Homotopy Theory (Pure and Applied Mathematics 8)

Homotopy Theory SZE-TSEN HU Wayne State University, Detroit, Michigan 1959 IDI ACADEMIC PRESS • New York and London ...

Author: Sze-Tsen Hu

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Homotopy Theory

SZE-TSEN HU Wayne State University, Detroit, Michigan

1959

IDI ACADEMIC PRESS • New York and London

COPYRIGHT © 1959, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London WIX 6BA

LIBRARY OF CONGRESS CATALOG CARD NUMBER:

Fifth Printing, 1971

PRINTED IN THE UNITED STATES OF AMERICA

59-11526

Preface The recognition of the branch of mathematics now called homotopy theory took place in the few years after the introduction of homotopy groups by Witold Hurewicz in 1935. Since then, with numerous advances made by various workers, it has been playing an increasingly important role in the expanding field of algebraic topology. However, there exists no textbook on the subject at any level except the extremely condensed Cambridge tract of P. J. Hilton entitled "An Introduction to Homotopy Theory." The present book is designed to guide a reader, who might be a beginning student or a newcomer to this branch of mathematics and who has a little knowledge of elementary algebraic topology, through the basic principles of homotopy theory. The author has aimed to provide the reader with sufficient detail for him to understand the fundamental ideas and master the elementary techniques so that he may be able to study the more advanced and more complicated results directly from the original papers. The main problem in homotopy theory is the extension problem as formulated in Chapter I and illustrated in Chapter II. The fiber spaces, which are of fundamental importance, are defined and studied in Chapter III. Homotopy groups are constructed and axiomatized in Chapter IV while the elementary techniques of computation are given in Chapter V. Chapter VI gives an introduction to the obstruction theory of continuous maps, and Chapter VII contains an account of the cohomotopy groups. In the next three chapters, one will find an exposition of the spectacular results obtained mostly by the French school after Leray's discovery of the spectral sequence. The techniques developed in these chapters are applied to compute the first few homotopy groups of spheres in the final chapter. As indicated in the second paragraph above, this book is by no means designed to be an exhaustive treatment of its subject; for example, the recent celebrated contribution of M. M. Postnikov is not included. Besides, homotopy theory is advancing so rapidly that any treatment of this subject becomes obsolete within a few years. At the end of each chapter is a list of exercises. These cover material which might well have been incorporated in the text but was omitted as not essential to the main line of thought. The inexperienced reader should not be discouraged if he cannot work out these exercises. In fact, if he is interested in one of the exercises, he is expected to read the papers indicated there. The bibliography at the end of this book has been reduced to the minimum essential to the text and the exercises. References to this bibliography are included for the convenience of the reader so that he can find more details

VI

PREFACE

concerning the material; the references are not intended to be a historical record of mathematical discovery. (These references are cited in the text by numbers enclosed in square brackets). Frequently, expository articles are preferred to the earlier original papers. Whenever no reference is given concerning some subject or in some exercise, it means only that the reader does not have to look for further details in order to understand the material or to work out the problem. Cross references are given in the form (II; 7.1), where II stands for Chapter II and 7.1 for the numbering of the statement in the chapter. A list of special symbols and abbreviations used in this book is given immediately after the Table of Contents. Certain deviations from standard set-theoretic notations have been adopted in the text; namely, 1:1 is used to denote the empty set and A\B the set-theoretic difference usually denoted by A—B. On the other hand, the symbol I indicates the end of a proof and the abbreviation if stands for the phrase "if and only if." Finally, for the algebraic terminology used in this book, the reader may refer to Claude Chevalley's "Fundamental Concepts of Algebra," published in this series. The author acknowledges with great pleasure his gratitude to Professor Norman Steenrod who has read several versions of the manuscript and whose numerous suggestions and criticisms resulted in substantial improvements. The author also wishes to express his appreciation of the friendly care with which Dr. John S. Griffin, Jr., and Professor C. T. Yang have read the final manuscript, of the many improvements they suggested, and of their help in the proofreading. It is a pleasure to acknowledge the invaluable assistance the author received in the form of partial financial support from the Office of Naval Research when he was at Tulane University and from the Air Force Office of Scientific Research while at Wayne State University. SZE-TSEN

Wayne State University, Detroit, Michigan

Hu

Contents

PREFACE

. .

v

,

LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS

. . x ii

CHAPTER I. MAIN PROBLEM AND PRELIMINARY NOTIONS 1 . • 1. Introduction . . 1 • • . 2. The extension problem . 1 3. The method of algebraic topology . 3 4. The retraction problem . . 5 5. Combined maps . . . . 7 6. Topological identification . . . 8 7. The adjunction space . . . 9 . 8. Homotopy problem and classification problem . 11 9. The homotopy extension property 13 . • • . . 10.Relative homotopy.. . 15 . . 11. Homotopy equivalences. . 17 12.The mapping cylinder • . . 18 • 13.A generalization of the extension problem. . 20 14.The partial mapping cylinder . 21 . 15.The deformation problem . 22 16.The lifting problem. . 24 17.The most general problem . . 25 Exercises. . . . 25 • • CHAPTER II. SOME SPECIAL CASES OF THE MAIN PROBLEMS 35 I. Introduction . . . . . 35 2. The exponential map p: R --> S1 . 35 37 3. Classification of the maps SI ---> S' . . 39 4. The fundamental group. . . . . 42 5. Simply connected spaces 6. Relation between 7 t i (X , x0) and Hi (X) 44 . 7. The Bruschlinsky group. 47 . . 52 8. The Hopf theorems. . , 9. The Hurewicz theorem . . . 56 • Exercises . . . . . 57 CHAPTER III. FIBER SPACES . . 1. Introduction . . 2. Covering homotopy property vii

. ,

. 61 . 61 61

viii

CONTENTS

3. 4. 5. 6. 7. 8. 9.

Definition of fiber space. Bundle spaces . . • Hopf fiberings of spheres . Algebraically trivial maps X -÷ 5 2 Liftings and cross-sections . Fiber maps and induced fiber spaces Mapping spaces . 10.The spaces of paths • • . 11.The space of loops . • . 12.The path lifting property . 13.The fibering theorem for mapping spaces . 14.The induced maps in mapping spaces. 15. Fiberings with discrete fibers . 16. Covering spaces . 17. Construction of covering spaces . Exercises. . . . CHAPTER IV. HOMOTOPY GROUPS. 1. Introduction . . 2. Absolute homotopy groups 3. Relative homotopy groups . 4. The boundary operator . 5. Induced transformations 6. The algebraic properties. . 7. The exactness property .. . 8. The homotopy property. . 9. The fibering property . 10.The triviality property . 11. Homotopy systems. . 12.The uniqueness theorem . 13.The group structures . 14.The role of the.basic point . 15.Local syStem of groups . 16. n-Simple spaces : Exercises. . .

. . • • • . . . .

. . .

. •

• -

. . . .

• .

• .

. 62 65 . . 66 . 68 . 69 . 71 73 . . 78 79 . . 82 . 83 . 85 86 . . 89 93 . 97 . 107 107 107 110 112 . 113 . 114 115 117 118 119 119 121 123 125 129 131 135

CHAPTER V. THE CALCULATION OF HOMOTOPY GROUPS. 143 • • 143 1. Introduction . • • • • • • • • . . 143 . 2. Homotopy groups of the product of two spaces 145 3. The one-point union of two spaces . . 4. The natural homomorphisms from homotopy groups to homo. 146 logy groups . . 150 5. Direct sum theorems . . 152 6. Homotopy groups of fiber spaces.

iX

CONTENTS

7. Homotopy groups of covering spaces 8. The n-connective fiberings . . 9. The homotopy sequence of a triple 10. The homotopy groups of a triad . . 11. Freudenthal's suspension Exercises. .

154 155 159 160 162 164

CHAPTER VI. OBSTRUCTION THEORY 1. Introduction •

175 175 175 176 178 180 181 182 183 184 185 186 187 188 190 191 191 193 193

2. The extension index 3. The obstruction en+1. (g) 4. The difference cochain . . 5. Eilenberg's extension theorem 6. The obstruction sets for extension 7. The homotopy problem . 8. The obstruction dn(f ,g ht) 9. The group Rn(K ,L f) 10. The obstruction sets for homotopy 11. The general homotopy theorem 12. The classi fi cation problem . . 13. The primary obstructions 14. Primary extension theorems. 15. Primary homotopy theorems 16. Primary classification theorems 17. The characteristic element of Y • • Exercises. .

•

;

.

;

.

CHAPTER VII. COHOMOTOPY GROUPS . • . 1. Introduction . • 2. 3. 4. 5. 6. 7. 8. 9:

205 205 205 206 208 209 214 216 219 220 222 222 224 226

The cohomotopy set am(X A) The induced transformations . . The coboundary operator The group operation in nm(X, A) The cohomotopy sequence of a triple . An important lemma The statement (6) . The statement (5) . ,

.

.

10. Higher cohomotopy groups . 11. Relations with cohomology groups 12. Relations with homotopy groups. Exercises. . CHAPTER VIII. EXACT COUPLES AND SPECTRAL . SEQUENCES. 1. Introduction

.

•

. .

229 229

X

CONTENTS

Differential groups . . . Graded and bigraded groups. . . Exact couples . . Bigraded exact couples . Regular couples . . . . 7. The graded groups R(W) and S(W) . 8. The fundamental exact sequence. 9. Mappings of exact couples . . 10. Filtered differential groups . . • 11. Filtered graded differential groups . 12. Mappings of filtered graded d-groups . Exercises. . . • • . 2. 3. 4. 5. 6.

.

.

.

. . •

. •

.

• .

. . . . .

229 231 232 234 236 . 238 . 240 . 242 . 244 . 245 . 248 . 249

CHAPTER IX. THE SPECTRAL SEQUENCE OF A FIBER SPACE 259 1. Introduction . . . . • • • • • 259 • 2. Cubical singular homology theory . . . 259 . 3. A filtration in the group of singular chains in a fiber space . 262 4. The associated exact couple . . • • 263 • 5. The derived couple. . . . 266 . 6. Homology with arbitrary coefficients . 269 7. The spectral homology sequence . • • 271 . 8. Proof of Lemma A . . . 272 9. Proof of Lemma B . . . . 274 10. Proof of Lemmas C and D • • 275 11.The Poincaré polynomials • • . . 277 12. Gysin's exact sequences. . - 280 13. Wang's exact sequences. . . 282 14. Truncated exact sequences . . 284 15.The spectral sequence of a regular covering space . 285 16.A theorem of P. A. Smith. 287 17. Influence of the fundamental group on homology and cohomology groups . . . . 288 . . . . . 18. Finite groups operating freely on Sr . . . 290 Exercises. . . • 292 ■

CHAPTER X. CLASSES OF ABELIAN GROUPS . 1. Introduction . . . 2. The definition of classes. . 3. The primary components of abelian groups 4. The '-notions on abelian groups . . . . 5. Perfectness and completeness 6. Applications of classes to fiber spaces. 7. Applications to n-connective fiber spaces . 8. The generalized Hurewicz theorem .

.

• . . .

. 297 . 297 . 297 . 298 . 298 . 300 . 300 . 304 . 305

xi

CONTENTS

9. The relative Hurewicz theorem 10. The Whitehead theorem - . . . Exercises.

. • .

•

• •

. . .

. 306 . 307 . 308

CHAPTER XL HOMOTOPY GROUPS OF SPHERES . 1. Introduction . . . 2. The suspension theorem . . . 3. The canonical map . . 4. Wang's isomorphism p * . . 5. Relation between p * and i# . . . • • 6. The triad homotopy groups . 7. Finiteness of higher homotopy groups of odd-dimensional . . spheres . 8. The iterated suspension . . . . 9. The p-primary components of n m (S 3) . 10. Pseudo-projective spaces . 11. Stiefel manifolds • 12. Finiteness of higher homotopy groups of even-dimensional . • spheres . . . . . . 13. The fi-primary components of homotopy groups of even. dimensional spheres . . . . 14. The Hopf invariant. . . • • 15. The groups n n ,i(Sn) and nni_2(S92). . 16. The groups 7rri+3(Sn) . . 17. The groups an 4(S) . . . 18. The groups nn_i,r(Sn), 5 Y denote any map. Then, by Tietze's extension theorem [L2 ; p. 28], g has an extension over X. See Ex. D at the end of the chapter. In the last example given above, the space Y = / has the property that the extension problem is always trivial regardless of the domain (X, A) provided that X is normal and A is closed. The class of spaces having this property are the solid spaces. Precisely, a space Y is said to be solid if every map g :A > Y of any closed subspace A of an arbitrary normal space X has an extension over X. -

-

Proposition 2.1.

Any topological product of solid spaces is solid.

3.

3

THE METHOD OF ALGEBRAIC TOPOLOGY

Proof. Let { Y o ltteM} be a collection of solid spaces and Y

Pt, Y„ denote the topological product of this collection, [1. 2 ; p. 10]. We are going to prove that Y is solid. Let A be a closed subspace of a normal space X and g:A -÷ Y any given map. Denote by pm : Y M, the natural projection of Y onto Yi, and set 5,,g .11 Since Y,„ is solid, gt, has an extension /I, : X -÷ Y 12 . Define a map f :X by taking /(x) f t,(x), (x E X).

Y

It is obvious that f is an extension of g over X. I Since the closed unit interval I is solid as noted above, it follows from (2.1) that any compact parallelotope, [L 2 ; p. 19], is solid; in particular, the n-cube in and the Hilbert cube P are solid, Their homeomorphs are likewise solid, hence the n-cell and the n-simplex are solid. 3. The method of algebraic topology

In the preceding section, we formulated the extension problem and gave examples in which the extension existed. It is natural to look for examples where the extension does not exist. The primary method of proving nonexistense is to apply homology theory and derive an algebraic problem from the geometric one and, finally, show that the algebraic problem has no solution. For this purpose, let us consider the triangle A X

of maps as described in the preceding section. In any homology theory satisfying the Eilenberg-Steenrod axioms [E—S; pp. 10-12], the maps f, g, h induce for each m the homomorphisms f * , g* , h* indicated in the following diagram: H Hm (A) g 11\

Hm (Y)

/1* Hm (X)

According to Axiom 2, [E—S; p. 11], the relation fh = g implies the commutativity relation f * h* =g* in the triangle of homomorphisms given above. Hence, the existence of an extension f: X Y of the map g:A -÷ Y gives a solution of a derived algebraic problem, namely, to find a homomorphism çb : Hm(X) H m (Y)

I, MAIN PROBLEM AND PRELIMINARY NOTIONS

4

such that the commutativity relation Oh* = g* holds. Thus, the existence of the homomorphism st. is a necessary (though not generally sufficient) condition for the existence of an extension of the given map g over X. On many occasions, this necessary condition provides us a method to show that a particular given map g :A -* Y fails to have an extension over X. For example, let us take X to be the unit n-cell En of the n-dimensional euclidean space Rn, A = Y to be the boundary (n-1)-sphere Sn -1- of En, and g:A -* Y to be the identity map on sa-1. We shall prove the following Proposition 3.1.

For each n>1, the identity map g:Sn --1 -*Sn --1 has no

extension over En .

Assume that there is some extension f: En --›- Sn -1 of g. We shall deduce a contradiction as follows. Assume n> 1 and consider the homology theory with the group Z of integers as the coefficient group. Take m = n — I, then we have Proof.

Hm(E

)

= 0,

Hm (S") gy, Z.

Since g is the identity map on Sn -1-, it follows from Axiom 1 that g* is the identity automorphism of Hm (Sn -1). Since Hm (S 1) 0 0, this implies that g* O. On the other hand, since Hm(E) , 0, the inclusion map h:Sn -4 c En induces h* = O. Thus, we obtain f*h* = 0 and g* O. This contradicts the relation ii.h. _-=, g* . In fact, the derived algebraic problem has no solution. It remains to dispose of the case n = 1. In this case, SnA consists of two points and hence is disconnected. On the other hand, En is connected and so is its continuous image f (En) Since f is an extension of the identity map g, we have /(E) = Sn-1 . This is a contradiction. 1 .

can also be proved from the derived algebraic problem provided that one uses the reduced homology groups, [E-S; pp. 18-19]. As an important application of (3.1), we shall give the following Note. The case n = 1 of (3.1)

Theorem 3.2.

(The Brouwer Fixed-Point Theorem). Every map f :En -*En

has a fixed point, that is to say, there exists a point x of En such that f(x) . x.

that f :En -)- En is free of fixed points. Then, we may define a map r : En' --> Sn -1 as follows. Let x E En . Since f has no fixed points, we have f (x) 0 x. Draw the line from f (x) to x and produce until it intersects Sn -1- at a point r(x). One verifies that the assignment x -* r(x) defines a continuous function r:En -* Sn -'. If x E S12-1, it is obvious from the construction that r(x) - x. Hence r is an extension of the identity map on Sn --1-. This contradicts (3.1). 1 This application of (3.1) shows that the negative nature of a "non-existence" Proof. Assume

4.

THE RETRACTION PROBLEM

5

theorem may not diminish its interest. A reformulation sometimes gives it a positive aspect. In the derived algebraic problem formulated above, one may of course use cohomology theory instead of homology theory. 4. The retraction problem

If Y . A and g = i is the identity map on A, then we obtain an important special case of the extension problem which will be referred to as the retraction problem. If i has an extension r:X ---> A, then A is called a retract of X, r is called a retraction of X onto A, and we will write r:X

A.

According to (3.1), the boundary (n-1)-sphere Sn --1 of En is not a retract of E. On the other hand, if X denotes the space obtained by deleting from En an interior point which may be assumed to be the origin 0 without loss of generality, then Sn -1 is a retract of X. In fact, a retraction r: X .D Sn --1 is given by r(x) —

Xn, )

(x i

lx I

.) • • • P

Ix I

for every point x = (x 1 , .. . ,x) of X — E\ O, where I x i denotes the distance between 0 and x. The same formula also gives a retraction r of kn \ 0 onto Sa-1 . For another example of retracts, let us consider the topological product X = A x B. Pick a point bo from B. Then A can be considered as a subspace of X by means of the homeomorphism h:A --> X defined by h(a) = (a, bo) for each a e A. This having been observed, it becomes clear that the natural projection X=A xB -4- A gives a retraction of X onto A. In particular, a meridian of the torus T2 -= .9- X .S1 is a retract of T2. Observe that if A is a retract of X then the extension problem becomes trivial regardless of the range Y. Indeed, we have the following Prdposition 4.1. A is a retract of X iff, , for any space Y, every map g:A ---)- Y has an extension over X.

Proof. If A is a retract of X with a retraction r:X

A, then gr:X ---> Y

is an extension of g. Conversely, assume that the condition holds and take Y . A. Then the identity map i on A should have an extension r: X -* A. I The retraction problem gives rise to a derived algebraic problem as follows. In any homology theory or cohomology theory satisfying the EilenbergSteenrod axioms, the inclusion map i:A c X induces for each m the homomorphisms

i *: . Hn,(A) --> II,„(X) , i* :Ilm(X) ---> Ilm(A).

6

I. MAIN PROBLEM AND PRELIMINARY NOTIONS

The derived algebraic problem is to determine whether or not there exist

homomorphisms m (X) H m (A) , v:Hm(A) Hm(X) such that 0i* and i*v are the identities on Hm (A) and Hm(A) respectively. The existence of the homomorphisms 0 and v is a necessary condition for A to be a retract of X. In fact, if r:X D A is a retraction, then ri is the identity map on A and hence 0 — r* and v r* are solutions of the derived algebraic problem. Furthermore, since ri* and i*r* are the identities, it follows that i* , e are monomorphosms, that r* , i* are epimorphisms, and that H.,(X), Hm(X) decompose into the following direct sums : Image i* ± Kernel Y* ,

Hm (X)

Hm(X) = Kernel

i* + Image r* .

If the coefficient group of the cohomology theory is a ring, then the cohomo0, 1, • , constitute a ring H*(X) with the cup logy groups Hm(X), rn product as multiplication. The inclusion i: A c X and the retraction r :X A induce the ring homomorphisms —

i* :H(X) H(A),

r*

:H(A)

H* (X).

Since ri the identity map on A, it follows that i*r* is the identity automorphism of the ring H* (A). Hence Y* is a monomorphism, i* is an epimorphism, and H*(X) decomposes into the direct sum

H*(X) = Kernel i* ± Image

7*

where Kernel i* is an ideal and Image 7* is a subring isomorphic to H* (A) under r* . These necessary conditions can be used to prove that a particular given subspace A of a certain space X fails to be a retract of X. For example, let X denote the complex projective space of complex dimension n > 1 and A a linear subspace,of X of complex dimension Y with 0 < Y < n. Then A is not a retract of X. To prove this fact, let us assume that there is a retraction Y: X D A. Then we obtain a ring monomorphism 7* H*(X) of the cohomology rings with integral coefficients. Let a and E denote generators of the free cyclic groups .1/2 (A) and H2(X) respectively. Sitirc-cr* is a monomorphism, there is a non-zero integer k such that r*(a) k4. Since n > we have an = 0. Since 7* preserves multiplication, we obtain

le This contradicts the fact that H2n (x)

=

r* (an)

0.

is a generator of the free cyclic group

Finally, let us give an important example of retract in the form of the following

5.

COMBINED MAPS

7

1/ (X, A) is a (finitely) triangulable pair, [E-S; p. 60], then the closed subspace L ,-- (X x 0) U (A X I) Proposition 4.2.

of the product space M—X XI is a retract of M. Proof. First, let us prove the special case where X is the unit n-simplex Lin of the euclidean (n + 1)-space, [E-S; p. 55] , and A is the boundary (n-1)-sphere of d i/ which is empty if n--- O. Then a retraction r:M L can be constructed geometrically as follows. Since I c R, it follows that M is a subspace of Li n X R. Then we define Y to be the central projection of M onto L from the point (e, 2) of dn X R, where c denotes the centroid of tin. This proves the special case. For a finitely triangulable pair (X, A), we may assume that X is a finite simplicial polyhedron and A is a subpolyhedron. Since a retract of a retract is also a retract, one can easily prove the proposition by induction on the number of simplexes in X but not in A and by the aid of the special case proved above. I A strengthened form of (4.2) will be given in § 10. Besides, (4.2) can also be generalized to some non-triangulable pairs; see Ex.0 at the end of the chapter.

5. Combined maps Frequently, a function is constructed by prescribing it on pieces of its domain. The purpose of this section is to give sufficient conditions, for the continuity of functions so constructed. Let { X, itteM} be a given system of subspaces of a space X, indexed by the elements of a set M, such that the union of all subspaces X ii ,,ct E M, is the whole space X. Let Dtt, .--- Xi, 11 X v for each pair of indices tt, v in M. For any given map g:X -÷ Y of X into a space Y, the partial maps gp . g I X t, are well-defined and satisfy the relation gti lDp„ --.. g„IDtt , for every pair of indices ,u,v in M. Hence, our problem in this section is to study the inverse of this process described as follows. Let us assume that, for each index iu e M, there is given a map /12 :4 --> Y such that 41 I Dp v = lv 1 D to for each pair of indices tt and y in M. Then we may define a function /: X --> Y by taking f I x, - ft. (itt G

m

).

