext(E, G) and h,: Ext(E, G) -+Ext(E, G').
3 THE EILENBERGSTEENROD AXIOMS
92
-
k
j
(d) If 0 holds in
C-+ 0 is a short exact sequence, then exactness
A -+ B-
0 + Hom(C, C)
k*
Hom(B, C)
h*
Ext(B, C)
-
-
j#
j*
Hom(A, G)
Ext(A, C)
--f
Ext(C, C)
0.
Since we have defined P ( X , A ; C) = Hom(S,(X, A), C), it is useful to adopt the following notation: if 4 is in P ( X , A ; C) and c is in S,(X, A), then the value of 4 on c is the element of G denoted by (4, c). Note that this pairing is bilinear in the sense that (41
+
and c1
($9
$2,
c> = ($*, c>
+ c,>
=
<$, c,>
+
($2,
c>
+ (4, 4.
In particular, for any integer n we have (4, nc)
= (n4, c).
Thus the pairing
produces a homomorphism P ( X , A ; G ) @ S,(X, A ) + G
for each n. This homomorphism will be studied in more detail in the next chapter. In this notation the boundary and coboundary operators are adjoint; that is,
(4, ac> == <s4, c>. (In a somewhat different setting this is called the fundamental theorem of calculus.) Furthermore, if f: (X, A) + (Y, B ) is a map of pairs, 4 E P ( Y , B; G) and c E S,(X, A), then
The cochain
4 E Sn(X, A ; C) is a cocycle if and only if (84,c ’ )
=0
for all c’ E S,+,(X, A ) ,
or equivalently, if
(4, aci> = 0. Thus, 4 is a cocycle if and only if 4 annihilates B,(X, A).
RELATED TOPICS
On the other hand, suppose that Then
4‘ E Sn-l(X, A ; G).
4 = 84‘
(4, c> = <4‘, c> =
(4’9
93
is a coboundary, where
ac>
so that if 4 is a coboundary, then 4 annihilates Z,(X, A ) . Now let x E H n ( X , A ; G) be represented by a cocycle 4 and y E Hn(X, A ) be represented by a cycle c. Then we define a pairing ( , ): H n ( X , A ; G) 0H,,(X, A )
--f
G
by ( x , y ) = (4, c). To see that this is well defined, let 4 be other choices for representatives of x and y. Then
+ 84‘ and c + dc’
This pairing is called the Kronecker index and may be viewed as a homomorphism a : Hn(X, A ; G ) -+Hom(H,(X, A ) , G). For example, consider the zero-dimensional cohomology of a space X . So(X; G) = Hom(So(X), G) and So(X) may be identified with the free abelian group generated by the points of X . Since any homomorphism defined on So(X) is determined by its value on the basis, we may identify So(X; G) with the set of all functions from X to G. Of course, B o ( X ;G ) = 0, so the cohomology may be determined by identifying the group of 0-cocycles. Note that 4 will be a 0-cocycle if and only if (84, c) = (4, d c ) = 0 for all c in S , ( X ) . This will be true if and only if $(a(l)) = $(a(O)) for every path (T in X , Therefore, we have identified H o ( X ; G) = Z o ( X ;G ) with the set of all functions from X to G which are constant on the path components of X . From this description we have the following. 3.11 PROPOSITION If X is a topological space and {Xa}acAis the de-
composition of X into its path components, then
H o ( X ;G ) =
n
G,,
a€A
the direct product of copies of G, one for each path component of X .
0
94
3 THE EILENBERG-STEENROD AXIOMS
There is a natural embedding of G in H o ( X ; G) defined by sending g E G into the function from X to G having constant value g. If p is a point and iz: X - p is the map of X to p, then
maps H o ( p ; G) = G isomorphically onto this embedded copy of G. As for the case of homology we define
A*(X;G) = H * ( X ; G)/im IC*, the reduced cohomology group of X with coefficients in G. Essentially all of the results we have established previously for homology carry over in dual form to cohomology. For example, we have the following.
THEOREM Let (X, A) be a pair of spaces and U 5 A a subset with 0 E Int A. Then the inclusion map of pairs
3.12
i: (A' - U, A - U )
-
(X, A )
induces an isomorphism
i*:
H * ( X , A ; G) 4 H * ( X
- U, A - U ; G).
PROOF We have observed that
i,:
S*(X - U, A
-
U)
-
SJX, A )
induces an isomorphism on homology groups (Theorem 2.11). The following exercise implies that i# is a chain homotopy equivalence. Thus,
i#:
S*(X, A ; G) --+ S*(X - U, A - U ;G)
is a cochain homotopy equivalence and i* is an isomorphism.
0
Exercise 4 Iff: C + D is a chain map of degree zero between free chain complexes, such that f induces an isomorphism of homology groups, then f i s a chain homotopy equivalence. [Hint: Take the algebraic mapping cone Cfof f (see page 152). Use the fact that Cfhas trivial homology to construct the homotopies. ]
RELATED TOPICS
95
3.13 THEOREM If S,g: (X,A ) + ( Y , B ) are homotopic as maps of pairs, then the induced homomorphisms ,f*, g * : H * ( Y , B ; G )
-
H*(X, A ; G)
are equal. PROOF
We showed in Theorem 2.10 that
are chain homotopic. Therefore
f #, g # : S*( Y , B ; G ) + S*(X, A ; G ) are cochain homotopic, and it follows that f * = g*. Exercise 5 Formulate and prove the Mayer-Vietoris sequence for singular cohomology. Exercise 6 State the Eilenberg-Steenrod axioms for cohomology and prove the uniqueness theorem in the category of finite CW complexes. Example We want to compute H*(Sn, x,; G ) . First note that Hk(So,x,; G) is isomorphic by excision to for k = 0 Hk(pt; otherwise.
{ 0"
As usual (Figure 3.1) we decompose the n-sphere into its upper cap E,"
L
I
\ -
S"
Figure 3.1
96
3 THE EILENBERG-STEENROD AXIOMS
and lower cap EPn with xoE Sn-' = E+" n E-?'. Let z be a point in the interior of JLn. Note that since the inclusions x, --* E+" and x,+ EPnare homotopy equivalences, the relative cohomology groups H*(E+", x,; C) and H * ( L n , x,; C) are both zero. Thus, in the exact cohomology sequence of the triple (Sn,E-", x,) we have an isomorphism
The point z may be excised from the pair (Sn,EPn)to give an isomorphism
Now the pair (Sn - z, E_" - z) may be mapped by a relative homeomorphism to the pair (I?+*, Sn-') so that we have an isomorphism
H k ( P - Z, E_" - Z ; C) * Hk(E+",P-'; C). Finally, the exact sequence of the triple (E+*,Sn--l, x,) yields an isomorphism Hk-l(Sn-l, x,; C) Hk(E+",Sn-'; C). Thus
Hk(S", x,; G )
-
Hk-l(Sn-', x,; G),
which completes the inductive step to prove that
Hk(Sn, x,; G ) =
G 0
for k = n otherwise.
All of these similarities between homology and cohomology might lead one to ask: why bother? There may be many answers to this question; we briefly cite only three: (i) When the coefficient group is also a ring, the cohomology of a space may be given a natural ring structure. (This is not true for homology groups.) This additional algebraic structure gives us another topological invariant. (ii) Cohomology theory is the natural setting for "characteristic classes." These are particular cohomology classes, arising in the study of fiber bundles, which have many applications, particularly to the topology of manifolds.
RELATED TOPICS
97
(iii) There are “cohomology operations,” naturally occurring transformations in cohomology theory, which have many applications in homotopy theory. In Chapter 4 we will define the ring structure in (i) while studying the relationships between homology and cohomology theory. The topics in (ii) and (iii) are more advanced and will not be dealt with here. Perhaps the best source for topic (ii) is Milnor [1957]. The original source for topic (iii) is Steenrod and Epstein [1962]; a more recent book is by Mosher and Tangora [ 19681. We close this chapter with an extension of the universal coefficient theorem which establishes the first basic connection between homology and cohomology groups. 3.14 THEOREM Given a pair ( X , A ) of spaces and an abelian group G , there exists a split exact sequence
0 + Ext(H,-,(X, A ) , G)
-
H ” ( X , A ; G) 5 Hom(H,(X, A ) , G)
Exercise 7 Prove Theorem 3.14.
0
-
0.
chapter 4 PRODUCTS
In this chapter we introduce the theory of products in homology and cohomology. Following the Kiinneth formula for free chain complexes, we state and prove the acyclic model theorem. This is applied to establish the Eilenberg-Zilber theorem and the resulting external products in homology and cohomology. When the coefficient group is a ring R, it is shown that the cohomology external product may be refined to the cup product, giving the cohomology group the structure of an R-algebra. This structure is computed for the torus by introducing the Alexander-Whitney diagonal approximation. Also, a cup product definition of the Hopf invariant is given. Finally, the cap product between homology and cohomology is defined in anticipation of Chapter 5.
Suppose that C = {Cn, a } and D = {Dn,a } are chain complexes. In Chapter 3 we discussed the formation of a new chain complex by tensoring a given chain complex with an abelian group. We now want to generalize this to give a procedure for tensoring two chain complexes to form a new chain complex. Define a chain complex C 0D by setting
98
KUNNETH FORMULA
99
The boundary operator on a direct summand
is given by the formula
To check that this gives a chain complex, note that a(a(c @ d)) = d(ac @ d
+ (-
=ddc@d+
+ (-l)”(c
1)PC
@ ad)
(--l)p-’ac@ad+
(-1)PacOad
@ aad)
= 0.
-
Since the elements (c @ d) generate C @ D,it follows that a o a = 0. Note that iff: C- C‘ and g : D D’ are chain maps between chain complexes, there is an associated chain map .f@g:
COD-C’@D’
characterized by f@ g(c @ d)= f(c) @ g(d). Now suppose that C is a free chain complex. The exact sequence a
0 + Z , ( C ) -2c, + B,-,(C) + 0, where LL is the inclusion, must split because B,t-l(C) is free. Thus, there exists a homomorphism
4: Cn+ Zn(C) which is just projection onto a direct summand, that is, q5 o LL = identity on Z,(C). We consider the graded groups Z,(C), B,(C), and H,(C) to be chain complexes in which the boundary operators are all identically zero. Denote by @J the composition di = n o 4,
where n is the quotient map. Then di is a chain map between chain complexes because q a c ) = n(ac) = 0 = a@@).
ZOO
4 PRODUCTS
4.1 THEOREM If C and D are free chain complexes, the chain map
induces an isomorphism
PROOF
Recall the exact sequence of chain complexes and chain maps
0+Z*(C)
-~a
c
B*(C) -+ 0,
where a has degree - 1. Since the sequence splits, we may tensor with the chain complex D and preserve exactness. This yields an exact sequence of chain complexes and chain maps
0- Z*(C) @ D
u 6 id
C@ D%
B*(C) 0D + 0
and thus an exact homology sequence
where (a @ id), has degree - 1 and d is the connecting homomorphism. On the other hand, the short exact sequence 0
-
B*(C) -% Z * ( C ) L H * ( C )
-
0
of chain complexes need not split. However, since D is free, exactness will be preserved in
o-+ B*(C) 0D
2 Z*(C) @ D
n Q id
H*(C) @ D + 0.
Passing to the homology groups of these complexes we have the long exact sequence
may be related in such a way that each rectangle commutes up to sign (see the following exercise). Now the proof of the five lemma (Exercise 4, Chapter 2) only required commutativity up to sign; hence, we apply the five lemma to conclude that (@@id)*: H,(C @ D)-% H,(H,(C) @ D) is an isomorphism. This completes the proof.
0
Exercise 1 Show that in the diagram in the preceding proof each rectangle commutes up to sign. This proposition reduces the problem of computing the homology of the chain complex C @ D to computing the homology of the simpler complex H,( C ) @ D.Note that if c 0d E Hp(C )@ D,, then 8(c @ d ) = (- 1)Pc @ ad, so that up to sign the boundary operator is just id @ 8: Hp(C)@ 0,+ Hp(C)0D,--l. Therefore, for p fixed HJC) @ D is a subcomplex of H,(C) a direct summand, and we conclude that Hn(H*(C)O D) =
0D,in fact
1P Hn(Hp(C) 0D)*
Now if two boundary operators differ by sign only, it is evident that they produce the same homology groups. Thus, we may assume that the boundary operator in the chain complex H p ( C )@ D is id @ 8. Note that the n-dimensional component of this complex is H J C ) 0Dn-p. Since D is a free chain complex, we are in a position to apply the uni-
102
4 PRODUCTS
versa1 coefficient theorem, Theorem 3.6, to the chain complex H,(C) @ D. Thus H,,(H,(C) O D)= H,(C)
O H,-,(D) 0Tor(Hp(C), H,,-p-lP)).
Summing these over all values of p , we have completed the proof of the Kunnerh formula for free chain complexes :
Example Suppose c E 2, (C) but c is not a boundary. Suppose further that r . c = dc’ for some c‘ E C,,, and some minimal integer r > 0, so that c represents a homology class of order r. Similarly let d E Z J D ) represent a homology class of order r so that rd = ad‘ for some d‘ E Dq+,. Then in (COD),*+, the element ( c ’ @ d - ( - 1 ) P c B d ’ ) is a cycle because a(c’ @ d - (- 1 )PC @ d ’ ) = dc’ @ d - (-
+ (-
ad
1)pf’~’
- (- 1 )’dC
O d’
I)% @ ad’
=
rc @ d - c
=
r(c@d- c@d)
rd
= 0.
In this way torsion common to H,(C) and H J D ) produces homology classes in Hp+q+l(C@ D). Given spaces X and Y, the Kunneth formula of Corollary 4.2 may be applied to the singular chain complexes S , ( X ) and S , ( Y ) to give the isomorphism
Hn(S*(X)0S * ( Y ) )=p
c
+ w
H p W
c
0H&Y) 0r+a-n-1
Tor(Hr(X), W Y ) ) .
We turn now to the problem of relating Hn(S,(X) @ S,( Y)) to H J X x Y), the homology of the Cartesian product of X and Y. The solution of this problem will be stated in terms of the acyclic model theorem, a useful tool in homological algebra. To put this result in its proper setting we require a number of definitions. A category C is (a) a class of objects,
ACYCLIC MODEL THEOREM
203
(b) for every ordered pair of objects a set hom(X, Y ) , of morphisms viewed as functions with domain X and range Y, such that whenever f: X - t Y and g : Y - t Z are morphisms, there is an element g o f in hom(X, Z). These are required to satisfy the following axioms : 1. Associativity: (h o g) o f = h o (g o f ). 2. Identity: For every object Y there is an element 1 E hom(Y, Y ) such that iff: X -+ Y and g : Y + Z are morphisms, then 1 o f = f andgoly=g.
Examples (i) The category whose objects are sets and whose morphisms are functions.
(ii) (iii) (iv) (v)
The The The The
category of abelian groups and homomorphisms. category of topological spaces and continuous functions. category of pairs of spaces and maps of pairs. category of chain complexes and chain maps.
If C and 9 are categories, a covariant functor T: C -t 9 is a function that assigns to each object X in C an object T ( X ) in 9 and to each morphism f:X - + Y a morphism T(f ): T ( X ) -t T ( Y ) such that (a) TUY) = h Y ) , (b) T(f08) = T(f1 O A functor K is contravariant if for f: X + Y, K ( f ) : K ( Y )
(a') (b')
lK(Y), K(fog) = m ) O K(1Y) =
wf).
-
K ( X ) and
-
Examples (1) The correspondence (X, A ) S,(X, A ) and [f': (X, A) -+ ( Y , B ) ] [f#:S,(X, A ) + S,( Y , B ) ] is a covariant functor from Category (iv) above to Category (v). -+
(2) The correspondence X - . H n ( X ;G) and [f: X + Y ]+ [f * : Hn(Y ;G ) H n ( X ; G)] is a contravariant functor from Category (iii) to Category (ii).
-t
Suppose that C and 9 are categories and T I ,T,: C 9 are covariant functors. A natural transformation t:TI -+ T, is a function which assigns to each object X in C a morphism r(X): T,(A'+ T,(X) in 9 such that -+
204
4 PRODUCTS
-
commutativity holds in
Ti(f)
T,(Y)
wheneverf: X-+ Y is a morphism in C. is a specified collecNow fix a category C. Suppose that % = tion of objects in C. % will be called the models of C. A functor T from C to the category of abelian groups and homomorphisms is free with respect to the models 9K if there exists an element e, E T(M,) for each a such that for every X in C the set
is a basis for T ( X )as a free abelian group. A functor T from C to the category of graded abelian groups is free with respect to the models % if each T, is, where T, is the nth component of T. 4.3 THEOREM (Acyclic model theorem) Let C be a category with models 9K and T, T‘ covariant functors from C to the category of chain complexes and chain maps, such that T, = 0 = T,’ for n < 0 and T is free with models 9R. Suppose further that H,(T’(M,)) = 0 for i > 0 and MaE %. If there is a natural transformation @:
Ho(T)+ Ho(T’),
then there is a natural transformation
4:
T+T‘
which induces @, and furthermore any two such homotopic.
4
are naturally chain
PROOF By the hypothesis To(M,) and To‘(M,) are the respective cycle groups in dimension zero. Thus, there are epimorphisms n and iz’ onto the homology groups:
To(Ma)2Ho(T(Ma))
.c T,’(*M,)
Ho(T’(M,))
ACYCLIC MODEL THEOREM
Since To is free with models %, there is for each
01
205
a prescribed element
eaoE To(Ma).So for each IX we choose an element #(euo)E To‘(Ma)such that 3t’ o #(eao) = @ o n(e2).
-
Let f: Ma X be a morphism in C. Then T ( f ) ( e u ois ) a basis element in T,(X) and we define # ( T ( f ) ( e a 0 )= ) T’(f)(#(eao)).This defines # on the basis elements of the free abelian group To(X) so there is a unique extension to a homomorphism
#:
T,(X) + To’(X).
To check that # induces the original @ on zero dimensional homology, we must show that the front face of the following diagram commutes:
I I
, I
1
I
The bottom face commutes by the naturality of @. The left and right faces commute since T ( f ) and T ’ ( f ) are chain maps. The back face and the top face commute by definition; thus, the front face must also commute. Since T, is free with models %, there is for each 01 a prescribed element e,‘ E Tl(Ma).From the above, #(de,l) is a well-defined element of T i ( M a ) . Moreover, since # induces @ on zero-dimensional homology, #(de,l) must be a boundary in To’(Ma).So let c E Tl’(Ma)with dc = $(de,l) and define #(e:) = c. Using the above technique we extend # to a homomorphism #: T , ( X ) - T,’(X) for each object X in C. Suppose # is defined in dimensions less than n, and consider the set {e,” I eatLE Tn(Ma)}given by the fact that T,, is free with models %. By the inductive hypothesis #(dea*) is a well-defined element of TA-l (Ma). Since it is a cycle and T’ is acyclic in positive dimensions, it is also a boundary. So define #(can) to be an element of T,’(Ma) whose boundary is #(dean). Once again # may be extended using the fact that T is free with models %. This defines # on T ( X ) for all objects X in C.
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4 PRODUCTS
Exercise 2 Show that for each X in C, 4: T(X) + T’(X) is a chain map, and for each morphism f: X - + Y, $ o T(f)= T ’ ( f )o 4. This defines the natural transformation 4: T + T’. Suppose now that T‘ is another such natural transformation, inducing @ on zerodimensional homology. For each object X in C we must construct a chain homotopy 3:T ( X ) T’(X), which is natural with respect to morphisms in C, having
4’: T+
-
a3
+ 3a = 4 - 4’.
We define 3 inductively. Suppose that it has been defined in dimensions ) a free abelian less than n, and recall that T,(X) has basis { T ( f ) ( e a n )as group. For n > 0 the element
4(ean)- 4’(ean) - 3(aean) is a cycle because
Since T‘ is acyclic in positive dimensions, this cycle must bound, so define 3(e,“) to be an element of ?‘;+,(Ma)whose boundary is ($(eafl) - $'(can) - 3(ae,)l)).Again we extend 3 to bc defined on T(X ) for all objects X in C by using the fact that T is free with models %. This same technique will work for the case n = 0 because the cycle $(eao) - #’(eao) must bound. This is a consequence of the fact that $ and 4’ induce the same homomorphism (@) on zero-dimensional homology. This 3 gives the desired chain homotopy and is natural with respect to morphisms of C, so the proof is complete. NOTE The technique in the proof of Theorem 4.3 is essentially the same as that used in Theorem 1.10 and Appendix I, although the former is in a more general context.
Exercise 3 Re-prove Theorem 1.10 as a corollary to the acyclic model theorem. We now want to apply this theorem to relate the homology of the chain complex S , ( X x Y) to the homology of S,(X) @ S,( Y).Let C be the cat-
EILENBERG-ZILBER THEOREM
107
egory of topological spaces and continuous functions. (This may easily be generalized to the category of pairs of spaces and maps of pairs.) Denote by C x C the category whose objects are ordered pairs ( X , Y ) of objects in C and whose morphisms are ordered pairs (f,f ’) of morphisms in C with, f:X - , X’ and f Y + Y’. Let 9R be the set of all pairs (up, d),p , q 2 0 in C x C where a‘; is the standard k-simplex. Define two functors from x C to the category of chain complexes and chain maps by I :
e
T(X, Y ) = S,(Xx Y )
and
T’(X, Y ) = S,(X)
0S,(Y).
Both of these functors are free with models 3.Furthermore, both have models acyclic in positive dimensions. The path components of X x Y are of the form C X D, where C and D are path components of X and Y, respectively. As a result there is a natural isomorphism
-
Ho(Xx y>
0
HO(S*(X)0S * ( Y ) )
because Ho(S,(X) @ S,( Y ) ) = Ho(X) @ No( Y ) by the Kunneth formula of Corollary 4.2. From the natural transformations @ and @-I we apply the acyclic model theorem, Theorem 4.3, in each direction to conclude that there exist chain maps $: S * ( X x Y ) S*(X) 0S * ( Y ) +
and
4: S * ( X ) 0S * ( Y )
-
S*(Xx Y )
that induce @ and @-I, respectively, in dimension zero. Thus, $ o 4 is a chain map from S , ( X ) 0S , ( Y ) to itself inducing the identity on zero-dimensional homology. But the identity chain map also has this property, so by Theorem 4.3, $ is chain homotopic to the identity. Similarly the composition 4 o $ is chain homotopic to the identity on S,(Xx Y). Therefore
04
$*: H * ( X x Y )
-
H*(S*(X) 0S * ( Y ) )
is an isomorphism with inverse $*. This completes the proof of the Eilenberg-Zilber theorem :
THEOREM For any spaces X and Y and any integer k there is an isomorphism
4.4
4*:
Hk(XX Y )--* Hk(S*(x) 0S*(Y)).
0
208
4 PRODUCTS
By combining Theorem 4.4 and Corollary 4.2 we have established the Kiinneth formula for singular homology theory : 4.5
THEOREM If X and Y are spaces, there is a natural isomorphism Hn(Xx Y )
for each n.
C
p + m
Hp(X) 0Hq(Y) 0
1
r+s=n-1
Tor(Hr(X), H J Y ) )
0
Suppose now that we have fixed a natural chain map
(6: S * ( X x Y )
-
S*(X) @ S*( Y )
far any spaces X and Y with the above properties. The composition
where the first homomorphism takes {x} @ { y } into {x @ y } , is called the homology external product. The image of { x } @ { y } under the composition is usually denoted { x } x { y } . From the Kiinneth formula we may conclude that this is a monomorphism for any choice of p and q. In fact the Kiinneth formula for singular homology may be restated as a split exact sequence
where the rnonomorphism is given by the external product. Our primary purpose now is to construct the analog of this in cohomology, that is, a product H p ( X ; G,)@ HQ(Y ; C2) -.+ H*+Q(XXY ; GI @ G2).
If u E S p ( X ; GI) and ,!IE SQ(Y ; G2),then a : S p ( X )-+ GIand fl: S , ( Y ) - G, are homomorphisms. Denote by a x @ the homomorphism given by the composition
Sp,(Xx Y )5 S , ( X )
0&( Y ) 3GI 0 G,,
where a @ B is defined to be zero on any term not lying in S J X ) 0S,(Y). Thus, axj3 E Sp+Q(XxY ; G,0G2).This defines an external product on cochains S”X; G,)@ S Q ( Y ;G 2 ) S”’p(Xx Y ; G1 @ G2).