We are concerned with the problem whether or not / is continuous. If M is finite and all the subspaces X,4 , tt e M, are closed in X, then the combined function f is continuous, Proposition 5.1.

Proof. Let F be any closed set in Y. Then it follows from the continuity of /1,, su e M, that /1,-1.(F) is a closed set of X. Since X, 4 is closed in X, this

8

1.

MAIN PROBLEM AND PRELIMINARY NOTIONS

implies that /.-.1-(F) is a closed set of X. Since M is finite, , . U pem/ i.-1(F)

is a closed set of X. Hence / is continuous. I A frequent application of this proposition is as follows. Let /,g: X x 1 . ----> Y be two given maps such that /(x,1) = g(x ,0) for every x E X. Then we may define a function h:X x I --> Y by taking: f /(x,2t), (x e X, 0 - Y (0 < t < I) by taking f t — / for every te I. Then ft :/c_—_ /. (2) To prove the symmetry, let ht :f .c._-, g be a homotopy connecting two given maps / e Q and g e Q. Define a homotopy k t : X -* Y (0 < t < 1) by taking kt — 14_4 for each t e I. Then kt : g _,--. f. (3) To prove the transitivity assume Ot :/ c_—_ g and vt :g c_-_- h. Define a system of maps X t : X --> Y (0 < t < 1) by taking (0 Y, (0 < t < 1), a given partial homotopy of f. Consider the product space M ,-- X x I and its closed subspace L = (X x 0) U (A x I). Define a map H:L -÷ Y by setting

H(x, t) —

f f (x), 1 ht (x),

if x E X, t — 0, if xEA., tE.T.

According to (4.2), there is a retraction r:M D L. Define a homotopy gt :X ---> Y, (0 < t Y a given map, and ht :A ---> 17., (o t < 1), a given partial homotopy of f. Consider the spaces M -, X x I and L . (X x 0) U (A x I), and define a map II:L -± Y a‘s in the proof of (9.2). By §2, .4 71 is solid. Since L is closed in the normal space M, the map H:L-->Y 4. has an extension G:M--->Jn. Let V . G -1 (U). Then V is an open neighborhood of L in M. Since the unit interval I is compact, one can easily prove that there exists an open neighborhood W of A in X such that W x I is contained in V. Since X is normal, it follows from Urysohn's lemma, [L 2 ; p. 27], that there exists a continuous real function Z: X -> I such that Z(X \ W) = 0 and X(A) ..-- 1. Define a map .17 :M ---> Y by taking

Y, (0 < t < 1), defined by gt (x) — F(x,t) for each x e X and t GI. Clearly go . f. 1

10. RELATIVE HOMOTOPY

15

Generalizations of (9.2) and (9.3) will be given in Ex.N and Ex.0 at the end of the chapter. Now let A be a given subspace of a space X and g:A ---> Y a given map. If A has the HEP in X with respect to Y, then the restricted extension problem in § 2 obviously depends only on the homotopy class of g. In this case, the restricted extension problem is equivalent to a broadened form, namely to ask whether there is a map f: X -÷ Y such that fh g, where h:A c X denotes the inclusion map. This new broadened form is apparently weaker than the original restricted form. However, once we reflect upon the concept of homotopy, we realize that the original form of the problem was formulated in too narrow a fashion. Hereafter, when we refer to the extension problem, we shall mean its broad — ened form unless otherwise stated.

10. Relative homotopy Let M be an abstract set and X, Y be two given spaces. Let { }, be two systems of sets indexed by tt G M such that X I, C X and Yt, a Y for each index IL e M. For the sake of brevity, we shall denote by { M } the detailed notation { X y , Y . 11.1 E M} whenever there is no danger of ambiguity. Consider the maps f: X --)-Y such that f(X li) c Y„ for each la e M. The totality of these maps will be denoted by

Q Yx{M} Yx

I

eM1.

For example, if M consists of a single element p, and if X I, ---- A, Y y = B, then D is the set of all maps f (X, A) (Y, B), see [E—S; p. 3]. For a second example, let M contain two elements kt and v. If X„ = A, Y ,2 = B are as }, Y,, yo 1 are singletons with xo e A, yo e B, then above and X, D is the set of all maps f: (X, A, x0) -4 (Y, B, yo). It is primarily these cases that the notation is intended to cover. Two maps f and g in Q are said to be homotopic relative to the system { M } if there exists a homotopy ht: X Y, (0 < t < 1), such that ho = f, h 1 = g and h E D for every t e I. In notation,

f

g rel { M } .

In this case, ht is called a homotopy connecting f and g in Q and is denoted by

ht :f

g rel { M}.

If is a single point and Y p = f(X,2), we may identifyu with X 14 If this holds for all iu E M, then M is identified with a subset of X. In this important case, we shall use the usual notation:

f

g rd l M, h t :f

g rel M.

gt


In particular, if M is the empty set, the notation will be simply /-----e. g and ht :f --... g as in §8. As in (8.1) one can easily prove that the relation of homotopy relative to { M } of the maps D is an equivalence relation. Hence, Q is divided into disjoint classes, called the homotopy classes relative to { M } or the homotopy classes in Q. The classification problem of the maps Q is to enumerate these classes in terms of known topological invariants. Let B be a subspace of Y which intersects Yt, for each pc G M. A map / E Q is said to be deformable into B relative to { M }, if there exists a map g:X-->B such that f -,- -2 ig rel { M } Y. In particular, B may consist where i denotes the inclusion map i:B of a single point yo in the intersection of the subspaces Y. Let 0 denote the constant map 0(X) . yo . Then 0 is in Q. A map / is said to be null-homotopic or inessential relative to { M } if /c_-_ 0 rel { M }, otherwise it is said to be essential relative to {M}. Now, let us define the important notion of deformation retract. A subspace A of a space X is said to be a deformation retract of X if there exists a deformation ht : X -4. X, (0 < t (Y, B) is said to be a homotopy equivalence and is denoted by f : (X , A) -_....- (Y, B) if f : X _ Y rel { A, B }. The pairs (X, A) and ( Y, B) are said to be homotopically equivalent, (X, A) _ (Y, B), if there exists a homotopy equivalence /: (X, A) _ (Y, B). The notion of homotopy equivalence introduced above is justified by the fact that the extension problem as broadened in § 9 depends essentially only on the homotopy type of the pair (X, A) and that of the space Y. This result can be precisely formulated as follows. Assume that (X, A) c_-_ (X', A') and Y Y' and let g:A -4- Y be a given map. Let h:A X and h' :A' X' denote the inclusion maps. By definition, there exist maps

such that composed maps 4/0, 00', v'v and vip' are homotopic to the corre-

18

L MAIN PROBLEM AND PRELIMINARY NOTIONS

A' and r :A' -9-A denote the maps sponding identity maps. Let X:A. defined by the maps 0 and 56' respectively, and let g' = ipgr :A' --> Y'. If the extension problem for g has a solution over X, i.e., if these exists a map /:X --> Y such that th g, then the extension problem for g' also has a solution over X'. In fact, /' = WO' : X' -> Y' satisfies the relation

l'h' = VA" = Vihr Vgr = g'.

Conversely, if there exists a map e' : X' -9- Y' such that e'h' extension problem for g has a solution over X. In fact, e

g', then the ---> Y

satisfies the relation

fL-2.1prg7 = v/vgrX

eh -= yle'011

g.

Y is a homeomorphism Examples of homotopy equivalences: (1) If /: X of X onto Y, then 1'1 is obviously a two-sided homotopy inverse of /. Hence homeomorphs are of the same homotopy type. (2) If X is a deformation retract of Y, then the inclusion map /: X C Y is a homotopy equivalence. According to the homotopy axiom and the algebraic axioms, a homotopy equivalence induces isomorphisms on the homology groups and the cohomology groups. Hence, homotopically equivalent spaces have isomorphic homology groups and cohomology groups. A property of spaces is said to be a homotopy invariant if it is preserved by homotopy equivalences. Almost all invariants studied in algebraic topology are homotopy invariants. The role of homotopy equivalences having been clarified, it is natural to search for two-sided homotopy inverses, if any, of a given map /: X -÷ Y relative to a given system { M }. For this purpose, it is sometimes helpful to observe the following fact. If a given map /: X -÷ Y has both left and right homotopy inverses relative to a given system { M }, then it has a two-sided one. In fact, if. g': Y X is a left homotopy inverse of f relative to { M } and if g":Y ---> X is a right homotopy inverse of / relative to { M }, then it is easy to verify that the composed map g = g ig": Y --> X is a two-sided homotopy inverse of f relative to { M }. 12. The mapping cylinder Let /: X -± Y be a given map. By the process of topological identification, we shall construct a space Mf which is called the mapping cylinder off. For this purpose, let us consider the topological sum

W

(X x /) U

of the spaces X x / and Y. If we identify (x, 1)

X x / with /(x) E Y for

12. THE MAPPING CYLINDER

19

every x E X, we obtain a quotient space Mf, the mapping cylinder of /, and a natural projection P.W ----> Mf.

One can easily verify that p maps Y homeomorphically onto a closed subspace poi of Mf. By means of this imbedding, Y will be considered as a closed subspace of Mf. On the other hand, the map g: X -->given by g(x) p(x, 0) for each x G X carries X homeomorphically onto the closed subspace p(X x 0) of Il/If. By means of this topological map g, X is imbedded as a closed subspace of Mf. Thus X and Y are considered as disjoint closed subspaces of Mf and will be called the domain and the range of the mapping cylinder Mf. The range Y of the mapping cylinder Mf of a map Y is a strong deformation retract of Mf.

Proposition 12.1.

f :X

Proof. Define a system of functions ht:Mf ht [p(x, s)]

p(x, s

Mf,

t

(0 < t < I), by taking

st)

ht(Y) = Y for every x c X, y c Y, s E I and t E I. By (6.1), it is easily verified that ht is a homotopy. It is obvious that ho is the identity map on Mf, h1 is a retraction of Mf onto Y, and ht Y is the identity map on Y for each t c T. Hence Y is a strong deformation retract of Mf. I

For any given map f: X are homotopic.

Proposition 12.2.

p/: X -4- Mf

Y, the maps g: X -4- Il/If and

Proof. Let ht : X -÷ Mf, (0 < t < 1), denote the homotopy defined by ht(z) = P(z, 1) for every x c X and t E.E. Then we have ho g and h 1 = Pt. Hence g p/ . I Two maps /: X -4- Y and f: X' -4- Y' are said to be homotopically equivalent if there exist homotopy equivalences 0: X X' and Y ••••,.. Y' such that vf PO. Let cy: X' X and yr: Y' Y denote two-sided homotopy inverses of 96 and v respectively. Then the three conditions V/

rO,

VT5b, V/95' r

are mutually equivalent. As an immediate consequence of (12.1) and (12.2), we have the following theorem which justifies the introduction of the notion of mapping cylinders. Theorem 12.3. Every map f : X -› Y is homotopically equivalent to an in-

clusion map, namely, g:Xc 1V11.


20

Finally, the following special case of mapping cylinder is of importance. If Y consists of a single point y, then the map 1:X ---> Y has to be the constant map f (X) = v. In this case, the mapping cylinder Mf is called the join of X to y or the cone over X and will be denoted by k(X). The point 2, is called the vertex of k(X). 13. A generalization of the extension problem The extension problem as broadened in § 9 suggests a generalized problem

which can be displayed diagrammatically by

where / and g are given maps and h must be found such that hg _.,-_—__ 1. However, this generalized problem is equivalent to the extension problem of the map f: X -÷ Y over the mapping cylinder Mg D X of the map g: X -÷ Z. More precisely, we have the following For any two maps f :X ---> Y and g: X -4- Z, the following two statements are equivalent: (i) There is a map h:Z -÷ Y such that hg c_._-, /. (ii) There is a map F: Mg '---> Y such that FqK_-.-_ f, where q: X c Mg denotes the inclusion mat. Proposition 13.1.

Proof. (i)

>

—

(ii). According to (12.1), there is a retraction r: Mg

defined by

Z

r(p(x, t)) -, g(x), (x e X, t EÏ).

Hence the map h: Z -÷ Y has an extension F = hr:Mg .-- Y. By (12.2), we have Fq — hrq _—_,z hrpg ,--- hg...,- , f, (ii) -4- (1). Let h—FIZ. Then, by (12.2), we have ,

hg = Fpg c.-2.Fq ny_ f . I The derived algebraic problem of this general problem is as follows. In any homology theory or cohomology theory satisfying the EilenbergSteenrod axions, the given maps f: X -÷ Y and g : X -÷ Z induce the homomorphisms /*, g* , f*, g* indicated in the following diagrams: t*

.1-1,,(xY) g\

1-1 m - (Z)

-÷11,,(Y) 51.

Hm(X)Y such that hg /.

14. The partial mapping cylinder The objective of the present section is to show that the generalized problem in § 13, which is so far the broadest form of the extension problem, is equivalent to a retraction problem, which is apparently the narrowest form of the problem. For this purpose, let us introduce the partial mapping cylinder Mg (X) of a given map g : A Y defined on a subspace A of a space X. Consider the topological sum "W = X U (A x I)

UY

of the spaces X, A X / and Y. For each a e A X, identify a with (a, 0) e A x I, and (a, 1) with g(a) E Y. In this way, we obtain a quotient space Mg (X) which is called the partial mapping cylinder of g over X and a natural projection p : -w mg(x).

Then one can easily see that p maps X and Y homeomorphically onto disjoint closed subspaces p(x) and p(Y) of Mg (X) respectively. Thus, X and Y will be considered as disjoint closed subspaces of Mg (X) and will be called the domain and the range of the partial mapping cylinder Mg(X). Lemma 14.1. If the range Y of Mg (X) is a retract of Mg (X), then there exists

a map f : X -> Y such that f A

Proof. Let r : Mg (X) Y be a retraction and define f :X-->Ybyf rIX. Consider the homotopy ht : A -> Y, (0 Y(0 Z be given maps. Let

W

Mf (Mg), V — k( Y),

then Yc W and Yc V. Define a map ç6: (W, Y)

w, 0(w) = (v, q(f (x), 1

t),

(V, Y) by taking

(if w Y), (if w M (if w — p(x, t), x E x tE

0,

where y denotes the vertex of the cone k(Y) and p : M9 U (X x I) U Y W V denote the natural projections. and q : Y x

There exists a map h : Z ---> Y such that hg : (W, Y) (V, Y) is deformable into Y. Theorem 15.1.

f if the map

Proof. Necessity. According to (14.3) there is a retraction 7: WD Y. Define two homotopies : (W, Y) (V, Y), (0 < 1 Y such that GO x I)c B and G(X x l)c Yo . Proposition 15.2. /:

The special case of the deformation problem where Yo B is of importance. In this case, we have nice necessary conditions as given in the following proposition the proof of which is immediate and hence omitted.

24


If a map /: (X, A) --->- (Y, B) is deformable into B, then it induces trivial homomorphisms on homology and cohomology groups, that is to say, f. . 0 and f* = O. Proposition 15.3.

In particular, if (X, A) . (Y, B) and if / is the identity map, we have the following definition. The pair (X, A) is said to be deformable into A if the identity map on (X, A) is deformable into A. If this is the case, then the inclusion map i:AŒ X is a homotopy equivalence. Furthermore, in case A has the HEP in X with respect to A, then the pair (X, A) is deformable into A iff A is a deformation retract of X.

16. The lifting problem

Consider a given map

p : E >B

of a space E onto a space B. Let X be a given space and / :X---)-B a given map. Then the lifting problem in the restricted form is to look for a map g : X -* E such that pg = f. The map g is called a lifting of / (relative to p). This problem is dual to the restricted extension problem of § 2. Dual to the homotopy extension property of § 9, we have the following important notion of covering homotopy property. map p : E --- > B is said to have the covering homotopy pyoperty (abbreviated CHP) for a space X, if, for every map g:X--->-E and every homotopy /t : X ---> B, (0 < t E, (0 < t < I), of g which covers ft, that is to say, pgt --- ft for every t e I. The covering homotopy property is basic to many constructions in homotopy theory. Fortunately, this important property holds in a great many of cases, namely, the various fiber spaces. See Chapter III. Now, if p : E -÷ B has the CHP for X, then the restricted lifting problem for / : X -÷ B obviously depends only on the homotopy class of /. In this case, it is equivalent to the broadened form, namely to ask for a map g: X --)- E such that pg .-2--_, f. Hereafter, when we refer to the lifting problem, we shall mean this broadened form unless otherwise stated. Once the lifting problem is broadened as above, the original condition that p is onto is no longer important. In fact, we have a generalized problem displayed diagrammatically by Definition 16.1. A

X

f

X4/

y

z

where f and h are given maps

and g must be found such that hg..c,-__, f. This general problem is dual to the generalized extension problem of § 13. If h is onto, then it is the lifting problem for f; if h is an inclusion map, then it is the deformation problem for / into the subspace 2 of Y. On the other

17. THE MOST GENERAL PROBLEM

25

hand, this general problem is equivalent to a deformation problem. More precisely, we have the following Theorem 16.2.

For any two maps f: X --)- Y and h: Z -± Y, the following

two statements are equivalent: (i) There exists a mpg : 2 C --> Z such that hg rz-._ /. (ii) The maP 75/:x- -÷ Mh is deformable into Z, where p:YcMh, denotes the inclusion map. Proof. (i) -± (ii). According to (12.2), we have ph :_-_.-. q, where q: Z c Mh denotes the inclusion map. Then, (i) implies that pi n_.-. phg f-_-_-, qg. Hence, pf is deformable into Z. (ii) -÷ (i). There is a homotopy gt : X -÷ Mh, (0 < t < 1), such that go ---, p/ and g1 (X) c Z. Let g: X -÷ Z denote the map defined by gl. By (12.1), there is a retraction r: Mh D Y such that r I Z = h. Let ft : X ---> Y, (0 < t S of f such that F(X) is contained in the convex hull of f(A) in S. Proofs are given in [Dugundji 1]. E. Kuratowski's imbedding

Let X be a given metrizable space and d a distance function which makes X a metric space. Replacing d by d 1(1 + d) if necessary, we may assume that d is bounded. A metric space X with a bounded distance function d is usually called a bounded metric space. 1. Let S denote the totality of bounded continuous real functions defined on X. Prove that S forms a Banach algebra with its norm defined by / I = supxEx f (x) 2. For an arbitrary point a E X, consider the bounded continuous real function fa GS defined by fa(x)

d(a, x)

for each x E X. Prove that the correspondence a fa defines an isometric map : X -± S. X is called Kuratowshi's imbedding of the bounded metric space X, [Kuratowski 1]. 3. Woidyslawski's theorem. The image X(X) of Kuratowski's imbedding Z: X S is a closed subset of the convex hull Z of X(X) in S. If X is separable, then so is Z, [Wojdyslawski 1]. F. Spaces obtained by topological identification

1. Miibius strip. Take a rectangle ABCD and identify the side AB with the side CD so that A goes into C and B into D as shown in the following figure: A

D

AC MN BD

Fig. 1. On the left, there is a rectangle ABCD with M and N as mid-points of two sides. The figure on the right is obtained by pasting the two sides of the rectangle together so that A goes into C, B goes into D, and M goes into N.

The quotient space X thus obtained is called the Möbius strip. Let the construction be so carried out that the mid-points M, N of A B and CD are identified. As a consequence the line MN will go into a Jordan curve

28


on the strip X. Prove that J is a deformation retract of X and that X \J is pathwise connected. 2. Torus and Klein bottle. Take the 1-sphere S1 represented as the complex numbers z with I z I = 1 and consider the product space W .51 X I. If we identify (z, 0) and (z, 1) for each z ES1-, we obtain from W a quotient space T 2 called the torus. On the other hand, if we identify (z, 0) and (z--', 1) for each z E 9, we obtain from W a quotient space U2 called the Klein bottle. In each case, the line 1 X / will go into a Jordan curve J on the quotient space. Prove that J is a retract of the quotient space but not a deformation retract of the quotient space. 5. Closed orientable surfaces. Consider a plane convex region W bounded by a 4g-sided regular polygon P. Let the sides of P be denoted by P = a lb ic id i «

agbgCgdg

in a certain positive sense. For each i 1, • • • , g, let us identify ai with ci and bi with di-1 , where ci -1 denotes ci with reverse direction. The quotient space Eg thus obtained is called the closed Orientable surface of genus g. Prove that the Betti numbers of Eg are Ro . 1,

RI. = 2g, R2 = 1.

4. Closed non-orientable surfaces. Consider a plane convex region W bounded by a 2n-sided regular polygon P. Let the sides of P be denoted by

P

a1b 1a2b 2 • • an b .

in a certain positive sense. Identify ai with bi for each i = 1, 2, • • , n. Thus we obtain a quotient space On, which is called the closed non-orientable surface of characteristic 2 — n. Prove that the Betti-numbers of Q7, are Ro = 1, R1 = n — 1, R2 = O. G. An application of the cone k(X)

Let us use the notation of § 10. Prove that a map f e Q is null-homotopic relative to { M iff there exists a map F: 1?(X) Y of the cone k(X) into Y such that FIX =f, F(v) yo, and F[k(Xti)] c yu for each ,/,/ e M. An important special case is as follows a map f Sn -÷ Y is homotopic to a constant map iff f has an extension F: En+1 Y. Using this, prove that the identify map on Sn is not null-homotopic. H. Extension properties

A subset A of a space X is said to have the extension property in X with Y can be extended over X. The respect to a space Y, if every map f : A subset A is said to have the neighborhood extension property in X with respect to Y, if every map f : A --- Y can be extended over some open set U of X which contains A.

EXERCISES

29

Thus, a space Y is solid,azyery closed subset A of any normal space X has the extension property in X with respect to Y. The subset A is said to have the absolute extension property in X, if it has the extension property in X with respect to every space Y. The subset A is said to have the absolute neighborhoo d. exten' sion property in X, if it has the neighborhood extension property in X with respect to every space Y. Prove the following two assertions: I. A subset A of a space X is a retract of X if it has the absolute extension property in X. 2. A subset A of a space X is a neighborhood retract of X iff it has the absolute neighborhood extension property in X. 1. Borsuk's extension theorems

By means of the results in the exercises D and E, prove the following two extension theorems: 1. A metrizable space Y is an AR iff every closed subset A of an arbitrary metrizable space X has the extension property in X with respect to Y. 2. A metrizable space Y is an ANR if every closed subset A of an arbitrary metrizable space X has the neighborhood extension property in X with respect to Y. As a consequence of the theorem 1, show that every convex subset of a locally convex linear space is an AR; in particular, every simplex is an AR and every euclidean space Rn is an AR. J. Separable ANR's and compact ANR's

Prove the following two theorems: 1. For a separable metrizable space Y, the following statements are equivalent: (i) Y is an ANR. (ii) A topological image of Y as a closed subset Z0 of any separable metrizable space Z is necessarily a neighborhood retract of Z. (iii) Every closed subset A of an arbitrary separable metrizable space X has the neighborhood extension property in X with respect to Y. 2. For a compact metrizable space Y, the following statements are equivalent: (1) Y is an ANR. (ii) Y is homeomorphic with a neighborhood retract of the Hilbert parallelotope (iii) Every closed subset A of an arbitrary normal space X has the neighborhood extension property in X with respect to Y. State and prove the analogous theorems for the AR's. It follows from the theorem 1 that, for a separable metrizable space Y,

30


the definitions of AR's and ANR's given in Ex.0 are equivalent to Kuratowski's modifications of Borsuk's original definitions. In fact, the statement (ii) in theorem I is used as the definition of ANR's by Kuratowski as well as by Lefschetz, [L 3 ; p. 58]. K. Operations on ANR's

Prove the following properties of ANR's: 1. Every open subspace of an ANR is an ANR. 2. Let Y be an ANR and B a closed subspace of Y. B is an ANR if B is a neighborhood retract of Y. 3. Let Y and Z be metrizable spaces. The product space Y x Z is an ANR if both Y and Z arc ANR's. 4. Let Y 1 and Y 2 be two closed subspaces of a rnetrizable space Y with Y . Y 1 U Y2. If Y 1 , Y2 , Y1 n Y2 are ANR's, then so is Y. [Borsuk 5. Let a metrizable space Y be covered by a collection { 17/2 I 1u E M } of disjoint open subspaces. If Kt, is an ANR for every p G M, then so is Y. 6. Let a metrizable space Y be covered by a countable collection { Y n I n--- 1, 2, • • • } of open subspaces. If Yn is an ANR for each n — 1, 2, • • • , then so is Y. [Hanner 1]. L. Locally finite simplicial polyhedra By induction on the number of simplexes and with the aid of the property 4 in Ex.K, prove that every finite simplicial polyhedron is an ANR and hence a compact ANR. [Borsuk 2]. Then, by means of the properties 1, 5 and 6 in Ex.K, prove that every locally finite simplicial polyhedron is an ANR. [Liao 1 Hanner 1]. fi. The general homotopy extension theorem

The notion of the HEP given in § 8 can be localized and apparently generalized as fo'llows. A subspace A of a space X is said to have the neighborhood homotopy extension property (abbreviated NHEP) in X with respect to a space Y, if every partial homotopy

ht : A ÷ , (0 < t Y , (0 < t < 1), -

over some open neighborhood V of A in X such that go = f V. However, if A is a closed subspace of a normal space X, then the NHEP is equivalent to the HEP. In other words, A has the HEP in X with respect to Y if A has the NHEP in X with respect to Y. By the aid of a Urysohn's

EXERCISES

31

characteristic function [L 2 ; p. 27], prove this general homotopy extension theorem. N. Borsuk's homotopy extension theorems

By means of Borsules extension theorem in Ex.I and the general homotopy extension theorem in Ex.M, prove the following theorems: 1. If Y is an ANR, then every closed subspace A of an arbitrary metrizable space X has the HEP in X with respect to Y. [H-W; p. 86]. 2. If Y is a compact ANR, then every closed subspace A of an arbitrary binormal space X has the HEP in X with respect to Y. As an application of these theorems, prove that every contractible ANR is an AR. O. Closed ANR subspaces in an ANR

Let X be an ANR, A a closed subspace of X, and T denote the subspace (X x 0) U (A x I) of the product space X x I. Prove that the following statements are equivalent, [Hu 3 ] : (1) A has the AHEP in X. (2) T is a retract of X X I. (3) T is an ANR. (4) A is an ANR. As a well-known special case, every subpolyhedron A of a locally finite simplicial polyhedron X has the AHEP in X. P. Relation between partial mapping cylinder and adjunction space

Let g : A --> Y be a given map defined on a given subspace A of a space X into a space Y. Consider the partial mapping cylinder Mg (X) and the adjunction space Z obtained by adjoining X to Y by means of g. There is a natural map T : Mg (X) ->- Z -

defined by

T-(u)

q(w), qg(a),

(if u p(w), w E X U Y), (if u = p(a, t) , acA,te /),

where p : X U (A X I) U Y g (X) and q:X U Y Z denote the natural projections. By means of these natural projections, Y can be considered as a subspace of both M g (X) and Z. Hence, T is actually a map (il/g (X), Y) into (Z, Y) such that T Y is the identity map on Y. Using the equivalence of (2) and (4) in the preceding exercise, prove that the map T (Mg (X), Y) -->- (Z, Y) is a homotopy equivalence if X is an ANR and A is a closed ANR subspace of X.

32


Q. Local contractibility of AN R's

A space X is said to be locally contractible if, for each x e X and every open neighborhood U of x, there is an open neighborhood V U of x which is contractible to the point x in U. By means of Kuratowski's imbedding, prove that every ANR is locally contractible. As a partial converse of this result, [Borsuk 2] proved that every locally contractible compact metrizable space of finite dimension is an ANR. R. Dominating polyhedra of ANR's

A space X is said to be dominated by another space D if there are two maps / : X -÷ D, g : D --). X such that the composed map g/ is homotopic to the identity map on X. In this case, D is called a dominating space of X. Prove the following assertions: 1. Every ANR has a dominating simplicial polyhedron. 2. Every separable ANR has a dominating locally finite simplicial polyhedron. 3. Every compact ANR has a dominating finite simplicial polyhedron. S. Some strong deformation retracts

Prove the following assertions: 1. If A is a strong deformation retract of a compact space X, Y is a Hansdorff space, and f: (X, A) ---> (Y, B) is a relative homeomorphism, then B is a strong deformation retract of Y. [Spanier 1; p. 208]. 2. In the relative n-cell (Y, B) obtained by adjoining En to B by means of a map g : Sn --1. --> B defined on the boundary (n — 1)-sphere Su --1- of En, the subspace B of Y is a strong deformation retract of the space obtained from Y by deleting an interior point of E. 3. In the product space Sm x Sn, the subspace Sm V Sn = (Sm x to) U (so x Sn), where so e Sm and to E Sn are given points, is a strong deformation retract of the space obtained from Sm x Sn by deleting one point which is not in Sm V S. 4. In the product space Si x Snl X Sn, the subspace T -,-- (Si x Sm. x 10) U (Si x so x Sm) U (r0 x Sm x Sn), where 1'0 e Si, so e Sm, to e Sn, is a strong deformation retract of the space obtained from Si x Sm x Sn by deleting one point which is not in T. 5. In the product space (S/ V Sm) x Sn, the subspace Sl V Sm V Sn is a strong deformation retract of the space obtained from (S/ V Sm) x Sn by deleting a point of St x Sn not in Si V Sn and a point of Sm x Sn not in

ExEkcisES

33

Sm V S. Generalize this result to the one-point union of a finite number of spheres. 6. If A is a closed subspace of a metrizable space X and if both A and X are AR's, then A is a strong deformation retract of X. 7. If A is a strong deformation retract of a metrizable space X, then

T . (X x 0) U (A x I) U (X x 1) is a strong deformation retract of the product space X x I. [Hu 2]. T. Relations between different notions of deformation retracts

Assume that X and A are ANR's and that A is a closed subspace of X. Prove that the following statements are equivalent, [Fox 1] : 1. The subspace A is a strong deformation retract of X. 2. The subspace A is a deformation retract of X. 3. There exists a homotopy h1 : (X, A) -÷ (X, A), (0 < t < 1), such that ho is the identity map on X and hi(X) c A. U. Expansion of the relativity of a null homotopy

Let X,Y be given spaces, Ac X, Bc Y given subspaces, and xo e A, Ye e B given points. Consider the maps Q . Yx { A, B; x o, yo }. According to § 10, Q consists of the totality of the maps f: X -4- Y such that f(A)c: B and / (x0)---,-- yo. Let f e Q and let 0 denote the constant map 0(X)---- y o. Prove that f.c.-_, 0 rel { A , B } implies f .c-_-, 0 rd l { A, B; xo, yo } provided that the following side conditions are satisfied: (1) The subspace A U k(x 0) of the cone k(A) has the HEP in k(A) with respect to B. (2) The subspace X U k(A) of the cone k(X) has the HEP in k(X) with respect to Y. These side conditions are satisfied, for example, if X is a finite simplicial polyhedron and A is a closed subpolyhedron of X. For more general cases see N and O. V. A general expansion theorem

Let X be a metrizable space and

Nc M

A cX

be closed subspaces. Let Y be a space and B a subspace of Y not necessarily closed. Establish the following Theorem. Let f, g: X --). Y be two given maps such that

/(A)c B, /1.111

OM,

g(A)c B.

34


Then /_.---4 g rel { A, B ;.N } implies / r_._-_ g rel { A, B; M } provided that the following conditions are satisfied, [Hu 4] : (1)N is a deformation retract of M. (2) The subspace P — (A x 0) U (M x I) U (A x 1) has the HEP in A x I with respect to B. (3) The subspace Q — (X x 0) U (A x I) U (X x 1) has the HEP in X x I with respect to Y. The conditions (2) and (3) are satisfied, for example, if X is a finite simplicial polyhedron and A, M are closed subpolyhedra of X.

CHAPTER II SOME SPECIAL CASES OF THE MAIN PROBLEMS

1. Introduction In the first chapter, we have described the main problems in homotopy theory. However, the examples and their solutions given there are essentially trivial cases. The present chapter takes up the first nontrivial cases. Answers will be obtained by using homology and cohomology groups. By means of the elementary properties of the exponential map p : R Si in § 2, we begin with the classification problem of the maps S1 -*SI in § 3. Then, the classification problem of the maps of S1 into an arbitrarily given space X is treated in § 4 and § 5 by studying the fundamental group. In this connection, the classical relation between the fundamental group and the first homology group is given in § 6. Dually, the classification problem of the maps X 5 1 is studied in § 7 by means of the Bruschlinsky group. we take up the higher dimensional sphere S. The Hopf theorems concerning the maps S n S n and Kn S n are proved in § 8. Dual to the Hopf theorem is the Hurewicz theorem, which is stated in § 9; the proof of this important proposition is deferred until Chapter V, since it seems to depend in an essential way on the elementary properties of the homotopy groups.

2. The exponential map p:R S' Represent the 1-sphere Si as the unit circle in the space K of all complex numbers, that is to say

Sl= {zeKlizi = 1}.

Therefore, Si is a compact abelian topological group with the usual multiplication of complex numbers as group operation. Consider the map p : R S' defined on the space R of real numbers by the formula p(x) exp (27Exi) = enxi, (x E R), where e denotes the base of natural logarithms and i the unit of imaginary numbers. This map p will be called the exponential map of R onto Sl. The continuity of p and the following proposition are obvious. The exponential map p is a homomorphism, that is, p(x + y) = p(x)p(y) for any x and y in R. The kernel p-1(1) is the subgroup Z of Proposition 2.1.

all integers. Since 75(x)

=

cos (27(x)

i sin (2ax), it is easy to see that 35

p is one-to-one on

36

IL SOME SPECIAL CASES OF THE MAIN PROBLEMS

the open interval (– i) of R. By a standard theorem in general topology, p maps every closed subinterval of (– homeornorphically and hence p is a homeomorphism on the open interval itself. Since p is a homomorphism, this implies that p is a homeomorphism on each translate of (– i). Therefore, for every open interval (a, b) with b — a < 1, p carries (a, homeomorphically onto an open subspace of Sl. This and (2.1) imply the following Proposition 2.2. For every proper connected subspace

every component of

By

p--1(u) homeomorphically

a path in a space

U of 9,

p

carries

onto U.

X, we mean a map

of the unit interval I into X. The points a(0) and a(1) are called respectively the initial point and the terminal point of the path a. They are said to be connected by a. If ci(0) = o-(1), then a is called a loop in X with o-(0) = o-(1) as its basic point.

For every path ci: I -÷S1 and every point X0 E R such that p(x 0) = cr(0), there exists a unique path 'r :I -÷R such that T(0) x o and x = Proposition 2.3. (The covering path property).

Proof. By the continuity of a and the compactness of I, there exists a partition < o to < t1 < • • • < of I such that the image of every closed subinterval /4 ti +1],O < i SI is the integer deg(f) uniquely determined by the relation f* (c) = deg(f)• c

38


for every c G H1 (S 1). Obviously, deg(f) depends only on the homotopy class [f] of f. Hence we may define the degree of a homotopy class a Ggr(S1- ; .3 1) by taking deg(a) = deg(f) where f e a. Thus we obtain a function deg : n(S1- ; SI)

Z.

We are going to show that deg mapsn(S 1 ,S 1) ontoZis a one-to-one fashion. The exponential map p R S 1 maps the open interval (0,1) homeomorphically onto the open subspace W SI \ 1. Let q: W ---> (0,1) denote the unique homeomorphism such that pq is the inclusion map W c SI. To prove that deg is onto, let n be any integer and let chn :S1 --->-

denote the map defined by 0(z) = zn for every z E Sl. One can easily see that 0. is also given by the following formula on(z)

(if z (ifz

1,

e

\ 1), 1 ES 1),

where En :I-4-R denotes the map defined by ,,,(t) = nt for every t E I. The following lemma is obvious and implies that deg is onto. Lemma 3.1. deg (On) = n.

To prove that deg is one-to-one, it suffices to establish the following Lemma 3.2. If the degree of a given map f :

.->5i is n, then f

On.

Proof. Pick 0 ER such that 5 (0) = f (1 ). Define a homotopy It: SI -÷ SI, (0 < t < I), by taking

it(2)

f(z)/(0t),

(zES 1-,teI).

I. Set g = fl. Then we have f g and g(1) Let a gy5 : I -÷51- . Since a(0) = 1 = p (0), it follows from (2.3) that there exists a unique path T I --> R such that r(0) 0 and pr = a. Since px(l) ci(l) = 1, r( 1 ) is an integer, say r( 1 ) m. Define a homotopy nt :1 -±R, (0 < t < 1), by taking nt(s) = T(s)

t4„,(5) — 1-r(s), (s GI, t G I).

Then no T, i m for every t e I. m and nt (o) = o, nt(l) Finally, define a homotopy gt .S 1 ---> Si , (0 < t < 1), by taking gt(z)

(pntq(z), 1

(if zE SI \ 1), (if z = 1 ES').

Then go = g and gl O.. This implies f g qf m . It follows that deg n and ) — deg (Om). Since deg (f) = n and deg (Om) = m, we obtain m hence f 95.. I Thus we have proved the following

14 THE FUNDAMENTAL GROUP

39

(Classification theorem). The homotopy classes n(.31 ; Sl) are in a one-to-one correspondence with the integers Z under the function deg: n(S1- ; S1)--> Z . The homotopy class of ,degreen is represented by the map On :SiTheorem 3.3.

To strengthen (3.3), let us make use of the fact that is a topological abelian group under the usual multiplication of complex numbers. The maps A form a ring with addition and multiplication defined as follows: If f, g : S1 ---> 9 are any maps, then f + g and 1g are the maps in A defined by (/ g)(z) (z)g(z), (fg)(z) = f [g(z)] for every z e Since the homotopy classes [/ ± g] and [fg] obviously depend only on the classes [f] and [g], this ring structure of A induces a ring structure in n(Sl; 9). Now, (3.3) can be strengthened by the following The function deg is an isomorphism of the ring n(S 1 ; Sl) onto the ring Z of integers. Corollary 3.4.

Proof. Let /, g EA. It suffices to prove that deg(/ ± g) — deg(f)

deg(g), deg(fg) = deg(/)deg(g).

Let deg(f) = m and deg(g) = n. Then f I) from the definitions that

Om +On ---

and g

sbn . It is obvious

Ont+n, 43/40/21-1 =

This implies the corollary. I

4. The fundamental group Let X be a space and let xo E X; let S2 denote the set of all maps of 9into X such that the point I of 9 is mapped at the point xo of X. X in D, the composition ip : For any map f : of f with the exponential map p of § 2 is a loop in X with its basic point at xo. This correspondence is obviously one-to-one. Hence, by identifying f with /p, we may consider D as the set of all loops in X with given basic point xo . In symbols, we have {/: / -> X /(0)

xo

/(1) }.

We may define a muitip/ticalion in D as follows. For any two loops /, g ED, their product fg is the loop defined by (f

(t)

f f (2t) , g(2t — 1),

(if 0 j denote the homeomorphism such that pq(z) = z for every z c W. Let U = f-1 (W) and define a map It : U -÷ R by taking h(x) — q/(x) for every x c U. Obviously we have h(x i.) = r i and ph . / I U. We are going to prove that g coincides with h in an open neighborhood of x i. in X.

44


Since X is locally pathwise connected and U is an open neighborhood of xi. in X, there exists an open neighborhood V c U of x1 in X such that every point of V can be connected to x, by a path in U. We shall prove that g(x) = h(x) for every x V. For this purpose, let x eV be given. Then there is a path n x such that 7)(0) — x l and 7i(1) = x. Since X is pathwise connected, there is a path E : I -÷ X such that (0) = x o and 4:(I) = xi . Let = $.27 : 1 x be the path defined in the obvious way. Set p = g and a — fn. Denote by 0 : I --> R the unique path such that 0(0) = r o and po = p. By the very construction of g, we have 0(1) = g(x 1) = r1 . On the other hand, we also have h(x 1) r i . Hence, we may define a path r: .2" -> R by taking (if 0 < é < 0(20, r(t) h21(2t — 1), (if < t H, (X). Let denote the generator of the free cyclic group HAS') corresponding to the counter-clockwise orientation of Sl. By the homotopy axiom, the element f* (t) in Hi (X) depends only on the class cc and therefore, we define h* by taking h(Œ) f * W. By an elementary property of induced homomorphisms in homology theory, [E-S; p. 36], it is easy to see that h* is a homomorphism. Let us consider the unit n-simplex ti n in the (n 1)-space Rn•-E-1- which consists of the points (to , t,, • • • , tn) such that t o -I- ti + • • • + t I and ti > 0 for each i = 0,1, • • • , n. Let j : I d i denote the homeomorphism t, t) for each t e I and let k : 71 1 -± S1 denote the map defined by j(t) = (1 p(t,) for every point (t o, ti) of A i , where p stands for defined by k(t o, tl) p. As a singular 1-simplex the exponential map in § 2. Then we have kj X is in the group Ci(X) of integral singular 1-chains. in X, T fk: Since OT 0, T is a singular 1-cycle. It is not difficult to verify that T represents the homology class Va.). Since .117 (X) is abelian, the commutator subgroup Comm {7c 1 (X, x0) 1 of 7C1(X, .X0) must be contained in the kernel of h* . In fact, we have the following

classical theorem. Theorem 6.1. if X is pathwise connected, then the natural homomorphism h* maps nj (X, x0) onto Hi(X) with the commutator subgroup of ni(X, x o) as its kernel. Hence 111(X) is isomorphic with the group nl (X, x 0) made abelian.

To prove this theorem, let us first carry out a preliminary reduction. Consider the singular complex S(X) of X, [E-S; p. 186], and denote by Sl (X) the subcomplex of S(X) defined as follows. A singular simplex T : Zi a -* X is in S l (X) iff T sends the vertices of Llq into the point xo. We shall call S i (X) the first Eilenberg subcomplex of S(X). Since X is pathwise connected, it follows from Eilenberg's reduction theorem, [Eilenberg 2; p. 440], that the inclusion cellular map 2 :S 1 = Si (X) S(X)

induces an isomorphism A:

11 1(X)-

On the other hand, let ac be any element da l (X, x0) and choose a representative loop f: X for at. Then the singular simplex T = fk : X is a 1-cycle in S i and hence determines a homology class it(oc) e Ill (S 1 ) which does not depend on the choice of f. The operation cc ,u(cc) defines a homomorphism

itt ni (X, x0) --> I/1(Si).

Obviously we have the relation h* = 2,kitt. Hence the theorem reduces to the following Lemma 6.2. The homomorphism tt maps n i(X, x o) onto Hi (S i ) with the commutator subgroup of nl (X, xo) as its kernel.

46


p

be an arbitrary element of H i (S 1). Choose a 1-cycle z of S i which represents 13. Since z E C 1 (S 1), it can be written in the form 2 .--= E aiTi Proof. To prove that p is onto,

let

i

---= 1

where T11 • • • , T. are 1-simplexes in S1 and a l , • - - , a. are integers. Since Ti maps the vertices of ZI, into xo, it is a 1-cycle in .51 and hence represents an element pi of H1 (.31). Then we have n

p = E aipi. I-1

On the other hand, the loop Tii : I -) - X represents an element ai of ni (X, xo). According to the definition of it, we have it(ai) — fi. Let a , ceice2. • •cc an Eni (X, X0)• n 1 2

Since p is a homomorphism, it follows that n

p(a) = E ai(cc) i=1

n

—1=1 E ailei — P.

This implies that tt maps n1 (X , x o) onto I/1 (S 1). To study the kernel of p, let us first note that, because of the cominutativity of INS 1), the kernel of ict contains the commutator subgroup Comm {n i (X, xo) } of nl (X, x 0). The quotient group ' ni *(X, x0)----- n i (X, x0)/Comm { nl (X, x 0)1 is commutative and is known as the group ni (X, x o) made abelian. We shall use the additive notation in ni* (X, x0) and consider the natural projection

y : n1 (X, x0) --> ni * (X , x 0). Since the kernel of p contains Comm { ni (X, xo) }, p induces a unique homomorphism p* : ni*(X, x 0) -)- H 1(S1) such that p = p* v . To prove that the kernel of iu is Comm { ni (X, xo) }, it suffices to show that it* is a monomorphism. Indeed, we will produce a left inverse x* for p*. Since ni* (X, x 0) is abelian and C i (S i) is the free abelian group generated by the 1-simplexes of S 1, we may define a homomorphism x : C 1 (5 1) --> ni *(X, x o) described as follows. Let T be any 1-simplex of S 1 . Then the loop Ti : I -4 X represents an element [Ti] of 1 (X, x0) and x is defined by taking 2-c(T) — We assert that x maps B 1 (S 1) into the zero element of ni.*(X, x 0). To prove this, let T be an arbitrary 2-simplex in S r Then T is a map T: ti 2 -4- X

7.