-
EXTERNAL PRODUCTS
209
PROPOSITION If a E SP(X; C,) and ,!?E SQ(Y;C,) are cochains and a x p E SP+Q(XxY ; G, @ C,) is their external product, then
4.6
+ (-1)Paxd,!?.
d ( n x p ) = (Sa)x/?
(This is the derivation formula for cochains.) PROOF The diagram
commutes since 4 is a chain map. Thus
On the other hand
Therefore, it is sufficient to check the behavior of these three homomorphisms on the image of 4. Let e @ c be a basis element of S , ( X ) @ S,(Y). Then (a @ B) o d will be zero on e @ c unless (i) e E Sp+,(X)and c E Sq(Y ) or (ii) e E S J X ) and c E Sq+l(Y). In the first case
In the second case ( a @ P > o d ( e @ c )= ( a @ p ) ( d e @ c + ( - l ) p e @ d c ) =
(- l)Pa(e)@ j?(dc)
=
(-l)p(a @ dp)(e 0c).
Since all three homomorphisms will be zero on any basis element not of the form (i) or (ii), we conclude that S(a x p ) = (da)x /? (- l)P(a x Sg). 0
+
110
4 PRODUCTS
4.7 COROLLARY This induces a well-defined external product on cohomology groups H P ( X ; GI) @ HQ(Y;Gz) + H P + ~ ( X Y X ; GI @ G,)
given by { a } X { B } = {crxB}. PROOF
This will follow immediately from three consequences of Proposi-
tion 4.6: (a) cocyle x cocycle is a cocycle; (b) cocycle x coboundary is a coboundary; (c) coboundary x cocycle is a coboundary.
+
If Scr = 0 = SB, then S(a x 8) = (6a)x B (- I)% x Sg lishes (a); (b) and (c) follow in similar fashion. 0
= 0.
This estab-
The product given by Corollary 4.7 is the cohomology external product.
Exercise 4 I f f : X ‘ + X and g: Y’ -+ Y are maps, { a }E HP(X; Gl), and { B } E Hq( Y ; Gz),show that ( f x g > * ( { a ) x(8)) = f * { a > x g * { B } in Hp+q(X’x Y ’ ; G, @ Gz). Let R be an associative commutative ring with unit. So there is a homomorphism p : R @ R --* R given by p(a @ b ) = ab. We now specialize the cohomology external product to the case where G, = R = G,. For 01 E P ( X ; R ) and j3 E 9 ( Y ; R ) define cr x /? E SP+Q(XxY ; R ) to be the composition
As before this induces a well-defined product on cohomology groups
H p ( X ; R) @ Hq( Y ; R ) + HP+q(Xx Y ; R ) by taking { a } @
{ B } into
,
{a x ,8}.
4.8 LEMMA Let { a }E H p ( X ; R) and {/I} E H Q ( Y ;R ) and define the map T : X x Y --* Y x X by T(x,y ) = ( y , x). Then T * : H P + ‘ J ( Y x XR; ) - + HP+q(Xx Y ; R )
has
T * ( { 8 ) X l { a } )= ( - l P ( { a }
x1
{PI).
EXTERNAL PRODUCTS
-
111
Define T’ : S,(X) @ S,( Y ) S,( Y ) @ S , ( X ) on a basis element e @ c, where e E S,(X) and c E Sq(Y ) , by PROOF
T’(e@c) = (-l)pQc@e.
Then consider the diagram
-
S*(X x y >
9
S * ( X )0S*( Y )
IT* -
S * ( Y X XI
\
(-1PLx 0 p
Ij
RORAR.
S * ( Y ) 0S * ( X )
It is evident that (- 1)Pqp o (a @ ,9) = p o (/?0a ) o T‘ since R is commutative. Restricting our attention to the rectangle, we observe that the composition 4 o T* is a chain map, since both 4 and T* are chain maps. We also claim that T‘ o 4 is a chain map. To establish this it is sufficient to show that T‘ is a chain map, so let e E S p ( X ) and c E Sq(Y).Then
+
T‘ o a(e @ c ) = T’(ae @ c (- 1)Pe @ ac) - (- l)(p-1)qc@ de + (- 1 )p+p(q-l)dc@ e -- (- I)(P-% -
0de + (-
1)mdc @ e.
On the other hand
a o T’(e @ c) = a((-
1)Wc @ e )
=
( - I ) p Q a c @ e + (-l)Pqfpc@de
=
(- 1)pQac@ e
+ (-
l)(p+% @ ae.
Since these two expressions are equal, we conclude that T’ is a chain map. Now if we check on zero-dimensional homology, it is evident that T‘ 0 4 and 4 o T* induce the same transformation. By applying the acyclic model theorem, Theorem 4.3, we conclude that these two chain maps are naturally chain homotopic. Therefore, the cohomology class represented by the composition p o (,9 @ a ) o 4 o T* is the same as the class represented by (- 1)Pqp o (a @ 8) o 4. In other words T*({B>x1 {a>>= (-1>pQ{a)X l
LEMMA If { a } E H * ( X ; R ) and g : Y’ Y are maps, then 4.9
{/3}
@>.
0
E H * ( Y ; R ) and
--f
(fxg)*({4 x 1
{B)) = f * W X l g * W .
f: X ‘ - X,
222
4 PRODUCTS
PROOF
4.10
This follows routinely as in Exercise 4.
LEMMA If { a }E H * ( X ; R),
0
(8) E H*(Y;R), and { y } E H*( W ; R),
then ( { a >x1 {BH
X I
Exercise 5 Prove Lemma 4.10.
(71 = { a > X
l
({PI X l {?I)*
0
Observe that if we take Y = point, then H * ( Y ; R ) = H"(Y; R) = R and the external product
H * ( X ; R ) @ H*(Y; R) -+ H * ( X x Y ; R) has the form
H * ( X ; R) @ R + H * ( X ; R). This gives H * ( X ; R) the structure of a graded R-module. Moreover, it follows from Lemma 4.9 that any map fi X' + X induces an R-module homomorphism f *: H * ( X ; R) --+ H * ( X ' ; R). For any space X let d : X + X x X be the diagonal mapping given by d(x) = (x, x). Then the composition HP(X; R) @ Hg(X; R) -+ HP+q(XxX ; R ) .-% Hp+P(X; R)
sending { a }@ { B } into d*({or} x { p } ) defines a multiplication in the R-module H * ( X ; A). This is called the cup product and is usually written { a >u (8) or just {a>. {PI. By applying the previous lemmas we may conclude the following important result. 4.11 THEOREM For R a commutative associative ring with unit, X a topological space, H * ( X ; R) is a commutative associative graded R-algebra with unit. Any continuous functionf: X ' + X induces an R-algebra homomorphism
f *: H * ( X ; R ) + H*(X'; R) of degree zero.
0
As a point of information, a graded R-algebra M = CkM kis commutative if given any homogeneous elements mp E Mp and m, E Mg, we have
-
mp m, = (- l ) P @ m p p
in MPM.
CUP PRODUCTS
123
It is important to observe that while all of the development of products so far has been in terms of single spaces for the sake of clarity, the same constructions may be duplicated using pairs of spaces and relative homology and cohomology groups. It is important to point out that in this context, the Cartesian product of pairs is another pair given by
NOTE
( X , A ) X ( Y ,B ) = ( X x Y, X x B
u A x Y).
Exercise 6 Is the connecting homomorphism in the long exact cohomology sequence for the pair ( X , A ) an R-module homomorphism? An R-algebra homomorphism ?
The essential tool used in defining the cup product of two cohomology classes is the composition of chain maps
S*(X)
dr
SdXX
More generally, suppose that such that
4
S*W) 0S*(X).
t:S , ( X ) + S , ( X )
@ S , ( X ) is a chain map
(i) t(a) = a @ a for any singular 0-simplex a ; (ii) t commutes appropriately with homomorphisms induced by maps of spaces. Then by applying the acyclic model theorem, Theorem 4.3, we see that any such z must be chain homotopic to 4 o d , . This implies that the cup product on cohomology classes is independent of the choice o f t as long as the stated conditions are satisfied. A chain map t with these properties is usually called a diagonal approximation. For use in later definitions and examples it will be helpful to have a specific example for t. The following is the Alexander- Whitney diagonal approximation. Given a singular n-simplex #: u" 4X in a space X , define the front i-face i+, 0 5 i 5 n, to be the singular i-simplex
&ro,. . . , t i ) = + ( t o , . . . , ri, 0,. . . ,O). Similarly let the hack j-face $j(t0,.
+j,
0 5 j 5 n, be the singular j-simplex
. . , t j ) = +(O,. . . ,o, t o , . . . , tj).
Then define
C i4
t(+> = i+i-n
for
+ a singular n-simplex in X .
@41
4 PRODUCTS
114
+
For example, if 4: cr2 -+ u2 is the identity, then t(4) = 0 @ 4 (0,1) @ (1,2) 4 @ 2 where 0 and 2 are the obvious 0-simplices and (0, 1) and (1, 2) are 1-simplices (see Figure 4.1). It is evident that Properties (i) and (ii) above are satisfied by t.The only mild complication is left as the following exercise.
+
Exercise 7 Show that the Alexander-Whitney diagonal approximation is a chain map.
t
Using this specific model for t, let us see exactly what the cup product looks like. Let ct E P ( X ; R) and 1E S ( X ; R) and 4 be a singular (p q)-simplex in X . Then
+
(a
u B, 4)
is the image of the composition
4 5 c i4 04j
a@B
4*4) 0B(4,) JL 4 A ) B(4,). *
Example We want to compute the cohomology ring of the two-dimensional torus T2= S x S'. Recall that H , ( P ; 2) = 2 0Z, and the generators may
Figure 4.1
(a)
Figure 4.2
HOPF INVARIANT
be represented by E and as generator the 2-chain
fl in Figure 4.2b. For H2(F'; 2 ) = Z we may use
4 - y , where (Figure 4 . 2 ~ )
= a,,
4U)= a,,
4(2)
= a2
~ ( 0= ) a,,
~ ( 1 )= a3,
w(2)
= a2
4(0)
225
and
(see Figure 4.2a). Using the universal coefficient theorem, Theorem 3.14, we see that H'(r?; Z) M Hom(H,(T2;Z), Z) and we choose as generators a, 8, where a(C) = 1,
a@) = 0,
B(C) = 0,
B(J)
= 1.
Now (a u B, 4
- Y> = (a,1 4 ) (B, 41) - (a,IY> - (8, Yl) = (a,3 (B, 18) - (a,18) (8, G) *
*
*
= 1-0 =
1.
On the other hand
-
Similarly ( B u /I,4 - y) = 0. Since H 2 ( F ' ;2) Hom(H2(T2; Z), Z) Z, we now have computed the cohomology ring H * ( T 2 ;Z). Thus, H * ( T 2 ; Z ) is the graded algebra over Z generated by elements a and B of degree 1, subject to the relations a2= 0,
= 0,
aB = -Pa.
NOTE This has the form of an exterior algebra on two generators. How about the cohomology ring of the n-torus P = S'x . x S'?
- .
Suppose that f: SZn-' -+ S is a map, n 2 2. There is a procedure for associating with such a map an integer H ( f ), the Hopf invariant off. This may be defined using the cup product in the following way: Let {a} and { B } be generators of the cohomology groups H2n-1(S2n-1; Z) and H n ( S n ;Z), respectively, represented by the cocycles a and B. Since { B } u { B } = 0, the cocycle B u /? must be a coboundary. That is, there exists a cochain
116
4 PRODUCTS
y E Pn-I(Sfl;Z ) with
By
=
p u p.
Since Hn(Pn-'; Z ) = 0, the cocycle f # ( p ) E Sn(Pn-* ; Z ) must be a coboundary, and there exists a cochain E in Sn-1(S2n-1;Z ) such that BE = f#(/?). Now E uf#(/?)and f # ( y ) are cochains in S2n-1(S2n-1* , Z ). Moreover B(E
uf " ( B )
-f # ( y ) ) = B ( E = BE
u B E ) -f " ( P u B ) u B E -f"(P) uf#(P)
= 0.
So we define H ( f ) to be the integer which when multiplied times { a } gives the cohomology class of E uf#(p)- f # ( y ) . That is
Exercise 8 (a) Let f: Pn-' Sn be a map of Hopf invariant k. If g: P n - I -,S2n-1 and g : Sn + Sn are maps of degree p , determine H ( g f ) and H ( f g ) . (b) If ir -+
s2n-1
-
pn-1
is a commutative diagram where f has Hopf invariant k,how are the degrees of h and h related? Let us give an alternate definition for H(f).Recall that Sn and S"-' may be given the structure of finite CW complexes, each having only two cells. Given a map f: Pn-l Sn, we denote by Sjn the space obtained by attaching a 2n-cell to Sn via f (see Chapter 2). Then S,. is a finite CW complex with three cells, of dimension 0, n, and 2n. Applying the technique of Theorem 2.21 we see that since n > 1, the cohomology of S,. is given bv Z for i = 0, n, 2n Hi(S,") = 0 otherwise. -+
Denoting by b E Hn(Sfn;Z ) and a E HPn(Sjn; Z ) a chosen pair of generators, we define H(f)to be that integer for which b2 = H(f). a in H2n(S,n;Z ) . Exercise 9 Show that the two definitions of H(f)are equivalent.
HOPF INVARIANT
I27
In order to show that H(f)is an invariant of the homotopy class o f f , we need the following result due to J. H. C. Whitehead.
-
4.12 PROPOSITION Iffo ,f, : Sp X are homotopic maps into a space X,then the identity map of X extends to a homotopy equivalence
Let { A } be a homotopy betweenf, a n d h and denote an element of DP+' by Ou, where u E Sp and 0 5 8 5 1. Given a radius in the attached disk in XJn (Figure 4.32), the inner half should be mapped onto the corresponding radius in Xr,. Then the outer half is used to trace out the path of the homotopy fromfi(u) tof,(u) (Figure 4.3b). Specifically then define the map h by PROOF
for x E X;
h(x) = x h(eu)
=
2eu
~(OU= ) f;-ze(~)
4;
for u E
sp,
0585
for u E
sp,
4 5 8 5 1.
Defining a similar map h' : XJl .+ Xfn, it is easily seen that the compositions h o h' and h' o h are homotopic to the respective identities. 0 4.13 PROPOSITION If H(f0) =
Wh).
fo,ft : Pn-' + Sn are
homotopic maps, then
-
-
Let h : S; S;l be the homotopy equivalence given in Proposition 4.12. If io: ( P SZn-'), (S;, Sn) and i,: (DZnSZn-'1 cs;, S"),
PROOF
9
(a)
Figure 4.3
228
4 PRODUCTS
denote the relative homeomorphisms corresponding to diagram
fo
and f,, the
is homotopy commutative, This homotopy is easily defined by setting gt(Ou) = h o i,((l - 4t)Ou).
This implies that the diagram of cohomology groups H2"(S;l, S")
h*
H y s ; , S")
is commutative. Thus, a choice of an orientation for DZndictates compatible choices of generators d, E H2n(S;l, S")
and
do E Hzn(S',
Sn)
and corresponding choices of generators a, E H2"(S$
and
a, E HZn(S')
such that h*(ul) = a,. Furthermore, if b, E Hn(S'l) and b, E Hn(S;) are generators corresponding to a chosen orientation of D", then since h is the identity on s",it follows that h*(b,) = bo. Therefore H(f,)
*
U, =
b,2 = (h*(bl))2 = h*(6,2) = h*(H(fl)
'
a,)
Wf,) . h*(a*) = Wf,) . =
a07
and
HOPF INVARIANT
119
If n is odd, the commutativity of the cup product implies that b2 = -b2, so that H(f)= 0. Thus, the Hopf invariant can only be nonzero for maps f:S4n-1 S2". NOTE
-
PROPOSITION For any n > 0 there exist maps from S4n-1to Fnof arbitrary even Hopf invariant. 4.14
PROOF
As a corollary of Exercise 8, it is sufficient to show that there
exists a map with Hopf invariant *2. Recall that S z n x S Z may n be given the structure of a finite CW complex having one 0-cell, two 2n-cells, and one 4n-cell (see Proposition 2.6). Furthermore, there is a map f:
S4n-1
-+
S2n
S2n
where S2"v Snis the 2n-skeleton of S Z n xSZn,such that S 2 n ~ S 2isn the space obtained by attaching a 4n-cell to S2" v SZnvia f. Define a map g : s 2 n v S2" 3 S2" by g(x, p ) = x, g(p, y) = y, where S2" v SZnis identified with (SnXp)
u ( p x S 2 " ) _c
sznxs2n.
From the commutative diagram
we see that g induces a map
Using this map g we want to prove that the Hopf invariant of g j i s f2. Let e E Ho(p), 1 E Ho(S2"),and c E H Z n ( S z nbe ) generators of these infinite cyclic groups. Then H4n(S2n x SZt1)is the infinite cyclic group generated by c x c and Hzn(SZn x S2")is the free abelian group with basis consisting of 1 x c and c x 1. As before let a E If4"($?) and b E HZn(X;) be generators of these infinite cyclic groups. SZn is the First, we must compute g*(b) E H2"(P"xPn).If j : p -+
220
4 PRODUCTS
inclusion, then both rectangles in the following diagram commute : Hygy)
22 HZn(S*n x S2") 2-H2yS2n) @ H O ( P )
I
(id x j)*
Hzn(S2n) 2H"n(S2nxp)
I
m (id)*@j*
+
H2n(S2n) @ HO(p)
Thus
i*(b) = f c x e
=
&(id)*(c)xj*(l)
or (idxj)*g*(b) = &(idxj)*(cx 1). This means that the element
2*(6) f c x 1 is in the kernel of (idxi)* for some choice of sign. Now the kernel of (id x j ) * in H2n(S2nx S2") is the infinite cyclic subgroup generated by 1 x c, so g*(b) f.c x 1 = m(l x c ) for some integer m. By the same argument, g*(b) & 1 x c = k(cx 1) for some integer k. These two properties together imply that with a proper choice of sign, we have g*(b)=fcxl&lxc. It can be easily checked that
( f c x 1 f 1 xc)2 = ( c x 1 ) 2 f 2(cx 1) = c2x1 =
f2cxc
*
(1 x c )
+ lxc2
+ (1 xc)2
f2CX c,
since c2 = 0. Finally, since
g* : H 4 y q y ) 4H4yS2n x S2n) is an isomorphism, b2
= g'*-'@*(bZ)) = g*-'((&cx
= g*-y i-2c
x c)
= +2a.
This proves that H ( g f ) = f 2 .
0
1 f 1 xc)')
CAP PRODUCTS
221
There remains the question of the existence of maps having odd Hopf invariant. Using the results of the next chapter we will be able to show that the Hopf maps SJ4S2 and S74 S4 each have Hopf invariant one. By using the Cayley numbers, one can define an analogous map S15 Sn of Hopf invariant one. Results of Adem [I9521 on certain cohomology operations imply that there can exist maps f:S4+l SZnof odd Hopf invariant only when tz is a power of 2. Finally, there is a deep theorem due to Adams [1960], that for tz f I , 2, or 4 there is no map f: S4"-' + S2" of odd Hopf invariant. An important consequence of this theorem is that the only values of n for which Wn carries the structure of a real division algebra are: n = 1 (real numbers), n 2 (complex numbers), and n = 4 (quaternions). [See Eilenberg and Steenrod, 1952, p. 3201. As a reference for further information on the Kopf invariant we recommend Hu [1959]. We cite only briefly one further result: two maps from S3 to S2 are homotopic if and only if they have the same Hopf invariant.
-
-
7
As the final topic of this chapter we introduce a variant of the cup product which will be useful in the following chapter. Let X be a space and R be a commutative ring with unit. If a E S p ( X ; R ) we may view a as a homomorphism of all S,(X) into R by setting it equal to zero on elements of dimension different from p . The composition
S,(X)
2 S,(X)
@ S,(X) 2.?% R @ S,(X)
when tensored throGghout with R yields R 0S,(X)
id8r
R @ S * ( X ) 0S * ( X )
pQid
id 0 a 0 id
R@R
0S * ( X )
R @ S,(X).
If c E S , ( X ; R) = R 0S,,(X), we define the cap product of a and c, a n c, to be the image of c under this composition. Note that a n c E S,-,(X; R ) .
For example, suppose z is the Alexander-Whitney diagonal and singular n-simplex. Then the above composition has
9 is a
If we interpret R @ S , ( X ) as the free R-module generated by the singular
222
4 PRODUCTS
simplices of X, then a n4
=
*
4n-p*
It is evident that this is closely connected to the cup product. To make this relationship specific, let a E SP(X; R ) , B E P ( X ; R) and 4 E S,+,(X; R), where 4 is a singular simplex. Then
= B(q4)
*
44,).
On the other hand ( a , B n 4)
=
- 4,)
=
B(,b)
-
Since this is true for all 4, it follows that for any c E Sp,(X; R),
(B u a, c)
(4.15)
=
( a , B n c).
Finally, we must determine the action of a on the chain a n c. To do this we evaluate an arbitrary cochain y on a(a n c), ( y , a(a n c)) = (dy, a n c) = ( a
u dy, c).
Suppose that a E S ( X ; R), c E S,(X;R) so that a n c E Sn,(X; R). Recall that d(a u y ) = 6a u y (-1)qa u dy
+
or a
u 6y = (-1)q(S(a u y ) - (6a) u y ) .
So by substituting into the previous equation we have ( y , a(a n c ) )
=
(-1)q
uy
-
da u y, c>
n dc) - ( y , 6a n c)] = ( y , (-1)q(a n ac - 6a n c ) ) . = (-1)q[(y, a
Since this is true for all cochains y, it follows that (-l)G(a n c ) = (a n d c ) - (da n c). From this derivation formula we conclude that the cap product on chain groups induces a well-defined product on homology groups which takes the form Hq(X; R) @ H,(X; R)-+ H,,(X; R) and sends { a } @ {c} into { a n c}.
CAP PRODUCTS
223
Exercise 10 Formulate and prove a statement showing that the cap product is natural with respect to homomorphisms induced by mappings of spaces. Exercise 11 A graded group {G,} is said to be of finite type if for each q, Gq is finitely generated. Prove the following theorem. 4.16 THEOREM (Cohomology Kunneth formula) Let G and G' be abelian groups with Tor(G, G') = 0. If H * ( X ; 2) and H,(Y; Z) are of finite type, then there is a split exact sequence
H p ( X ; G ) 0Hq(Y; G')2 H n ( X x Y; G 0G')
0+ P+q-n +
C
Tor(Hp(X; G), Hq(Y; G'))
p+q-n+l
-
0.
0
Exercise 12 Let X and Y be spaces and R be a commutative ring. If u, E HP(X; R), 24, E Hq(X; R), u, E H r ( Y ; R), and u2 E H 8 ( Y ;R), then in Hp+Q+r+s(X~ Y; R) we have (u1
x ul)
u (u, x u2) = (-
l)"(u,
u 242) x (u1 u u2).
Exercise 13 If u E HP(X; R), u E HQ(Y ; R ) , p , : X x Y + Y are the projection maps, then
pl:X X Y
-
X,
and
u x u = PI*@) u pa*(u). Exercise 14 If u1 E H p ( X ; R ) , u, E H g ( Y ; R), z1E H,(X; R), and E H,(Y; R), then in Hm+n-p-q(Xx Y; R) we have
2,
(ulx u,) n (z,x z2)= (-
l)q(m-p)(ul
n zl)x (u, n z,).
Exercise 25 Show that the cap product may be extended to relative homology and cohomology groups of a pair to give products of the form
chapter 5 MANIFOLDS AND POINCARE DUALITY
This chapter deals with some of the basic homological properties of topological manifolds. Since the main result is the Poincart duality theorem, we begin with a simple example to establish an intuitive feeling for this classical result. This is followed by material on topological manifolds and a detailed proof of the theorem. The approach used follows the excellent treatment of Samelson [1965, pp. 323-3361 and proceeds by way of the Thom isomorphism theorem. Several applications of the theorem follow, including the determination of the cohomology rings of projective spaces and results on the index of topological manifolds and cobordism. Before proceeding with the general approach, let us see how the theorem may be motivated from an example. Briefly, the Poincart duality theorem will say that if M is a compact oriented n-manifold without boundary, the ith Betti number of M is the same as the (n - i)th Betti number for 0 5 i 5 n. In the following example we will indicate how such a correspondence arises. Suppose we are given a portion of a triangulated surface K as shown in Figure 5.1. By taking the first barycentric subdivision (see Appendix I) we arrive at a new triangulation K', as shown in Figure 5.2. If u is a vertex in this new triangulation, define the star of u in K' to be the union of all open cells in K' that contain u in their closure. Thus, star(A; K ' ) is the 124
INTRODUCTION
225
open 2-cell shown in Figure 5.3, whereas star(o,; K’) is the open 2-cell shown in Figure 5.4. Given a simplex 0 in K, we define its dual cell o* in K‘ by c* =
n star(”; K’), U
Figure 5.2
I I
.. .