THE BRUSCHLINSKY, GROUP

47

which sends the vertices of z1 2 into xo. Let TR, T('), T(2) denote the 1-dimensional faces of T; then we have

n(aT) = n(T(0) — T( 1) + T(2)) n( 7'( 2) T(0)— T(L)) x(T( 2)) x(T (°)) x(T( 1)) v ( [T(1)i] v( [T(0)7] ) v( [T( 2)1. ]) v( [T(2)i] [T(°)1.] [T(1)/] -1) = because of the relation [T(i)j] [T(2)f] [T(c)i] since T is defined throughout A. Since ./3 1 (S 1) is generated by all OT with T running over the 2-simplexes of S 1 , this proves our assertion. Thus "c induces a homomorphism

x* : H1 (5 1) --)-7( 1*(X, x0). Finally, x*y* is the identity. To prove this, let cc* be any element of ni *(X, x0). Choose ccEn l (X, x0) with v(cc) --- cc*. Let f: X be a representative loop for cc. Then the singular simplex T = fk :Z1 1 X represents the class itz(a) and hence we have

elev(a) =n */-1 (a)= v( [7' i])

v([1

v(UPD = v(cc)

cc* ,

where p denotes the exponential map. This implies that la* is a monomorphism. An important consequence of (6.1) is the fact that, for any pathwise connected space X, the fundamental group n I (X) completely determines the integral singular homology group I1 1 (X).

7. The Bruschlinsky group Let (X, A) be an arbitrary p9,ir consisting of a space X and a subspace A of X which may be empty. Let us consider the set

W=

f (X, A) (S 1-, 1) }

of all maps f of the pair (X, A) into the pair (Si, 1), where SI denotes the unit circle in the complex plane as in § 2. Since Si is an abelian group under multiplication of complex numbers, we may define an addition in W as follows. If f, g are any maps in W, then f + g is the map in W defined by

(f

g) (x) = f(x) g(x)

for every x G X. In this way, W becomes an abelian group. The homotopy class (relative to A) of the map f + g depends only on that of / and that of g. Hence the set '(X, A) of all homotopy classes of the maps of W forms an abelian group under the addition defined as follows : For any cc, j3 in n1-(X, A), we have cc +

p=

[f

+ g],

f G cc,

gE

II. SOME SPECIAL CASES OF THE MAIN PROBLEMS

48

This abelian group 7d-(X, A) will be called the Bruschlinsky group of the pair (X, A). If A is empty, then it will be denoted by ni-(X) and called the Bruschlinsky group of the space X. See [Bruschlinsky 1]. For example, we have TOM

7C1 (.51 , 1)t'•:/-'

arl (51-, 1)

Z.

To determine the structure of gri-(X , A), let us assurhe throughout the remainder of the section that the pair (X, A) is triangulable. Thus, we assume that X is a finite simpliCial complex, [E–S; p. 56], and that A is a subcomplex of X. Under these assumptions, let us consider the integral cohomology group H1 (X , A) of the pair (X, A) and construct a natural honornorphism

h* : 70(X , A) –> H1(X , A)

as follows. Let t denote the generator of the free cyclic group Hi(S1 , 1) determined by the counter-clockwise orientation of Si. For an arbitrary element a e 0-(X, A), pick a map : (X, A) (S', 1) which represents a. induces a homomorphism

1* :

1) –> I-P(X , A)

which depends only on a according to the homotopy axiom of cohomology theory. Hence we may define h* by taking h* (a) = f *( t). It remains to verify that h* is a homomorphism. Let a En'-(X, A). By means of the homotopy extension property (I; § 9), (S1, 1) which sends all one can easily see that a contains a map : (X, A) vertices of X into the point 1 of Sl. Such a map / determines an integral cochain cl(/) E 0-(X, A) as follows. Leto. = vov i be any 1-simplex of X and let A:I –> o- denote the linear map which sends 0 to vo and 1 to v 1. The composed map f A : I .S 1 is a loop in and hence the degree of f A is a welldefined integer deg (/ A) which depends on f and a. Then the cochain cl(/) is defined by [0-(/) ] (o-) deg (f A). This cochain cl(/) is a cocycle. To prove this assertion, let r = v0viv 2 be an arbitrary 2-simplex. Then we have

Or

o.0 — c11

Let Ai : I Gri, i follows that

0. 2 ,

Go = 7117.12, 0'1 =

v0v 2 , 472 =

0,1,2, denote the linear maps as described above. It = [ci(i)](ar) = [c l(i)](ao) — [01 (i)](ai.) = deg(/ Ao) — deg(/ Al) + deg(/ A 2) = 0

{c1 (i)i(a2)

since / is defined throughout 2. This proves that eV) is a cocycle. By the definition of the induced homomorphism /*, it is not difficult to see that this cocycle cl(/) EZI(X, A) represents the element /*(t) E .r-p(x , A). Now let 13 be another element of 7d- (X , A) and pick a representative map

7.

THE BRUSCHLINSKY GROUP

49

g: (X, A) -÷ (.31, 1) of /3 which sends all vertices of X into 1. Then the map f + g represents cc fl and sends all vertices of X into 1. For every 1-simplex vovi, we have a deg ( (I g) A) = deg(P) deg(g)).

It follows that c1 (f + g) = cl(f) h*(ct

ci(g) and hence

f3) = h*(a)

h*(f3).

This proves that h* is a homomorphism and completes its construction. Theorem 7.1.

The natural homontorphism h* is an isomorphism of '(X, A)

onto .f-11(X , A).

Proof. Let z E Z1 (X, A) be arbitrarily given. To prove that h* is an epimorphism, it suffices to construct a map /: (X, A) -0. (Si, 1) which sends all vertices of X into 1 and such that e(f) z. For each n 0, 1, • • • , dim X, let K n denote the subcomplex of X which consists of all simplexes in A together with the simplexes in X \ A of dimension not exceeding n. We are going to construct a sequence of maps

f„ :

-÷ Sl, n = 1, 2, • • • , dim X,

as follows. To construct fi , let us take for each 1-simplex a = vovi in X \A a loop go. : I -÷ such that

ga(0) = 1 =g(1), deg (g,)

z(a).

: I --> a denote the linear map which sends 0 to vo and I to v 1 . Then we define fl : K, S 1 by taking 1 (if x e Ko), 11(x) --- g0, 41 (if xeo-eK, \A). Let

(x ) ,

Next, let us construct the map f2 : K2 Sl. Let r = v0v 1v 2 be an arbitrary 2-simplex in X \A and ao = v1v2 , al = v0v2 , a 2 = vov 1. Then fl I Or is a loop in Sl of degree z(co)

z(c i)

z(a2) = z(Or) = (62) (T) = 0

since z is a cocycle. By (3.3), f1 jOr is homotopic to the constant map. Hence it follows from the homotopy extension property that / I ar has an extension : r -> S'. Then we define /2 by the following formula: / 2(x) = i(z) i1 t h_r(x),

(if x E K1), (if xer E K2 \A).

Now we shall complete the construction of the sequence { by induction. Let In > 2 and assume that int _i Km_l SI- has already been constructed. Let 0 be an arbitrary m-simplex in X \A , and vo be a vertex

50


of 0. Since nt > 2, the (m — 1)-sphere 00 is simply connected by (5.2). Then, according to (5.3), there is a unique map fo : 00 ---> R such that

I(v) — 0 , Pie — fm -i 1 00 where p : R -÷ Si- denotes the exponential map in § 2. Since R is solid, the map to has an extension k6 : 0 -->- R. Then we define fm by setting fm (x )

_

(if x e Km_ i ),

(ifxe0EK.,\A).

1 Pko (x ),

This completes the construction of the sequence { f.}. Since in f Kn_ i. = f, we obtain a map f: X -÷ S1 such that f 1K. = f.. In particular, f (K0) — 1 and f 1 K 1. = fi . By the construction of fi , this implies that OW — z. Hence, h* is an epimorphism. To prove that h* is a monomorphism, let a be any element of n-1.(X, A) with h*(a) = O. Pick a representative map f: (X, A) --->- (5 1,1) of a which sends all vertices of X to 1. Then the cocycle OM is a coboundary of X modulo A and hence there is a cochain c° e Cy(X, A) such that OM = dc°. For each n = 1,2, - • • , dim X +1, let J. denote the subspace of X x I defined by

(X x 0) U (Ku_ j_ x I) U (X x 1).

To prove that f is homotopic to the constant map 0(X) = 1 relative to A, let us construct a sequence of maps

/7. : J. ---> Sl,

n = 1,2, • - - , dim X + 1

as follows. To construct Fi. , let us take for each vertex y in X \A a loop I --->- S1 such that

1 = Œv (1),

deg („) = c°(v).

Then we define Fl : J1 --). S1 by taking

1) Fi (x, t) = {$„(t),

(if x e A or if t = 0), (if x --, y), (if t . 1).

Next, let us construct the Map F 2 : 1 2 -± SI. Let a -- vov i , be any 1-simplex in X \ A. Then the partial map Fi. 10(o. x I) on the boundary 19(a X I) of a x I is a loop in 51 of degree

c° (v].) — [c1 ( in (a) — c° (vo) — OM (a) — {c1 (/)] (a) = O. Hence Fi f 0(a x I) has an extension following formula:

F 2(y) = { F1(Y)1

na :a x I -*SI. (if

We define F2 by the

y e,f1),

(ifyea x I).

Now we shall complete the construction of the sequence { F. } by induction. Let ni > 2 and assume that Fm _ i. : J.,,, -> SI- has already been

7.

THE BRUSCHLINSKY GROUP

51

constructed, Let -r be any (ni 1)-simplex in X \ A. Since in > 2, the (ni 1)-sphere a(r X I) is simply connected. Hence, it follows just as before, that Fni_ l ia(r x I) has an extension C, : r x Sl. Then we define Fm : 1. SI. by setting

Fm- (Y),

YG (if yEr X I). ( if

This completes the construction of the sequence f F7, 1. Since En I In_ 3 En_ 1 , we obtain a map F:XxI-4-S 1 such that F I J = F. In particular, we have F(x, 0) = 1, F(x, 1) = /(x), F(a, t) = 1 for every x e X, a c A and t e I. This implies that f is homotopic to the constant map 0(X) = 1 relative to A. Hence a. 0 and h* is a monomorphism. I Since H1 (X y A) is effectively computable, the theorem (7.1) solves the classification problem for the maps (X, A) —> (51, 1). In particular, if we take A = o, then it gives a solution of the classification problem for the maps X -÷ S1 . On the other hand, it also solves the homotopy problem and the extension problem in the form of the following corollaries. Corollary 7.2. Two maps

/ g: (X, A) (S 1,1) are homotopic (relative to A)

i ff f *(1) = g*(1). This corollary is an immediate consequence of (7.1). In particular, let us take A = D. The inclusion map j : SI c (51 , 1) induces an isomorphism I/1 (S', 1) I/ 1 (S') and x j*(t) is the generator of the free cyclic group Hl(S 1) determined by the counter-clockwise orientation of S'. Then (7.2) gives the following corollary as a special case. Corollary 7.3. Two maps f, g:

X -±S' are homotopic iff f *(x)

g*(x).

: A ---> 9- can be extended over X iff the element 1*(x) of 111 (A) is contained in the image of the homomorphism i* : I-11(X) .111 (A) induced by the inclusion map I : A c X. In fact, if a is an element of 1-11-(X) such that i* (a) = * (x), then / has an extension g: X -± .31 with g*() Corollary 7 4.

A map

Proof. The necessity of the condition is obvious. For the sufficiency, it suffices to establish the second assertion. By (7.1), there exists a map k: X ---> such that (ik)* (t) = a. Then' we have (iki) * ( 1) = i* (ik) * ( 1) i* * = /* ( ) = / *Mt) = (ii) * (t). By (7.2), this implies that iki 17 and hence hi f . According to the homotopy extension property, there exists an extension g: X Sl of f such that g k. Then we have

g*(x) = k* (x) = k*i* (t) = WO* (t) = a. I Following the definition of triangulable pairs in [E—S; p.60], we have assumed above that X is a finite simplicial complex. However, the proof of

52


(7.1) is so arranged that it extends to the case that X is an infinite simplicial complex and A is any subcomplex of X provided that Whitehead's weak topology is used in X. Hence all results in this section are true for the infinitely triangulable pairs (X, A) defined in the obvious way. On the other hand, these results can be extended to more general pairs (X, A) by using various Cech cohomology theories. Some of these generalizations will be given as exercises at the end of the chapter.

8. The Hopf theorems With some dimensional restrictions on the triangulable pair (X, A), the nice results of the preceding section for S1 can be generalized to higher spheres. For this purpose, let us first consider some preliminaries. For every n > 1, let St& denote the boundary n-sphere of the unit (n + 1)simplex LI — zi n,+1 . For each m — 0, 1, • • • , n + 1, let /1(m) denote the m-th (n-dimensional) face of J. Then Sn is the union of the n-simplexes A(0', /1(1), 1). The n-dimensional cocycle sb of Sm defined by sb (A (0)) . I and•,21(n+ #(4 WO) — 0, m — 1, • • • , n + 1, represents a generator x of the free cyclic cohomology group Hn(S"). For any map f: X -÷ Sm, the element f*(x) of Hm(X) depends only on the hornotopy class [fj of f and will be called the degree of the map f or of the class [f]. In particular, if X = SI z , then the element f*(x) of lin(Sr6) determines a unique integer deg(f) such that /* (x) -,.. deg(f)se. In this case, it is this integer deg(f) which is traditionally known as the degree of the map f : Sn --->- S. If n — I, this definition of deg(f) is obviously equivalent to the one given in § 3. Let •vc, denote the leading vertex of Sn. For each q — 0, 1, • • • , n + 1, let Ii) denote the union of all d ( n) with m 0 q. Then the inclusion maps

$ : Sn

(Sn, yo), 17q : (Sn, vo)

(Sn, r(o)

induce isomorphisms $*, 4 on the cohomology groups of positive dimensions. Let , ,_. $*-1(;,) , lq .„. ne _1 ( ,). Then t and 2,q are generators of the free cyclic groups Hn(Sn, yo) and Hrir(Sn, 17(Q )) respectively. Next, let us consider the unit n-simplex AI and its boundary (n sphere 071 n. The n-cocycle y of Li n given by y(z1 n) . 1 represents a generator it of the free cyclic group lin(Zi n, aLl n). For any map f: (A s, ad n) -÷ (Sn, yo), the element /*(t) determines a unique integer deg(f) such that f* (t) — deg(f) • itt This integer deg(f) will be called the degree of the map f. For each q = 0,1,- - • , n + 1, let (J., aA n) -4- (Sn, FM) denote the map defined by the order-preserving one-to-one assignment of vertices of An into 4 (q) • Then it is not difficult to verify that :

(8.1)

Ce (2,q) = (— 1)11#

8.

THE HOPF THEOREMS

53

Now, let f: Sn' —> Sn be a map which sends the (n — 1)-dimensional skeleton of St& into yo. Then, by (8.1) and a theorem in cohomology theory, [E-S; p. 37, Theorem 14.6c], one can easily deduce the following relation n+1

(8.2)

deg(f) = E (— 1)q deg(gq). q=0

Finally, let us prove the following 8.3. For every integer m, there exists a map fm: (z1, 2,, OLI n) --->-(Sm, y o) with deg(fm) = m. Lemma

Proof. If m = 0, then the constant map f(z1 ) = 2,0 is obviously of degree O. Next, assume m > O. Take a sufficiently fine simplicial subdivision K of Li n so that we may pick m mutually disjoint closed n-simplexes al , • • ., chit contained in the interior of A n. Let the vertices um of ai be ordered in such a way that the orientation of

ai = (uii, N2P" •, uin+i) agrees with that of 21 n . We now define hn to be the unique simplicial map of K into Sn' which sends uij into the vertex vg of Sn for each i = 1,- • • , In and each f = 1,• • • , n + 1 and sends every other vertex of K into yo. Then it is easily seen that f ra(0/1,) = yo and deg(fm) = in. Finally, let A denote a linear homeomorphism of An which interchanges a pair of vertices of J. and leaves other vertices fixed. Then deg(fm,A) — m. I An organization of the Hopf theorems is as follows. H". (Homotopy). If a map f : Sn --> Sn has degree deg(f) = 0, then / is hornotopic to the constant map 0(Sn) = vo. Theorem

(Extension). Let (X , A) be a triangulable pair with dim(X \A) Sn be a given map. If there exists an element a e ffn(X) such that i*(oc) = f* (e), where i:Ac X, then f admits an extension g: X --› Sn' such that g*(x) = oc. Theorem En.

C". (Classification). If X is a triangulable space with dim X < n, then the assignment f --> f*(x) sets up a one-to-one correspondence between the homotopy classes of the maps f: X -> Sn and the elements of the cohomology group Hn(X). Theorem

The converse of En is trivial: if f admits an extension g: X --> Sn, then there exists an element oc e H( X) such that i*(a) = f*(x). In fact, we have a Since H 1 is a special case of (3.2), these theorems can be established inductively by proving H" -

En

• an+1

H

,,

11'n

t

C ol

for every n >1. Also, it worth noting that H n is a special case of C", C' is a special case of (7.1), and El is a special case of (7.4),


54

simplicial complex and A a subcomplex of X. We may assume that f sends the (n — 1)-dimensional skeleton of A into the point yo of Sn, for otherwise we may replace f by a homotopic map which satisfies this condition by applying the method of simplicial approximation Proof

of

Hi' = En. Let X be a

and observing that every proper subspace of Sn is contractible in Sn. Under these ass.umptions, we may define an n-cochain cn(f) of A as follows. Let a be any n-simplex of A and Aa : A n --->- a denote the linear homeomorphism which preserves the order of vertices. Then P... is a map of (21,/, ad.) into (Sn, yo) and hence we may define the eochain c(f) by taking

[cn(f)] (a)

deg(f)

for every n-simplex a of A. By (8.2), one can easily prove that cn(f) is a cocycle. By the definition of the induced homomorphism f*, it is not difficult to see that cn(f) represents the cohornology class f*(x) G H(A). Assume that oc is an element of fln(X) such that i*(oc) = /*(n). Then it follows that there exists an n-cocycle zn of X which represents Oc and satisfies the relation znocr)= [On (a) for every n-simplex a of A. Let K denote the union of A and the n-dimensional skeleton of X. We shall define a map h : K Sn as follows. Let a be any n-simplex of K which is not in A. By (8.3), there is a map

: (An,

OLI n) -4- (Sn, y o)

with deg(f) = zn(o-). Let 4, : Li n a denote the linear homeomorphism which preserves the order of vertices. Then we define h by setting

h(x) =

(x) iisrAcr-1 (z),

(if x G A), (if x G ci e K \A).

Finally, we are going to construct an extension g: X Sn of k as follows. Let T be any (n 1)-simplex of X \A and 2,, : Z1,,, 1 r the linear homeomorphism which preserves the order of vertices. Denote k, = kA, I S. By (8.2), we have deg (kx)

E (— 1)izn(0)) = zn(ar) = azn(r) = O. 0

According to Theorem Hn, this implies that k is homotopic to the constant map O. Then it follows from the homotopy extension property that k, has an extension gr dn-H. -÷ 5n Then we define g: X -÷ Sn by taking g(x)

k(x)

(if x K), (if xe X G X \A).

Since g is obviously an extension of f, it remains to verify that g*(x) oc. According to the construction of g, it can be seen that g maps the (n — 1)dimensional skeleton of X into yo and that c(g) =zn. Since c(g) represents g*(n) and zn represents oc, this implies g *() = oc. I

8.

THE HOPE THEOREMS

55

H n+1 . Let f: Sn -a Sn+1 be a given map with deg(f) = O. We Proof of En are going to prove that / is homotopic to a constant map. Using the method of simplicial approximation, we may assume that f is a simplicial map of a triangulation ,K of S92-0- into a triangulation J of Pick an (n 1)-simplex a (uo, u l , • » , un+1) of J such that the cocycle defined by 56(a) = 1 and OW 0 for every (n 1)-simplex r of J other than a represents the generator ;,e of fin+1 (Sn+1). Let M denote the subcomplex of K consisting of the closed (n + 1)simplexes ot K which are mapped into a by the simplicial map f. We may assume that no two of these (n 1)-simplexes have a common vertex, for otherwise we could replace K and J by their second barycentric subdivisions K" and J", and take for a an (n 1)-simplex of j" none of whose vertices is also a vertex of J. Under these assumptions, we have Ha(M) = 0 for every q> O. The simplicial map f : K induces an (n 1)-cocycle /#(0) of K which represents the element f*(x) = deg(f) = of Hn+1(K). Hence there exists an n-cochain c of K such that v c5c. The simplicial inclusion map p:m- K induces the following two cochains of M

c i 0(c),

= P4t(1P) — 0/ 4(95) — (IP) 4 *-

The relation v = dc in K implies that v i = àc i in M. Next, consider the leading n-face cs(°) = (u1 , ••, un +1 ) of the (11 1)-simplex a and let -00 denote the n-cochain of J defined by 00(0)) — 1 and 00(7) = 0 for other n.-simplexes induces an n,-cochain co — r of J. Then the simplicial map ft: M = (/p)* ( 0) of M. Since qfi is the coboundary of 00 on the subcomplex ci of / and fit' maps M into a, it follows that v i = 6c0. Then b(c i — co) = vl — yi =0 and hence c1 — co is an n-cocycle of M. Since H(M) — 0, this implies that c i — co is a coboundary of M. Let Mn denote the n-dimensional skeleton of the complex M. Then f defines a simplicial map X of Mn into the n-sphere Oa. By the construction given above, it is clear that c o is a cocycle of Mn and represents the degree of the map X. Since cl — co is a coboundary, the degree of X is also represented by the cocycle c l of Mn. Let N (K\M) U Mn and consider the inclusion map q : N OE K. Since 6c(r) v(T) = 0 for any (n 1)-simplex r which is not in M, it follows that c 2 q4t(c) is a eocycle of the complex N. Since c 2 is an extension of c 1 , we may apply Theorem En to conclude that Z has an extension p: N ---> Oa. Define a map g : Sn+1 ---> Sn+1 by taking

g(x) = ç f (z) p(x),

(if (if x

EM),


Since / maps N into the space W = (,/ \ U aa which is solid, the maps g. Next, since f f N and p are homotopic in W modulo Mn. This implies f the image of g is contained in ci, it follows that g is homotopic to a constant O. I map. Since Sn+1 is pathwise connected, this implies that f g Cn. Let X be a simplicial complex with dim X 0 is a given integer. We proceed to extend ft* over the n-dimensional - skeleton Xn of X. Let a be any closed n-simplex of X. Then ft* has been defined on the boundary au for every te[tt„ta+1]. Let M = a x [t,„ ti4+1 ] and consider the subspace N = (a x la) U (au X [t,„ tp+]])

of M. According to (I; 4.2), these exists a retraction p: M N. Let 0: N --->E denote the map defined by 0(x, t) = ft*(x). Then we may extend ft* over a by setting it * (x) Oua Ut(x), vu 0 .95u;1 0 p(x, t)] for every x G a and t E [ta, ta ., 1 ]. This completes the construction of / t*. I The preceding theorem is a corollary of the following general theorem which is known as the covering homotopy theorem of bundle spaces. Theorem 4.2. If a maP b: E B has the

BP, then it has the CHP

for

every paracompact Hausdorff space.

For a proof of (4.2), see [Huebsch 1] and also [S; p. 50 ] . 5. Hopf fiberings of spheres

Among the early examples of bundle spaces were the three fiberings of spheres p S2 1 _>. S , — 2, 4, 8 discovered by Hopf [2] in 1935. We shall examine the first of these (the case n 2) in detail here, and show in § 6 that it may be applied to the classification problem for the maps f: X --> S2, where X denotes a triangulable space of dimension not more than 3. To construct the fibering for n = 2, let us represent S3 as the unit sphere in the space C2 of two complex variables, that is to say, 53 consists of the points (z 1 , z2) in C2 such that 2 1*- 1

.Z2•2"=•-• 1'

5.

HOPF FIBERINGS OF SPHERES

67

Let S2 be represented as the complex projective line, that is to say, as pairs [21, 22] of complex numbers, not both zero, with equivalence relation [z1 , z 2] ("•••1 [Az 1 , )22] where A I O. Then the Hopf map 75: S 3 ±S2 is defined by /7(2 1 , 22) --- [2 1 , 2 2] for each (2 1 , 2 2) e S 3. The continuity of p is obvious. Since any pair [21 , 22] can be normalized by dividing by (ztit + 7 7 maps S3 onto S2. To prove that S3 is a bundle space over S 2 relative to p, let us represent S1 as the set of all complex numbers A with A 1. Consider the points a = [1, 0] and b = [0,1] of 52 and the open sets -

-

S2 \ a, V = S 2 \b. Then U and V cover 5 2 . Every point in U can be represented by a pair [z, 1]. Hence we may define a map Ou of U x S1 into S3 by taking

95u ([z,

],

+ 1 V2-21 + 1 for each [z, 1] G U and each A E Sl. One can easily verify that Ou maps homeomorphically onto 75---1 (U) and that Nu(u, cl) u for each Ux u E U and ci e D. Hence Ou is a decomposing function. Similarly, we can construct a decomposing function Ov. This completes the proof that Sa is a bundle space over 5 2 relative to the Hopf map p. If (21 , 2 2 ) e S3, then one verifies immediately that the fiber p --1 [21 , 22] consists of all the points (Az I , 222 ) with A e Sl. Hence the fibers are just great circles of 5 3 . In this way the 3-sphere is decomposed into a family of great circles with the 2-sphere as a quotient space. The Hopf fiberings p :57 S 4 and p : -÷ 58 are constructed in an analogous fashion from the quaternions and the Cayley numbers respectively; a concise and clear description may be found in [S; pp. 105-110]. In these fiberings, the fibers are 3-spheres and 7-spheres respectively. The Hopi maps are all essential ; in fact, this is a consequence of the following If a sphere Sn is a fiber space over a base space B which contains more than one point, then the projection p:592, B is an essential map. Proposition 5.1.

Proof. Assume that p were inessential. Then there exists a homotopy ht : Sn B (0 < t S2 , one can solve the classification problem of the algebraically trivial maps : X -÷ S 2 of a 3-dimensional trianguiable space X into 52. Let X be any given triangulable space. For an arbitrary map F : X Sa, pF : X S 2 is algebraically it is obvious that the coMposed map / trivial and that the homotopy class of / depends only on that of F. any given triangulable space X, the assignment F --> f F sets up a one-to-one correspondence between the homotopy classes of the maps F: X S 3 and those of the algebraically trivial maps : X S2. As an immediate consequence of (6.1) and the Hopf classification theorem C3 in (II; § 8), we have the following Proposition 6.1. For

Theorem 6.2. The homotopy

classes of the algebraically trivial maps f: X ---> S 2 of a 3-dimensional triangulable space X into the 2-sphere S2 are in a one-to-one correspondence•with the elements the integral cohomology group H3(X). For any ix G I-13(X), the homotopy class which corresponds to oc contains the map / = pF : x 9, where F : X S 3 is a map with oc as its degree. In particular, if X -= S 3, then every map : X -÷ S 2 is algebraically trivial. Hence, we have the following Corollary 6.3. The homotopy classes of the maps : S 8 --> S2 are in a one-to-

one correspondence with the integers. For any integer n, the homotopy class which corresponds to n is represented by the composition f pF : S 3 S 2 of the Hop/ map p : s3 S 2 and a map F : Sa -9..53 with deg (F) n. The proof of (6.1) consists of the following two lemmas.

If a map : X 5 2 is algebraically trivial, then there exists a map F : X S 3 such that pF f. Proof. Since S3 is a bundle space over 5 2 relative to the Hopf map p 53 -÷ S2, we may choose a collection co ={ U } of decomposing neighborhoods U which covers S 2 . Taking a sufficiently fine triangulation of X, we may assume that X is a simplicial complex such that the given map f carries every closed simplex a of X into some decomposing neighborhood U a e co. Let XIII denote the m-dimensional skeleton of X, and let /m = f Xm. For each m = 2, 3, - • , we shall construct a map Fm : Xm S3 such that pF. = fm , Fm+ , I X711 = Fm . First, let us construct F2. Since the given map / is algebraically trivial, so is / 2 . According to the Hopf classification theorem C 2 of (II; § 8), /2 is homotopic to a constant map which can be lifted. Hence it follows from the CHP that there is a map F2: X2 ---> S3 such that pF 2 = / 2 . Next, assume that n > 2 and that Fm has been constructed for every Lemma 6.4.

7.

LIFTINGS AND CROSS-SECTIONS

69

an < n. Let o. be an arbitrary closed n-simplex.of X and choose a decomposing neighborhood U„ e w which contains f (a). Let Oa : Uo. x Sl -÷ p-1 (Ucr), va : U, x S 1 --.- S 1 denote the decomposing function and the natural projection respectively. Define a map Œc, : Oa --->- S1 by taking for every x e am By (II; § 7), a has an extension nci : a -÷ Sl. Then we define a map F.: )02' -). S3 by taking I F 2,_1 (x), (if x e Xn--1), F.(x) —

(if xeceXn). Obvionsly we have pFn, . in and F. I Xn --3- = Fn _i. This completes the inductive construction of the maps F., in = 2, 3, • • .. Finally, define a map F : X -.)- S 3 by taking F 1 X 2n = F. for every in >2. Then we have PF = /. 1

t OcrEin(x), na(x)],

Lemma 6.5. If F, G: X -÷ S 3 are two maps such that pF -_-_-, pG, then F ... G. Proof. Since pF c_.- pG, it follows from the PCHP that F is homotopic to a map F' such that pF' — pG. Hence we may simply assume that PF = pG. Consider S3 as the group of quaternions q with g . 1. Then the fiber which contains the quaternion 1 is a subgroup 9 of 53 and the other fibers are cosets of SI- in 53. In fact, in the usual representation q . x i + x2i + xai + x4ij — x i + x21 + (x3 + x4i)f

= zi + z2i where 2.1 --= x i + x2i and 2. 2 . x 3 + x4i, the multiplication is based on the rules /2 -- — 1 and zi = ji. Then 51- is the subgroup defined by 2.2 :------- 0 and the right cosets of SI are the fibers of the Hopf map p : sa -÷ S2. Define a map H : X -* S 3 by taking H(x) . F(x) • [G(x)] -1 for every x E X. Since pF = f G, F(x) and G(x) are contained in a coset of Sl- and hence H carries X into a proper subspace 51 of 53 • Then it follows that there exists a homotopy Ht: X -4-5 3, (0 < t < 1), such that H o = H and H1 (X) = 1. Define a homotopy jt : X -± S3, (0 < t E maps X homeomorphically onto n(X) with p ;.e(x) as its inverse. Therefore, a cross-section x:X-÷E is considered intuitively as lifting the subspace X of the base space B up into E. If E is a bundle space over B relative to p E --> B and U c B is a decomposing neighborhood with decomposing function and natural projection

Ou:UxD-->p-i(u), then, for any point e in p-i(u), there is a cross-section x e : U -> E given by i8 (u) Ou(u, d, d VuOu -1 (e) for each u G U. If u = p(e), then xe(u) = e. Thus, in bundle spaces, local cross-sections always exist. However, the existence of a global cross-section, i.e. a cross-section over the whole base space B, is a rather strong condition on the structure of the fiber space. In fact, if a global cross-section x:B-÷E exists, then, in any homology theory,satisfying the Eilenberg-Steenrod axioms, the projection p :E--->13 and the cross-section x:B-÷E induce for each m the homo-

morphisms

p * : ll",,,(E)

x * : Ho(B) 11.(E). Since pn is the identity on B,p *x * must be the identity on Hm (B). It follows that m * is a monomorphism, that p * is an epimorphism, and that Hm(E)

decomposes into the direct sum

11.(E) = Kernel

p * + Image x * .

An immediate consequence of this necessary condition is that the Hopf fiberings in § 5 do not have global cross-sections. Dually, one can deduce necessary conditions for the existence of a global cross-section in terms of cohomology :

Hm(E) = Image p* + Kernel x*.

8.

FIBER MAPS AND INDUCED FIBER SPACES

71

These conditions (both those on the groups Hm(E) and those on the groups H(E)) resemble those for retracts. This is no accident: if x is a cross-section, then the image of x is a homeomorph of B and is a retract of E. Let x : X --,- E be a given cross-section. Corresponding to the extension problem of maps in (I; § 2), we have the extension problem of cross-sections to determine if x can be extended over the whole base space, that is to say, whether or not there exists a cross-section x* : B —> E such that x* I X . x. This extension problem of cross-sections is a generalization of that of maps in (I; § 2). Indeed, if E=B XD where D is a given space, then E is a bundle space over B with projection p : E -4- B defined by p(b, d) = b. For a given map f : X -->- D on a subspace X of B, we have a cross-section ni: X -±E defined ' by xf(X) /(x))

---- (x, for each x e X. Then it is obvious that f has an extension over B iff the crosssection xf can be extended throughout B. Similar to the classification problem for maps in (I; § 8) is the classification problem of cross-sections. Let K denote the set of all cross-sections x : B .-› E. Introduce an equivalence relation ,---, in K as follows; For any two crosssections f, g E K, f ,----i g iff there exists a homotopy ht: B -÷E, (0 < t < 1), such that ho = f, h, = g, and ht e K for every t e I. Then the classification problem of cross-sections is to enumerate the classes of K divided by this equivalence relation ,----i. An argument similar to that used above for the extension problem shows that this classification problem of cross-sections is a generalization of that of maps in (I; § 8).

8. Fiber maps and induced fiber spaces Let p:E --›13 and p' : E' -÷ B' be any two fiberings. A map F : E --)- E' is said to be a fiber map if it carries fibers into fibers. Precisely, F is a fiber map if, for every point b in B, there exists a point h' in B' such that F carries p—i-(b) into p` -1-(Y). Now let F : E .-- E' be a given fiber map. Then F induces a function f : B -± B' defined by f (b) = fi'F 5 -1 (b) for every b E B. For any arbitrary set U in B', we have

1 -1-(U) Hence f is continuous if p is either open or closed. If this is the case, f is called the induced map of the fiber map F. In particular, if E is a bundle space over B relative to p, then p is obviously open and so f is continuous. The following rectangle is commutative: F

->. E'

Iv J`

-÷ B'

72

III. FIBER SPACES

Some special cases of fiber maps are important. First, let us take pi : E' -9-B' to be the trivial fibering over B, that is to say, B' = B, E' = B, and pi is the identity map. In this case, the projection p: E -443 is a fiber map and its induced map is the identity map on B. Second, let us take p : E --> B to be the trivial fibering over a space X, that is to say, B . X, E . X, and p is the identity. In this case, every map F: X --* E' is a fiber map with f . X .--). B' as induced map. This suggests the following extension of the lifting problem in § 7. Let p : E -+ B and pi : E' ---> B' be two fiberings. By a lifting of a given map f : B -9- B', we mean a fiber map F : E ---> E' which induces f. Hence, a map F: E -+ E' is a lifting of f: B --> B' if piF = it,. The lifting problem for f: B -). B' is to determine whether or not f has a lifting F: E -÷ E'. As we have seen in the special case of cross-sections, the answer to this problem is not always affirmative. However, for a given fibering pi : E' --> B' and a given map f: B -9- B' of a given space B into B', we can construct a fibering j5: E --* B together with a lifting F: E -> E' of f as follows. Let E denote the subspace of B x E' given by E . { (1), e') e B x E' I f (b) . pr(e 1 )} and let p : E -> B denote the natural projection defined by p(b, e') = b. Let F : E -9- E denote the map defined by F(b, e') = e'. We are going to prove that j5: E -->- B is a fibering and that F is a lifting of f. By the preceding construction, we have lp = piF. Hence it remains to prove that p : E .-.> B is a fibering. Let 0 : X ---> E be a given map of a triangulable space X into E and he : X -->B, (0 < t < 1), be a homotopy of the map y =p ? 5. Let

= Fek : X -->. E` , k t = f he : X ---> B', (0 < t < 1). Then k g is a homotopy of the map ko = i ha = 095 = P'FO = 75 '6* Since p' : E' -9- B' is a fibering, there exists a homotopy kt* : X --> E', E, (0 B constructed above is said to be induced by f; the lifting F: E ---> E' of f will also be said to be induced by f. Note that, for each b e B, F maps the fiber j5-'(b) homeomorphically onto the fiber pi-i(bi), where b' = f (b). Finally, it is straightforward to verify that, if E' is a bundle space over B', then so is E over B.

9.

MAPPING SPACES

73

Finally, if x' : B' ---)-E' is a cross-section, then the map x : B --). E defined by x(b) — (b, x' f (b)) is also a cross-section. We shall call x the induced cross-section of x' by f. The following special case will be used in the sequel. If B is a subspace of B' and f: B .--› B' is the inclusion map, then E can be identified with p'-'(B) and p with p' 1 B in an obvious way. In this case, the induced fibering p : E -4- B will be called the restriction of p' : E' --)- B' on B. Hence we have the following E is a fiber space over a base space B with projection p : E —> - B and if A is any subspace of B, then p1(A) is a fiber space over A with p f pi(A) as projection. Proposition 8.1. If

9. Mapping spaces Let X and Y be arbitrarily given spaces and denote by 40 . yx the totality of maps of X into Y. There are various ways of topologizing Q, but we will be concerned only with the cornpact-open topology, [Fox 3]. It is also called the h-topology, [Arens 2], and the topology of compact convergence, [B; HI] and [lg. For any two sets K c X and W c Y, let M(K, W) denote the subset of Q defined by M(K,W) — {feQlf(K)c W}. M(K, W) will be called a subbasic set of Q if K is compact and W is open. The compact-open topology of Q is defined by selecting as a subbasis for the open sets of D the totality of the subbasic sets M(K, W) of Q. According to the usual definition of a subbasis, every subbasic set is open in Q and every open set of D is the union of a collection of the finite intersections of subbasic sets. Throughout the present book, mapping spaces are understood to be topologized by their compact-open topologies, unless otherwise stated. If X is a Hausdorff space and { U } is a subbasis for the open sets of Y, then the totality of the sets M(K, U), for K a compact subset of X and U e { U }, constitutes a subbasis for the compact-open topology of Q [Jackson 21. Lemma 9.1.

Proof. It suffices to show that if K is a compact subset of X and W an open subset of Y, and if f e M(K, W), then there exist compact subsets K1 , • • - , Km of X and members U 1,- • - , Um in {U } such that f e M(Ki , U1)

ri • • •

n M(K„,,, U.) c M(K, W).

Let x e K. Since f (x) E W, there are a finite number of sets in { U }, say Uf, • • • , U,I,z, such that f (x) e Uf n - • • n

rgx c W.

FIBER SPACES

74

Since / is continuous, there is a neighborhood Gz of x in X such that

f(G x )c UT()

• • n U.

As a compact Hausdorff space, K is regular. So there is an open neighborhood Hz of x in K such that the closure K z ---- Hz is contained in G. The collection { I-Ix IxEK} is an open covering of the compact space K, and hence there are a finite number of points in K, say x l, •, xq, such that K = Hz, U • • • U H zq . In the subscripts and superscripts involved above, we shall simply replace xj by f,f =1,• • • ,q. Now the sets K 1 , • • , Kg are compact. Moreover,

/(Ki) c RCA U n • • • n U q

Hence we have

fen i=1

nj [fl

w, (i =

M(Kj, WI)].

• =1

Suppose that g E D is contained in the set on the right-hand side of the preceding formula. If x E K, then x is in some Hy and hence is in Kf. Therefore

g(x)

Un • • n Ni c W.

Thus g EM (K, W), and so

q ni tE 11 [f) M(K i ,Uii)]c M(K,W). z=1 Next, there is a natural function co : x X defined by co( /, x) (x) for each / evaluation of the mapping space D. Proposition 9.2. // X

Y

D and each x E X. co will be called the

is a locally compact regular space, then the evaluation

co of D is continuous.

Proof. Let fED, xe X, and an open set W of Y which contains / (x) be arbitrarily given. Since/is continuous, the inverse image f 4-(W) is an open set containing x. Since X is regular and locally compact, there exists an open neighborhood V of x such that the closure V is compact and is contained in ---1 (W). Then U M(V,W) is a subbasic open set of D which contains /. U = M(V , W) implies that co maps U x V into W. This proves the continuity of co. I For the necessity of the local compactness of X in (9.2), see Ex. J at the end of the chapter. For each point y E Y, let j(y) denote the constant map in D which maps X into the single point y. The assignment y --->- 1(y) defines a function

9.

MAPPING SPACES

75

called the natural injection of Y into D. It can be easily verified that j maps Y homeomorphically onto a subspace f( Y) of D. Furthermore, if Y is a Hausdorff space, then j(Y) is closed in D. For any given point a e X, let denote the function defined by py,( f) = f (a) for each f e D. Clearly, pa, is a map and sends the subspace j(Y) of S2 onto Y. We shall call pa the projection of D onto Y determined by the given point a E X. From the definitions given above, we observe that pa/. is the identity map on Y and ipa is a retraction of D onto its subspace j(Y). Hence we have the following

Proposition 9.3. The natural injection f maps Y homeomorphically onto a retract j(Y) of D. Next, let us consider three given spaces T, X, Y and the mapping spaces yX o = yXxT , QT . For each map 0 : X x T Y in 0, define a function 0(0) :T taking [0(0)(t)1(x) #(x, t), (t E T,xE X) .

S2 by

0(0) is said to be associated with 1). Proposition 9.4. The associated /unction 0(0) of

GO

is continuous.

v = 0(0) and let U = 111(K, W) be any subbasic open set in D. It suffices to prove that vr-'(U) is an open set of T. Let to be any given point in ip--1 (U). By definition, we have Proof. Let

K x t o .95 - '(W) The continuity of implies that 95 -1 (W) is an open set of X x T. Hen.ce 95 -1 (W) is the union of a collection of open sets of the form G, x Hp , where G, and Hp are open sets of X and T respectively. Since K is compact, K Xt o iq contained in the union of a finite number of these open sets, say G I x H I , G2 X 1-12,• • • ,

x Hn

with to G Hi for each i = I, 2,• • • , n. Then

H 11 1 n

H2

n•••n

Hn

is an open set of T containing to and is contained in v -1 (U). Therefore, v -i(U) is an open set of T. I As a consequence of (9.4) the assignment 0 0(0) defines a function

0:0 -÷T, when continuous, this function will be called the association map; that this is usually the case is shown by the following Proposition 9.5. If

T is a Hausdorff space, then 0 : (15

VI is continuous.

76

III. FIBER SPACES

Proof. Since T is a Hausdorff space and the totality of the subbasic sets M(K, W) constitutes a subbasis of Q, it follows from (9.1) that the subsets

M[L,M(K,W)] Oper tp(L) M(K,W) form a subbasis for P, where L runs through the compact subsets of T, K runs through the compact subsets of X, and W runs through the open subsets of Y. It follows clearly from the definition of 0 that

6 -1 { M[L, M(K, W) ]} M(K X L, W). Since K X L is compact, M (K X L, W) is open in 0. Since { M [L ,M(K ,W)] } is a subbasis of VI, it follows that 0 is continuous. I It is obvious from the definition that 0 carries 0 into P in a one-to-one fashion. In general, 0 is not onto. However, we have the following Proposition 9.6. The evaluation co of

S2 is continuous if, for every space T,

The function 0 : qi -›-V-1 is onto. Proof. Necessity.

Let

defined by Let 0 — (DX

v G ¶ and

consider the map Z:XxT -* Q x X

X(x, t) = (v(t), x), (x E X, t E T). E

P. Since

[0(0)(0] (x) = 0(x, t) = cox(x,t) = w[y(t),

= [y(t)](x)

for every t E T and x E X, we have 0(0) = y. Hence 0 is onto. Sufficiency. Assume that the condition holds. In particular, select T = S2 and take ip to be the identity map on D. Then there is a map 0 E 0 with 0(0) ---- /p. Since

0(x, f) = [V(i)](x) = 1(x) = w(/ , x)

for every x e X and f E Q, the continuity of 0 implies that of co. I As an immediate consequence of (9.2) and (9.6), we have the following If X is a locally compact regular space, then the function û: P -->W sends 0 onto W in a one-to-one fashion. Corollary 9.7.

X and T are Hausdorff spaces, then the association map o : 0 --->Y1 is a homeomorphism of 0 onto a subspace of VI [Jackson 21. Proposition 9.8. If

Since 0 is one-to-one and continuous by (9.5), it remains to prove that 0-1 is continuous on 0(0) c T. For this purpose, it suffices to prove that, if / is a compact subspace of X x T and W is an open set in Y, then the image 0 [M(J, W)] is an open set of 0(0). Let ip E 0[M( j, W)] be arbitrarily given. Choose a 0 e W) with 0(0)= y. Let JA, and IT denote the projections of Ji X and T respectively. For each point z = (x, t) EJ, choose an open neighborhood Uz of x in Jx and an open neighborhood Vz of t in hi, such that Proof.

56(Uz

Vz ) c W.

9.