..
Figure 5.3
I
I I / I
Figure 5.4
126
5 MANIFOLDS AND PO IN CAR^ DUALITY
where u ranges over the vertices of cr. For example, the dual of the vertex A is =r(A; K ’ ) , the closure of Figure 5.3, while the dual of the 2-simplex ABC is the vertex uo. Similarly the dual of the 1-simplex A B is the 1-cell joining uo and u l . Note that while the dual of a simplex need not be a simplex, it is a cell in the complementary dimension. As a specific example we take the boundary of a 3-simplex, a triangulated surface homeomorphic to S2 (Figure 5.5). This surface may be viewed as a finite CW complex having four 0-cells, six 1-cells, and four 2-cells. By taking the dual cells of each of these simplices we get a corresponding C W decomposition for the same space as shown in Figure 5.6. Here we have four 2-cells (A*, B*, C*, D*),six 1-cells ( A B * , . . . , CD*),and four 0-cells (ABC*, . . . , BCD*). To compute the Betti numbers of these complexes we use the cellular chain complex of Theorem 2.21. Recall that if Y is a finite CW complex and ni Ca+1( Y ) -%Ci( Y ) Ci-I(Y )
-
is a portion of the chain complex, then the ith Betti number M Y ) = .i(Y) - Yi+l(Y) - Yi(Y), where ai( Y) is the number of i-cells in Y and y j ( Y) is the rank of the image of aj. Denoting by X and X * the two structures above we may make the following comparisons: ao(X)= 4 given by A , B, C, D a I ( X )= 6
a2(X*) = 4
given by A*, B*, C*, D*
a,(X*) = 6 A*
A
B*
Figure 5.5
Figure 5.6
TOPOLOGICAL MANIFOLDS
given by A B , . . . , C D
4 X )= 4 given by ABC, . . . , BCD
given by AB*,. . . , CD* u,(X*)
=4
given by ABC*,. . . , BCD*
YOV)=0
Y3(X*) =
y l ( X ) = 3, basis
y 2 ( X * ) = 3, basis
given by A - B, A - C , A - D
227
0
given by d(A* - B*),
d(A* - C* ), d ( A * - D* 1 y 2 ( X ) = 3, basis
given by d(ABC - ABD), d(ABC - ADC), d ( A B C - BCD) Y3(W =
0.
y,(X*)
=
3, basis
given by ABC* - ABD*, ABC* - ADC*, ABC* - BCD*
yo(X*) = 0.
Putting this information together, it is evident that p o ( X ) = p 2 ( X * ) = p l ( X ) = /?,(A'*) = 0, and , b 2 ( X ) = p o ( X * ) = I . It may be helpful to keep this sort of geometric picture in mind as we develop the algebraic techniques necessary to establish the theorem in its general setting. In RJ1define the half-space U n to be the set of all points (x, ,. . . , x,) such that s,,2 0. A topological n-manifold is a Hausdorff space M having a countable basis of open sets, with the property that every point of M has a neighborhood homeomorphic to an open subset of Hn. The boundary of M , denoted d M , is the set of all points x in M for which there exists a homeomorphism of some neighborhood of x onto an open set in M n taking x into {(x, ,. . . , x,) 1 x, = 0} = d R n c Ha. Exercise I Let h be a homeomorphism of an open subset U of W n onto an open subset of Nn. If x E U n dWn, then show h(x) E dH". It follows immediately from this exercise that if x E d M , then all homeumorphisms from open sets about x to open sets in W n must map x into dH". M is an n-manifold without boundary if d M = 0, or, equivalently, if each x E M has a neighborhood homeomorphic to an open set in Wn.A closed n-manifold is a compact n-manifold without boundary.
228
5 MANIFOLDS AND POINCAR6 DUALITY
Examples ( I ) Any open subset of R" is obviously an n-manifold.
(2) For each point x E Sn,stereographic projection from - x is a homeomorphism from S" - { - x } onto Rn.This gives S" the structure of a closed n-manifold. (3) For each point y E RP(n) pick a point x E S" with n ( x ) = y, where n: S n - t R P ( n )
is the identification map. Let i : (Dn- Sn-l) -t S" be the inclusion of the open hemisphere centered at x. Then jz o i is a homeomorphism of an open subset of Rn onto an open set about y. Therefore, RP(n) is a closed n-mani fold. (4) Let GL(n) denote the set of all real n x n matrices having nonzero determinant. By ordering the entries we may view GL(n) as a subspace of Rn2 and give it the induced topology. Under this identification, the determinant function Rn2 R is continuous and has GL(n) as the inverse of the open set R - (0). Thus, GL(n) is an open subset of Rn2 and hence is an n2-manifold without boundary. ---+
( 5 ) The Mobius band, formed by identifying the two ends o f a rectangle so that the indicated arrows coincide (Figure 5.7), is obviously a 2-manifold with boundary.
5.1 LEMMA If U is an open subset of R", then H i ( U ) = 0 for i 2 n.
Before proceeding with the proof, we point out a slight generalization of the chain complex in Theorem 2.21. If (A', A ) is a finite CW pair, the groups H P ( P u A , xp-1 u A ) PROOF
form a chain complex whose homology is H,(A', A ) . Note that if every cell of X of dimension greater than p is contained in A , then Hi(X, A ) = 0 for i > p .
i
Figure 5.7
TOPOLOGICAL MANIFOLDS
129
Let { z } E H , ( U ) be an homology class represented by an i-cycle z , i 2 n. The image of each singular simplex in U is a compact subset. Since z is a finite linear combination of singular i-simplices, the union of the associated images forms a compact subset X G U . Define E = Inf{ll x - y 1 I x E X , y E Rn - U } . Note that E > 0, since X is compact, Rn - U is closed, and X n (Rn - U ) = 8. Since X is compact, there exists a large simplex Sn in Rn such that X is contained in the interior of 5". From Appendix I we know that there exists an integer m with mesh SdmSn< E . Now consider SdmSn as a finite CW complex under the simplicia] decomposition. Let K be the subcomplex of SdmSnconsisting of all faces of simplices which intersect X (Figure 5.8). Note that by construction
A portion of the exact homology sequence of the finite CW pair (SdmSn,K ) has the form
By our previous comments, Hi+l(SdmSn,K ) = 0. Also Hi(SdmSfl)= 0 since the space is a simplex, hence, a convex subset of Rn (see Theorem 1.8). Therefore, H , ( K ) = 0.
Figure 5.8
130
5 MANIFOLDS AND POINCARE DUALITY
Since z was a cycle in X,it is also a cycle in K.The fact that H i ( K ) = 0 implies that z bounds an (i 1)-chain in K. But this (i I)-chain also lies in U ;hence, z bounds a chain in U and {z} = 0. 0
+
+
5.2 LEMMA If M is an n-manifold without boundary, then H , ( M ) = 0 for i > n. PROOF Let z be an i-cycle in M, i > n. Then as in Lemma 5.1 we associate with z the compact subset X c M which is the union of the images of the singular simplices which make up z. There exists a finite collection U,,. . . , Urnof open sets in M , each homeomorphic to an open set in P, with X c (J Uj.(Open sets of this type will usually be called coordinate neighborhoods or coordinate churls.) We will show that {z} = 0 by proving that Hi(U Uj)= 0 so that z must bound in (J Uj. Proceeding by induction on the number of coordinate neighborhoods, it is true for m = 1 by Lemma 5.1. Suppose that
There is a Mayer-Vietoris sequence for the space has the form
((Ji-, Uj)u
U,+,, which
The term on the right is zero by Lemma 5.1 since the space is an open subset of a coordinate chart; similarly Hi(U,+J = 0. It follows from the inductive hypothesis that Hi((Jj+=\Uj)= 0. This completes the inductive step. This result tells us that the nontrivial homology of such a manifold all occurs in dimensions less than or equal to the dimension of the manifold. When the manifold is connected but not compact, this result may be refined to show that the top-dimensional homology group (dimension of the manifold) must also be zero. To establish this, we need the following lemmas. 5.3 LEMMA Let U be open in P and U E Hn(Rn,U). If for every - U,the homomorphism induced by inclusion pE
j,:
has j p (a) = 0, then a
Hn(Rn, U)-+ H n ( P , Wn - p ) = 0.
TOPOLOGICAL MANIFOLDS
I32
PROOF The connecting homomorphism for the exact sequence of the pair
(P, U ) gives an isomorphism d : ll,(.JP, U ) 5 H,-I(U). We will prove that a = 0 by establishing that d ( a ) = 0 in H,-l(U). So let b = d(a). Once again, since the “image” of a cycle representing b is a compact subset of U, there exists an open set V with V compact c U and an element b‘ in H,-,(V) with i*(b’) = b, i : V - U the inclusion. Let Q be an open cube containing V and define K = Q - Q n U (Figure 5.9). For each point p in I? there exists a closed cube P containing p such that P n V = 0. From the diagram
in which the rectangles commute, it is evident that the image of b‘ under the homomorphism
H,-l(V
-
H,-l(R” - P)
is zero. A finite number of such closed cubes cover
Figure 5.9
K, say PI,. . . ,P,.
132
5
MANIFOLDS AND POINCARfi DUALITY
Now suppose that the image of b' under the homomorphism
is zero (this is certainly true when k = 0, since Q is contractible). Denoting Qk = Q - (Pi u . . u Pk),consider the Mayer-Vietoris sequence relating Qk and P - Pk+l:
The first group is zero by Lemma 5.1. The images of 6' in the two direct summands are both zero; hence, the image of 6' in Hn-,
-
K-dQ- ( p i u
- - - u p,))
(Inclh
L d QnW
is zero. Since the inclusion of V in U factors through Q n U,it follows that the image of 6' in Hn-l(U), that is, b, is zero. Hence, a = 0. 0 5.4 LEMMA If x and y are points in the interior of a connected
n-manifold M, then there is a homeomorphism h: M to the identity, having h(x) = y .
-+
M , homotopic
Note that since M is locally homeomorphic to Hn,M is locally pathwise connected. That is, each point of M is contained in a pathwise
PROOF
h.
A X
Figure 5.10
TOPOLOGICAL MANIFOLDS
233
connected open set. This implies that the path components of M are both open and closed. Since M is connected, there can only be one path cornponent, so M must be pathwise connected. Now if M is connected, so is M - dM = the interior of A4 (see the following exercise). So let p : [0, 11 -+ M - dM be a path from x to y. This compact subset of M - a M may be covered by a finite number of open sets, each homeomorphic to the open unit disk in Rn. Denote these disks by U , , . . . , Urnand the corresponding homeomorphisms by h,, . . . ,h,. Let x = x o , x , ,. . , , xk = y be a collection of points on the path with the property that for each j , the segment from x j to xj+, is contained in some Ui (Figure 5.10). The desired homeomorphism may now be constructed inductively. It is sufficient then to show that for 0 5 j < k there is a homeomorphism h : M + M , homotopic to the identity, with h(Xj) = X j + l . So suppose x j and x , + ~are in Ui.Define a homeomorphism
The inverse of g is given by g-'(w)
=
W
l+lwl.
Let gh,(xj) = ( u l , a 2 , .. . ,a,) and ghi(xj+,) = (bl, b 2 , .. . ,bn). Define the translation function
f: P+P
Moreover, f is homotopic to the identity via the homotopy
Thus, we have a homeomorphism for each t
234
5 MANIFOLDS AND POINCARB DUALITY
that takes h,(xj) into hi(xj+,) when t = 1. Note further that, for each t , this may be extended to a homeomorphism from Dn to Dn by defining it to be the identity on the boundary (see Exercise 3). Now define a homotopy h,: M M by
-
Then each ht is a homeomorphism, h, is the identity and h,(xj) = x ~ + ~ . This completes the inductive step, so that by composing maps and homotopies we may give a homeomorphism homotopic to the identity taking x into y. 0 We actually have proved something stronger than the conclusion of the lemma. If f, g : X-+ Y are homeomorphisms between topological spaces, then f is isotopic to g if there exists a map F : X x [0, 11 Y such that NOTE
-+
F(x, 0) = f ( x ) ; (2) 1) = g(x); (3) for 0 5 t 5 1 the map x onto Y.
(1)
m,
--+
F(x, t ) is a homeomorphism of X
The construction used in the theorem makes it evident that the map h is isotopic to the identity. Exercise 2 If M is a connected n-manifold, show that the interior of M , M - a M , is also connected. Exercise 3 Let (a, ,. . . ,a,) be a point in Rn and define fi JP JP by f ( x l , .. . ,x,) = (x, a , , . . . ,x, an).Using the map g : Dn- Sn-' JP defined by g ( z ) = z / ( l - I z I), show that the homeomorphism
+
+
--f
--f
may be extended to a homeomorphism Dn-+ D n by defining it to be the identity on the boundary.
THEOREM If M is a connected, noncompact n-manifold without boundary, then H,(M) = 0.
5.5
TOPOLOGICAL MANIFOLDS
135
-
First note that if p is any point in M , the homomorphism Hn(M,M - p ) , induced by the inclusion, is identically zero. To check this let {z} E H , ( M ) be represented by the cycle z and let C C M be a compact subset which supports the cycle z. First suppose p E M - C, so that {z} is in the image of the homomorphism Hn(M - p ) + H f l ( M ) Then . by the exact sequence of the pair ( M , M - p ) it follows that k , ( { z } ) = 0. If p E C, then select a point q with q E M - C. Such a point must exist, since C is compact and M is not. By Lemma 5.4 there is a homeomorphism h : M - M , homotopic to the identity, with h ( p ) = q. The restriction of h will then yield a homeomorphism from M - p to M - q. From the commutativity of the diagram PROOF
k,: H , , ( M )
ke
H,(W
Hfl(M,M - P )
I-
h+ = id
and the fact that k*’({.z})= 0 we conclude that k , ( { z } ) = 0. Now let {z} be an arbitrary homology class as above and cover the compact set C with a finite number of coordinate neighborhoods U , , . . . , Uk,where each Ui is homeomorphic to an open disk in Rn. Denoting Vj = (Ji-, Ui, we will show that { z } = 0 by proving that { z } is a boundary in V k , that is, Hn(Vk) = 0. The argument proceeds by induction on k. For k = I the result follows from Lemma 5.1. So suppose Hn( V,) = 0 and consider the Mayer-Vietoris sequence for the union V,,, u Urn+,= V,,,:
Hn(V,)0Hn(u,,,+i)
Hn(J‘m+1)
-+
--
Hn-i(Vm n Urn+,> K-l(V,)
0~fl--I(U,+l).
Now the first term is zero by the inductive hypothesis and Lemma 5.1 and H,-,(U,,,) = 0 since Urn+,is homeomorphic to an open disk. Thus, to prove that Hn(Vm+,)= 0 it is sufficient to show that the homomorphism
b : H,-,(V,,, n urn+,> -,Hn-,(Q is a monomorphism. So suppose that i,(/?) = 0. Then there exist elements p’ E Hn(U,,,+, , Vrnn U,,,) and /?” E Hn(Vrn,V,,, n Urn+,)such that A,(/?’) = /?= A,(p”), where d and A , are the respective connecting homomorphisms.
,
5
236
MANIFOLDS AND POINCARE DUALITY
Hn -I Setting p = i,.(j3‘) - i2.(p’),observe that A @ ) = 0, where d is the connecting homomorphism of the pair ( Vm+l,Vrnn Vrn+,). Thus, there exists an cr E Hn(Vrn+l)with j*(a) = p. Let p E Urn+,- V, n Urn+,. Then by the remarks at the beginning of the proof, ip(a) = 0. Thus
0 = I*(i*(a)) = ,*
= ,*(il*(B‘> - i 2 * ( B ” ) ) =
f*i1.(p) - ,*i2.(/?”).
Now since p g V,, lei2. factors through H,(M - p, M - p ) = 0, so f,i2,(/?”) = 0. Hence, l*ilt(/3’)= 0. This implies that j p ( p )= 0. It follows from Lemma 5.3 that B’ must be zero, hence = 0 and i* is a monomorphism. Therefore, H,( = 0 and we have completed the inductive step. 0 5.6 COROLLARY Let M be a closed connected n-manifold. If z E H,(M) and p E M such that
ip: H,(M) -+ H,(M, M - p ) has i&)
= 0,
then z = 0.
= 0, there must be an element z‘ E H,(M - p) with z being the image of z’. However, M - p is not compact, so by Theorem 5.5
PROOF Since i&) H,(M - p)
= 0.
Thus, z‘ and also z must be zero.
0
5.7 COROLLARY If M is a connected n-manifold without boundary, then either
or (ii) H,(M) = 2, and for every p E M the homomorphism ip: H,(M) H,(M, M - p) is an isomorphism. (i)
H,(M)
--f
= 0,
ORIENTATION
I37
-
From Corollary 5.6 we have that ip is a monomorphism. Since H,,(M, M - p ) = 2, it follows that either H,,(M) = 0 or H,(M) Z . So suppose H,,( M ) f 0 and let z E H,, ( M ) and II’ E H , ( M , M - p ) be generators for the respective infinite cyclic groups. Then i,.(z) = f rn . w for some positive integer rn. We must show that m = I . Note that the same proof as for Theorem 5.5 may be given to show that for any abelian group G,H,,(M;G) = 0 for M a connected noncompact manifold without boundary. Then consider the diagram
PROOF
Hn(W
1
0Z m
ip.
0 id
ff,,(M, M - P> 0Z m
a, mono
as
I
mono
where the vertical monomorphisms come from the universal coefficient theorem. The commutativity of the square implies that ip @ id is a monomorphism. But (i,.
@ id)(z 01) = i p ( z ) 01 = f rn
so that z @ 1 = 0. This only happens if m morphism. 0
- w @ 1 = f w @ m = 0, =
1. Therefore, ip* is an iso-
Let M be an n-manifold without boundary. For each p E M let T,
=
H,,(M, M - p )
2
and for coefficients in 2,
T,(Z,)= H,(M, M
Figure 5.11
-p ; Z,)
= z,.
138
5 MANIFOLDS AND POINCARE DUALITY
Define the set
We want to introduce a topology on the set 3. To do so requires the notion of a proper n-ball. A proper n-ball in M is an open set V E M such that there exists a homeomorphism of Dn onto V taking Sn-' onto P - V. For example, the interior of the region shown in Figure 5.11a fails to be a proper n-ball while the interior of the region in Figure 5.1 1b is a proper n-ball. Exercise 4 Show that if M is an n-manifold without boundary, then
the collection of all proper n-balls in M forms a basis for the topology on M. Now if V is a proper n-ball in M and p E V, then there is an isomorphism j P : Hn(M, M - V) As a basis for the topology on
Ua.v
M
Tp.
3 we take the sets
=
{ iP(4I P E V )
as V ranges over all proper n-balls of M and a ranges over all elements for H , ( M , M - V). For example, the choice of a generator for H,(M, M - V) dictates, via the isomorphisms j P , a generator for each T p , and these selected generators form a sheet in 3 which is homeomorphic to V. Since this may be done for either generator of Hn(M, M - V) we see that the generators of the Tp form two disjoint sheets each homeomorphic to V. More generally, if t:3 .+ M is the natural projection, it is evident that t is a local homeomorphism. Each component of 3 is a covering space of M with either one or two sheets. In particular the generators of all the Tp form either a double covering of M or two distinct simple coverings. The restriction of t to this subset of 3 is the orientation double covering of M. In the case that there are two distinct simple coverings we say that M is orientable. An orientation of M is a map s: A4 ---* 3 with t o s = identity on M and s(p) a generator of Tp for each p E M.Of course, an orientable manifold has two possible orientations.
- aB is a 2-manifold without boundary. Prove that M is not orientable by showing that the
Exercise 5 Let B be the Mobius band so that M = B
ORIENTATION
139
domain of its orientation double covering is homeomorphic to the annulus S I X (0,1). Exercise 6 Let M be an n-manifold without boundary and let
domain of its orientation double covering. Then show that able n-manifold.
fi be the
fi is an orient-
Following the same procedure for the groups Tp(2,), the generators form a simple covering of M so that there always exists a unique 2,-orientation. If M is a closed n-manifold, a fundamental class on M is an element Z E H n ( M ) such that
ip.: H n ( M )
-
Hn(M, M - p)
=
Tp
has i&) a generator of Tp for each p E M. A cycle representing z is a fundamental cycle. 5.8 LEMMA Let M be a closed n-manifold and U be an open subset of M. If an element x E Hn(M, U ) has j,(x) = 0 for all p E M - U, where jp.: H,(M, u) Tp, then x = 0. +
PROOF First suppose that M - U is contained in some coordinate neigh-
borhood W. Then consider the commutative diagram H,(w, W n u ) ~ ” . H , ( w ,w - p )
-I
H,(M, M - P) = Tp
where the vertical homomorphisms are excision isomorphisms. For each p E M - U = W - ( W n U ) , it follows that j,(x) = 0 if and only if h , kills the preimage of x. Then by Lemma 5.3 the preimage of x must be zero, which implies that x = 0. For the general case, since M - U is compact, we express M - U as the union of a finite number of compact sets, each contained in a coordinate neighborhood. We proceed by induction on the number k of such compact sets. In the previous paragraph we have proved the result for k = 1. For the inductive step we use the Mayer-Vietoris sequence Hn+,(M, U‘ u U ” ) + Hn(M, U’ n ZJ”)
--+
Hn(M, U ’ ) @ H,,(M, U”),
I40
5
MANIFOLDS AND POINCARfi DUALITY
where U‘and U“ are open sets in M , and the fact that H,+,(M, U‘ u U”)=O, which follows by Lemmas 5.1 and 5.5 and the exact sequence of a pair. 0 5.9 LEMMA Let M be a closed orientable n-manifold with orientation s: M + 3. Then there exists a class z E H , ( M ) such that i,(z) = s ( p ) E Tp for all p E M. From our previous observations we know that for each p there exists a proper n-ball V, about p and an element x p E H,(M, M - ),?I such that if q E V,, j p ( x p ) = s(q). The technique then is to piece together such proper n-balls, using a Mayer-Vietoris sequence and the compactness of M , to construct the desired global homology class. Since M is compact, there is a finite collection Vl,.. . , Vk of proper n-balls which cover M. Suppose that there is an element
PROOF
z, E H,(M, M - ( V ,
u
* * *
u 7,))
such that
h&fI)= s(q) for all q E sequence
rl u . - - u
H,(M, M
+
-
Pm. Then
(Vl u
Hn(M, M
H,(M, M
* * .
u
consider the relative Mayer-Vietoris
r,,,))
. u V,)) 0H,(M, M - V,+1) - (Vlu . . * u 7,) n 7m+l). - (V1 u
* *
Starting with the element z, - x,+~ in the direct sum, let w be its image in H,(M, M u .. u Vm) n This implies that for all qE U **. U n 7m+l, &(w) = 0. Now by Lemma 5.8 it follows that w = 0. Let zm+l E Hn(M, M u * u F,+l)) be the element which is mapped into z, - xm+, and note that ~,JZ,+~) = s(q) for all q E 7, u . . U rm+, . This completes the inductive step and the desired class z is 2,. 0
(rl
(rl r,)
rn,+,).
(rl -
All of these results are summarized in the following theorem expressing the precise relation between orientation and fundamental class :
5.10 THEOREM If M is a closed connected orientable n-manifold with orientation s: M + 3, then there is a unique fundamental class z E H , ( M ) such that i&) = s(p) for each p E M.
241
THOM ISOMORPHISM THEOREM
PROOF This follows immediately from the previous lemmas. From Lemma 5.9 we have the existence of such a fundamental class z. If z' is another such class, then for each p E M
so that z - z'
=0
by Corollary 5.6. This proves uniqueness.