MAPPING SPACES

77

As compact Hausdorff spaces, jx and IT are regular. Hence we may shrink Uz and Vz a little bit so that O(Kz x Lz ) c 147, where Kz denotes the closure of Uz in jx and Lz that of Vz in /T. The collection { (Uz X Vz)nfIZEJ} is an open covering of the compact space J. Hence there is a finite number of points in j, say z i ,• • • , zn , such that j c (Uzi x Vz i) U • • • U (Uz„ X

For the subscripts in the notations of the various sets involved above, we shall simply replace zi by i, i = 1,• - • , n. Now the sets Ki and L i , i — 1 ,• - • , n, are compact. Moreover, [v(Li)] (4------- ck(Ki x Li) c W

for each i = 1,—, n. Hence tz

0(0)

n { f) M[L i, M(K i,

4: (z) =

(x, t) = [X(t)] (z )

TV)] }. 2:=1. Suppose that X c P is contained in the set on the right-hand side of the preceding formula. Since X G 0(0), there is a E 0 with X = 0(e). If z = (x, t) e J, then z is contained in some Ui x Vi and hence in Ki x Li. Since X is in M[Li, 111(K1, W)] and since x e Ki, t e Li, we have VE

e W.

This proves that

(,j) c W. Therefore, E e M(J, W) and hence X =-- 0(e) G 0[M(J, W)]. Thus, we obtain .

VG

0(0)

n { ir---n-3.M[Li, M(Ki, W)]). c 0[M(j ,

W.)].

This proves that 0 [lf(j, W),] is an open set of 0(0). i As an immediate consequence of (9.7) and (9.8), we obtain the following Theorem 9.9. Let X and T be Hausdorff spaces and Y any space. If X is locally compact, then the association map 0 is a homeomorphism of the mapping space 0 — Yxx T onto the mapping space P = (3(x) T.

Hereafter, when the assumptions of (9.9) are satisfied, the two mapping spaces will be identified by means of the homeomorphism O. In symbols, we have yX x T =. (yX)T .

This will be called the exponential law of mapping spaces. Now let us go back to (9.3) and look for a sufficient condition that j(Y) be a strong deformation retract of D. Such a condition is given by the following Proposition 9.10. If X is a locally compact regular space and contractible to a point a G X, then j( Y) is a strong deformation retract of D.

III. FIBER SPACES

78

(9.2) the evaluation co : Q x X Y is continuous. Since X is contractible to the point a E X, there exists a map h:X X I --> X such that h(x, 0) = x, h(x, 1) --- a, (x E X); h(a, t) = a, (I E I). Define a map sb : X xiQx / Y by taking 0(x, f, t) w[f, , h(x,t)], (x e X, f E D, t E I). According to (9.4), the associated function v= 0(0) :Q X /—>Q is continuous. Define a homotopy Xt : Q --> D, (0 < t < I), by setting Proof. By

Xt(i) EQ, E Then it is easily verified that X0 is the identity map, X 1 — j, and Xt( f) = fcr every / E f(Y) and t E I. I Since it is easily verified that Xtriral (Y)] c iral (Y) , (ye Y, te/), we have proved the following

f

X is a locally compact regular space and contractible to a point a e X, then, for each y E Y, the subspace p;1 (y) is contractible to The point i(y). Proposition 9.11. If

10. The spaces of paths

Let Y denote a given space. By a path in Y we mean a map : I —> Y of the unit interval / = [0, 1] into Y. The points o-(0) and o-(1) are called respectively the initial point and the terminal point of the path o, and are said to be connected by the path or. The relation that two points a and b can be connected by a path in Y is obviously symmetric, reflexive and transitive; hence the points of Y are divided into disjoint classes, called the path-components of Y. We shall denote by Cy the path-component of Y which contains the point y E Y. Y is said to be pathwise connected if it consists of a single path-component and hence every pair of points in Y can be connected by a path. The totality of paths in Y forms a space Y/ with the compact-open topology defined in the previous section. This space D together with certain of its subspaces is of fundamental importance in homotopy theory as well as in the functional topology of Morse, [M1 and MO. By a generalized triad (Y; A, B), we mean a space Y together with two subspaces A and B. (Y; A, B) is said to be a triad if the intersection A fl B is non-empty. For a given generalized triad (Y; A, B), let us denote by ; A, B1

II. THE SPACE OF LOOPS

79

the subspace of D which consists of the paths o- in Y such that o.(0) e A and o.(1) E B. The following particular cases are of importance. If A = Y and B consists of a single point y e Y, then we shall denote the subspace [Y; Y, y] of D simply by Q. It is called the space of paths in Y with a given terminal point y. A loop in Y is a .path : I -÷ Y such that o.(0) = o-(1). The point o.(0) = o.(1) will be called the basic point of the loop o-. The set of all loops in Y forms a subspace A of Q which will be called the space of loops in Y. If both A and B of a given generalized triad (Y ; A, B) consist of the same single point y e Y, then we shall denote the subspace [Y; y, y] of Q simply by A. It is obviously a subspace of A and will be called the space of loops in Y with a given basic point y. Clearly we have

Ay = A n

Qy.

The projections po, Y determined by the points 0, I as in the preceding section shall be called the initial projection and the terminal projection respectively. If Y is a Hausdorff space, then the continuity of pa and p, implies that the subspaces A, A y , Dy of Q are all closed. Let by denote the degenerate loop y (I) = y. By (9.11), we have the following Proposition 10.1. Qy is contractible to the point by .

Intuitively, a contraction of Qy to the point by is obtained by pushing all paths of Qy along themselves simultaneously to the terminal point y.

11. The space of loops Let Y denote a given space and yeY a given point. The space Ay of loops in Y with y as basic point has a central role in the study of homotopy groups; therefore, we shall study its important properties in the present section. There is a natural multiplication defined in A y as follows. For any pair of loops f, g e Ay, the product fg e Ay is the loop in Y defined by

(fg) (t)

(

1(.20, g(21— 1),

(0 < t X be representative loops of a, /3 respectively. Since xo is an idempotent, we may define a loop h I -4- X at xo by taking

h(t)

f (t)g(t)

for each t e I. This loop h represents an element y of n 1 (X , x 0) which obviously depends only on a and p. It suffices to prove that y cc13 and y = 13cc. To prove y = 099 , we may assume that the representive loops f, g have been so chosen that f (t) x o for each t > and g(t) x o for each t < I. It follows that (if 0 < t < f f (I) h(t)

(ifj Z a given map. For each f E Yx , the composition Of is in Zx . The assignment f Of defines a yX zX function which is obviously continuous and will be called the induced map of 0 on Yx into Zx . Let C denote the category composed of all spaces and all maps, [E—S; p. 110]. The operation Y Yx and 0 0x , where X is a given space, defines a covariant functor of C into itself, [E—S; p. I 1 1]. Furthermore, if we consider the natural injection j and the projection pa determined by a given point a e X, the comrnutativity relations

= Ate 1.0 el, obviously hold in the following rectangles of maps: .„,„ Pa

Y X ckx

r7

iPa ZX

Throughout the remainder of the present section, we are concerned with the important special case that X = I. Consider a given map 0 : Y Z. Let y be a given point of Y and denote z ç(y) E Z. As above, we shall use the following notation: Dy [Y; Y, y], A= [Y; y, y] ; D, = [Z; Z, z], A,= rZ;z, z]. Then the induced map 01 obviously sends Dy into Qz and Ay into A. More generally, 01 maps [Y ; A, B ] into [Z ; (A), 5b(B)] for any given subspaces A and B of Y. According to § 11, there are natural multiplications defined both in Ay and in A. It follows from the definition that 0/( k ) 0/(.00/ (g) (14.1) for any E Ay and g E A. Since 0/ is a map, it sends the path-components of Ay into those of A. Hence, (14.1) shows that 0/ induces a homomorphism of the group of path-components of Ay into that of A z which is the induced homomorphism 0,, of 1 (Y, y) into ni (Z, z). Now let Y, Z be given spaces, A C Y, B C Z given subspaces, and yo E A, zo E B given points. Let • U = [Y; Y, yo], C [Y ; A, y o], V = [Z ; Z , z o], D = [Z ; B, and denote by ito E C and vo E D the degenerate loops /40 (/) 2'0 respectively. Let q :V Z

yo and vo(/)

denote the initial projections. We are going to establish a covering map theorem which will be important in the next chapter.

m.

86

FIBER SPACES

Theorem 14.2. If p:

U -÷ Z is a map such that y(C) C B and y(u0) ---- zo, then there exists a map y* : U -÷ V such that qy* ----- y, y*(C) c D, and V * (uo) = vo. Proof. According to (9.2), the evaluation w: U x I --1- Y is continuous. Define a map a:IxIx U —,. Y by setting a(s, t, /)---, co(/, s + 1 — st), (s El, t e /, /E U). By (9.3), the associated function X: i. x U -÷ U defined by [X(t, /)](s) --- a(s, t, /), (t el, / e U, s e /), is continuous. Let E — vX : I x U -->- Z. By (9.3), the associated function v* : U -)- V de fi ned by

[v*(/)](t) — (1, /),

(t e ./-, t e U),

is continuous. It remains to verify the relations qv* --- v, v*(C) c D and

vo.

From the definition of X, one can easily see that X(0, /) = / for every / e U. Hence we have

X(° , f) = V(i) for every / e U. This proves that qv* = v. v*(C) C D is an immediate consequence of qv* = v and v(C) c B. To check v*(u0), = yo we first note that X(i, u0)----- u0 for every t e I. Then we have qe(t) ' Ce(/)](°) —

(t, uo) = VX(t, uo) --a v(u) = zo. for every t € I. This implies v*(uo) = yo. I If we removed the condition v*(uo) = yo from the conclusion, then (14.2) would be easier to prove. By (13.1), the map q :V -÷ Z has the PLP and hence also the ACHP. Since U is contractible, an application of the CHP for U gives a lifting v* : U -÷ V of the map v: U ---> Z . That *(C) c: D is obvious but the condition v (u0) = yo does not necessarily hold. The lifting v* constructed in the proof of (14.2) may be called the canonical lifting of v. 15. Fiherings with discrete fibers Motivated by the results concerning the exponential map p : R ---> Si in Chapter II, we are going to establish similar results for fiber spaces with discrete fibers, i.e. in which all fibers are discrete. Let E be a given fiber space over B relative to p :E-÷B with discrete fibers.

For each path a : I —> B joining b0 to b l and for each eo e p --1 (b0), there exists one and only one (covering) path Gr* : I ---> - E such that a*(0) — eo and pa* — a. Lemma 15.1.

Proof. The path a may be considered as a homotopy of the partial map a I O; hence, the existence of a* is an immediate consequence of the covering homotopy property. To prove the uniqueness, let a*, a4t :I -± E be any two paths in E such

15. FIBERINGS WITH DISCRETE FIBERS

87

that pa* -- a = po4 and a*(0) — eo = a4(0). Let s e I be arbitrarily given. It remains to show that a*(s) = a#(s). For this purpose, let us define a map g : I --> E by taking

f a*(s — 2st), g(t) = t a#(2st — s),

(if 0 < t < I), (if i < t B has a homotopy /, :I--->B, (0 defined by r a(s —2st + 2rst), (if 0 " Mn,n-k• 5. Mn, iis essentially the (n — 1)- dimensional real projective space Pn-1 and Mn , i , the (n 1)-sphere Sn -1 . 1. Elementary properties of mapping spaces

Let D denote the mapping space Yx with the compact-open topology. Prove the following assertions : I. If Y is a To-, Tr, T 2-, or regular space, then so is D. Conversely, if r2 is a To-, T1-, T2-, or regular space, then so is Y.

III. FIBER SPACES

102

2. If X is a locally compact Hausdorff space and if X, Y are both separable (being separable means having a countable basis), then so is D. On the other hand, if D is separable, then so is Y. 3. Assume that X is a compact metrizable space and Y is a metrizable space. Then D is an ANR iff Y is such; similarly, Q is an AR iff Y is such. J. Admissible topologies

A great variety of topologies may be introduced into the set D = Yx making it a space. We shall denote by D r the space obtained by topologizing D with a topology T. A topology r of S2 is said to be admissible if the evaluation co : D, x X Y is a map. Thus, by (9.1), the compact-open topology of D is admissible provided that X is a locally compact regular space. Prove the following assertions, [Arens 1] : I. Any admissible topology of Q contains every open set in the compactopen topology of D. 2. If X is a completely regular space and Y is a T1-space containing a nondegenerate path, then a necessary and sufficient condition for the compact-open topology to be admissible is the local compactness of X. K. The topology of uniform convergence

Let Y be a rnetrizable space and let d be a given bounded metric consistent with the topology of Y. There is a natural metric d* defined on D Yx by

d*(f , g)

supx ex cl[f (x), g(x)]

for each pair of map f, g E D. The metric d* determines a topology of D called the d* topology, or the topo/ogy of uniform convergence with respect to d. Prove the following assertions : 1. The d*-topology of S2 is admissible. 2. If X is compact, then the compact-open topology of D coincides with the d*-topology induced by any given bounded metric d on Y. The word "bounded" in this assertion might have been omitted. 3. If X is a completely regular Hausdorff space and Y a metrizable space containing a nondegenerate path, then a necessary and sufficient condition for the compact-open topology of D to coincide with the d*-topology induced by a given bounded metric d on Y is the compactness of X [Jackson 1]. 4. The d*-topology of D depends not only on the topologies of X and Y but also on the metric d chosen for Y. Construct a few examples. -

L. Maps on topological products

Consider three given spaces T, X, Y and the mapping spaces:

Q=

yX

q

= yXxT

i

In § 9, we have defined the association O : assertions :

_Q

T.

T. Prove the following

EXERCISES

103

1. If both X and T satisfy the first countability axiom, then 0 sends 0 onto W., [Fox 3].

2. If X and T are Hausdorff spaces satisfying the first countability axiom, then 19 is a homeomorphism of 0 onto Vir and hence the exponential law yXXT = (yX)T holds in this case. M. Borsuk's fibering theorem

Let X be a compact metrizable space, A a closed subspace of X, and Y a compact ANR. Consider the mapping space E = Yx and the subspace B of the mapping space YA consisting of the maps g:A---> Y which can be extended over X. Prove that E is a sliced fiber space over B relative to the natural projection p:E--->B defined by p(/) = LA. for every f e E. By Exercises C and I, show that this fibering has a unified slicing function. [H; p. 173]. N. Change of the boundary sets

There are changes of the boundary sets A and B without changing the homotopy type of the space [X; A, B] of paths. Because of symmetry, we may study only the changes of the terminal set B. Let B 1 and B 2 be two subspaces of X. A deformation of B 1 into B 2 in X is a homotopy ht : B 1 ---> X, (0 < t < 1), such that ho is the inclusion map and hi (B i) c B 2 . Such a deformation ht induces a map h*: [X; A, B 1 ] [X; A, B 2] described as follows: for each f E [X; A, B 1], g = 114*(f) is given by f (2t), (0 < t < i),

g(t)

0 and a map f of the closed interval [0, a] into Y. The points f (0) and f (a) are called the initial point and the terminal point of f respec-

III. FIBER SPACES

10 4

tively. A curve f: [0, a] Y is said to be closed if / (0) = 1(a); in this case, f (0) is called the basic point of the closed curve f. Now, let rdenote the set of all curves in Y. To topologize r, let us consider the subspace J. of the real line R consisting of the real numbers a > 0 and the space D = Y-T of all paths in Y. Define a function çt : r by taking OM = (a, ay) for every curve f : [0, a] Y, where a1 : Y is the path given by Cf (t) f (at) for each t e I. This function 4) carries r onto a subspace 95(r) of j x Q in a one-to-one fashion. We topologize r in such a way that 95 becomes a homeomorphism. Prove that (J X Q) \ 95(r) is the subspace 0 x [S2 \ j( Y)] where Y --> Q denotes the natural injection : Y --> Q of § 9. Since I = [0, 1], every path in Y is also a curve in Y. The topology of r defined above permits us to consider Q as a subspace of r Construct a homotopy h r -÷ r, (0 < t < 1), such that the following conditions are satisfied: (i) ho is the identity map on P. (ii) h1 is a retraction of r onto D. (iii) kt(f) = f for every f E Q and t E I. (iv) For every f G r, the initial point and the terminal point of the curve ht(i) are the same as those of / for each t E I. 1ff: [0, a] Y and g [0, b] ---> Y are two curves in Y such that f (a) = g(0) , then we may define a product f .g : [0, a + b] ---> Y by taking :

[i•g] (t)

( I)

3

g(t — a),

(if 0 < t < a), (if a f.g is defined for every pair f,ge ey with the trivial , curve e: [0, 0] -÷ y as a two-sided unit. Hence, if Y is a Hausdorff space, then ey is a mob with e as a two-sided unit in the sense of Wallace [1].

r

P. Generalized covering spaces

A space E is called a generalized covering space over a space B relative to a map p E B if the following conditions are satisfied: (GCS1) p maps E onto B. (GCS2) For each b e B, there exists an open neighborhood U of b in B such that p--1(y) can be represented as a disjoint union of open sets in E each of which is mapped homeomorphically onto U by p. Prove that every bundle space E over B relative to p : E B with discrete fibers is a generalized covering space over B relative to p.

EXERCISES

105

Let E be a generalized covering space over B relative to p : E -* B. Prove the following assertions: 1. E is a sliced fiber space over B relative to p with discrete fibers. 2. If B is connected, then E is a bundle space over B relative to p with discrete fibers. 3. If E is connected and B is locally connected, then E is a covering space over B relative to p. 4. Let q : X -÷ Y be a given map of a space X into a space Y. If g : X ---> E is a map and ft : Y -÷ B, (0 < t < 1), a homotopy with /0q = pg, then there exists a unique homotopy gt : X --> E, (0 < t < 1), of g such that f tq = pgt for every t E I and that gt is stationary with f t. In particular, p : E ---›- B has the ACHE'. This assertion is also true for sliced fiber spaces with totally pathwise disconnected fibers, [Griffin 1]. Q. Local homeomorphisms

A map p : E -- > B of a space E onto a space B is called a kcal homeomorphism of E onto B if every point x of E has an open neighborhood which is mapped homeomorphically by p onto an open neighborhood of p(x) in B. Prove that, if p : E -÷ B is a local homeomorphism of a regular Hausdorff space E onto a space B such that p--1(b) is finite for every b e B, then E is a generalized covering space over B relative to p. R. The covering spaces of the torus

As an exercise to determine all equivalence classes of the covering spaces over a given space B, let us study to case where B is the torus Sl- x Sl. According to (II; Ex. A), the fundamental group ni. (B) is a free abelian group with two free generators a and b. Among the subgroups of nl(B) there is a doubly indexed system Gm , n , (in, n = 0, 1, 2, • • • ), where Gm , n is the subgroup _generated by am and bn. Hence, G0 ,0 = 0 and G1 ,1 = Prove the following assertions: 1. Every covering space over B is regular. 2. Corresponding to G0, 0, we have the universal covering space E — R 2 — R x R over B = Si- x Ss' relative to the projection p 0 ,0 : E --->- B defined by po,o(x, y) — (px, py), where p: R ---> Si- denotes the exponential map of (II; § 2). 3. Corresponding to G0 , n , n> 0, we have the covering space E = R x 51 over B — 9 x 9 relative to the projection po,,,: E -). B defined by 5o , ( x, Z) = (Px, zn) for each x E R and z G SI- . Similarly, we may get the covering space corresponding to Gm„0, ni> O. 4. Corresponding to G,n , n, m > 0, n > 0, we have the covering space E = S1- x 51 over B = S1- x .3 1- relative to the projection pm ,. : E --> B defined by pm,. (u, y) = (um, vn) for each (u, y) G E.

io6

III. FIBER SPACES

S. Maps of the torus into the projective plane

Consider the torus T and the projective plane P. Pick arbitrary basic points to e T and po e P, and study the maps f: (T, t o) ---> (P, po). The fundamental group n l (T , to) is a free abelian group with two free generators a and b, while n1 (P, po) is a group of order 2 generated by C. Hence, there are four possible homomorphisms

hm , n, : 7r 1 (T, to) --->n1 (P,

po),

with m, n running over 0 and 1, defined by hm , n (a) = me,

hm , n (b) = ne.

By considering T as the unit square with the opposite sides identified, construct for each (m, n) a map

/m,„ : (T, to) -> (P, po) such that (1m , 9, ) * — h.,.. Hence, the homotopy classes of the maps f: (T, t o) --->- (P, po) are divided into four disjoint collections Cm , n such that / e Crn„. iff f* hmoi . Next, prove that, for each (m, n), the homotopy classes in the collection Cm , n are in a one-to-one correspondence with the homotopy classes of the maps (S 2, so) --> (P, po) and hence with the integers, —

T. Maps of a surface into a surface

Let X and Y be any two closed surfaces other than the sphere and the projective plane. Study the classification problem of the maps /: X -> Y as follows. Pick arbitrary basic points xo e X and yo G Y. Two homomorphisms h, k : ni (X , x 0) ->ni (Y , y o) are said to be equivalent (notation : h ,-..-!. k) if there exists an element b E n l (Y, y) such that k(a) — b -1 . h(a)-b holds for every a en 1 (X, x 0). Thus the homomorphisms h : 7C 1 (X, z 0) ---> n i ( Y, yo) are divided into disjoint equivalence classes. Let C denote the set of these equivalence classes. Prove that the homotopy classes of the maps it : X --> Y are in a one-toone correspondence with the equivalence classes C by establishing the following assertions : I. Every map / : X -> Y is homotopic to a map g:X---> Y such that g(x 0) = yo. 2. For any two maps f, g : X -> Y satisfying f (x0) = yo = g(x0), f ny_ g implies f * fL-_, g* . 3. For any he C, there exists a 'map /: X --> Y such that /(z0) =y0 and /* =h. 4. If f: X ->. Y is a map with f(z0) = yo and h e C is equivalent to f *, then there exists a map g : X --> Y such' that g(z 0) = yo, g ,-.-,-, f and g* = h. 5. If /, g : X ---> Y are two maps such that f(x0) — yo = g(x0) and f,1, then / ._,--2. g.

CHAPTER IV

HOMOTOPY GROUPS 1. Introduction The basic problem which led to the discovery of "homotopy groups" was to classify homotopically the maps of an n-sphere Su into a given space X. In the case n = 1, this was facilitated by pinning down a base point to obtain a group structure as in (II; § 4). The same trick was found to work in higher dimensions; in fact, if we pinch the equator of Sn to a point, we obtain two n-spheres with one point in common. If n> 1, there is a rotation of Sn which gives a homotopy interchanging the two hemispheres. This implies the striking feature that the group is abelian. The relation between homotopy and homology groups and the existence of relative homology groups H„(X , A) led quickly to the relative homotopy groups z n,(X , A, x0), giving a system highly analogous to homology theory. But it differs in several ways : no(X, x o) and 7c 1(X, A, x o) are not ordinarily groups; 1 (X, x o) and 7c 2 (X, A, xo) are not usually abelian; and the excision property for homology does not hold for homotopy. A very important fact in the minds of those involved in the development of the theory was the result of Hopf in 1930 that7r3(S 2) is infinite. This showed that the groups express some very deep topological properties of spaces. A second important fact is that the definition of nn, is not effectively computable; and there was no definition available which led immediately to effective computations as in the case of homology groups of complexes. Successful calculations in special cases came slowly. It is only recently that methods have been found which apply to a reasonably broad variety of cases. These are the subject of much current research. The objectives of the present chapter are to define the groups and related homomorphisms, to establish their main general properties, and to show that certain of these properties are characteristic.

2. Absolute homotopy groups Let (X, xo) be a given pair consisting of a space X and a point xo in X. Let n0 (X, x o) denote the set of all path-components of X. The path-component of X which contains xo will be called the neutral element of 7t0(X, xo) and will be denoted by O. As in (II; § 4), we shall denote by n i (X , x0) the fundamental group of X at xo. For each integer n> 1, the definition of the n-th (absolute) homotopy 107

Io8

Iv.

HOMOTOPY GROUPS

group an (X , x o) is strictly analogous to that of the fundamental group. We replace the unit interval / by the n-cube In, i.e. the topological product of n copies of I. Every point I e In is represented by n real numbers t = (11, • . • ,tn), ti e .T, (1 = 1, 2, • • - , n), called the coordinates of t. The number ti is called the i-th coordinate of t. An (n — 1)-face of In is obtained by setting some coordinate ti to be 0 or 1. The union of the (n — 1)-faces forms the' boundary dIn of In; it is topologically equivalent to the unit (n — 1)-sphere Sn'. Consider the set Fn . Fn(X, xo) of all maps f: (/n, 0/n) -->- (X, x0). These maps are divided into homotopy classes (relative to 0/n). We shall denote by nn (X , x0) the totality of these homotopy classes. We shall also denote by [f] the class which contains the map / and by 0 the class which contains the unique constant map do(In) . xo. Topologize Fn by means of the compact-open topology as in (III; § 9) ; then nn(X, x0) becomes the set of all path-components of the space F. We may define an addition (usually non-commutative) in Fn as follows. For any two maps /, g in Ft, their sum f + g is the map defined by u

+ g) (t) . ( i i( n2ti, t2,• - • , Ili), gk htl — 1, 4) * * in), '

if 0 < t1 < i, if + < t1 < 1,

for every point t = (t1 ,• • • , tn) in In. Obviously, f + g is in Fn. The homotopy class [/ + g] clearly depends only on the classes [f] and [g]. Hence we may define an addition in nn (X, xo) by taking

[t] + [g] = U -I- gl. Just as in (II; § 4) for the fundamental group, one can easily verify that this addition makes an (X, x0) a group which will be called the n-th homotopy group of X at xo. The class 0 is the group-theoretic neutral element of nn (X, x0), and the inverse element of [f] is the class [/0], where 0 : In -›- In denotes the map defined by 0(t) = (1 — t1, t 2 ,• • • , tn)

for every t = (t1 , t 2 ,• • • , tn) in I. If the boundary 0/n of In is identified to a point, we get a quotient space which is topologically equivalent to an n-sphere Sn' with a given basic point so e Sn. It follows that one might equally well define an element of ;r„(X, x0) as a homotopy class (relative to so) of the maps f: (Sn, so) --4- (X, x0). Since the two halves of Ps, defined by the conditions t, < i•- and ti > i- respectively, correspond to two hemispheres of Sn, one can clearly see how to define the group operation of 7rn (X, xo) from this point of view. For details, see [Hu 4 11. Since, when n > 1, there exists a rotation of Sn which leaves so fixed and interchanges the two hemispheres, this definition suggests the following striking property of nn (X, x0) which however will be proved in a different way.

2. ABSOLUTE 110 MOTOPY GROUPS

109

Proposition 2.1. For every n> 1, orn (X , x 0) is an abelian group.

Proof. As in (III; § II), one can prove that Frl -1 is an H-space with the constant map do e F"--1- as a two-sided homotopy unit. Hence, by (III; 9.9), we have we have xin _ xi Then one can easily see that nn(X, x0) and i (Fn- , do) are essentially the same group. Hence an (X , x 0) is abelian. I In the preceding proof, we have incidentally obtained an interesting result: 721,(X,

z0) = gr1 (F4-1, do).

Hence every homotopy group of a space can be expressed as the fundamental group of some other space. This relation can be used to define higher homotopy groups in terms of fundamental groups; indeed, Hurewicz used this definition when he introduced these groups in 1935. One can say more: let p be any positive integer less than n and let q = n p. By (III; 9.9), we have xin _ x/Px/q (x/P)/(l . —

By means of this relation, it is easy to deduce the following result :

p

nn (X, x 0) = nq (F29 , do),

q

n,

where do denotes the constant map do (iP) = xo. In particular, when p = 1, FP becomes the space W of loops in X with basic point xo and do the degenerate loop at x o . Thus we have proved the following proposition which will be used in the sequel. Proposition 2.2. nn (X, x0)

7141 _ 1 (W, do).

Finally, the following proposition is an immediate consequence of the fact that In is pathwise connected. Proposition 2.3. If X 0 denotes the path-component of X containing x o, then

nn (X 0, x0) = Ten (X , z0), n> O. Examples. If X is a contractible space, thenn n (X, x0) = 0 for every n > O. Next, by (II; 7.1) we have

n0 (S1, 1 ) = O, n1( 9, I) nn (51-, 1) = 0,

Z

if n > 1.

On the other hand, by means of (II; § 8), one can prove that orm,(Sn, so) = 0,

(if m < n),

nn(Sn , so)

Z.

Finally, by (III; 6,3), we deduce the result a3(S 2, so)Ft,' Z.

IV. HOIVIOTOPY GROUPS

3. Relative homotopy groups The objective of the present section is to generalize the notion of homotopy groups in § 2 by defining the relative homotopy groups n„(X, A, x0). By a triplet (X, A, x0), we mean a space X, a nonvacuous subspace A of X, and a point xo in A. If x o is the only point of A, then the triplet (X, A, x0) will be simply denoted by (X, x 0) and may be considered as a pair consisting of a space X and a point xo in X. Let n> 0 and define the n-th relative homotopy setn n (X, A, xo) as fbllows. Consider again the n-cube I. The initial (n 1)-face of In defined by 0 will be identified with In -1 hereafter. The union of all remaining (n I)-faces of in is denoted by J'. Then we have In-1 n jn-l. In-1 U In-1 , aIn-1 oin By a map : (In Jn1, jn-r,) ---> (X, A, x0), we mean a continuous function from In to X which carries In-1 into A and Jn- 1 into xo . In particular, it sends ain into A and atn-1- into xo . We denote by Fn Fn(X, A, xo) the set of all such maps. These maps are divided into homotopy classes (relative to the system { In -1 , A; Jn-1 , zo } ), We shall denote by nn (X, A, xo) the totality of these homotopy classes. We shall also denote by [1] the class which contains the map / and by 0 the class which contains the constant map d0 (I) = xo. Topologize Fn by means of the compact-open topology; then n„(X, A, x 0) becomes the set of all path-components of the space F. If n> 1, we may define an addition (usually non-commutative) in F. For any two maps /, g in Fn, their sum f + g e Fn is defined by the formula given in § 2 for the absolute case. The homotopy class [f g] depends only on the classes [f] and [g] and hence we may define an addition in gr,,(A ,X ,x 0) by taking [g] = [/ + g]. As in § 2, one can verify that this addition makes nn (X, A, x0) a group which will be called the n-th relative homotopy group of X modulo A at xo . The class 0 is the group-theoretic neutral element of nn (X, A, x0), and the inverse element of [/ ] is-the class f/0], where O: In In denotes the map defined in § 2. If xo is the only point of A, then we have Fn(X , A, x0) = Fn(X, xo). Hence, in this case, 74,(X , A, xo) reduces to the absolute homotopy group x0) defined in § 2. If j71 -1 is pinched to a point so , (In, In -', J12-1) becomes a con figuration topologically equivalent to the triplet (En, Sn -1 , so) consisting of the unit n-cell En , its boundary (n 1)-sphere Sn ---1 , and a reference point so e It follows that one might equally well define an element of 7En (X, A , x0) as a 4; so , x0 ) of the maps of homotopy class (relative to the system { (En, 5n-1 , so) into (X, A, x0). Since, when n > 2, there exists a rotation of

3. RELATIVE HOIVIOTOPY GROUPS

III

En which leaves so fixed and interchanges the two halves of En, we see that nn,(X, A, x0) is abelian for every n > 2. This commutativity property is also an immediate consequence of the next proposition. n2(X , A , xo) is in general non-abelian. Next, let us introduce the notion of the derived triplet of a given triplet which will be frequently used in the sequel. Let

T ---- (X, A, x0)

be a given triplet. Consider the space of paths X' ---- [X ; X, xo] and the initial projection

p : x' --> x

as defined in (III; § 10). Let

A' ----.-- p--1(A) = [x ; A, xo]c X' and denote by x'0 E A' the degenerate loop x'0 (/) = xo. Thus we obtain a triplet T' = (X', A', x'0),

called the derived triplet of T, and a map

p:

(X', A', x'0 ) -÷ (X, A, x0),

called the derived protection over (X, A, x0). Proposition 3.1.

For every n> 0, we have A, x0)

—

nn 1 (A

',

x '0),

Proof. If n = 1, this is obvious since each side can be considered as the set of all path-components of A'. Assume n> I. Since Kin = (2CI) in-1 by (III; 9.9), it follows that Fn(X, A, x0) ---=-. Fn-4-(A' , x'0).

Hence nn(X , A, x0) andm 1(A', x'0) coincide set-theoretically. It is also clear that the group structures are the same for any n> 1. I Another consequence of (3.1) is that every relative homotopy group can be expressed as an absolute homotopy group and hence as a fundamental group.

Since din is pathwise connected for n> 1, the following proposition is obvious. Proposition 3.2. 11 X 0 denotes the path-component o/ X containing xo

and A o that of A, then A, xo) = nn(X 0, A 0, xo), n> 1.

Finally, the following proposition will be used in the sequel. If a Enn(X, A, x0) is represented by a map / e Fn(X, A, x0) such that /(In) c A, then a. — 0. Proposition 3.3.

112

IV. HOMOTOPY GROUPS

Proof. Since f E Fn(X, A, xo) and f(In) a A, we may define a homotopy ft E Fn (X, A, x0),0 - (X, A, x0)

for oc. Applying (II; Ex. C), we can easily prove that we may free eo from the image of f by means of a suitable homotopy of / relative to { 1 n -1 , A ; In", x0 }. Since A is a strong deformation retract of X \ eo , this implies that there exists a homotopy f t : (1m, Im --1, Pm -l) -)- (X, A, x 0),. (0 . (X, A, x0). If n = 1, f(I 1 ) is a point of A which determines a path-component fi en,_ 1 (A, x0) of A. If n > 1, then the restriction f 1 In-1 is a map of (In-1, all") into (A, xo) and hence represents an element fl E n,,,_..1 (A, x0). - Obviously, the element i5 Enn _ I (A , x0) does not depend on the choice of the map / which represents the given element a enn (X, A, x0). Hence we may defi ne the transformation 0 by setting ô(oc) = fi. Hereafter, 0 will be called the boundary operator. The following two properties of d are obvious from the definition. The boundary operator nn (X, A, x 0) into that of 7Ln ... 1 (A , x 0) . Proposition 4.1.

Proposition 4.2.11n

a

sends the neutral element of

> 1, then the boundary operator 0 is a hamomorphism.

5,

INDUCED TRANSFORMATIONS

113

5. Induced transformations Let (X, A, x0) and (Y, B, y o) be given triplets. By a map of (X, A, xo) into (Y, B, yo), we mean a continuous function X to Y which carries A into B and xo into yo. Consider such a map f: (X, A, x0) -* (Y, B,Yo)• Since / is continuous, it sends the path-components of X into those of Y. Hence, / determines an induced transformation

i* : 760( X, x0) -+N(Y, Yo) which obviously sends the neutral element of 0(X, x0) into that of zo(Y, yo). Now let n > O. For any map sbeFlt(X, A, xo), the composition /0 is in Fn(Y, B, yo) and the assignment 5b --> to defines a map Fn(X, A, xo) -÷ Fn(Y , B, yo). The continuity of /4# implies that 1# carries the path-components of Fn(X, A, xo) into those of Fn(Y, B, yo). Hence it determines an induced transformation /* : an,(X , A, x0) -->yrn (Y , B, Yo) which obviously sends the neutral element of nn(X, A, x0) into that of an( Y, B, yo)If n=.-- 1, A = xo, B = yo, or if n > 1, then nn (X, A , xo) and nn ( Y, B, Yo) are groups. For any two maps 95, v in Fn(X, A, x0), one can easily see that in Fn(Y , B, yo). Hence it follows that 1* is a homomorphism. Thus we have established the following two properties of 1* .

n ,--.-- 0, A = x o, B = yo, or if n> 0, then the induced transformation f * sends the neutral element of 2r,„(X, A, x0) into that of an (Y , B, y o). Proposition 5.1. If

If n = 1,A = zo, B = yo, or if n> 1, then the induced transformation f* is a homomorphism. Proposition 5.2.

In the case of (5.2), f * will be called the induced homomorphism. Now, consider the derived triplets (X', A', z' 0), (Y', B', y' 0) of the given triplets together with the derived projections

p : (X', A', x'0) --)- (X, A, x0), r: (Y', B', y 10) --).

(Y, B, yo).

The given map f: (X, A, x 0) --)- (Y, B, yo) considered as a map from X into Y induces a map

f : 2C/ ---› V

yo, it follows that f according to (III; § 14). Since f (A) c B and carries X' into Y', A' into B', and x'0 into y'o. Hence /I defines a map /' : (K', A', x' 0) -÷ (Y', B', y'0)

IV. HOMOTOPY GROUPS

114

fp and which will be called the derived map which satisfies the relation rf of f. Let y /' (A', x'o)• Then induces : z n _ i(A

x'0)

y'0).

Under the identifications in (3.1) it is obvious that 1* Finally, the boundary operator O in § 4 is essentially a special case of induced transformations. In fact, for any triplet (X, A, x0), consider the pair (A', x') of the derived triplet and the restriction q = p I (A', x'0) of the derived projection p. Then q induces

....1 (A' ,z'0)

q* :

zn-i (A '_co).

Under the identification in (3.1), it is obvious that

a = q* .

6. The algebraic properties

a,

and the The homotopy groups nn (X, A, x0), the boundary operator induced transformations defined in the previous sections possess seven fundamental properties which will be given in this and the next few sections. By the definition of induced transformations, the following two properties are obvious.

If f: (X, A, x0) (X, A, x0) is the identity map, then /,, is the identity transformation on nn (X, A, xo) for every n. Property I.

Property II.

I/ f : (X, A, xo) -->- (Y, B, yo) and g: (Y, B, yo)

(Z, C, zo) are

maps, then for every n >0 we have (gf) * = Hence, for any given n, the assignment (X, A, xo) -->g(X, A, xo) and -.> f* defines a covariant functor. The next property gives a relation between the boundary operator and the induced transformations. It is an obvious consequence of their definitions. I/ f: (X, A, xo) (Y B, yo) is a map and if g : (A, x0) -÷ (B, yo) is the restriction of f, then the commulativity relation Of *= g*0 holds in the following rectangle for every n> : Property III.

nn (X, A, xo)

a -±

if*

2-cnCY B,

,

a

zo) 1g* 4, 21n-1(B yo)

Note that Property III is also an easy consequence of Property II. To see this, let us consider the derived map

f' : ( )( ' , A' , x' 0)

(Y', B', y'0)

of f as well as the derived projections

p:

, A', x' 0)

(X, A, x0), r: (Y', B', y'0)

(Y, B, y0).

7.

THE EXACTNESS PROPERTY

115

Let = /' I (A', x'0), q =p I (A', x' 0) and s = Y I (if, y'0). After the various identifications described in (3.1) and § 5, the preceding rectangle reduces to the following form : x'0) q * grn _1 (A, xo) f*

g*

s* -÷ an-1(-131 ,y ro) Yo) Since /' satisfies the relation fp, we have sf = gq. Hence, Property II implies that the commutativity relation s*-/* = g*q* holds in this rectangle.

7. The exactness property Let (X, A, xo) be any given triplet. The inclusion maps

: (A, x 0)

(X, x o),

: (X, x0)

(X, A, xo)

induce transformations i* and f* for each n > O. Together with the boundary operators 0 , they form a beginningless sequence: , A, (X. x o) 1L+ 2, nn (A, x o) •n i (X , , X 0) _a_>. no(A , x 0 ) --> no(X xo)

vx.+1 (X, A, x 0) • •

.

x 0) .J, t •••

which will be called the h,omotopy sequence of the triplet (X, A, x0) and will be denoted by n(X,A,x o). Every set in (X, A, x 0) has a specified element called its neutral element and every transformation in n(X, A, x o) carries the neutral element into the neutral element. We define the kernel of a transformation in 7( (X, A, x 0) to be the inverse image of the neutral element. Such a sequence is said to be exact if the kernel of each transformation coincides exactly with the image of the preceding transformation. Property IV. The homotopy sequence of any triplet (X, A, x 0) is exact.

The proof breaks up into the proofs of the following six statements: (I) f *i* = 0, (2) 0i* = 0, (3) i*a = 0. (4) If Ot G nn,(X, x 0) and f * cc = 0, then there exists an element /3 G 741 (A , x0) such that 413 = oc. (5) If cc E 7c.(X, A, xo) and 0at = 0, then there exists an element 18 G nii(X,x0) such that j*I3 = oc. (6) If oc G nn _i (A , x o) and i*cc — 0, then there exists an element 13 Enn (X, A, xo) such that 013 — at. In the preceding statements, the symbol 0 denotes either the neutral element of the set involved or the transformation which sends every element into the neutral element. Proof of (1). For each n> 0, let a e 7rn (A, x o) and choose a map (A, xo) which represents cc. Then the element f *i* (oc) in nn (X, A, xo) is represented by the composition fit E Fn (X, A, x0). Since obviously ii/(//t) c A,

it follows from (3.3) that f *i*cc = O. Since a is arbitrary, this implies j*i* = O. I

IV. IIOMOTOPY GROUPS

116

Proof of (2). For each n> 0, let a ea- (X, x0) and choose a map / e Fn(X, xo) which represents a. Then the element Of*a is determined by the restriction O. Hence Of *= O. I jf f I In -1 • Since f = xo, we have 0f cc Proof of (3). For each n> 0, let a Enn (X, A, xo) and choose a map E Tn(X, A, x0) which represents a. Then the element Oa is determined Define a homotopy gt : -->- X, O < t < 1, by the restriction g = f by setting gt (t1,• - • , tn _l) = Ri p - • • , tn _i , t).

xo, and gt c Fn-1 (X, x0) if n> I. This implies Then go = g, g1 (In -1 ) i*Occ = O. Hence i*0 = O. I Proof of (4). Choose a map f E Fn(X, x0) which represents a. The condition fcc 0 implies that there exists a homotopy /t : In --> X, 0 < t < 1, such x o , and f t GFn(X, A, x0) for each t G I. Define a homothat /0 = /, f i (In) topy gt : In -->X,0 A , O < t < 1, such that go = fi fn -i , g 1 (In -1) xo, and xo for every te I. Define a partial homotopy ht :01-n 4 A, gt(din -1 ) O< t < 1, by setting (s c In-1, t E I), gt (s), ht(s) = te G I x 0) -

-

-

Since ho = f j 0/n, it follows from the homotopy extension property that the homotopy ht has an extension ft : In > X, 0 < t < 1, such that /0 /. Since /i (din) b 1 (0I) = xo, fi represents an element 18 in 7rn (X, x0). Since ft e Fn(X, A, x0) for t El , it follows that j*,e .,__ cc. For the remaining case n = 1, oc is represented by a path f: I -->- X such that f (0) EA and f (1) = xo. The condition Oa = 0 means that f(0) is contained in the same path-component of A as xo. Hence there exists a homotopy ft : I ÷ X, 0 < 1 < 1, such that /0 = /, /t(0) E A, ft(1) = xo , and /1 (0) = xo. Then, /1 represents an element 18 G7C 1 (X, z o) and the homotopy ft implies that /*# =- cc. 1 -

-

Proof of (6). First, assume that n > 1. Choose a map f e Fn 1 (A, x0) which represents cc. Then the condition i * a implies that there exists a homotopy ft : -->- X, O < t < 1, such that fo = /, fi (in -1) = zo and it(O/n -1-) = xo for every t e I. Define a map g: In ÷ X by taking -

-

g(t p . • • , tn-i, in) = ft,t(ti, " t1 -1).

8.

THE HOMOTOPY PROPERTY

117

This map g is in Fn(X, A, x 0) and represents an element a of grn (X, A, x0). Since g 1 In -1 = f, we have 013 — a.. For the remaining case n = 1, cc is a path-component of A. The condition i* cc = 0 means that cc is contained in the path-component of X which contains xo. Pick a point x from cc. Then there exists a path /: I -÷ X such that 1(0) — x and f(l) — xo . This path / represents an element 18 of gri (X, A, x0). Since f(0) e a, we have op cc. I .

8. The homotopy property

Consider any two given triplets (X, A, xo) and (Y, B, yo), and any two given maps /, g: (X, A, x 0)-->-(Y , B, y o). We recall that / and g are said to be homotopic (relative to { A, B ; xo, yo } ) if there exists a homotopy ht:(X,A,x 0) --> (Y,B, y o), 0 nn(Y , B , 3 1 0)

are equal for every n. Proof. Let cc Egr„(X, A, x0). It suffices to prove that / * a =-- g*cx. First, assume that n> O. Choose a map 0 e Fn(X, A, x 0) which represents oc. Then the elements foc and g*.oc are represented by the compositions /0 and g o respectively. The composition ht0 of 0 and a homotopy ht : /..r--.' .--. g gives a homotopy connecting 4 and gy5. Hence f * cc = g* cx. It remains to settle the case n . 0, A = xo, and B — yo. Here a is a pathcomponent of X. Pick a point x e a. Then /* a and g* a are the path-components of Y containing the points /(x) and g(x) respectively. Let ht : f L-__-, g. Define a path a : I -* Y by taking a(t) = ht (x) for each t e I. Since a joins /(x) to g(x), it follows that /*oc — g*oc. I We recall that a map f: (X, A, x 0) -->- (Y, B, y o) is said to be a homotopy equivalence if there exists a map g : (Y, B, y o) ---> (X, A, xo) such that gf and fg are homotopic with the identity maps on the triplets (X, A, xo) and (Y, B, yo) respectively. As an immediate consequence of the properties I, II and V, we have the following B. yo) is a homotopy equivalence, then the induced transformation f * sends nn (X , A, x 0) onto nn,(Y , B, yo) in a oneto-one fashion. Corollary 8.1. // f : (X, A, x 0) ---> (Y,

The significance of (8.1) is that nn (X , A, xo) depends only on the homotopy type of (X, A, x0). In particular, if A is a strong deformation retract of X, then i* sends A, x0) onto an(X, x0) in a one-to-one fashion for every n .,.> O. Hence, by Property IV, this implies that nn (X,A,x 0) = 0 for every n > O.