0
Exercise 7 Let M , be a closed orientable n,-manifold and M , be a closed orientable n,-manifold. Then show M , x M , is a closed orientable (n, 11,)manifold.
+
Our procedure for proving the PoincarC duality theorem follows the style of Milnor [I9571 by first establishing a form of the Thom isomorphism. So for the present, we assume that M is a closed, orientable n-manifold with an orientation s: M + 3. Then by the above exercise, M x M is a compact, orientable 2n-manifold. Define maps zl , n,:M x M + M by projection onto the first or second coordinate, and for any p E M
Ip,rp: M+ M x M by &)
=
( p , x), r&)
the diagonal d(x)
= (x, p ) . Finally, denote by
= ( x , x)
and note that
It, o d = z, o
A
= identity
on M .
5.11 LEMMA Let V be a proper n-ball in M with p
E
V corresponding
to the origin in Dn.There is a homeomorphism
0 : n;'(V)= V x M + n ; ' ( V ) such that V); (i) zl= n,o 8 on all of ql( (ii) 8 o d = rp on V ; (iii) (nz0 8 0 I&(s(q)) = s(p) for all q E V.
This states that V x M may be deformed in such a way that the first coordinate is unchanged (i), and the diagonal over V is transformed into the level set V x { p } (ii). Furthermore this is done in such a way as
NOTE
142
5 MANIFOLDS AND POINCAR6 DUALITY
PROOF Denote by
h: p + D n the homeomorphism taking p into the origin. Define a homeomorphism
A: (Dn- dDn)XD"-t(Dn-aDn)XDn as follows: for x E Dn - aDn, y E aDn, map the entire segment from ( x , x ) to ( x , y ) linearly into the segment from (x, 0) to ( x , y ) (Figure 5.12). Now define (4,q') if q'$ V for q' E V. e(q' ") = (q, h-l(A(h(q), h(q')))
{
For fixed q E V, as q' E V approaches aV, O(q, q') approaches (q, 4'). It follows that 0 is a well-defined homeomorphism with the desired properties. 0
D"
Y
-
(
D" Figure 5.12
-
aD"
)
THOM ISOMORPHISM THEOREM
243
For any open set U G M denote by U x the pair ux =
(n;l(U), .;'(U)
- d ( M ) ) = (UX M , U x M - d ( M ) )
and in particular M X= ( M x M , M x M - d ( M ) ) .
5.12 LEMMA
-
(i) Hi(MX) = 0 for i < n ; (ii) H o ( M ) H , ( M X ) under the homomorphism sending the O-chain represented by p E M into the relative class represented by C(S(P)), where
IF
H,(M x M , M x M - d ( M ) ) .
: H,(M, M -p )
PROOF First consider the statement of the lemma with a proper n-ball V replacing M. The homeomorphism 8 of Lemma 5.11 induces
8:
V X =
( V X M , v ~ M - - ~ ( M ) ) - .V ~ ( M , M - - ) ,
since d(V) is taken into Vxp. Therefore, the induced homomorphism on the relative homology groups is an isomorphism. Applying the Kunneth formula of Theorem 4.5 to Vx ( M , M - p), Statement (i) follows for V x . Consider the composition H,(V,
v-
-
q ) L H , ( V x M , V x M - d(M)) e.
7 Hn(V x ( M , M - PI) 11
HO(V 0HnW,M
n¶*
%(M, M - PI.
- P)
From Lemma 5.11, Part (iii) we know that n,,8*lP(s(q))= s(p). The vertical isomorphism follows from the Kunneth formula, where Ho( V ) @ H , ( M , M - p ) is the infinite cyclic group generated by {q} @ s ( p ) . These two isomorphisms imply that Part (ii) holds for V replacing M. Finally, we again use an inductive procedure to extend to the general case. Suppose that U and V are open sets in M such that the lemma holds for U , V, and U n V. There is a diagram of Mayer-Vietoris sequences
+ H o ( U n V ) - + H o ( U ) @ H o ( V ) + H o ( U u V)+O+
...
144
5 MANIFOLDS A N D POINCARG DUALITY
where the vertical homomorphisms are those described in Part (ii). Note that if p and q are in the same path component of U, then it follows from Lemma 5.1 1, Part (iii) that f,(s(p)) = /q,(s(q)). Applying the five lemma (Exercise 4, Chapter 2) completes the inductive step and, since M is compact, the proof is complete. 0 We are now ready to prove the following important theorem which has many applications in algebraic topology. 5.13 THEOREM (Thom isomorphism theorem) For a compact oriented n-manifold M without boundary, there is a cohomology class U E H n ( M X ) such that for any coefficient group G the homomorphism @*:
H k ( M ;G ) -+ Hn+k(MX;G )
given by @*(x) =
u u nl*(x)
is an isomorphism. Here the cup product has the form H k ( M x M ; G ) @ H n ( M xM y M X M - A ( M ) ; 2) -+
H ' + k ( M ~M , M XM - d ( M ) ;G).
The class U E H n ( M X )is called the Thom class of the topological manifold M.
NOTE
PROOF We prove the theorem for G = 2. The general case follows by applying the universal coefficient theorem. Since H,(MX) = 0 for i < n, it follows from the universal coefficient theorem that H i ( M X )= 0 for i < nand H n ( M X ) Hom(H,(MX), Z). From Lemma 5.12 we have a natural isomorphism of H n ( M Xwith ) H,(M), hence also of Hom(H,(MX), Z) with Hom(Ho(M), Z). Then we define U E H " ( M x ) to be the class corresponding to the augmentation homomorphism
-
under these isomorphisms. In particular then it follows from Lemma 5.12 that for all p E M the Kronecker index ( U , /&(P)))
For any open set V
cM
=
1.
we denote by U,E H n ( V X )the restriction of
THOM ISOMORPHISM THEOREM
the Thom class U. There is a cap product Hn( v x M ,
vxM
-
d ( M ) ) @ Hk( vx M ,
vx M
- d( M ) )
-
I45
vx M )
sending U , @ a into U, n a. For any a E H k ( V x )define @*(a) to be the V ) given by @*(a)= nlr(U,, n a). Thus element of Hk-n( @*:
H*( V")
-
H*( V )
is a natural homomorphism of degree -n between graded groups. Now restrict to the case of V being a proper n-ball. Under the isomotphism of Lemma 5.1 1
e*:
H ~ V (XM , M - p ) )
-
H~VX),
the element e*-'(U,) may be identified via the Kiinneth formula with 1 @ y E H o ( V )@ H n ( M ,M - p ) , where y is a generator of the infinite cyclic group H n ( M , M - p ) . By the naturality of the cap product
SO if w E H,,(M, M - p ) is a generator with ( y ,
0 )
= 1,
then
n14& n 4 = B, where P @ w
corresponds to a under the isomorphisms
-
Therefore, @*: Hk(V x ) V ) is an isomorphism for each k. If Vl G Vz are open sets in M , the restriction of UV2is U ? , . This fact, together with naturality and the Mayer-Vietoris sequence, may be used in the manner of Lemma 5.12 to extend the result inductively to an isomorphism @*: Hk(M") HkJM).
z
By returning to the chain level we can define the adjoint of @*; this yields a homomorphism @*:
Hi(M) + Hi+n (MX).
146
5 MANIFOLDS AND POINCARB DUALITY
Applying the universal coefficient theorem, we see that Qi* is also an isomorphism. Finally note that
for any x and y, so that Qi*(x) = U
u nl*(x). 0
Example As an aid to understanding this important theorem, consider the following simple example: Let M = S1so that M x M is the two-dimensional torus. Recall from Chapter 4 the determination of the cohomology ring structure in M x M. As before we denote generating I-cycles in M x M by ti and p, and their dual cocyles by Q and (Figure 5.13). It is not difficult to see that M x M - A M has the homotopy type of S1, as is demonstrated in the following deformation (Figure 5.14). Now given an orientation for M = S', there is a generator d E H , ( M ) such that i&) = s(p) for all p E M. Thus, from the commutative B
Figure 5.13
MXM-AM Figure 5.14
POINCARE DUALITY THEOREM
147
diagram
we see that I,(s(p)) = Ipip(6) = i*Ip(6). With a possible change in sign resulting from the choice of orientation, we have /,(a) = E. Thus, the Thom class U E H ' ( M X ) will have the property that 1 = (U,I,s(p))
=
(U,i*(fi))
= (i*
U, C),
From the exact cohomology sequence of the pair MX, H'(MX)
i*
H ' ( M x M ) -L H'(Mx M - AM),
it is not difficult to argue that i* is a monomorphism and j * is an epimorphism. Furthermore, H 1 ( M xM ) is free abelian with basis elements a and B, and the kernel of j * is infinite cyclic generated by a - @. This uniquely determines the Thom class U corresponding to the given orientation. Changing the orientation of M changes the sign of U. Finally consider the Thom homomorphism
@*:
H'(M)
+
H*(MX).
If a is the generator dual to 6, @*(a) = U u nl*(a). Now in H ' ( M x M ) we have (nl*(a),rnE
+ no) = (a, nl.(mE + n&>
= m,
-
so that nl*(a) = 01. Using the isomorphism H2(MX) follows that
i*
H 2 ( M xM ) , it
i*(@*(u)) = i*(U u nl*(a))= i * ( U u a) = i * ( U ) u a =(a-/?)ua=-Bua,
which is a generator of H 2 ( M xM ) . Thus, @*(a) is a generator of H 2 ( M X ) and @* is an isomorphism. At this point we need another very basic property of topological manifolds.
148
5 MANIFOLDS AND POINCA@
DUALITY
5.14 THEOREM If M is a closed topological n-manifold, there exists a topological embedding of M in Rk for some large value of k. Furthermore, under this embedding there exists an open set U about M such that M is a retract of U, that is, there exists a map r : U-. M such that rlM is the identity. PROOF See Appendix 11.
0
5.15 LEMMA For M a closed manifold, there exists a neighborhood N of d ( M ) in M x M such that zllNand n21Nare homotopic as maps from N to M. Applying Theorem 5.14 we embed M in Rk and let U be a neighborhood of M which retracts onto M. Since M is compact there exists an E > 0 such that for any points x, y E M having distance (in Rk)between x and y less than E , the segment from x to y lies in U. It is evident then that any two maps into M with the distance between corresponding points less than E are homotopic in U via the obvious homotopy. Applying the retraction r moves the homotopy into M. Now the projection maps nland nz coincide on d ( M ) in M x M. Again using compactness, there must exist a neighborhood N of d ( M ) in M x M such that the distance between nlINand n z J Nis less than E . It follows then that these restrictions are homotopic in M. 0 PROOF
5.16 LEMMA Define t: M x M -+ M x M by t ( x , y ) = (y, x) and note that t induces a map of pairs t: MX -+MX. Then for x E H * ( M X ;G), t * ( x ) = (-1)"x.
PROOF
First let V be a proper n-ball in M and consider the diagram
Hn(M x M , M x M - d ( M ) )5 Hn(M x M , M x M - d ( M ) )
where the vertical homomorphisms are induced by the inclusion map. Note that Hn(V X V, V x V - d (V)) is infinite cyclic since ( V x V, V x V - d ( V ) ) has the homotopy type of (D", P-l).Furthermore, i * ( U ) is a generator of this group and t*(i*U) = ( - l Y i * ( U ) . Thus, we conclude that t * ( U ) = (-1)"U.
POINCA&
DUALITY THEOREM
219
Now let N be a closed neighborhood of d ( M ) satisfying the requirement of Lemma 5.15. We may further require that N be invariant under t. Then the diagram
H * ( M ) ” I l - H * (M x M )
H*(MX)
is commutative where the composition across the top is @*. Recalling that n2 = n1o t we have
which by Lemma 5.15 =
(-lYj*(U)
u (n&)*(x)
=
(-l)”j*(@*(x)).
Since j * and @* are isomorphisms, the result follows.
0
5.17 PROPOSITION If M is a closed n-manifold, then H , ( M ) is finitely generated.
Using Theorem 5.14 we embed M in a high-dimensional euclidean space Rmso that some open set Nabout M in Rmadmits a retraction onto M , r: N + M. Choose a large rn-simplex sm in Rm so that M is contained in its interior. By the results of Appendix I there exists an integer k so that mesh Sdksmis less than the distance from A4 to Rm - N. Let K be the union of all simplices in Sdksmwhose closures intersect M. Then K is a finite CW complex, M G K C N and the retraction r restricts to a retraction r: K M. By Proposition 2.23 H , ( K ) is finitely generated and by Corollary 1.12 H , ( M ) is isomorphic to a direct summand of H,(K); hence, H , ( M ) is finitely generated. 0 PROOF
--+
We are now ready to prove the main theorem of this chapter. 5.18 POINCARE DUALITY THEOREM For M a compact connected orientable n-manifold without boundary, with orientation s: M -+3 and
I50
5 MANIFOLDS AND POINCA& DUALITY
associated fundamental class z, the homomorphism
D : H k ( M ;G ) ---* H,-p(M; G), given by D ( x ) = x n z, is an isomorphism for each k. PROOF Let 2 be a cycle in S , ( M ) representing
2.
Then there is a homo-
morphism of chain complexes
S'(M; G ) + S,-k(M; G )
D,:
given by cap product with 2. Note that D, commutes with coboundary and boundary operators up to sign and induces D on cohomology and homology. Now let R be a ring with unit, x, y E H * ( M ; R ) and a, t9 E H * ( M ; R). Then there are the elements a x P E H , ( M x M ; R)
and
x x y = nl*(x)u n , * ( y ) ~H * ( M x M ; R).
If U E H n ( M X )is the Thorn class of M , denote by zf the class i*( U ) , where
H n ( M X -+ ) H n ( M xM )
i*:
is induced by inclusion. Let E be a chosen generator for H,(M). The homomorphism i,: H , ( M x M) -+H,(MX) takes E X z into l&(p)). Thus (5.19)
(-I)n(O,
ZXE)
=
(0,EXZ)
=
(i*( U),E X z )
=
( U , /,(S(P)>>
= 1.
If x E H r ( M ; R) and y (5.20)
E
H 8 ( M ;R), then we claim that
0 u ( x x y ) = (-1YU u ( y x x ) .
-
In order to give this meaning we must view o a s an element of H n ( M x M ; R ) . This is done by using the coefficient homomorphism 2 R given by taking 1 into the unit of the ring R. Then in the diagram H n ( M X ;R ) @ H r f 8 ( M XM ; R )
I
U
H n t r f 8 ( M XR; )
i* @ id
H n ( M xM ; R ) @ H r + 8 ( M M ~ ; R ) 2H n + r + a ( MM~; R ) ,
POINCARG DUALITY THEOREM
252
we have (-I)"(/
u ( x x y ) = t*(U u ( x x y ) ) = t*(U) u f * ( x x y ) = (-l)"+'#U u ( y x x ) .
Therefore
it u ( x x y ) = i * ( U ) u id(xxy) = i*(U u ( x x y ) ) - (-l)"i*(U u ( y x x ) ) = (-1)"o u ( y x x ) _.
which proves Equation (5.20). Finally, for x E H k ( M ;R) and a E H k ( M ;R) we have
(it, D ( x ) x a ) = (it, ( x n z ) x a ) = (it, ( x x 1) n ( z x a ) ) = ( ( X X I ) u it, ( z x a ) ) . Then by Equation (5.20)
(0, D ( x ) x a ) = ((1
u it, z x a ) = (it, (1 X X ) n ( z x a ) ) XX)
= (-1)~(0,2x(x,a)
*
E).
It follows from Equation (5.19) that this last expression is equal to ( - l ) n ( n - k ) ( ~a, ) . 1. We summarize these statements in the following important equation :
(5.21)
( x , a) = (- l)n(n-k) ( 0, D ( x ) x a>.
For the special case R = 2, , where p is a prime, the universal coefficient theorem becomes an isomorphism,
Applying this to the above equation we see that x # 0 implies D ( x ) # 0. Therefore D : H * ( M ; Z,) -+ H * ( M ; 2,) is a monomorphism. By Proposition 5.17 these are finite-dimensional vector spaces, so since their dimensions must be the same, D must be an isomorphism.
252
5 MANIFOLDS AND POINCAR6 DUALITY
To extend this result to more general coefficient groups we use the method of "algebraic mapping cylinders" [Eilenberg and Steenrod, 19521. Recall that D, is a homomorphism D,:
Sk(M)
--+
S,-,(M)
and a boundary operator
Then we check that
a}
Hence, { C, , defines a chain complex C. There is a short exact sequence
that determines a short exact sequence of chain complexes (the second homomorphism will be a chain map up to sign only, but this will be sufficient for our purposes). Therefore, we have a long exact sequence of homology groups. What is the connecting homomorphism? Let y E Sn-m+l(M)with Sy = 0. Pick the element (y, 0) E C, that projects onto y. Then J(Y, 0) = (--6Y, D,(Y))
=
(0,D#cv))
and the element in S,-,(M) having this as its image is D,(Y). Thus the connecting homomorphism for the sequence is D and the sequence has the form
- - --+ H,-"(M) *
.-%H,(M)
+ H,(C) + Hn-m+'(M)
D
H,-,(M)
+*
- ..
MANIFOLDS WITH BOUNDARY
253
Now we may identify S k ( M ;2,) with S k ( M )0Z , , so that for each prime p we have a long exact sequence * *
+
Hn-”(M;Z,)
D
w
H,(M; 2,) 4H,(C; 2,) + Hfl-,+l(M; 2,) -+ * * * .
In the sequence, D is an isomorphism wherever it occurs. Hencs, H,(C; Z,) = 0 for all integers m and primes p . But since H,(C) it finitely generated, it follows from the universal coefficient theorem thae H,(C) = 0 for all m. Thus, the first sequence shows D to be an isomorphism for integral coefficients. This same technique shows D to be an isomorphism for G any finitely generated abelian group. Finally, to extend to the general case, the fact that H , ( M ) is finitely generated implies that H * ( M ; G) is the direct limit of { H , ( M ; G’)} where G’ ranges over the finitely generated subgroups of G. Since D commutes with coefficient homomorphisms, we conclude that D is an isomorphism for general abelian groups G. 0 NOTE The essence of the proof is the relationship between the Thom class and the duality homomorphism. Specifically, if x is a cohomology class and a is an homology class, the Kronecker index of x on a is, up to sign, the same as the Kronecker index of the “restricted” Thom class 0 on the external homology product D ( x ) x a.
Since all manifolds are orientable when the coefficient group is Z , , we may duplicate the previous proof to establish: 5.22 THEOREM For M a closed connected n-manifold with Z,-fundamental class z2 E H,,(M; Zz),the homomorphism
D : H k ( M ;2 2 ) -
H,+(M; ZZ),
given by D ( x ) = x n z z , is an isomorphism.
0
We now turn to the relative case, that is, the duality theorem for a manifold with boundary. Therefore, let M be a compact manifold with boundary b’M. The structures that were aefined previously, the local homology groups T, and the orientation covering 3 with projection t, may still be defined for points p E M - b’M. We define ( M , b’M) to be orientable if there exists a continuous map s: M - b’M + 3 with t o s = identity and s ( p ) a generator of T, for each p in M - d M . One of the most useful tools in studying manifolds with boundary is the “collaring theorem” which states that there is a neighborhood of
154
5 MANIFOLDS AND POINCAR6 DUALITY
the boundary which resembles a collar, that is, it is homeomorphic to the Cartesian product of the boundary and an interval. In its topological form it is due to Brown [1962].
5.23 TOPOLOGICAL COLLARING THEOREM If M is a topological manifold with boundary dM, then there exists a neighborhood W of d M in M such that W is homeomorphic to d M x [0, I ] in such a way that d M corresponds naturally with d M x 0. PROOF See Appendix 11.
0
If M is a manifold with boundary, define the “double” of M to be the manifold A formed by identifying two copies of M along 8M. Exercise 8 Show that (MI d M ) is orientable if and only if
A? is orientable.
Exercise 9 Show that if (M, d M ) is orientable, then d M is an orientable manifold without boundary. Is the converse true?
5.24 THEOREM If ( M , d M ) is a compact connected orientable n-manifold with orientation s, then there exists a unique fundamental class z E H,(M, d M ) such that for each p E M - d M , j&) = s(p). Furthermore, if d : H,(M, d M ) -+H,-,(dM) is the connecting homomorphism, then d(z) is a fundamental class of d M , that is, it restricts to a fundamental class on each component of d M .
Figure 5.15
MANIFOLDS WITH BOUNDARY
155
is orientable, there exists a fundamental class 2 E H n ( M ) such that &(i) = S(p) for allp E M . Then we define z to be the image of 2 under the composition
PROOF Since I@
-
Hn(M)
H n ( ~M, - ( M - d M ) )
-
Hn(M, dM),
where the second homomorphism is the inverse of an excision isomorphism. This gives the existence of the desired fundamental class z. Let D be a proper (n - 1)-ball in dM. If W is a collar for d M in M (Theorem 5.23), then under the homeomorphism W d M x I there is an n-cell E corresponding to D x I (Figure 5.15). For any point p in the interior, Do, of the (n - 1)-cell D in d M we have the following diagram:
-
Hn(M,a M )
I
A
Hll-l(dW
i*
i*#
Hn(M, M
,
I
Hn-,(M
-EO)
k* +
A
ki' -
EO).
Hn(E, d E )
-1.
Hn-1
in which each rectangle commutes and the horizontal isomorphisms follow by excision. If q is a point in EO, the factorization
HAM, a
w
j,*
HAMY M - 4 )
H,(M, M
- EO)
and the fact that &(z) is the generator s(q) imply that i*(z) is a generator for the infinite cyclic group Hn(M, M - EO). Thus, there exists a generator z' for Hn(E, dE) such that k,(z') = i*(z). From the diagram k,'d(z') = i,'d(z) and hence the images of d(z) and d(z') in the infinite cyclic group H,-,(M - E O , ( M - EO) - p) must coincide. Since the image of d ( ~ ' is ) a generator, so is the image of d ( ~ ) . Therefore, jp.(d(z)) is a generator of H,_,(dM, d M - p ) and d(z) must be a fundamental class for dM. To prove uniqueness, note first that if W is a collar for d M , then both M and M - d M are homotopy equivalent to M - Wo. Thus
H,(M)
-
H,(M
- dM) =0
256
5 MANIFOLDS AND P O I N C A a DUALITY
by Theorem 5.5. So suppose z and w are fundamental classes in Hn(M,d M ) corresponding to the orientation s. Since O(z) and d(w) are fundamental classes in Hn-,(dM) corresponding to the orientation induced by s on d M , the restrictions of d(z) and d(w) to each component of d M must agree by Theorem 5.10. Thus, z - w is in the kernel of A . By exactness and the fact that H n ( M ) = 0 it follows that z = w. 0
5.25 POINCARGLEFSCHETZ DUALITY THEOREM Let ( M , d M ) be a compact orientable n-manifold with fundamental class z E H,(M, d M ) . Then the duality maps
given by taking the cap product with z are both isomorphisms.
fi let Ml and M2 denote the two copies of M (Figure 5.16). There exists a two-sided collaring N of d M in fi.That is, N is homeomorPROOF In
phic to d M x I, where I = [-1, 11, with d M correspoding to d M x (0). Note that d N is homeomorphic to d M x 81. For i = 1, 2 consider the following diagram: Hk(M)*
j*
Hk(Mi
~
1
nj,(z) N)KHn-k(fi,
&-,(Mi
u N, a(Mi u N ) )
/
Mi*l - NO)
w excision
where Diis defined to make the triangle commute. Since j is the inclusion map, the rectangle commutes by the naturality of the cap product;
aM
N,
ni Figure 5.16
MANIFOLDS WITH BOUNDARY
257
that is, Mj*(x)
n z ) = x n j*(z).
On the other hand, if a E H , , ( I , 31) is a generator, and E is the element of HO(I) having E n a = a, then we have the following diagram:
1
Hk(dMx I )
Hk(dM) _Inii
xo
=
Hk(N)
k
nmxo)
AHn.+(&, k
-
Hn-&1(aM) 7 H,_k(dMX ( I , az))
=
&-NO)
excision
H,-,(N, a N )
where, once again, D is defined to make the diagram commute. The rectangle commutes since (x x E ) n ( A z x a) = (x
n d z ) x ( E n a) = (x n dz) x a.
Note further that since LIZ is a fundamental class for aM, it follows from Theorem 5.18 that 0 is an isomorphism. These homomorphisms may be used to connect the following MayerVietoris sequences :
-
. . . + H (fi)
HyM,
u N ) 0Hk(M2 u N )
where b is given by taking the cap product with 2, the fundamental class for & which is associated with z. It can be checked that each rectangle commutes up to sign. Since D and b are isomorphisms, it follows by the five lemma (Exercise 4, Chapter 2) that D,0D, is an isomorphism. Going back to the first diagram, the fact that D, is an isomorphism implies that
is an isomorphism for each k.
258
5 MANIFOLDS AND POINCARB DUALITY
Finally, it follows from the diagram
H y a M ) + P ( M , a M ) + H"M)
* * * 4
-
--+
* * *
and another application of the five lemma that D : H'(M, d M )
is an isomorphism.
Hn-,(M)
0
Exercise 10 Suppose M is a compact connected oriented n-manifold with dM = M I u M 2 ,the disjoint union of two closed (n - 1)-manifolds. If z E H , ( M , d M ) is a fundamental class, show that there is a suitably defined cap product which yields an isomorphism H~(M, MI)
Hn-k(M,
~
2
)
given by capping with z. In the remainder of this chapter we will give a number of immediate applications of the PoincarC duality theorem. We make no attempt to be complete in this sense, because many of the known facts about manifolds are related to this theorem.
LEMMA If M is a closed connected oriented n-manifold, then H,-,(M) is free abelian.
5.26
PROOF Suppose this is not true. Since H,-,(M) is finitely generated, it
must contain a direct summand isomorphic to Z , for some integer p > 1. Thus Hn(M)= Z and Hn-,(M) = Z , @ A for some abelian group A. By the universal coefficient theorem
-
Hn(M; Z p ) = Hn(M) 0 0Tor(Hn-l(M), Zp) Z, @ Z, @ Tor@, Z,). Now from our previous observations we know that the inclusion map induces a monomorphism H , ( M ; Z,)
--+
H,(M, A4 - X; Z , )
Z,
APPLICATIONS
159
for any point x E M . This implies the existence of a monomorphism
which is impossible. Thus, H,-,(M) is free abelian.
0
It follows immediately from Lemma 5.26 that if M is a closed, connected, oriented n-manifold, EXt(H,-,(M), 2 ) = 0
so that H n ( M ) = Hom(H,(M), Z ) ,
which is infinite cyclic. If z E H , ( M ) is the fundamental class corresponding to the given orientation, define a E H n ( M ) to be the "dual" of z in the sense that ( a , z ) = 1, so a is a generator for H n ( M ) . For any integer q define a pairing
by sending x @ y into the integer (x u y , z ) . Note that if r . x = 0 in H Q ( M )for some integer r # 0, then (r x) u y = r (x u y ) = 0; hence, x u y = 0 because H n ( M ) is infinite cyclic. Similarly x u y = 0 if y has finite order. On the other hand, suppose that X E W ( M ) does not have finite order. From the universal coefficient theorem the homomorphism
sending x into the homomorphism w + (x, w } , must take x into a nontrivial homomorphism. Thus, there exists an element w E H J M ) with (x, w} # 0. Furthermore this is a split monomorphism, so that if x generates a direct summand of Hq(M), then there exists an element w E H , ( M ) with (s,w) = 1. Now by the Poincark duality theorem there is an element y E Hn-'J(M) with y n z = w. Then
and so x
u y # 0. This completes the proof of the following:
5.27 PROPOSITION If M is a closed connected oriented n-manifold and A , c Hq(M) is the torsion subgroup, then there is a nonsingular dual
5
260
MANIFOLDS AND POINCAR6 DUALITY
5.28 COROLLARY If a E H2(CP(n))is a generator, then as EH"(CP(~)) is a generator for 1 5 k 5 n. PROOF CP(n) is an orientable, compact, connected 2n-manifold whose cohomology is given by for rn even, 0 5 m 5 2n HW(CP(n)) = otherwise.
{ 0"
We prove the result by induction on n. It is obviously true for n suppose it is true for n - 1 2 1, and consider the inclusion i:
CP(n - 1)
5
=
I , so
CP(n)
of the finite subcomplex which contains all cells of CP(n) except the one cell of dimension 2n. From the exact sequence of the pair *
.-+
H"(CP(n), CP(n
- 1))
-+
H"(CP(n))
2HZk((@P(n- 1))
-,H2k-1 (CP(n), CP(n
- 1))
+ * * *
we see that i* is an isomorphism for 2k < 2n. Since i*(a) generates H2(CP(n - l)), the inductive hypothesis implies that [i*(a)lk generates Hzs(CP(n - 1)) for all k < n. Now i* is a ring homomorphism, so ak must generate H"(CP(n)) for k < n. Finally, by Proposition 5.27 there is an element b E H2n-2(CP(n)) with a u b generating H"(CP(n)). This b must generate Hzn-2(CP(n)) so that b = &an-]. Therefore, a u b = f a n is a generator of HZn(CP(n)).This completes the proof. 0 Note that this completely describes the structure of the cohomology ring of CP(n).
5.29 COROLLARY H*(CP(n)) is a polynomial ring over the integers with one generator a in dimension two, subject to the relation an+' = 0. 0 Now let R be a field and M be a closed, connected, oriented manifold. As we observed previously H n ( M ; R ) s Hom,(H,(M; R),R )
APPLICATIONS
and ,FI”(M;R )
-
R,
H,,(M; R )
-
I62
R.
Denote by zfl E H,,(M; R) a generator as an R-module. Then a slight variation of the PoincarC duality theorem states that the homomorphism
given by sending a into a n z R ,is an isomorphism. The technique of Proposition 5.27 may now be used to prove the following. 5.30 PROPOSITION The pairing H * ( M ; R) @ Hn-Q(M;R ) -* R given by sending x @ y into (x u y , zH) E R is a nonsingular dual pairing. 0
5.31 COROLLARY If M is a closed, connected n-manifold, then
is a nonsingular dual pairing.
0
5.32 COROLLARY If a E H’(RP(n); Z , ) is a generator, then an:generates H k ( R P ( n ) ;Z , ) for 1 5 k 5 n. Thus, H*(WP(n); Z , ) is a polynomial ring over Z , with one generator a in dimension one, subject to the relation an+’ = 0- 0 Similar arguments may be used to compute the cohomology ring of quaternionic projective space HP(n).
5.33 COROLLARY H*(HP(n)) is a polynomial ring over the integers with one generator a in dimension four subject to the relation an+’ = 0. 0 With these results we may now establish the existence of certain maps having odd Hopf invariant (see Chapter 4). 5.34 COROLLARY There exist maps S3 + S2, s7 having Hopf invariant 1.
--f
S4, and
S15+ Sa
PROOF Let f : S3 + S2be the Hopf map of Chapter 2. Recall that S,2 is homeomorphic to CP(2). So if b is a generator of H2(S,2)and a is a generator of H4(Sf2),then bB = &a by Corollary 5.29. Thus, H ( f ) = *l. Now the results of Exercise 8 (i) of Chapter 4 indicate how to alterf, if necessary, to give a map with Hopf invariant 1.
262
5 MANIFOLDS AND P O I N C A e DUALITY
The cases 5" -+ S4 and S15-+SS follow by applying the same approach to the quaternions and the Cayley numbers, respectively. 0 In order to develop some further applications we must introduce some basic facts about bilinear forms. Let V be a real vector space of finite dimension. A bilinear form
is nonsingular if @(x, y )
=0
for all y in V implies x
= 0.
Exercise I I Show that this is equivalent to requiring that @(x, y ) = 0 for all x in V imply y = 0. The form @ is symmetric if @(x, y ) = @(y, x); it is antisymmetric if @(x, y ) = -@byx), for all x and y in V. Example Let V = W 2 and denote its points by (x, y). Define
This is a nonsingular, antisymmetric bilinear form. Given any bilinear form @ on V x V we may write @ uniquely as @ = @' W , where @' is symmetric and @" is antisymmetric. To see this we set @/(x, r) = +[@(x,Y ) @b,41
+
+
and @'/(x, Y ) = 3[@(x, v) - @(AX I 1 '
Let @ be antisymmetric on V x V. Then @(x, x) = 0 for all x E V . If x1 E V has @(xl,y ) # 0 for some y, then obviously there exists an element y1 in V with @<XlY
Yl) = 1*
Define Vl to be the subspace of V given by Vl = {x E V I @(x, xl)
=0
and @(x, yl)
= O}.
This linear subspace may be identified with the kernel of the linear transformation 8 : v+w=
APPLICATIONS
263
given by O(x) = (@(x, x,), @(x,n)).Note that since O(x,) = (0, I ) and O(yl) = (-1, 0), the transformation is an epimorphism. Therefore, the dimension of Vl is equal to dimension V - 2. By repeating this process using the subspace V, to produce a subspace V,, and so forth, we will eventually either exhaust V or reach a subspace with the property that the product of any pair of its vectors is zero. Thus, there is a basis for V of the form
for which @(xi, yi) = I have product zero.
=
-@(yi, xi), and any other pair of basis vectors
5.35 LEMMA If @: Vx V + R is nonsingular and antisymmetric, then the dimension of V is even. PROOF This follows from the above, since s = 0.
0
5.36 COROLLARY If M is a closed, oriented manifold of dimension 4k 2, then z ( M ) is even.
+
PROOF
Recall that the Euler characteristic is given by 4k+2
x ( M ) = C (-l)i dim Hi(M; W). i-0
By the universal coefficient theorem this may also be expressed as the sum 4k+2
x(M) =
(-l)i dim Hi(M; R). i-0
Since M is closed and oriented, the PoincarC duality theorem implies Hi(M; R)
H,k+2-i(M;R)
-
Hom(H4k+2-i(M;R), W).
Therefore, dim H i ( M ; W) = dim Hgk+,-'(M;R). As a result the entries in the second sum are paired up, except in the middle dimension, so that
C i*2k+1
is even.
(-l)i dim Hi(M; R)
264
5 MANIFOLDS AND POINCARJ? DUALITY
Finally note that there is a bilinear form
defined by @(x,y ) = (x u y, zB) E R. By Proposition 5.30 this is nonsingular, and since x and y are both odd dimensional, it is antisymmetric. Thus, by Lemma 5.35 the dimension of HZk+'(M;R) is even and x ( M ) is even. c] 5.37 COROLLARY If M is a closed manifold of dimension 2k then x ( M )= 0.
+ 1,
Since Hi(M) is a finitely generated abelian group for each i, we may write H , ( M ) Ai 0Bi 0ci,
PROOF
-
where A i is free abelian of rank r i , Bi is a direct sum of si cyclic groups of order a power of two, and Ci is a direct sum of cyclic groups of odd order. Note that 2k+l
X ( M ) = 2 (-1)iri. i-0
By the universal coefficient theorem dim H d M ; Z , )
= dim(Hi(M) @ Z,) =
(r
+ + Si)
+ dim(Tor(Hi-l(M), Z,))
(Si-I).
Thus 2ktl
Pk+l
(-IIidim Hi(M; Z , )
=
i-0
1 (-1),[ri + si + siAl]
i-0
c
2k+l
=
=
X(M).
i-0
On the other hand, by the PoincarC duality theorem
Since i and 2k
+ 1 - i have different parity, these appear in the sum with
APPLICATIONS
165
opposite signs. Therefore 2k+l
x ( M ) = C (-l)i dim H i ( M ; 2,) = 0.
0
i=O
This result is obviously false for even-dimensional manifolds since 1. The vanishing of the Euler characteristic is a useful fact in differential geometry, as is seen in the following basic theorem: a closed differentiable manifold M admits a nonzero vector field if and only if x ( M ) = 0. Thus, Corollary 5.37 implies that any odd-dimensional, closed, differentiable manifold admits a nonzero vector field. We shall return to this subject in Chapter 6. NOTE
x ( S ~=~ 2, ) x(RP(2n))= 1, and z(CP(n))= n
+
Now suppose that 0:V x V - t R is symmetric. Then since @(x-ty , x
+y)
=
@(x,x)
+ W x ,Y ) + @(Y, Y )
it follows that if @ is nontrivial, there exists an x, E V with @ ( x , , x,) # 0. We may as well assume @(x,, x,) = f1 . Consider the homomorphism a: V - R
given by U ( X ) = @(x,x,). This is an epimorphism since a ( x l ) = f l , SO if Vl is the kernel of a, the dimension of V , is one less than the dimension of V . By applying the same analysis to Vl to give an element x 2 , and continuing the process, we produce a basis for V which may be renumbered so as to have the form x, ,. , . , x,, x,+, ,. . . , x,+*, x,+,+~,. . . , x r f s f t , where 1
-I 0
if l s i s r if r < i 5 r s if r + s < i < r + s + t ,
+
and any other pair has product zero. Exercise 12 Show that the numbers r and s are invariants of the symmetric form @; that is, that they are independent of the various choices made.
The signature of a symmetric form @ is the integer r - s. If @ is an arbitrary bilinear form, then we write Q, = @' + @" with @' symmetric and
166
5 MANIFOLDS AND POINCARfi DUALITY
0”antisymmetric. We then define the signature of @ to be the signature of @’. Let M be a closed, oriented n-manifold. Define the index of M , denoted t ( M ) , as follows: (i) T ( M )= 0 if n f 4k for some integer k ; (ii) if n = 4k, let t ( M ) be the signature of the nonsingular symmetric bilinear form
Let M be a closed, connected oriented 4k-manifold. Define a bilinear form Exercise 23
Y: H * ( M ; R) x H * ( M ; R)+ R
by “(4Y ) = ((x
u Y)4‘
9
ZR) E
R,
where (x u y)lk is the 4k-dimensional component of x that the signature of Y is t ( M ) .
u y.
Then show
Exercise 14 Let M, and M2 be disjoint, closed, connected oriented manifolds.
(a) Show that the manifold M, x M , may be oriented in such a way that
t ( M , x M,)
=
t ( M , ) - t(M,).
(b) If M , and M , have the same dimension, show that
Note that a change in the orientation of a manifold merely changes the sign of its index. It is evident that the index of CP(2k) is f l , depending on the choice of orientation. Thus, it follows from the above exercise that there exist 4k-dimensional manifolds of arbitrary index for all values of k. The final question we would like to consider is the following: given a closed topological manifold M, when does there exist a compact manifold W with M = a W? Of course we must require that W be compact since M is always the boundary of M x [0, 1). Our first result gives a necessary condition for M to be such a boundary.
167
APPLICATIONS
5.38 THEOREM If W is a compact topological manifold with d W then x ( M ) is even.
=
M,
If the dimension of M is odd, then x ( M ) = 0 by Corollary 5.37. Thus we assume that the dimension of M is even so that the dimension of W is odd. Now consider the manifold W x I (see Exercise 15), where I = [0, 11. We have PROOF
d ( W x I ) = M x I u W x d I = M x I u Wx(0)u Wx(1). Define U = d( Wx I) - W x (1 } and V = d ( W x I ) - W x (0). Note that U and V are open subsets of d ( W x I ) and W , U, and V all have the same homotopy type, whereas U n V has the homotopy type of M . The Mayer-Vietoris sequence for U and V becomes H,+,(d( Wx I ) ) 2Ha(M) LHi( W ) 0Hi( W ) 5 H;(d(W x I ) ) , where each group is finitely generated and zero in dimensions greater than the dimension of W . From the exactness we see that rank(Hi(M))
=
rank(image hi+,)
+ rank(imagefi),
+ rank(image gi), rank(Hi(d( Wx I)))= rank(image gi)+ rank(image hi).
rank(Hi( W ) 0Hi( W ) ) = rank(imagejJ
By multiple cancellations it follows that
C (-l)i rank(Hi(M)) - C ( - l ) i rank(Hi(W) @ H i ( W ) ) + C (-ly rank(Hi(d(Wx I)))= 0. Since a( WX I) is an odd-dimensional, closed manifold, we have x(d( Wx I))= 0 by Corollary 5.37. Therefore
x(M) = 2
- C (-ly
rank Hi(W)
=2
- x(W).
0
Exercise I5 Suppose M , and M, are topological manifolds. Then show that M , x M , is a topological manifold with
d(M, x M,)
=
(dM,)x M,
u M , x (dM,).
Note that as an immediate consequence of Theorem 5.38 we have many manifolds which are not boundaries of compact manifolds, for example, RP(2k) and CP(2k).
I68
5 MANIFOLDS AND POINCAR6 DUALITY
A necessary condition for a closed manifold M to bound a compact oriented manifold is that the index of M be zero. In order to prove this we will need the following: 5.39 LEMMA Suppose @ is a symmetric, nonsingular bilinear form on a vector space V of dimension 2n, and { x , ,. . . , x n }is a linearly independent aixi, C bjxj) = 0 for any coefficients a , , . . . ,a,, set in V such that b l , . . . , b,. Then the signature of @ is zero.
@(x
PROOF In the decomposition described previously, it is evident that t must be zero since @ is nonsingular. We must show that r = s = n. We will prove inductively that r 2 n ; a similar argument establishes s 2 n, from which the conclusion follows. For n = 1, there exists an element y1 in V with @ ( x l ,y,) # 0. Then
@(Yl
+ ax1 Yl + ax,) = @ ( Y l , y,) + 2a@(x,, Y,) Y
so by choosing
+
+
we have @(y, a x , , y, ax,) = 1 and r 2 1 = n. Now suppose the assertion is true for vector spaces of dimension 2(n - 1). Define a homomorphism 0: V+Rn by 0 ( z ) = (@(xl,z ) , . . . ,@ ( x n ,z ) ) . If 0 is not an epimorphism, then the dimension of the kernel of 0 is 2 n 1. On the other hand, we may extend the linearly independent set to a basis {xl,. . . ,x,, ol,. . . ,w n } for V and define 0': V+R"
+
by 0'(z) = (@(wl,z), . . . ,@(con,z)). The dimension of the kernel of 0' is 2 n ; hence ker 0 n ker 0'f 0. But this cannot happen since @ is nonsingular, so 0 must be an epimorphism. Let y , E W ( l , O , . . . ,0). As before, there exists a real number a with @(yl ax,, y1 ax,) = 1. Now define
+
+
!P:
v+w2
APPLICATIONS
269
by Y(z) = ( @ ( x 1 ,z ) , @(yl,2)) and note that Y is an epimorphism. If V’ is the kernel of Y, then the restriction of @ to V’ is a nonsingular form and ( x z , . . . ,x,} is a linearly independent set in V’ satisfying the hypothesis. Thus, by the inductive hypothesis there exist vectors q 2 , . . . , q, in V’ with @(qi, q j ) = dij. The collection y, a x , , q 2 , . . . , q, then shows that rzn. 0
+
THEOREM If M is a compact oriented (4n boundary, then the index of d M is zero.
5.40
+ 1)-manifold with
Denote by @ the symmetric nonsingular bilinear form on H z n ( d M ;R).We will show that the signature of @ is zero by proving that the image of j * : H z n ( M ;R)+ H 2 ” ( d M ;R)
PROOF
-
is a subspace of half the dimension of H z n ( a M ;R)on which @ is identically M is the inclusion. zero, where j : d M Let zR E H4n+l(/M,a M ; R ) be a fundamental class and take d z R E H4,(i3M; R ) to be the fundamental class given by the image of Z, under the connecting homomorphism. If j * ( a ) and j * ( @are two elements of H z n ( d M ;R) in the image of j * , then
So @ is identically zero on the image of j * . Consider the commutative diagram
as in the proof of Theorem 5.25, where D and D’ are PoincarC duality isomorphisms. Then since D( j*(u)) = A ( D ’ ( u ) ) ,it follows that the image of j * is isomorphic to the image of A . Thus, the dimension of the image of j * is the same as the dimension of the kernel of j , .
170
5 MANIFOLDS AND POINCAR6 DUALITY
On the other hand, since R is a field, the universal coefficient theorem gives a commutative diagram H%(M; R)
Hom(H,,(M; R),W)
1
1
H2fl(dM;W) -% Hom(H,,(dM; R),R)
in which the horizontal maps are isomorphisms. Then it is easily checked that the dimension of the image of j * is equal to the dimension of the image of j , . Putting these together we have 2 . dim im j *
=
dim ker j ,
= dim
+ dim im j ,
H,,(dM; R)
= dim H"(dM;
R).
Thus, the image of j * is a subspace of H2n(dM;R) of half the dimension. It follows from Lemma 5.39 that the index of d M is zero. 0 Note that Theorems 5.38 and 5.40 give certain necessary conditions for closed manifolds to be boundaries of compact manifolds of one dimension higher. These conditions are more closely related than may be readily apparent. 5.41 PROPOSITION If Mfl is a closed oriented manifold, then
t ( M ) = x ( M ) mod 2. This is clear if the dimension of M is odd since x ( M ) = 0 = t ( M ) . If dim M = 2 mod 4, then by Corollary 5.36 x ( M ) is even, hence congruent to t ( M ) mod 2. If the dimension of M is 4k, then x ( M ) = dim H2k(M;R) mod 2. On the other hand, t ( M ) = r - s, where r s = dim H"k(M; R). Thus x ( M ) - r ( M ) = 2s = 0 mod 2. 0 PROOF
+
From considering such examples as 9"or CP(n) it is apparent that the congruence in Proposition 5.41 cannot be replaced by equality. "Index" invariants of this type for manifolds are very important in algebraic and differential topology. Of particular interest is their connection with analysis, which arises from the analytical interpretation of cohomology
APPLICATIONS
I71
groups via the theory of Hodge and de Rham. Much significant progress has been made in recent years in relating geometrical invariants to such analytical invariants as the indices of differential operators. The results of Theorems 5.38 and 5.40 introduce us to another area of considerable current interest. Let M" be a closed, oriented manifold. Mfi is said to bound if there exists a compact, oriented manifold Wn+l and an orientation-preserving homeomorphism of M n onto d Wn+l.Note that it is essential to require Wn+' to be compact, as M n will always bound M n x [0, 1) if [0, 1) is properly oriented. Two closed, oriented n-manifolds Mln and MZnare oriented cobordant, M I n - Mzn, if the manifold given by the disjoint union MIn u -M,n bounds a compact (n 1)-manifold Wn+l,where -M2n is the manifold given by reversing the orientation on M2n. The manifold Wn+l is called a cobordism between Mln and This defines an equivalence relation on the class of closed oriented n-manifolds. To see this, note that M n M n because Mn u - M n is homeomorphic to the boundary of M n x [0, 1 ] by an orientation-preserving homeomorphism. To establish transitivity we glue two cobordisms together (Figure 5.17). That is, if B Wn+l= M I nu - Mznand d Vn+l = M2n u -M3n then by identifying Wn+land Vn+l along the common copy of Mzn we get a compact oriented manifold with boundary oriented homeomorphic to
+
-
M,n
u
-M3R.
Let [ M n ] be the equivalence class represented by M n . Denote the set of equivalence classes by We define an additive operation in by setting [ M I n ] [M,"]= [ M I nu M z n ] , the equivalence class of the disjoint union. This gives the structure of an abelian group in which - [ M n ] = [-M"] and the additive identity is the equivalence class of those
+
Figure 5.17
272
5 MANIFOLDS AND POINCARB DUALITY
manifolds which bound. The graded group
n-0
may be given the structure of a commutative graded ring by defining [M,n] [M,m] = [Minx MZm],the unit being the class of a positively oriented point in 9lX,STOP. is the oriented topological cobordism ring. If in the previous discussion we omit all references to orientation, there result the unoriented topological cobordism groups 91zop. Denoting the unoriented equivalence class of the closed manifold Mfl by [ W ] , , it is apparent that 2 . [MflI2= 0 since M n u M n bounds Mfl x [0, 13. Thus the unoriented topological cobordism ring 9QoPbecomes a 2,-algebra. Define mappings Y: 97,LQ;OP .+ Z and Y,: %Top + Z , by Y([Mn]) = r(Mn), the index of Mn, and Y2([Mn],) = x(Mfl) reduced mod 2. By Theorems 5.38 and 5.40 these are well-defined functions of the respective cobordism classes. Furthermore, since both invariants are additive over disjoint unions and multiplicative over Cartesian products, Y and Y, are ring homomorphisms. A closed, oriented 0-manifold is a finite collection of points, each given a positive or negative orientation. It bounds if and only if the “algebraic” sum of the points is zero. Similarly an unoriented 0-manifold bounds if and only if it consists of an even number of points. Thus, Y : 91X,SToP -+ Z and Y,: %fop + Z , are both isomorphisms. Note that since any closed 1-manifold is homeomorphic to a finite disjoint union of circles, it follows immediately that 9 y O P = 0 = %:TOP.
-
Exercise 16 From the classification of closed 2-manifolds, compute 9lZop and 91(29TOP. It is evident that since each WP(2n) has Euler characteristic equal to one, !P2:%c,T,.P + Z , is an epimorphism for each n. Similarly each CP(2n) has index f1 so that Y: 91izop-+ Z is an epimorphism for each n. The structure of these rings has remained a mystery for some time. Recently the ring 91x’fophas been determined in all dimensions f 4 by Brumfiel et al. [1971], using results of Kirby and Siebenmann [1969]. As an excellent further reference in this area we recommend Stong [1968].
chapter 6 FMED-POINT THEORY
We are interested in studying the behavior of continuous functions on manifolds with particular interest in detecting the presence or absence of fixed points or coincidences. This is a classical problem, so it may prove enlightening to take a brief look at some of its early development. During the 1880s, Poincart studied vector distributions on surfaces. For an isolated singularity of such a distribution he assigned an index which was an integer (positive, negative, or zero). A vector distribution may be interpreted as a map of the surface to itself by translating a point via the vector based at that point. Here the fixed points of the map are the singularities of the distribution. Thus, summing the indices of the isolated singularities was the first step toward “algebraically” counting the fixed points of a map. Poincart proved that if the surface is orientable of genus p and the distribution has only isolated singularities, then the sum of the indices is 2 - 2p. At the beginning of the twentieth ‘century, Brouwer defined the degree of a mapping between n-manifolds. This allowed him to prove his fixed point theorem for mappings of Dnas well as to extend from 2 to n dimensions the definition of the index given by Poincart. One of his important results was: i f f and g are homotopic mappings of an n-manifold to itself and both f and g have only a finite number of fixed points, then the sum 173
274
6 FIXED-POINT THEORY
of the indices of the fixed points is the same for both functions. Since every mapping can be deformed into one with only a finite number of fixed points, this produces a homotopy invariant for the “algebraic” number of fixed points. In 1923 Lefschetz published the first version of his fixed-point formula. Let M be a closed manifold and fi M - + M be a map. Then for each k there is the induced homomorphism on homology with rational coefficients
For each k we may choose a basis for the finite-dimensional, rational vector space H k ( M ;Q) and write& as a matrix with respect to this basis. Denote by tr(&) the trace of this matrix. If we define the Lefschetz number off by m
then L ( f ) is independent of the choices involved and hence is a welldefined, rational-valued function ofJ It is evident that L( f ) depends only on the homotopy class off. To see how this is connected with the earlier work of Brouwer, consider the case of a closed orientable manifold. Lefschetz proved the following: for each E > 0 there is an &-approximationg to f (here we are assuming a metric on the manifold) such that (i) g has only a finite number of fixed points, and (ii) for each fixed point x of g, g takes some neighborhood of x homeomorphically onto some other neighborhood of x . If xl,. . . ,x, are the fixed points of g, denote by al ,. . . ,a, the local degrees of g at these points in the sense of Brouwer. Then Lefschetz showed that
Now for
E
small, f and g are homotopic; hence, f *
=g,
and we have
This implies that L( f ) is always an integer and leads to the celebrated Lefschetz fixed-point theorem: if L( f ) # 0, then f has a fixed point. The idea of the proof is as follows: in the product space M x M consider the diagonal d(M) (Figure 6.1). Denote by G ( g ) the graph of the function g. The points of d(M) n G(g) correspond to the fixed points of g. The
INTRODUCTION
275
previously stated process of approximating f by g corresponds to making slight changes in G ( f ) in order to put it into “general position” with respect to d ( M ) . Here we see that the ai have the proper interpretation, determined by that particular intersection of the graph with the diagonal. Considering d ( M ) and G ( g ) as n-dimensional chains in M x M , Lefschetz computed their intersection number and showed it to be the trace formula. As a special case we may takefto be the identity map. Then L ( f ) = x ( M ) , the Euler characteristic of M . If M is a connected, differentiable manifold which admits a nonzero vector field, we may interpret this as before as a map homotopic to the identity but having no fixed points. Thus, L(f) = 0 and this implies x ( M ) = 0. The classical theorem of Hopf is the converse of this, that is, that if x ( M ) = 0 then M admits a nonzero vector field. Generalizing the previous, iff and g : M , M , are maps between closed oriented n-manifolds, a coincidence off and g is a point x E M , such that f ( x ) = g ( x ) . Geometrically, if G ( f ) and G ( g ) are the graphs of the respective functions in M , x M , , their points of intersection correspond to the coincidences. From the diagram
-
Hq(M1; 8)
f.
H“-q(M1;Q)
M
Figure 6.1
Hq(M2;
Q)
H”-9(M2;Q)
276
6 FIXED-POINT THEORY
where the vertical homomorphisms are Poincark duality isomorphisms, we define
-
0,: q M 1 ; Q)
Hg(M1;
Q)
to be 0, = ,ug*v-tf,. Then the coincidence number off and g is given by
uf, g) = c (-- 11, tr(@,). n
P O
As before, L ( f , g ) is the intersection number of G ( f ) and G(g), so if L ( f , g ) # 0, then f and g have a coincidence. Note that if M , = M , and g is the identity, then L ( f ,g ) = L( f ). In this chapter we will prove these major results in the framework of the previous chapters. We will do this by first defining the coincidence index and the fixed point index and establishing their basic properties. By introducing certain characteristic cohomology classes we establish the link between these indices and the corresponding coincidence numbers and Lefschetz numbers. In the process we encounter the Euler class and show that when evaluated on the fundamental class it yields the Euler characteristic. The principal tools used are the Poincark duality theorem and the Thom isomorphism theorem. We close with some applications and observations. It should be pointed out the spaces we consider, closed, oriented manifolds, could be made much more general. Similar techniques may be applied in the nonorientable case by using twisted coefficients. Many of the theorems are valid for such spaces as euclidean neighborhood retracts [Dold, 19651. We impose these restrictions both for the purpose of continuity with the previous material and so that the reader may easily grasp the fundamental ideas involved. Many of the techniques of this chapter have been evolved from the excellent papers of Dold [1965] and Samelson [1965].
Let M , and M , be closed, connected, oriented n-manifolds with fundamental classes zi E H,(M,), and corresponding Thom classes U i € H n ( M i x M i ,M i x M i - d ( M , ) ) ,
i = 1, 2.
Suppose W is an open set in M I and
are maps for which the coincidence set C = {x E W I f ( x ) = g(x)} is a compact subset of W.
COINCIDENCE INDEX
277
By the normality of M I there exists an open set V in M I with C E V c V c W . Define the coincidence index of the pair (f,g) on W to be the integer 1; given by the image of the fundamental class z1 under the composition
=
H,(M2x M,, M z x M,
- d ( M , ) ) = 2.
Here the map (f,g): W + M , x M z is given by (f,g)(x) = (f(x), g(x)), and the identification
is given by sending a class a into the integer ( U , , a). That this is an isomorphism follows from the fact in Equation (5.19) that for p E M , , u, I p ( 4 P ) ) ) = 1. It must first be shown that this definition is independent of the choice of the open set V. Suppose V ’ is another open set with C C V‘ C Y’C W. Then consider the following diagram:
<
9
Hfl(M1, MI - V )
w
Hfl(W,
w - V)
Here, as in Chapter 5, MZXdenotes the pair ( M , x M , , M , X M , - d ( M , ) ) . Since each triangle and rectangle commutes, the images of z1 across the top and across the bottom must be the same. This shows that I; is independent of the choice of V .
-
Exercise I Let W‘ be another open set in M I and f ’ and g‘: W ’ M , be maps such that f =f ’ and g = g’ on W n W‘ and
C’= {x E W’ I f ’ ( x ) = g’(x)} is equal to C. Then show that
278
6 FIXED-POINT THEORY
This exercise tells us that the coincidence index is completely determined by the behavior of the functions around the coincidence set. In this sense it may be viewed as a local invariant. It is of particular interest then to see how the later results will amalgamate these local invariants into a global invariant. u Wk is a disjoint union of open Now suppose W = W, u W , u sets and denote by Ci the compact set C n Wi and by fi and gi the restrictions off and g to Wi . 6.1 LEMMA
That is, the coincidence index is additive. PROOF For each i choose an open set Vi such that Ci G Vi 5 pi E Wi and set V = Uf& Vi. Then the result follows from the commutativity of the following diagram:
More generally, for any W let C = C , u - .. u Ckbe a decomposition of C into disjoint compact sets. Then by repeatedly using the normality of M I we can find a disjoint collection of open subsets of W, denoted W,,. . . , W,, such that Ci E Wi for each i. Setting W ' = U Wi we can apply Lemma 6.1 and Exercise 1 to conclude that
6.2 LEMMA If C
=
0, then
I E = 0.
PROOF Suppose f and g have no coincidence points in the open set W. Then for any V, the map ( f , g ) : (W, w -
n-+( ~
, x ~ z , ~ z d(M2)) x ~ z
I79
COINCIDENCE INDEX
can be factored through the pair ( M , x M , - A ( M , ) , M , x M , - d ( M , ) ) so that the induced homomorphism ( f , g ) * must be zero. 0 . 6.3 COROLLARY If I E f 0, then f and g have a coincidence in W. 6.4 LEMMA Suppose and denote by
fE
and g,: W -+ M , , 0 5 t 5 1, are homotopies
ct = {x E w I for 0 5 t 5 1. If D
=
0
U, C,is a
h ( X ) = gt(x>)
compact subset of W , then
for 0 5 t 5 1 give a homotopy of maps of pairs; hence
(h, go)* = (fi
9
a)*
and
W
W
Ifo,go = If*.gl.
0
Exercise 2 Suppose M,’ and M2’are closed, oriented m-manifolds and f ’ and g ’ : W ‘ + M,’ are maps, where W‘ is open in MI’. If
C‘= {Y E W‘ I f ’ ( Y ) = g ’ ( Y ) ) is a compact subset of W‘, show that the coincidence index Z ~ ~ ~ isx g , defined and is equal to (Z&,).
(ZE)-
As a special case of this construction we may take M , = M 2 (denoted by M ) and g = identity on the open set W . Here a coincidence off and g is merely a fixed point off. For this reason the coincidence index is denoted I,”’ and called thejxed-point index o f f on W. For convenience we restate the previous results in terms of the fixed-point index.
ZEd
6.5 LEMMA Iff ’: W‘ + M is a map from an open set W‘ in M such that f = f ’ on W n W‘ and the fixed-point sets off and f ’ are the same compact subset of W n W ‘ , then I”, = IF’. 0 6.6 LEMMA If W = W,u W, v subsets of M , then
.. u
W, is a disjoint union of open
280
6 FIXED-POINT THEORY
6.7 LEMMA If
# 0, then f has a fixed point in W. 0
6.8 LEMMA If fi: W
D=
{XE
--+
M is a homotopy for which the set
WI f t ( x ) = x
is compact, then Z I = Z z .
for some O S t l l }
0
For most of the cases we will consider, the open set W will be the entire manifold Ml. In this case we may choose the open set V to be Ml so that the coincidence index becomes the image of the class z1 under the homomorphism (f,g)*: Hn(M1) Hn(MzX) 2. +
When this occurs, the coincidence index is denoted by If,gand the fixedpoint index by Z,,
Example Suppose MI and M , are closed, connected, oriented n-manifolds, p E M , , f ( M , ) = p, and g has the property that g*: Hn(M1)
--+
Hn(M,)
is given by g,(zl) = m 2,. We want to determine the coincidence index If,g. To do this consider the following diagram:
The definition off allows us to factor (S,g)* through the upper rectangle. The commutativity of the other portions follows by using the natural identifications. Thus
In particular, if Ml = M , and g is the identity, then If = 1.
LOCAL DEGREE
181
As another example, suppose M, = M , = M and f: W M has a single fixed point p E W. We want to give another interpretation of Ifwfor this case. First, working in euclidean space, let Dn denote the closed unit disk in Rn. Define a map F: D n x D n + D n -+
by F(x, y ) = & ( y- x). This may be viewed geometrically in P x Rn as first taking the element of (0) x an which is equivalent to (x, y ) modulo the linear subspace A(Rn) and then multiplying by 3 (Figure 6.2). Note that this map induces a homotopy equivalence of pairs
F : (D" x Dn, Dn x Dn - A ( D n ) )+ (D", Dn - 0).
To see this define j : Dn -+ D n x Dn by j ( w ) = (0, w). The homotopy h,: Dn -+ Dn given by h,(w) = w/(l t) has h,(w) = w , h,(w) = F o j ( w ) , and h,(Dn - 0) E Dn - 0 for 0 5 t 5 1. On the other hand, define (1 - t ) x y - t x g,: D n x D n + D n x D n by g t ( x , y ) =
+
(
+t) ,
+
It is easily checked that both coordinates lie in Dn. Then go(& Y ) = ( x ,v),g,(x, Y ) = (0,4(Y - X I )
=j
O
F(x, Y )
and g t ( D n x D n- d ( D n ) ) E D n x D n - A ( D n )
Thus, j is a homotopy inverse for F.
10;
Figure 6.2
x
19"
for 0 5 t 5 I .
282
6 FIXED-POINT THEORY
Let Y be a closed, proper n-disk in M containing p such that the homeomorphism h: Y - . D" takes p into the origin. There exists an open, proper n-disk V in M such that p E V, E W n Y and f ( 7) G Y. Denote by k : 8+ D" the homeomorphism and note that k restricts to a homeomorphism of the boundaries k: d 8 3 Sn-'.We may assume that Dfl is oriented and both k and h preserve orientations. Define a map 4: Sn-' + Sn-' by taking the following composition: S'"1
(f.id)
dr-
YxY
- A(,)=
D n x Dn - d(D"):
D"
-
O:S?l-i,
where the last map is given by projecting radially from the origin. 6.9
PROPOSITION The degree of 4 is Ifw.
is given by the image of the fundamental class z of M under the composition
PROOF As we have observed, the chosen generator for Hn-,(Sn-')
H , , ( M )+ H , ( M , M -
v)
-
H,(v,
a ~.+a ) H " - , ( ~ V-% ) H,,-~(S''-~),
Note that in computing the fixed point index off on W we may choose the open proper n-disk V . Consider the following commutative diagram:
H f l ( M ) - + H f l ( M , M -V ) zHv(W, W - V ) -
(fib
H,l(MX) m
I I
Z
incl.
Hn(t7,dV)(f'i)tH,(YX Y , Y x Y - O ( Y ) ) (hXh)r
H,( Dn x Dn, D" x Dn - A (D"))
c
Hn(Dn,Dn - 0 )
Note that since the chosen generator of H,(Mx), /&(p)), arises from the orientation of the "vertical space" at (p, p ) E A (M)and the equivalence F collapses onto this vertical space, the vertical composition on the right takes I,(s(p)) into the chosen generator for Hn-l(Sfl-l). Now all of the
LEFSCHETZ COINCIDENCE THEOREM
183
vertical homomorphisms are isomorphisms, so if the composition across the top takes z into m . /,,(s(p)), then q5 must have degree m. 0 NOTE As pointed out in the introduction, this is an important step: identifying the local degree off at an isolated fixed point (degree of 4) with the local fixed-point index.
Having established these basic properties of the local index, we turn our attention to the corresponding global invariants. As before, we assume that M I and M , are closed connected oriented topological n-manifolds with fundamental classes z , and z2, respectively, and that f and g : M , M, are maps. Using the coefficient homomorphism Z A Q,we denote by Fl and 5, the images of z , and z2 in rational homology. That is
-
fl =
E*
F2 = E* (z2)E H n ( M 2 ;Q).
and
(zl) E H , , ( M , ;Q)
Consider the following diagram of groups and homomorphisms: H,(M,; %
I
-
Q)
1,
H,(M,;
Q)
I),
Hn-q( MI ;
Q)
H*-q(
M,;Q)
where D, and D, are the Poincart duality isomorphisms corresponding to TI and 4 , respectively. For each q define
-
0,: H , ( M , ; Q)
H , ( M , ; Q)
by 0, = D, o g* o D;l of*. Then the Lefschetz number or coincidence number of the pair (5 g) is defined to be the rational number L ( f , g ) = C (-1)qtrBq, 4
where tr 0, means the usual trace of 0, as a linear transformation from the finite-dimensional rational vector space H p ( M 1 ;Q)to itself. There is an alternate definition which will prove to be useful. For each q let ; d,,-,,: Hn-q(M2;Q)+ H n - Q ( M 2 Q) be given by
d,,-,= D,'
0
f,
0
D, og*.
184
6 FIXED-POINT THEORY
Then we define
The relationship between these two definitions is given in the following exercises.
Exercise 3 Show that tr 0,= tr dn-,.Hence, conclude that
&(I;g) = (-- 1)nUL g ) . Exercise 4 Show that L(I; g ) = (- I ) n L ( g , f ) . Recall that we have chosen a Thom class
u, E H”(M2 x M,, M2 x M ,
- d( M , ) )
corresponding to the fundamental class z2 in the manner of Chapter 5. From the composition
we define the Lefschetz class or coincidence class of ( I ; g) to be
Here d denotes the diagonal map. Note that the composition (fx g) 0 d is the map we have previously written as (f, g). Let gf,gbe the image of &f,o in rational cohomology under the coefficient homomorphism. We now want to establish a relationship between the Lefschetz number and the Lefschetz class of ( I ; g). To do so we will first establish two lemmas. Select a homogeneous basis { x i } for H*(M2;Q)and denote by {ai} the basis for H,(M,; Q) dual to { x i } under the Kronecker index. Define another basis { x i } for H*(M,; Q)by requiring that D 2 ( x [ ) = ai and let {a;} be the basis for H,(M2; Q) dual to {xi’} via the Kronecker index. Using the duality isomorphism D , we may define similarly related bases {yi} and { y ; } for H * ( M , ; &) and {bi} and {bi’} for H,(Ml; Q). Suppose now that
for some rational coefficients Bjr and yik.
LEFSCHETZ COINCIDENCE THEOREM
PROOF
285
Expanding g,(bj’) in terms of the basis {ak‘}we have g*(bj’) =
c A,
*
a;
k
for some rational coefficients 1,. Then note that Yij
(1 Yikykl, bj ) = (g*(xi’), bj’) k = (xi‘, g,(bj‘)) = xi’, C Akjak‘ = A i j . ( k ) =
xi
Thus, g,(bi) = yijai’, or, in other words, the coefficient of y i in g*(xi’) is the same as the coefficient of ai‘ in g , ( b i ) . Using the same approach, we can show that f*(bi) =
7
hi *
To prove the lemma, we first expand f* o D1 o g*(xi’) in terms of the basis {ai} and note that the coeficient of aj is given by
Thus
186
6 FIXED-POINT THEORY
6.11 LEMMA If d: Ml + MIx M , is the diagonal and 2, E H n ( M l ;Q) is the fundamental class, then
d,(t,)
=
. bix bi‘.
C (-l)(dimbd(dimbi’) i
PROOF This follows from the equations
THEOREM The Lefschetz class g,,g and the Lefschetz number L ( f , g) of the pair (f,g) are related by the equation 6.12
G,J,2,) = Uf, g). Since the {xi’} and {ai’} are dual bases under the Kronecker index, we may compute the Lefschetz number by
PROOF
L(J g ) = C (-17
tr6,
r
=
1 (-
1)dim
Z{’
(&Xi’),
I)dim
2t‘
(D;’of*
Ui’)
i
=
C (-
o
D1o g*(xi),a ; ) .
i
[The sign change, while bothersome, works out nicely, since dimf*(bi)
= dim
bi,
dim g&’)
= dim
bi’
and the sum of the two dimensions in each case is always n.]
LEFSCHETZ COINCIDENCE THEOREM
287
By Lemma 6.11
Therefore, L ( f ,g ) = (g,,,,, TI).
0
This relationship enables us to prove the following very important theorem :
6.13 LEFSCHETZ COINCIDENCETHEOREM The coincidence index of the pair ( L g ) on M I is equal to the Lefschetz number of ( f , g ) ; that is If,, = g).
uf,
PROOF
Recall that the coincidence index is the integer given by Recall
where the isomorphism is given by sending a class a into the integer (U,.a).
and the image of this in the rationals is just
by Theorem 6.12.
0
Note first that this may be viewed as an “integrality” theorem. That is, the Lefschetz number L(,hg ) is, by definition, a rational number in general. However, its identification with the coincidence index guarantees that it will be an integer.
-
6.14 COROLLARY Iff, g : MI M , are maps between closed oriented manifolds for which L ( f , g ) f 0, then f and g have a coincidence.
288
6 FIXED-POINT THEORY
PROOF This follows immediately from the fact that z,,g
and applying Corollary 6.3.
=
L(.L g) # 0
0
This is a convenient result, since in many cases the Lefschetz number is easier to compute than the coincidence index. Before proceeding with some applications, we examine a few special cases of the coincidence theorem. First suppose M , = M , = M and g is the identity map. As before the coincidence index If,,,is written I, and called the fixed-point index. Similarly the Lefschetz number L ( f , id) is written L( f ). For each k define
6.15 LEMMA
C (-l)k
tr cDk = t ( f ) .
k
PROOF
Recall that
t(f )= L(f,id) = C (-l)k tr ak, k
Thus,
@k
=
Ok for each k, and the result follows. 0
The Lefschetz class &,,id is written &, and, as before, its image in rational cohomology will be denoted by F,. With these definitions we have the following immediate consequences of the previous results. 6.16 LEFSCHETZ FIXED-POINT THEOREM Iff: M + M is a map of a closed, oriented n-manifold to itself, then I, = L( f ). Thus, if L ( f ) # 0, then f has a fixed point. PROOF
As in Theorem 6.13 we have
If = (€,
2) = L( f ).
Then the second conclusion follows from Lemma 6.7. In a further simplification we may take g and f both to be the identity on M. In this case the Lefschetz class is denoted by &M and called the
APPLICATIONS
289
Euler class of the oriented topological manifold M . The reason for this name is readily apparent since the Lefschetz number of the identity map is the Euler characteristic, that is ,!,(identity)
( - l ) k tr(idk)
= k
=
C (-l)kdimHk(M;Q) k
Thus, as a special case of the coincidence theorem (Theorem 6.13) we have established the following: 6.17 COROLLARY The value of the Euler class of M on the fundamental class of M is equal to the Euler-PoincarC characteristic of M . That is
( b ,z> = X W ) .
0
Note that the definition of the Lefschetz number L ( f , g ) is only dependent
on the homotopy classes of the maps f and g. Thus, we can observe the following corollaries: 6.18 COROLLARY If L ( f , g) # 0, g' is homotopic to g and f ' is homotopic to f, then g' and f ' have a coincidence. 0 6.19 COROLLARY Iff: M + M has L( f ) f 0, then any map homo-
topic to f has a fixed point.
0
6.20 COROLLARY If % ( M )# 0, then any map
to the identity must have a fixed point.
f: M - M homotopic
17
We now proceed with a number of applications of these theorems. First we give several fixed-point theorems due to Brouwer analogous to his theorem for the n-disk (Corollary 1.18), although slightly less well known. 6.21 COROLLARY Iff: Sfl --+ Sn is a map of degree m # (- l)n+l, then f has a fixed point.
- -
I f f has degree m, then the trace off,: Hfl(Sfl;Q) Hfl(sl"; must also be m. Since the trace of f*:Ho(Sn; Q) Ho(Sn;Q) is 1, we have
PROOF
=
1
+ (-l)n.
m.
I90
6 FIXED-POINT THEORY
Now since m # (-l)n+l we have that L ( f ) # 0 andfmust have a fixed point by Theorem 6.16. 0 Note that the antipodal map A : Sn --* Sn does not have fixed points, but, as we saw in Corollary 1.22, the degree of A on Sn is (-lY+l.
+
+
6.22 COROLLARY Iff: RP(2n 1) + RP(2n 1) is a map such that f,: Hzn+l(RP(2n+ 1); Q)+ H2,+,(RP(2n 1); Q) is multiplication by m # 1, then f has a fixed point.
+
Note from the universal coefficient theorem that the rational homology groups of RP(2n 1) are given by
PROOF
+
Thus
So if m # 1, L( f )# 0 and f has a fixed point.
0
To see that the restriction in the theorem is necessary, consider the following function on RP(2n 1). First write S2n+1 in complex coordinates as P n + I = { (zl,. , . ,zn+J E @n+l I C I zjl2 = 1}. Let
+
), i = 1/ - 1 . Note be given by g(zl, . . . ,z,,) = ( i zl,.. . ,i . z ~ + ~where that g o g is the antipodal map A and o A = A 0 g. Thus, there is asso1) -+ RP(2n 1) for which g2 = identity. ciated with a map g : RP(2n If g has a fixed point, then there must be a nonzero zk = a b i such that either i zk = z, or i . zk = -zk. But neither of these can happen, hence, g does not have a fixed point. Note that RP(2n 1) is a closed orientable manifold having the same rational homology groups as a sphere of the corresponding dimension. Such manifolds are called rational homology spheres. It is evident that corollaries of the type of Corollaries 6.21 and 6.22 will hold for any rational homology spheres.
+
-
+
+
+ -
6.23 COROLLARY If f: C P ( n )
-
n is even, or ( 2 ) f *: H2(CP(n)) (1)
292
C P ( n ) is a map for which either
H*(CP(n))
is multiplication by m # (-1), PROOF
-
APPLICATIONS
then f has a fixed point.
Recall that H*(CP(n);Q) Q [ t ] / P + ' ,
where t E H*(CP(n); Q) is the image of an integral generator under the coefficient homomorphism. Thus, the trace of
-
f *: H2k(CP(n); Q) HZk(CP(n); Q) is ink for 0 5 k 5 n. This implies that the trace of
f:"':
Hzk(CP(n), &)
-
H2k(CP(n); @)
is also mk for 0 5 k 5 n. So we have
+
Note that if n 1 is odd, this number must be nonzero. On the other hand, if n 1 is even, it can only be zero when rn = - 1. Therefore, under the hypotheses of the corollary, L( f )# 0 and f has a fixed point. 0
+
Note that for n = 1, the antipodal map on CP(1)= S2 has no fixed points. Here, of course, rn = -1. Exercise 5 For a general odd integer n, define a map 8 CP(n)+ C P ( n ) that does not have fixed points. In the same manner we may establish the following. 6.24 COROLLARY Iff:
-
HP(n) H P ( n ) is a map for which either
(1) n is even, or ( 2 ) f *: H 4 ( R P ( n ) )+ H 4 ( H P ( n ) ) is multiplication by rn # - 1,
then f has a fixed point.
0
192
6 FIXED-POINT THEORY
Let us now investigate the situation for maps of the torus
n-fold
In this case it is necessary to change coeficients to an algebraically closed field, so let i: Q + C denote the inclusion of the rationals in the complex numbers. Using the coefficient homomorphism on homology and cohomology, the previous theorems could eesily be established ushg complex coefficients. Recall that H*(Tn; C) = E(C;xl, . . . , xn), the exterior algebra over C on n generators, all of dimension 1. Thus,f*: H1(Tn;C ) -+ H1(Tn; C) is a linear transformation on an n-dimensional complex vector space. Since C is algebraically closed, there exists a basis { y , , . . . ,y,} for H1(Tn;C ) with respect to which the matrix A = (aij) of,f* is upper triangular. This basis retains the property that
In Hl(Tn; C) denote by z,, . . . ,z, the dual basis. Then
= ski.
xi
Thus, f*(zi) = x k akizk,and the trace off:? is given by aii. C) dual to the product yi A y j . Denote by zi A z j the element of H,(Tn; Then {zi A zj I i < j } is a basis for H,(Tn; C) and
The last equality follows from the fact that i < j ; hence, aji = 0. Thus, the trace off&z)is given by
APPLICATIONS
Similarly we find that for 1 5 k 5 n the trace off )-
i,<
,
...
ai,il *
* *
293
Lk)is given by
aikjk.
This implies that the Lefschetz number off has the form
Therefore, we have established the following result. 6.25 COROLLARY Iff: 7"'- Tnis a map for whichf*: H1(Tn)+H'(Tn) does not have + 1 as an eigenvalue, then f has a fixed point. 0
It is evident that many maps of the torus exist without fixed points. For example if fz:P-'-+Tn-l is any map and fl: S' + S1 is a nontrivial rotation, then f, x fz: Tn+ T" has no fixed points. Exercise 6 Let f and g : Sn + Sn be maps of degree m and k , respectively. Determine L ( f ,g ) . Exercise 7 (a) Let f, g : S2n+ CP(n) be maps, n > 1. Show that I,,v = 0. = m for any (b) Do there exist maps f, g : CP(n) SZnsuch that integer m?
-
Exercise 8 Suppose that M is a closed, connected, oriented n-manifold with fundamental class z E H,(M). If f: M -P M is a map for which f*(z) = k . z for some integer k , then show that
Uf,f 1 = k X ( W . *
The coincidence theorem gives an indirect, but appealing approach to the following basic result.
+
6.26 FUNDAMENTAL THEOREM OF ALGEBRA If f(z) = zn an-lzn-l . . a,z a, is a nonconstant, complex polynomial, then f(z) has a root.
+
+
+
Denoting by C the complex numbers, we view f as a map from C to C. Note that I f(z) --* 00 as 1 z I + co;hence, we may extend f to a
PROOF
I
294
6 FIXED-POINT THEORY
map of the one-point compactification
by settingf(m) = 03, where 03 denotes the north pole. Similarly the map g : C+ C given by g(z) = 0 may be extended to S2 by setting g ( m ) = 0, 0 being identified with the south pole. Then a coincidence off and g will be a root of the polynomial f(z). This situation corresponds to that of the example following Lemma 6.8, so that the coincidence number L ( f ,g ) = k, where
f*: H2(m
+
H2(m
is multiplication by k. Certainly, if there is any justice, the degree of f should be n. To prove this, define the contracting homeomorphism r: C+ D2 - S1 by r ( z ) = z/(l 1 z I). Note that r-l(w) = w / ( l - I w I). There is a uniquely defined map f making the following diagram commute:
+
f c-c
We want to extend f to a map from D2to D2. So let wo E S' and let w approach wo through values in the interior of D2: f(W) = r(f(r-*(w)))
so that lim f ( w ) w+wo 1w1a
won
= -= won.
I won I
-
This implies that we may extendfto be defined on all of D2 by setting f(w,) = won.Note that the mapping r-l may be extended to a map h: D2 S2
APPLICATIONS
195
and the fact that on S1,f'(eie)= cine, we conclude that the degree off must be n. Therefore, L ( f , g) = n and f must have a root. 0 Exercise 9 In the above setting, suppose that zo is a root o f f of multiplicity k. Show that there exists an open set U about zo such that the local coincidence index Z l o is k.
The proof of Theorem 6.26 and, particularly, the accompanying exercise demonstrate that coincidence theory is a natural way to study problems of this type. As another application of Theorem 6.16 we can prove the PoincartHopf theorem on the sum of the indices of a vector field. For this purpose we must assume that our closed oriented manifold M n is differentiable. Let u be a smooth vector field on M n such that the singularities (zeros) of u are isolated points of Mn. As observed before, we may associate with u a map f:M + M , homotopic to the identity, having as its fixed-point set the singularities of u. If x is an isolated singularity of u, one defines the index of u at x , i, as follows. Select a coordinate neighborhood U of x ,
Figure 6.3
Figure 6.4
296
6 FIXED-POINT THEORY
homeomorphic to an open n-disk, which contains no other singularities of v. Within U choose an (n - 1)-sphere about x. At each point of this sphere the associated vector of u must be nonzero. Transferring this into Rn and normalizing the vectors defines a map from Sn-' to Sn-l.The degree of this map is i,. For example, on a two-dimensional manifold a singularity of index -1 is shown in Figure 6.3, while the index in Figure 6.4 is f2. An excellent reference in this area is Milnor [1965]. It is intuitively clear that the index i, is equal to the local degree o f f at the fixed point x as defined prior to Proposition 6.9. We may use this fact to establish the following classical theorem. 6.27 POINCARl?-HOPF THEOREM If u is a smooth vector field with isolated singularities on the closed oriented differentiable manifold Mn, then the sum of the indices of u is the Euler characteristic of M; that is
From the observations above, the sum of the local indices of u is the same as the sum of the local degrees o f f a t its isolated fixed points. By Proposition 6.9 this is the sum of the local fixed-point indices off. The additivity of the fixed-point index (Lemma 6.6), together with Theorem 6.16, implies that this sum is L ( f ) . But since f is homotopic to the identity, L ( f > = X(M). 0
PROOF
NOTE The theorem of Poincart mentioned in the introduction to this chapter is a special case of Theorem 6.27. Specifically, a closed surface of genus p has Euler characteristic equal to 2 - 2 p ; hence, this must also be the index sum. Having strayed this far afield, we may consider one more connection with differential topology and geometry. On a smooth manifold M we may define a cochain complex using the differential forms of M and the exterior derivative. The homology groups of the complex are the de Rham cohomoiogy groups of M , denoted H*(M, d). There is a natural transformation into cohomology with real coefficients @:
H*(M, d ) + H * ( M ; R)
that may be described as follows. Suppose that M has been smoothly triangulated and w is a smooth k-form on M. Then @ associates with w the
APPLICATIONS
197
function from the k-simplices of M into R,whose value on a given simplex is the integral of (1) over that simplex. The famous de Rham theorem states that @ is an isomorphism under which the exterior product in H * ( M , d ) corresponds to the cup product in H * ( M ; R).For a highly readable account of this, see Singer and Thorpe [ 19671.
Let M be a closed, connected, oriented, smooth 2-manifold endowed with a Riemannian metric. Then the volume element vol is a smooth 2-form on M and the curvature K is a smooth function associated with the Riemannian connection on M . The classical Gauss-Bonnet theorem then states that if the 2-form K vol is integrated over the magifold M , the result is 2n . x ( M ) . In other words,
-
1 , .
The connection between these results and the previous is that integrating a 2-form over the manifold M corresponds under @ with taking the Kronecker index with the fundamental class. It follows by Corollary 6.17 that the cohomology class represented by the 2-form (1/2n)K* vol is assigned by @ to the Euler class EM of 44. As a final application let M be a closed, oriented n-manifold. A p o w on M is a one-parameter group of homeomorphisms of M . Specifically, a flow is a function 4: R x M - M which is continuous and satisfies
+
(i) 4(tl tz > x ) = $ 0 1 (ii) 4(0,x ) = x
9
4 c t 2 9 XI),
for all t,, t, E R and x E M. Note that for each t E R this defines a homeomorphism # t : M - + M by r#t(x) = #(t, x) because r#rl = + - t . A point xo E M is a fixed point of the flow if r#t(xo) x,, for all r E R. Flows arise naturally on closed differentiable manifolds as the parameterized curves of a given vector field. 6.28 THEOREM If M is a closed oriented manifold such that x ( M ) # 0,
then any flow on M has a fixed point. PROOF For any to e
R the homeomorphism
k,:
A,+
M
198
6 FIXED-POINT THEORY
is homotopic (actually isotopic) to the identity. So L(4,J = L(identity) = has a fixed point. x ( M )# 0, and Now for each positive integer n denote by F, the fixed-point set of #]/2n. It follows from the additivity of the parameter that F, will be fixed by 4,,,/2n for any integer m. F,, is also compact since it is the inverse image of d ( M ) under the composition
For each positive integer n we have F,,, C F, because
Thus, { F , } is a nested family of nonempty, compact subsets of M which must have a nonempty intersection F. This set F is fixed by for any dyadic rational r. Since these are dense in R, the continuity of 4 implies that each point of F must be a fixed point of the flow 4. 0 Exercise 10 A flow on a manifold is the same as an action of the additive group of real numbers on the manifold. Using the techniques of this chapter, what results can you derive concerning actions of pathwise connected groups on closed oriented manifolds (for example, the additive group Rn or the multiplicative group S’)? Exercise 11 Letfand g be maps from S3to 9. Show that iffis not homotopic to g, then f and g must have a coincidence. It should be pointed out that, although the fixed-point techniques we have developed can be very useful, they are still inadequate to solve many problems. As a specific example we cite the “last geometric theorem” of PoincarC.
Figure 6.5
APPLICATIONS
199
Suppose that we have an oval billiard table as in Figure 6.5, on which a single ball is rolling. Do periodic orbits with k bounces per period exist for every k 2 2? That the answer is yes was conjectured by Poincari and proved by Birkhoff [1913]. If we orient the boundary curve, then the initial motion from the cushion is determined by the initial point x and the angle 0 of projection measured from the forward pointing tangent (so 0 5 0 5 n). This set of initial motions given the product topology is an annulus A = S'x [0, n]. Now define F: A - + A by taking each initial motion onto that which follows the next bounce of the orbit. The conjecture may be stated by saying that Fkhas a fixed point in the interior of A for all k 2 2. In solving the problem, the following theorem was proved: Any mapping G: A + A has two fixed points in the interior of A if (i) G is a homeomorphism leaving every point of the boundary circles fixed ; (ii) the G image of a radial I-cell wraps at least twice around the annulus ; (iii) G preserves areas. It is interesting to note that this problem could not have been solved by our techniques because the Lefschetz number in this case is zero. In closing we consider briefly the question of the existence of a converse to the Lefschetz fixed-point theorem. For differentiable manifolds the Hopf theorem states that if x ( M )= 0, then M admits a nonzero vector field, hence a map without fixed points which is homotopic to the identity. For topological manifolds there are a number of recent results by Brown and Fadell [1964], and others. As representatives of these results we state the following two theorems. 6.29 THEOREM Let M be a compact, connected topological manifold.
Then (i) M admits maps close to the identity with a single fixed point; (ii) x ( M ) = 0 if and only if M admits maps close to the identity without fixed points. 0
300
6 FIXED-POINT THEORY
6.30 THEOREM If M is a compact, simply connected topological maniM is a continuous map with L(f)= 0, then there exists fold and f: M a map g homotopic to f such that g is fixed-point free. 0 --+
For a deeper, more comprehensive study of fixed-point theory see Brown [to appear].
appendix I
The purpose of this appendix is to give a proof of Theorem 1.14. The proof requires the development of the subdivision operators on the chain groups. This fundamental technique is at the basis of essentially all of the computations and applications we will be able to make. If C G Rn is a bounded set, then the digmeter of C is given by diam C = lub{ll x - y 1 x, y E C } . If = {Ci} is a family of bounded subsets of Rn,then mesh C = lub{diam Cd}.
I
e
1.1 PROPOSITION If sn is an n-simplex with vertices a,, a , , . . . , a,, then diam sn = max{ll a<- aj 1 i, j = 0,.. . , n).
I
c
Let x = C tiai and y = ti'ai be points in sn. First fix x and allow y to vary. We want to show thet PROOF
201
202
APPENDIX I
Repeating the above, letting x vary gives
Let sn be an n-simplex with vertices a,, a,, . . . , a,. The barycenter b(sn) of s" is the point in s" given by
It is not difficult to show that for any i with 0 5 i 5 n, the points
span an n-simplex. We now define the barycentric subdivision Sd(s") inductively on the dimension of the simplex. First set Sd(so) = so for any zero simplex so. Suppose now that Sd is defined on any simplex of dimension (n - I), so if tn-' is any (n - 1)-simplex, Sd(t"-') is a collection of (n - 1)-simplices geometrically contained in tn-'. Denote by 9" the collection of all (n - 1)-faces of sn and define Sd(Sn) =
u Sd(tn-').
tn-l&n
Then Sd(sn) will consist of all n-simplices of the form (b(s"), t o , . . . , fn-l), where ( t o , . . . , f,J is an (n - 1)-simplex in Sd(Sn) (Figure I. 1).
A-
Figure I.1
APPENDIX I
203
1.2 PROPOSITION If K is a collection of n-simplices, then mesh Sd(K) 5 (n/(n
+ 1)) mesh K.
PROOF Proceeding by induction on n, for n = 0 both sides are zero. So
suppose the result is true for any collection of (n - 1)-simplices. Let t n be an n-simplex in Sd(K). Then t n = (b(s"),u,, . . . , where sn E K and u o , . . . , u , - ~ are vertices of an (n - 1)-simplex w in Sd(Sn). Let sn-l be the (n - 1)-simplex in Sn containing w . By Proposition 1.1 diam t n = max{ 11 ui - uj 11, First consider the terms of the form
11 ui - b(sn)11 } . I[ ui - uj 11. We know that
11 ui - uj 11 5 diam w 5 ((n - l)/n) diam sn-l
+
by the inductive hypothesis. Now since x / ( x 1) is an increasing function and the diameter of a subset is less than or equal to the diameter of the set, we have n-1 n diam sn-l < -diam s f l . n - n + l
+
Hence, any term of the form 11 ui - uj [I is less than or equal to (n/(n 1)) diam sn. For the terms /[ ui - b(sn)11 recall that if uor, . . . , u,' are the vertices of sn then b(sn)= (l/(n 1)) C uir. Each vertex ui is a point in sn so that 11 ui - b(sn)11 5 1 uj' - b(sn)11 for some j by the proof of Proposition 1.1. Then
+
where the sum now has n terms. But then
n diam sn. n f l
5-
Thus, all terms will satisfy the desired inequality and the proof is complete. 0
204
APPENDIX I
1.3 COROLLARY If K is a collection of n-simplices, let Sdm(K) = Sd(Sdrn-l(K)) be the iterated barycentric subdivision. Then for an n-simplex and any E > 0 there exists a positive integer m such that
mesh Sdm(sn)< E . PROOF This follows immediately from Proposition 1.2 and the fact that
With these basic properties of the subdivision operator on simplices in mind, we now want to define an analogous operation on singular simplices. If C and C' are convex sets, a map f: C + C' is ufJine if given x, y E C, O s t l l , then
f((1 - o x + 04 = (1
-M x )
+ tfdv).
It follows from this that if x o , . . . ,x P € C and t o , . . . , tp are nonnegative with C ti = 1, then
f ( C tixi) = c tif(Xi).
If C is convex, define A,(C) E S,(C) to be the subgroup generated by C. Denoting by u,, u1,. . . , u, the all affine singular n-simplices +: a, vertices of a,, for any affine +: a, C let xi = +(ui). Then we can denote 4 by x,,x, . . . x,. In this notation it is evident that ---f
--+
di(X,X,
*
x,) = (xo
* * *
xi-1xi+,
-
*
x,).
Thus, a(A,(C)) E A,-,(C) and {A,(C)} = A,(C) is a chain complex. We now define a chain map 9 d ' : A,(C) A,(C) which is the algebraic analog of the subdivision operation. The definition is given inductively on the dimension n. For n = 0 let 9 d 'be the identity. Suppose now that it is defined up through dimension n - 1, and let = x,x, - - x, be an affine singular n-simplex in C. The burycenter of 4 is the point --f
+
For any point 6 E C define a homomorphism
-
APPENDIX I
205
This is called the cone on b for obvious geometrical reasons. Finally, define for any affine singular n-simplex 4 (see Figure 1.2)
Yd’(4,= ~ t J ( + d 9 4 W ) . 1.4
PROPOSITION d 0 9 d ’= Y d ’0 d.
It is sufficient to check this on some affine singular n-simplex x,. Let b = b(4). Certainly the formula is true in dimension n = 0, so assume that it holds in dimension (n - l),
PROOF
4 = xoxl . - -
a.9d’(xo
* * *
x,) = dC*( 9 d ‘ a ( x o
-
* *
x,)).
We may split up the boundary on the right into those terms containing b and those not containing b, d 9 d ’ ( X O*
* *
x,) = Y d ’ d ( x 0 *
*
’
x,) - c,(a.Yd’d(x,
* * *
x,)).
But the second term here must be zero, for by the inductive hypothesis a9d’a(X0 ’
Therefore
*
- x,)
a%’
=9 / a a ( x o
=.wid.
* * *
x,)
= 0.
0
Thus, Y d ’ :A,(C) -+ A,(C) is a chain map of degree zero. Since the homology should not be affected by subdividing simplices, it is reasonable to expect that 9 d ’is chain homotopic to the identity. To verify that this is indeed the case we must define a homomorphism T ‘ : A,(C)
4J
Figure 1.2
-
A,+l(C)
A-
-
206
APPENDIX I
such that
aT’
+ T’a = 9/- 1.
We define T‘ inductively on n. Since for n = 0, 9 d ‘ is the identity, we take T‘ to be zero. Now suppose T’ is satisfactorily defined on all chains of dimension less than n and let 4 be an affine singular n-simplex. Note that
a ( . z , q - 4 - ~ ’ a 4=) [ a . n r =0
- a - ( 9-~ 1’ ~’a)a14
since 9 d ’ a = a 9 d ’ and
aa = 0.
Then set
T’(gb) = C b ( g ) ( Y d ‘ $
- 4 - T‘a4).
To compute aT’(4) we split it up into that part containing b(#) and that part not containing b(4). In other words,
a ~ ’ ( 4=) (.zJ’+
-4 -
- 4 - T’a4). - eb(+)a(.z=q
But from the above computation, the second term must be zero and T has the desired property. Iff: C -+ C‘ is an affine map between convex sets, then
f#(A,(C)) E A,(C’) and f # commutes with both 9’d’ and T‘. We want to use the above homomorphisms to construct a degree-zero chain map 9 d : S,(X) + S,(X)
-
for any space X,and show that it is chain homotopic to the identity. Using the same technique as in the proof of Theorem 1.10, let y: un be a singular n-simplex. There is the induced homomorphism Y#:
Now the element define
t, E
X
Sn(~n)
+
S,(u,) given by the identity map is in A,(u,). So
9 4 y ) = .Yc/ Y # ( d = Y#Yd’(t,).
This just has the effect of subdividing the simplex by subdividing in its domain, which is convex. Similarly, set
T ( y ) = TY#(%)
= ly#T’(%).
APPENDIX I
1.5
PROPOSITION aT
PROOF
207
+ Ta = Y d - 1.
This follows immediately from the same properties of T' and
Yd'.
We are now ready to give a proof of the theorem.
-
THEOREM If Gu is a family of subsets of X such that Int covers X , then the chain map i : S , ( X ) S,(X) induces an isomorphism 1.14
i,:
Kl(Q(X))
- fwo.
We will construct a chain map 0:S , ( X ) --+ e ( X ) such that @ . i is the identity and i @ is chain homotopic to the identity. Let 4: u , ~ X be a singular n-simplex. The family ?1 = {+"(U)l U E Gu} has Int "II covering a,. Since a, is compact, there exists a 6 > 0 such that if C c u, and diam C < 6, then C is contained in #-'(U) for some U. By Corollary 1.3 there exists an m p 0 with
PROOF
-
-
mesh Sdman < 6. This will imply that Y d m
4E
S,"ll(X).
Now for any singular simplex 4 in X let m(4) be the least integer for which . Y d m ' q 5 E S,"ll(X).
Note that for 0 5 i 5 n, m(4) 2 m(a,+). Recall that aT Ta = 9 d - 1, so for any positive integer k we have
+
dT9'dk-l
+ T9'd"'d
= .Ydk - .Ydk-',
Adding a sequence of these together gives dT( 1
+
* * *
+ 9'dk-l) + T(1 +
* * *
+ S?'dk-')d = Y d k- 1.
208
APPENDIX I
From looking at the summation we conclude that
@(+IE %'Yx)* To consider @(+) as an element of Sn(X)we apply the mapping i. The above manipulation shows that dS+Sd=io@-
hence,
io@
4 E &$(A'),
1;
is chain homotopic to the identity. On the other hand, if then m(f$)= 0 and @ o i is the identity. 0
appendix I1
The purpose of this appendix is to prove two of the basic theorems on topological manifolds which were used in Chapter 5. The first theorem states that any closed topological manifold may be imbedded in some euclidean space Rn and that the imbedded manifold is a retract of a neighborhood in Rn. The second theorem states that the boundary of a compact topological manifold admits a collaring. The first result is an excellent example of a “folk” theorem. That is, a result which is well known and may be proved in a variety of ways, but which is dificult to locate in the literature or trace to its true origin. The imbedding technique we use is due to Dold, whereas the approach to the retraction property was suggested by Bing. The second result is of more recent vintage. A collaring theorem for differentiable manifolds was proved by Milnor, and the analog for topological manifolds, by Brown [1962]. The proof we present here is a recent one due to Conelly [1971].
II.1 THEOREM If M is a closed topological n-manifold, then M can be imbedded in euclidean space Wk for some k. 209
210
APPENDIX I1
PROOF Let B, , . . . , B,,, be a collection of proper open n-balls in M which cover M. For i = 1, . . . ,m denote by hi:
B i + S n - {Y>
a homeomorphism onto the complement of the north pole. We can extend each hi to a map hi: M - + S"
Now define the map
by i ( x ) = (h,(x), h,(x),
. . . ,h,(x)).
Then i gives the desired imbedding.
0
NOTE In general this is not a very economical way to imbed the mani-
fold. That is, the dimension of the euclidean space is much higher than is generally necessary. For example, the covering of a circle by two proper 1-balls will produce the imbedding illustrated in Figure 11.1 (actually in R4). Suppose now that i: M n + Rk is an imbedding of a closed topological n-manifold. Denote by s a large k-simplex in Rkcontaining M in its interior. We want to triangulate the complement of M in s in a particular way. Denote by Sd the barycentric subdivision operator defined in Appendix I and let s1 = Sd(s), a simplicia1 complex that is a finite union of k-simplices. Now examine each closed k-simplex in sl. To those which intersect M we apply the operator Sd. Those which do not intersect M are left intact. The resulting simplicia1 complex is denoted s,. By continuing this process, we produce a sequence of finite simplicial
Figure II.1
APPENDIX I1
222
complexes {sn}, each a finite union of closed k-simplices, with the property that s, subdivides s, whenever m 2 n (Figure 11.2).
II.2 LEMMA This process defines a triangulation of s - M. In other words, for every point x of s - M there is an integer m such that each k-simplex containing x in s, remains intact in s, for all m' 2 m. PROOF Let x E s - M. Since M is compact, the distance from x to M
is some positive number such that
E.
By Corollary 1.3 there is a positive integer m mesh Sdm(s) < E .
There are two possibilities: (a) at some stage s1, I < m, all of the closed k-simplices containing x were disjoint from M ; or (b) at each stage s l , 1 < m ysome closed k-simplex containing x intersected M.
Figure 11.2
212
APPENDIX I1
In the first case, for all stages beyond SZ the triangulation around x remains unchanged. In the second case, each k-simplex of s,,,-~ that intersects M will be a k-simplex of Sdm-l(s). Thus, the k-simplices of s, containing x will either be k-simplices of s,,,-~ that do not intersect M, or barycentric subdivisions of those that did, hence k-simplices of Sd"(s). In either situation it follows that the k-simplices of s,, that contain x will be disjoint from M. Therefore, in each si,j 2 m, the triangulation about x remains constant. In this way we define the triangulation of s - M. 0 We now want to use this triangulation to define inductively a collection E Nk= N of s together with a map r : N + M. of subsets No C Nl C Initially we take No to be the union of M with all the vertices of s - M. If y E M, define r(y) = y . If x, is a vertex of s - M, define r(xJ to be some point of M for which dist(xo, r(xo)) = dist(xo , M). Suppose then that Ni-l and r: Nidl M have been defined. Let a be a closed i-simplex in s - M; a will be contained in Ni if both of the following requirements are satisfied:
- -
-
(a) the boundary of a is contained in Ni-, ; (b) the map r : da -+ M can be extended continuously over a. The space Ni will be the union of NiPl together with all such closed i-simplices a. To define r: Ni + M it is sufficient to define r on each a so that it is compatible with the previous definition on the boundary. Let A
=
(6 E R 1 there is a mapf: a
-
M , f l a a = r, and diam(imagef)
=
S}
For a an i-simplex in N i , A is a nonempty set that is bounded below. Let a be the greatest lower bound for A. Now define r: a + M by choosing a map that extends the restriction of r to aa and satisfies a 5 diam(r(a)) 5 2a. Note that if a = 0, we may take r to be the constant map taking a into r(da). This completes the inductive step so that N = Nkis a well-defined subset of s containing M and r : N + M is a function which is the identity on M. It remains to be shown that r is continuous and N is a neighborhood of M in Rk.First we present some preliminary lemmas. E > 0 there exists a 6 > 0 such that if x is a of s M and dist(x, M) < 6, then the mesh of the set of k-simplices point of s - A4 containing x is less than E.
11.3 LEMMA For any
APPENDIX I1
223
+
-
Let E > 0; choose a positive integer rn so that (k/(k diam(s) < E . Define K , to be the union of all closed k-simplices of s, that do not intersect M. K , is a compact subset of s. Let 6 dist(M, &), or, if K , is empty, take 6 = 1. Then if dist(x, M) < S, each closed k-simplex of s, that contains x must intersect M. Thus, each of these simplices must lie in Sdm(s) and their mesh IS less than or equal to mesh Sdm(s)5 ( k / ( k diam s < E . The same inequality will obviously be true for the set of k-simplices of s - M containing x. [7 PROOF
+
-
II.4 LEMMA If S is a p-simplex, K is a convex set, and fi 8s + K is a map, then f can be extended over all of S. PROOF Let b ( S ) be the barycenter of S and select a point w E K . Every point x of S has a unique representation in the form x = ty (1 - t) b ( S ) , where y is a point of 8s and 0 5 t 5 1. For each such point x define
+
-
This is well defined since K is convex; f is continuous and extends the original definition o f f on 85. 0
II.5 COROLLARY If S is a p-simplex and B is a proper n-ball in a topological manifold, then any m a p 8 8s + B can be extended over all of S. PROOF Since B is a proper n-ball, there
h : B - . Dn
is a homeomorphism
- Sn-1,
a convex subset of Rn. We may compose f with h and apply Lemma 11.4 to give the desired extension. 0 11.6
THEOREM The function r: N - . M defined previously is a continuous retraction of a neighborhood of M in Rk onto M. PROOF From the construction of r it is apparent that r is continuous at each point of s - M. Thus, it is sufficient to check continuity at a point y E M. So let E > 0 and denote by B(y, E ) the ball of radius E about y = r(y) in Rk. We construct inductively a collection of open sets V,,, Vl ,. . . , Vk+, in Rk about y. Let V,, = B(y, E ) . For the inductive step, suppose that Vl-l
214
APPENDIX I1
has been defined. Let V1be an open subset of VI-, in Rkcontaining y having the following properties : (i) Vl E B(y, 6l), where B ( y , 561) E Vl-,; (ii) Vl n M is a proper n-ball about y ; (iii) if x E Vl - My then the mesh of the set of k-simplices of s - M containing x is 5 dl. That Requirement (iii) may be satisfied follows from Lemma 11.3. Now let x E Vk+,, an open set about y. If x E M ythen r ( x ) = x E V, = B(y, E ) .
So suppose x 4 M and let S be a k-simplex of s - M containing x. By Requirements (iii) and (i) all of the vertices of S must lie in Vk and must be mapped by r into the proper n-ball Vk n M. By Corollary 11.5 each 1-simplex a in S admits a map into Vk n M extending the restriction of r to aa. Thus, each set a is contained in N. Furthermore, since the diameter of the image of this extension is less than 26k, the diameter of r(a) must be less than 4dk. The fact that r(a) intersects Vk n My together with Requirement (i), implies that r(a) C Vk-,n M. Thus, the image of the l-skeleton of S under r is contained in Vk-,n M, a proper n-ball. We may now apply the same argument to the 2-simplices of S. Continuing inductively, we find that the image of the k-skeleton of 85 under r, that is, r ( S ) , is contained in Vk-k = V, = B(y, E ) . In particular r ( x ) E B(y, E ) and r is continuous at y. To see that N is a neighborhood of M in Rk,note that in the above argument, Vk+lis an open set about y E M on which r is completely defined. Hence, Vk+,c N and y is an interior point of N. 0
Exercise I Make the necessary modifications in the preceding proofs to show that the results hold as well for compact manifolds with boundary. Exercise 2 Prove a similar imbedding and retraction theorem for noncompact manifolds.
Finally, we turn to the collaring theorem for topological manifolds. As we stated previously, the proof given here is an intuitively appealing one due to Conelly [1971].
II.7 THEOREM (Topological collaring theorem) Let Mn be a compact topological manifold with boundary a M = B. Then there exists an open
APPENDIX I1
215
set U in M , containing B, and a homeomorphism
h : U - + B x [ O ,1) such that h(x) = (x, 0) for all x E B. PROOF The idea of the proof is as follows. Since B = d M we can find
about each point of B an open set which looks like a portion of a “collar”; that is, B is locally collared in M . We attach a collar to the boundary of M and then use the local collaring to push the added collar into the manifold (or pull the manifold out over it), so that the added collar becomes the desired open set U. By using the topological properties of euclidean half-space W”, we can show that for any point x E B = d M there is an open set U, in B about x and an imbedding h,: 0,x [O, 11 + M such that for any x’ E az, h,(x’, 0) = x’. Now B is a closed subspace of the compact manifold M ; hence, B is compact and there exist a finite number of open sets U , ,. . . , U,,,in B and imbeddings
hi: OiX [O,I ]
-+
M
such that
hi(x, 0)
=
for all x
x
and B
=
u
in
Oi
Ui.
i
Since B is compact Hausdorff, it is also normal; hence, there exist open sets Vl,. . . , V, covering B such that Vi c Ui for each i. Define M+ to be the space formed from the union M u ( B x [- 1,Ol) by identifying x E B E M with ( x , 0) E B x [-1, 01 (Figure 11.3). For BX [-I,O]
Figure 11.3
APPENDIX I1
216
each i let hi:
uix [-I,
I ] +M+
be the function given by
Since these agree on the intersection, each hi is a well-defined imbedding. We now use these maps hi to define inductively a family of imbeddings gi: M + Mf and maps fi: B + [-1, 01, i = 0, 1,2,. . . , m, satisfying the following: (a) (b) (c) (d)
(uisi
gi(M) contains M u P ~ [-I, x 01); for any x E B, gdx) = ( x , f i ( x ) ) ; f,(x> = -1 ; for any x E B, { x } x [ f i ( x ) ,01 c gi(M).
The imbeddings gi correspond to the consecutive stages of pushing the collar into the manifold while the functions fikeep track of the location of the boundary of M at each stage. It follows that g, will be a homeomorphism of M with M+ taking x E B to (x, - 1 ) in Mf. This will give the desired collaring of B. Define go: M + Mf to be the inclusion and set f , ( B ) = 0. Inductively, suppose gi-l and fi-* have been defined. Consider
hTl(&,(M))
c
OiX
[-1, 11
(for example, the shaded region in Figure 11.4). We want to define an
-1
fi.,(x)
Figure 11.4
0
1
APPENDIX I1
217
imbedding #i:
h;’(g,-l(M))-
QX
[-l7 13
by pushing to the left along the fibers until #i(h:l(gi-l( Vi)))= Pix {- 1 }, but requiring that C$i be the identity on (0,- Ui)x [ - I 7 11 u U i x {l}. Thus, C$i represents a “pushing out” operation inside this local collar which will not effect the rest of the manifold. To do this, we want a map Ai: [-1, 1 1 such that Ai(X) =
ui2f,-l(x) + 1
if x E if x E
Pi
Ui- Ui
and Ai(x) 5 2f,-’(x) $- 1 for all x E 0,. Since Pi and ui - Ui are disjoint closed subsets of a normal space, we can find a map satisfying the stated condition on these two subspaces using the Tietze extension theorem. Taking the minimum of this map and 2L-,(x)i- 1 produces the desired map A i . Now define C$i by
The behavior of the map #i may be described as taking each interval { x } x [ t ( A i ( x )- l ) , Ai(x)] linearly onto { x } x [ - l 7 Ai(x)], recalling that 3(Ai(x) - 1) 5f,-l(x) with equality holding on Pi (Figure 11.5). We may now use C$i to alter giPlto produce g i . Specifically
gi(x> ==
(hi#ih;l)gi-l(x)
Fixed by 0, Graph
4-2
Graph A,
Figure 11.5
if x E gi-l(M) n hi(Di x [- 1 11) otherwise.
228
APPENDIX I1
Both gi and gyl are continuous. We define fi: B [-I, 01 by setting f i ( x ) = n ( g i ( x ) )where iz is the projection from B x [- 1,0] onto the second factor. This completes the induction step and the homeomorphism g,,, gives the required collaring of B. 0 --+
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INDEX
A Acyclic model theorem, 104 Adams, J . I?., 33, 121, 219 Adem, J . , 1 2 1 , 2 1 9 Alexander-Whitney diagonal approximation, 113 Algebraic mapping cylinder, 15 2 Antipodal map, 3 1
B Back j-face, 1 1 3 Barycenter. 202 of affine singular simplex, 204 Barycentric subdivision, 202
Betti number, 7 3 Bilinear forms, 162-166, 168 antisymmetric, 162 nonsingular, 162 signature of, 165 symmetric, 162 Bilinear mapping, 74 Bing, R . H., 209 Birkhoff, G. D., 199, 219 Boundary, 5 Boundary operator, 5 Brouwer, 1,. E. J . , 28, 38, 173 fixed-point theorem, 28 Brown, M., 154, 209, 219 Brown, Robert F., 199, 2 0 0 , 2 1 9 Brown, Ronald, 6 9 , 219 Brumfiel, G., 172, 219
233
234
INDEX
C
Cap product, 121 Category, 102 Cellular map, 69 Chain complex, 6 homology of, 6 subcomplex of, 50 quotient of, 50 Chain homotopy, 13 Cobordism, 171 oriented, 171 unoriented, 172 Coboundary operator, 88 Cochain complex, 88 map, 89 homotopy, 89 Cocycles, 89 Coefficient group, 86 Cohomology group, 89 singular, 90 de Rham, 196 Coincidence, 175 class, 184 index, 177 number, 176,183 theorem, 187 Collaring theorem, 154,214-218 Commutative graded R-algebra, 112 Cone, 205 Conelly, R., 209,214,219 Connecting homomorphism, 21 Convex, 1 Convex hull, 1 Coordinate neighborhood, 130 Cup product, 112 Curvature, 197 Cycle, 5
D Deformation retract, 19 strong, 56 Degree of antipodal map, 31 of chain map, 6 local, a t fixed point, 182 of map of manifolds, 173 of map of spheres, 28
de Rham. 171 theorem of, 197 Derivation property, 109 Diagonal, 41 Diagonal approximation, 113 Diameter, 201 Directed set, 33 Direct limit, 34 Direct product, 9 Direct sum, 9 Direct system, 33 Disk, 27 Dold, A., 176,209, 219 Double of M, 154 Dual cell, 125
E Eckmann, B., 33,219 Eilenberg,S., 85, 121, 152, 219, 220 Eilenberg-Steenrod axioms, 85 uniqueness theorem, 86 Eilenberg-Zilber theorem, 107 Epstein, D. B. A., 97,220 Equivalence relation, 10,40 generated, 42 graph of, 41 Euclidean half-space, 127 Euler characteristic, 73 Euler class, 189 Exact sequence, 19 long, 2 1 short, 19 split, 55 Excision theorem, 54 Ext, 91 External product cohomology, 110 homology, 108
F Face, 3 Fadell, E., 199,219 Finite CW complex, 62 Finite CW subcomplex, 65 Five lemma, 5 3
INDEX
Fixed-point index. 179 Fixed-point theorem, 188 Flow, 197 fixed point of, 197 Free abelian group, 4 Free resolution, 80 Front i-face, 113 Functor, 103 contravariant, 103 covariant, 103 free with respect to models, 104 natural transformation of, 103 Fundamental class, 139 of manifold with boundary, 153 Fundamental theorem of algebra, 193 Fundamental theorem of calculus, 92
I Index of fixed point, 173 of manifold, 166 of vector field, 195 Integral coefficients, 8 1 Interior, 22 Invariance of domain, 38 Isotopy, 134
J Jordan-Brouwer separation theorem, 37
K G Gauss-Bonnet theorem, 197 Generalized homology theory, 88 Graded abelian group, 5 of finite type, 123
Kirby, R., 172,220 Kronecker index, 93 Kiinneth formula for chain complexes, 102 for singular cohomology, 123 for singular homology, 108
L H Hodge, W. V. D., 171 Horn, 88 Homology groups, 6, see also Singular homology group Homology theory, 85 generalized, 88 Homotopy, 15 equivalence, 18 inverse, 18 of map of pairs, 52 type, 18 Hopf, H., 29 Hopf invariant, 115-121 for Hopf maps, 16 1 Hopf map, 48 Hu, S. T., 121, 220 Hurwitz and Radon, 33
Lefschetz, S., 50, 174, 220 class, 184 coincidence theorem, 187 fixed-point theorem, 174, 188 number, 174, 183 Long exact sequence, 21
M Madsen, I., 172, 219 Manifold, 127 Map of pairs, 52 Mayer-Vietoris sequence, 25 Mesh, 201 Milgram, R. J., 172, 219 Milnor, J., 97, 141, 196, 209, 220 Models, 104 Morphism, 103 Mosher, R. E., 97,220
235
236
INDEX
0 Orientation, 138 double covering, 138 of manifold with boundary, 153
P Pair of spaces, 5 1 Path components, 10 Pathwise connected, 8 Poincark, H., 173, 198 duality theorem of, 124, 149-153 last geometric theorem of, 198 relative form, 156 Poincar6-Hopf theorem, 196 Poincark-Lefschetz duality theorem, 156 Projective space real, 42, 46 complex, 47 quaternionic, 48 Proper n-ball, 138
Q Quotient function, 41
R Rank of finitely generated abelian group, 72 Rational homology sphere, 190 Relative homeomorphism, 59 Relative homology grovp, 5 0 Retraction, 19 Right exact functor, 79
S
Samelson, H., 124, 176,220 Segment, 1 Siebenmann, L., 172,220 Short exact, 19 Simplex, 1 ordered, 2 singular, 3 standard, 3
Singer, 1. M., 197,220 Singular chain, 4 of (X, A ) , 51 Singular cohomology group, 90 reduced, 94 of sphere, 96 Singular cohomology ring of complex projective space, 160 of quaternionic projective space, 161 of real projective space, 161 of torus, 114 Singular homology group, 5 additive property of, 11 with coefficients in G, 81 of complex projective space, 49 of generalized torus, 49 of point, 8 of quaternionic projective space, 5 0 of real projective space, 72 reduced, 56 relative, 50 of sphere, 27 topological invariance, 11 Singular simplex, 3 affine, 204 Skeletal chain complex, 67 Skeleton, 63 Sphere, 26 Spherical class, 49 Split exact, 5 5 Star of vertex, 124 Steenrod,N. E.,85,97, 121, 152,219,220 Stong, R., 172,220 Suspension, 30
T Tangora, M. C., 97, 220 Tensor product, 75 of chain complexes, 98 Thom class, 144 Thom isomorphism theorem, 144-147 for circle, 146 Thorpe, J. A., 197,220 Topological cobordism ring, 172 Topological manifold, 127 Tor, 80 Torus, 46
INDEX
U
Volume element, 197
Universal coefficient theorem, 84, 97 V Vector field, 3 3 Vertex, 2
W Wedge, 46 Whitehead, J . H. C., 117
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Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Paul A. Smith and Samuel Ellenberg
Columbia University, New York
1 : ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) BAER.Linear Algebra and Projective Geometry. 1952 2 : REINHOLD 3: HERBERT BUSEMANN A N D PAUL KELLY.Projective Geometry and Projective Metrics. 1953 BERCMAN AND M. SCHIFFER. Kernel Functions and Elliptic Differential 4 : STEFAN Equations in Mathematical Physics. 1953 BOAS,JR. Entire Functions. 1954 5 : RALPHPHILIP BUSEMANN. The Geometry of Geodesics. 1955 6: HERBERT CHEVALLEY. Fundamental Concepts of Algebra. 1956 7 : CLAUDE Hu. Homotopy Theory. 1959 8: SZE-TSEN Solution of Equations and Systems of Equations. Third 9: A. M. OSTROWSKI. Edition, in preparation Treatise on Analysis : Volume I, Foundations of Modern Analy10 : J. DIEUDONN~. sis, enlarged and corrected printing, 1969. Volume 11, 1970. Volume 111, 1972 Curvature and Homology. 1962. 11: S. I. GOLDBERG. HELCASON. Differential Geometry and Symmetric Spaces. 1962 12 : SIGURDUR Introduction to the Theory of Integration. 1963. 13 : T. H. HILDEBRANDT. ABHYANKAR. Local Analytic Geometry. 1964 14 : SHREERAM L. BISHOPAND RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 15 : RICHARD A. GAAL.Point Set Topology. 1964 16 : STEVEN Theory of Categories. 1965 17 : BARRYMITCHELL. P. MORSE.A Theory of Sets. 1965 18 : ANTHONY CHOQUET. Topology. 1966 19 : GUSTAVE 20: 2. I. BOREVICH AND I. R. SHAFAREVICH. Number Theory. 1966 21 : Josh LUISMASSERA A N D J U A N JORGE SCHAFFER. Linear Differential Equations and Function Spaces. 1966 22 : RICHARD D. SCHAFER. An Introduction to Nonassociative Alegbras. 1966 23: MARTINEICHLER.Introduction to the Theory of Algebraic Numbers and Functions. 1966 24 : SHREERAM ABHYANKAR. Resolution of Singularities of Embedded Algebraic Surfaces. 1966 25 : FRANCOIS TREVES.Topological Vector Spaces, Distributions, and Kernels. 1967 D. LAXA N D RALPHS. PHILLIPS. Scattering Theory. 1967. 26 : PETER 27: OYSTEINORE.The Four Color Problem. 1967 28: MAURICE HEINS.Complex Function Theory. 1968
29: R. M. BLUMENTHAL A N D R. K. GETOOR. Markov Processes and Potential Theory. 1968 30 : L. J. MORDELL. Diophantine Equations. 1969 31 : J. BARKLEY ROSSER.Simplified Independence Proofs : Boolean Valued Models of Set Theory. 1969 32 : WILLrahr F. DONOGHUE, JR. Distributions and Fourier Transforms. 1969 33: MARSTON MORSEA N D STEWART S. CAIRNS.Critical Point Theory in Global Analysis and Differential Topology. 1969 WEISS. Cohomology of Groups. 1969 34: EDWIN 35 : HANS FREUDENTHAL A N D H. DE VRIES.Linear Lie Groups. 1969 FUCHS. Infinite Abelian Groups : Volume I, 1970. Volume 11, 1973 36: LASZLO 37 : KEIONAGAMI. Dimension lheory. 1970 3 :PETER L. DUREN.Theory of HQSpaces. 1970 39 : BODOPAREIGIS. Categories and Functors. 1970 40 : PAUL L. BUTZERA N D ROLFJ. NESSEL.Fourier Analysis and Approximation : Volume 1, One-Dimensional Theory. 1971 PRUCOVEEKI. Quantum Mechanics in Hilbert Space. 1971 41 : EDUARD An Introduction to Transform Theory. 1971 42: D. V. WIDDER: 43: MAXD. LARSEN A N D PAUL J. MCCARTHY. Multiplicative Theory of Ideals. 1971 44: ERNST-AUGUST BEHRENS. Ring Theory. 1972 45 : MORRISNEWMAN. Integral Matrices. 1972 Introduction to Compact Transformation Groups. 1972 46 : GLENE. BREDON. ~~REUB STEPHEN , HALPERIN, A N D RAYVANSTONE. Connections, Curva47 : WERNER ture, and Cohomology: Volume I, De Rham Cohomology of Manifolds and Vector Bundles, 1972. Volume 11, Lie Groups, Principal Bundles, and Characteristic Classes, in preparation 48 : XIA DAO-XING.Measure and Integration Theory of Infinite-Dimensional Spaces : Abstract Harmonic Analysis. 1972 49 : RONALD G. DOUGLAS. Banach Algebra Techniques in Operator Theory. 1972 50 : WILLARD MILLER, JR. Symmetry Groups and Their Applications. 1972 5 1 : ARTHUR A. SACLE A N D RALPHE. WALDE.Introduction to Lie Groups and Lie Algebras. 1973 52 : T. BENNYRUSHING.Topological Embeddings. 1973 53 : JAMESW. VICK.Homology Theory : An Introduction to Algebraic Topology. 1973 54: E. R. KOLCHIN. Differential Algebra and Algebraic Groups. 1973 Iir preparation GERALD J. JANUSZ. Algebraic Number Fields SAMUEL EILENBERC. Automata, Languages, and Machines : Volume A H. M. EDWARDS. Riemann’s Zeta Function WAYNE ROBERTS A N D DALEVARBERC. Convex Functions