IV, HOMOTOPY GROUPS

118

9. The fibering property

Consider two given triplets (X, A, x0) and (Y, B, yo) and a given map f: (X, A, x0) --> (Y , B, yo). We are concerned with the notion of fibering as defined in (III; § 3). Property VI. If f

formation

: X --> Y is a fibering and A = /'(B), then the trans14, :nn (X, A, x o) ->nn(Y,B,Yo)

sends 7c.(X , A, x 0) onto an (Y , B, y 0) in a one-to-one fashion for every n > 0. Proof. To

prove that /* is onto, let a be an arbitrary element of arn.(Y, B, yo). By § 3, a is represented by a map 0 : (In , in-1 , in-1) _÷. (1 7 . , B, y0). Since j n-1 is a strong deformation retract of In, it follows from (v) of (III; 3.1) that there exists a map y : In ---> X such that / y . 0 and y(jn -i) . xo. Since A = f -4 (B), fy = 95 implies that y(/27-4) c A and hence we obtain a map y : (In, In- ', in-') -->- (X, A, x 0).

This map y represents an element 13 of an,(X , A, x0). Since ly ----. 0, we have /43 = a. This proves that 1* is onto. To prove that /4, is one-to-one, let a and 13 be elements of 7c„(X , A, x0) such that /cc = /O. Choose representative maps 0, y : (In, In -1-, Jn -1-) ---> (X, A, x0)

for .a and 16 respectively. Since /*a — /443, the composed maps /0 and fy represent the same element of nn,(Y, B, yo). Hence, there exists a map F: (In x I, In-I- x I, in-- 1- x I) ---> (Y, B, yo)

such that F(z, 0) = /0(z) and F(z, I) = fy(z) for each z e In . Consider the closed subspace " T = (In X 0) U (j n-1- X I) U (In X 1)

of In x / and define a map G : T ---› X by setting G(z, I) — (zo, V( 2),

(z e /91, t = 0), (z G In -1, t e I), (ze In, t--.-- 1).

Then we have fG =FIT. Since T is clearly a strong deformation retract of In X I, it follows from (y) of (III; 3.1) that G has an extension G* : In x I --› X such that /G* = F. Since F maps In-1 X I into B and A = f -1 (B), the condition /G* = F implies that G*(In -l- X I) c A. Hence we obtain a map G*: (In X I, In -' X I, Jn -1- X I) --> (X, A, x0)

II. HOMOTOPY SYSTEMS

119

with G*(x , 0) = 96(z) and G*(z , 1) — ip(z) for each z E I. This proves that 56 and ip represent the same element of an (X , A, x0). Hence oc ---- fl and 1* is one-to-one. I In homology theory, the corresponding fibering property is false in general. Instead of this, we have the famous excision property which does not hold in homotopy theory. As a special case, let us consider the derived projection 5 : (X', A', x' 0) ---> (X, A, x 0). By Properties VI and IV, p* and 0 in the following diagram A, x o) .,__P' __L'

(X' , A', x' 0) 1 _,.

7Ln _ 1 (A t ,

x'0)

are both one-to-one and onto. Furthermore, if (X , A, x 0) and mi _ 1 (A ', x'0) are identified as in (3.1) one can easily see that p * = 6. On the other hand, the identification in (3.1) may be considered as being effected by the oneto-one correspondence Z --- ON' : 7c,,(X, A, x 0) --)- 7c,2,_ 1 (A' , xto) which will be called the natural corresPondence. 10. The triviality property

If X is a space which consists of a single point xo, then, for each n„ the constant map f (./91) — xo is the only map of in into X. Hence we have the following Property VII. //X is a space consisting of a single oint x0, then 7(11(X , ) C 0) =- 0 for every n > O.

This property plays the role similar to that of the dimension property in homology theory. Since it apparently has nothing to do with the choice of the dimension n in vtn (X, x 0), we propose to call it the triviality property. 11. Homotopy systems

In the preceding sections, we constructed geometrically the homotopy groups nn (X , A, xo) and established seven basic properties of these groups. In next few sections, we shall show that they are characteristic; in fact, these seven properties, stated in a certain apparently weaker form, together with 0 (X, x 0) for all pairs (X, x 0) determines all nn (X,A, xo) for all triplets (X, A, x 0) up to one-to-one correspondence. A homotopy system H — { vr, a, * } consists of three functions 7c, a and *. The first function 7( assigns to each X, A, x 0). The triplet (X, A, x0) and each integer n > 0 an abstract set second function 0 assigns to each triplet (X, A, x 0) and each integer n > 0 a transformation

a

:

747,(X ,

A, x 0) ---> 741 _1(A ,

120

IV. HOMOTOPY GROUPS

where, in the case of n 1, 7c0(A, x0) denotes the set of all path-components of A as in § 2. The third function * assigns to each map : (X, A, x0) --)(Y, B, yo) and each integer n> 0 a transformation f* : 7c(X, A, xo)

B, yo).

Furthermore, the system H must satisfy the following seven axioms: Axiom I. p f: (X, A ) xo)

(X, A, x0) is the identity map, then f, is the identity transformation on nn (X, A, x 0) for every n > O.

Axiom II. If f: (X, A, x0) -÷ (Y, B, y 0) and g: (Y,B,y 0)

2.0) are

maps, then for every n > 0 we have (gf) *

Axiom III. if f : (X, A, x0) --> (Y B, yo) is a map and g: (A, x0)

(B, yo)

is the restriction of f, then the commutativity relation Of* = g*0 holds in the following rectangle for every n > 0:

n(X , A , xo)_ 19 7r,,, _1 (A , x o) 1*

g*

Y

7(91(1( B, yo)

where, in the case of n

zn-1(B, 1/0) 1, g* :7(0 (A, x0) -->no (B, yo) denotes the induced

transformation in § 5. Let (X, A, x0) be any given triplet and consider the inclusion maps

i (A, x0)= (X, x0), j : (X, x0)= (X, A, x0). The transformations i* , j* and ô form a beginningless sequence as in § 7 which will be called the homotopy sequence of the triplet (X, A, xo) in the system H.

Axiom IV. The homotopy sequence of any triplet (X, A, x0) is weakly exact. This means that, if nn,(X , x 0) -= 0 for all n > 0, then a sends nn,(X , A, x0) onto nn _i(A , xo) in a one-to-one fashion for every n> O. Axiom V. If the maps 1,g: (X, x0) -> (Y, yo) are homotopic, then 1* for every n > O.

g*

Axiom VI. If p : (X', A', x' 0)

(X, A, x0) is the derived projection over (X, A, x0), then p * sends 7c1,(X' , A', z' 0) onto 7c,,(X , A, x 0) in a one-to-one fashion for every n > O.

Axiom VII. If X is a space consisting of a single point xo, thenn„(X, x 0) for every n> O. Since the derived projection : X' -± X is a fibering by (III; 13.4), it follows that the axioms I-VII are weaker than the properties respectively. Hence, if we neglect the group operation in nn,(X , A, xo), the three functions a, a, * as defined in §§ 3-5 constitute a homotopy system. This proves the

12. THE UNIQUENESS THEOREM

121

existence of homotopy systems. One can also easily construct a homotopy system by induction and without using any geometrical representation, (see Ex. A at the end of the chapter).

12. The uniqueness theorem { n' , O', H } are said Two homotopy systems II ={ n, 0, * } and H' to be equivalent if there exists, for each triplet (X, A, x0) and each n > 0, a transformation lin, : (X, A, x 0) --->n'n (X, A, x 0) —

satisfying the conditions: (El) h. sends nn (X, A, x0) onto n'n (X, A, xo) in a one-to-one fashion. (E2) For each triplet (X, A, x0) and each n > 0 the commutativity relation hn _ i O = O'hn holds in the following rectangle:

nn (X , A, x0)

nn _1 (A, x 0)

4. 7C ' 7,(X,

A, x 0)

2r t°'n _1 (A X 0) ,

where, in the case of n = 1, ho denotes the identity map. (E3) For each map f: (X, A, x 0) ---> (Y, B, yo), the commutativity relation flth n holds in the following rectangle: lin t*

A, x 0) f* nn (Y , B, yo) hn Y

n'.(X, A, x0) f#

hn

4. „(Y, B, y o).

A collection of transformations h ,={ hn } satisfying the conditions (El) through (E3) is called an equivalence between the homotopy systems H and H' and is denoted by h H H. Theorem 12.1.

Any two homotopy systems are equivalent. [Milnor 1].

H n, a, * } and H' ={n', O' } be any two homotopy systems. We are going to construct an equivalence h : H H' as follows. Let n > 1 and assume that we have already constructed the transformations Jim : n.,(X , A, x0) --> ilm,(X, A, x 0) Proof. Let

for each m < n and each triplet (X, A, x0) such that the conditions (El) through (E3) are satisfied. Let us construct hn as follows. Let (X, A, x0) be any triplet. Consider the derived projection : (X' ,A' , 0) ---> (X, A, x 0). By Axiom VI, 75 * sends nn (X' , A', x'0) onto n.(X , A, x 0) in a one-to-one fashion and analogously for p# , According to (III ; 10.1), X' is contractible to the point x'0. Hence, by Axioms I, II, V and VII, we obtain

IV. HOMOTOPY GROUPS

122

x'o) = 0 for every m > 0. By Axiom IV, a 7rm (X' , 0) = 0 and sends grn (X' , A,' x' o) onto an _ 1 (A', x'0) in a one-to-one fashion, and analogously for a'. By our assumption of induction, hn _1 sends nr,... 1 (A', x'0) onto n'n _i (A' , x '0) in a one-to-one fashion. Hence we may define a transformation hn :(X, A, x 0) ---> n (X , A, xo) by taking

hn =

where, in the case of n = 1, ho denotes the identity map. Since (El) is obviously satisfied, it remains to verify (E2) and (E3). To check (E2), let q = p I (A', x'0). Then we have

d'hn =

= q#0'0'-ihn _1015,71-

qlthn _lap

hn _id

by Axiom III. This proves that hn satisfies (E2). To check (E3),. let (Y ,B, yo) be a second triplet and f: (X, A, x o) (Y, B, y o) be any given map. Let Y:

(Y', B' , y'0)

(Y B, yo),

/' (X' , A' , 0)

(Y' ,

, 31' 0)

denote respectively the derived projection over (Y, B, y o) and the derived map of f. The relation fp IT is satisfied. Set f = f' I (A', x'0). Then we have

fithn

= Y 11 ' -1

y o,

111hn-laK1

l ap

rj

-i hn--1.40;1

l ay;v * hn t*

by Axioms II and III. This proves that hn satisfies (E3). Thus we have., completed the inductive construction of an equivalence {h n } between H and H' which will be called the natural equivalence

h : H c H' The natural equivalence is the only possible equivalence between H and H' as will be seen in Ex. B at the end of the chapter. The uniqueness theorem (12A) shows that the homotopy system constructed geometrically in §§ 3-5 is essentially the only homotopy system. As a consequence of this, it follows that the set of seven properties in §§ 6-10 is equivalent to the set of seven axioms in § 11 which are apparently weaker. In fact, one can deduce the seven properties right from the axioms without using any geometrical representation of the sets nn (X, A, x0). The details of the proof will be left to the reader as an exercise. See Ex. C at the end of the chapter.

13. THE GROUP STRUCTURES

123

13. The group structures In the last two sections, we have shown that, apart from the group structures in the homotopy sets 2-4,(X, A, x 0), the homotopy system constructed in §§ 3-5 is completely characterized by the seven axioms in § 11 and the sets n0 (X, x0). To complete the axiomatic approach, it remains to determine all the possible group structures that can be introduced into the essentially unique homotopy system I/ For this purpose, let us first consider the sets n i (X, x0) in H. According to the uniqueness theorem (12.1), we may assume n1 (X, x 0) to be the underlying set of the fundamental group of X at xo . The product in ni (X, x 0) as defined in (H; § 5) will be called the customary product which is denoted by juxtaposition. The reverse of this product, denoted by a dot, is defined by

floc, (x , Eni.(X, xo)). For any map f: (X, x 0) (Y, yo), the transformation f* : v i (X , x 0) n i (Y, yo) is a homomorphism under the customary product as well as its reverse. These are the only group structures in all 7c 1 (X, x 0) such that f* is a homomorphism for every map f. In fact, we have the following There are exactly two ways of introducing a group structure into the sets n i (X, x 0) in such a way that f * is a homomorphism for every map f: (X, x 0) -± (Y, yo). These two group structures are defined by the customary product and its reverse respectively. Lemma 13.1.

Proof. Assume that there is a new product, denoted by a o j9, in each n 1 (X, x 0) such that, for each pair (X, x 0), n l (X, x 0) is a group under this product and that f* is a homomorphism with respect to this product for every map f: (X, x 0) -÷ (Y, yo). It suffices to prove that either oc o oci8 for any a, 13 En l (X, x0) of every (X, x 0) or a o = pa similarly. Let Z be the space which consists of two circles, intersecting at a single point zo. According to (II; Ex. A5), z 1 (Z, zo) is a free group on two generators a and b. For any two elements oc, /3 of n 1 (X, x 0), obviously there is a map f: (Z, zo) —> (X, xo) such that f(a) = oc and f * (b) . Since f* is a homomorphism under the new product, we have f * (a o b) o p. In terms of the custom* group structure of n 1 (Z, z 0), a o b is equal to some word w(a, b) of the free group. Since 1* is also a homomorphism under the customary product, we have f * (a o b) = f* [w (a , b)] = w[/ (a), f* (b)] = w(ct, f?). This implies that a o fl = w(a., (3). Thus it remains to prove that either w(a,b) = ab or w(a,b) = ha.

IV. HOMOTOPY GROUPS

124

For this purpose, let us first prove that the word w(a, b) has the following two propeties : w(a, 1) =--- a, w(1, b) = b, (1)

w [a, w(b, c)] = w [w(a , b), cj,

(2)

where (2) is an identity in the free group on three generators a, b, c. To prove (1), note that the identity element 1 of a 1(Z, zo) can be defined as the image of the homomorphism

a 1 (zo, z0) - . 1 (Z, 2'0) induced by the inclusion map 1: (zo, zo) (Z, zo). It follows that the new product must have this same identity element. Hence w(a, 1) =aol =a, w(1,b) =lob—b.

To prove (2), choose X to be the space which consists of three circles tangent to each other at the same point x0 . Then, as in (II; Ex. A5), 2r1(X, .X0) is a free group on three generators a, b, c. Since a o b = w(a,b) and b o c = w(b , c), the associative law for the new product implies (2). Finally, we shall complete the proof by showing that, if a reduced word, w (a , b) in the free group on two generators satisfies the conditions (1) and (2), then either w (a, b) = ab or w (a, b) -- ba. The proof is a long but easy exercise in the manipulation of reduced words sketched as follows. Let w(a, b) be a reduced word which satisfies (1) and (2). By (1), w(a, b) atm. More generally, it is impossible to have w(a, b) = amibni- • • amkbnkamk +1

with non-zero (positive or negative) integers m 1 , n i, • • • , Ink, nk, and mk+1. To see this, let us assume that w(a,b) were of this form with k > 1. Then, by (1), one can easily see that the reduced words of (w (a , b) )nti and (w (b , c) )711 would be of the same form, say, (w(a, b))mi = aPibqi. • • aP ibq ia2) i +1, (w (b , c)

)ni =

bricsi • • • bricsibri-1-1,

with non-zero integral exponents. By (I), i > 1 and i > L Then we would have w[a, w(b, cil = amibrlei • • • , w [w(a, b), c] --= aP 1 big]. aP2 • • • as their reduced words. This contradicts (2). Similarly, w(a, b) bm and it is impossible to have w (a , b) = bmiani• • • bmkankbmk +1

with non zero integral exponents. Next, assume that w(a, b) = amibni• - • amkbnk -

14. THE ROLE OF THE BASIC POINT

125

with non-zero integral exponents. If m i O. If n i 0 and ni > O. Then we have w[a, w(b, c)]

w[w(a,b), c]

amibmicni- • • ,

- • amkbnk- • •

1 and hence w(a, b) as their reduced words. By (2), it follows that k amibni. Then, (1) implies that m i = 1 and n i ----- 1. Consequently, we have w(a, b) = ab. Similarly, we can prove that, if w(a, b) = bmiani• • • bmkank ba • This exhausts all

with non-zero integral exponents, then w(a , b) possible cases. I

There are exactly tWo ways of introducing a group structure into the sets n„(X, A, x 0), n >2, and n i (X , x 0), in such a way that the transformations O and f * are homomorphisms. These two group structures are defined respectively by the customary group operation given in § 3 and its reverse. [Milnor 1]. Theorem 13.2.

Proof. Let us denote the customary group operation by juxtaposition and assume that there is a new group operation, denoted by cc o 13, in the sets 7cn (X, A, z 0), n > 2, and 1 (X, x0) such that the transformations O and f * are homomorphisms. We have to show that cc o is equal to cc/3 or 13a. By (13.1), this is true if a and 13 are in 1 (X, x0). To prove the theorem by induction on n, consider the natural one-to-one correspondence X = Op;' : nn (X , A, x 0) -->nn _1 (A' ,

0)

of § 9. Since X is a homomorphism for both group operations, we have X(ct o 13)

X((x) o X (f3 ),

Z(ccfl)

Z(Gc) ,C(M

for any two elements cc, /3 in z.(X, A, x 0). By the inductive hypothesis, X(cc) o X(13) is equal to X(x) X(p) or X(p)X(a). Hence X(oc o /3) is equal to X(ocp) or X(/3x). Since X is one-to-one this implies a o i3 = cc/3 or 13a. 1 The significance of (13.2) is that the group structure in the essentially 7( , O , * } is also essentially unique. unique homotopy system H This completes the axiomatic approach.

14. The role of the basic point In the notion of the homotopy groupsn n (X, x0) and 7c7,(X , A, x 0) the basic point xo is explicity used in any geometrical construction of these groups. The objective of the next few sections is to study the role played by the basic

126

IV. HOMOTOPY GROUPS

point, to compare the homotopy groups with various basic points, and to free these groups from the basic point wherever it is possible. Let us consider a given space X and two given points xo , x1 connected by a given path cr : / -÷ X, o(0) . xo, a(1) . x l . By the definition in § 2, we have no (X, xo) — P(X) = 7( 0 (X, x1)

as the set of all path-components of X. Moreover, since xo and x 1 are contained in the same path-component of X, the neutral element of 7( 0 (X, x 0) is the same as that of 7( ca , x 1 ) . Let us denote by ao : no (X,x 1) --x 0 (X, x 0) the identity map on 7c0(X, x 1) = 7c0 (X, x0). Theorem 14.1. For each n > 0, every path a : I -›- X gives in a natural

way an isomorphism

cr. :n.(X, x 1) ,..-d, 74,(X, x 0), x o — a(0),

x1

which depends only on the homotopy class of the path ci (relative to end points). If a is the degenerate path a(I) ---- xo, then 0.7, is the identity automorphism. If a, T are paths with T(0) = a(1), then (ar). = anTn . Finally, for each path o: I --> X and each map f : X -÷ Y, we have a commutative rectangle nn (X, x l)

a'_>. 7cn (X, x 0)

f*

f*

Y

nn ( Y, y 1)

Tn

—> grn(17,

yo),

whére r = fa, yo — f (x 0) and y i = /(x 1). As an immediate consequence of (14A), we deduce the following Corollary 14.2. The fundamental group n l (X, x 0) acts on 7c.(X, x 0), n, >1,

as a group of automorphisms.

To prove (14.1), let us construct a. as follows. Let a be any element of an (X, x i) and choose a representative map f: (In , a PI) --> (X , x l) for a. The geometrical idea of the construction is to pull the image of DIn along the path ci back to the point xo with the image of In being dragged in an arbitrary way. The map obtained after this homotopy represents an element i6 of 7c.(X, x0) which depends only on a and. the homotopy class of a. Then, we define an (a) — 16. The details are as follows. First, let us prove that there exists a homotopy ft : In ---> X, (0 < t < 1), of f such that ft(I) = a(1 — t) for every t E f. For this purpose, define a partial homotopy ckt : 01-7/ -÷ X, (0 < t < I), of f by taking Ot (01-3i) = a(1 —t)

14. THE ROLE OF THE BASIC POINT

127

for each t e I. By (I; 9.2), 0/n has the AHEP in I. Hence the homotopyçbt has an extension ft : In -9- X, (0 (X, x 1) are maps homotopic relative to a, x: I -›- X are paths homotopic relative to end points, and It, gt: In -4- X, Gx0.

IV.

130

HOMOTOPY GROUPS

(LSG3) If a is the degenerate path o-(/) xo , then a4 is the identity automorphism on Gxo . (LSG4) If two paths cr,r:I-->X are equivalent, i.e., if a, T have the same end-points and are homotopic with end-points held fixed, then * II • (LSG5) If two paths a, : I -÷ X are consecutive, i.e., if a(1) = then (at') n = 0474. According to (14.1), the collection of the homotopy groups { nn (X, xo) X0 E X ), for a given space X and a given integer n> 0, forms a local system of groups in X. Similarly, the collection of the relative homotopy groups {n n(X, A, x 0) I x o e A }, n> 1, forms a local system of groups in the subspace A of X. As an easy consequence of (LSG3)—(LSG5), we deduce as in the proof of (14.1) that every all is an isomorphism. Hence, if X is pathwise connected, then all the groups Gx , x e X, are isomorphic. Since the elements of the fundamental group 1 (X, xo) are homotopy classes of the loops [X ; x o , x o] with end-points held fixed, we deduce as a direct consequence of (LSG3)—(LSG5) that, for each x o E X, „(X, xo) acts as a group of operators (or automorphisms) on GA) in the sense defined as follows. A multiplicative group H is said to act as a group of operators on an additive group G, (or, simply H acts on G), if, for every h e H and every g E G, an element hg E G is defined in such a way that 45

h(g„

g 2) = hg,

hg2 , h 2 (kg)

(h 2111)g, lg = g

where g, gl, g 2 c G, h, h1 , h 2 e H are arbitrary elements and I E H denotes the neutral element. Applying this to the local system of groups (n.(X, x 0) I x o E X }, where n > 0, we obtain (14.2) restated as follows. For each xo e X, the fundamental group n i (X, x o) acts on the n-th homotopy grout , n7,(X, x 0) as a group of operators. Proposition 15.1.

In the special 'case n I, one can easily see that, for any two elements g and h in n i (X, x0), h acts on g as follows :

(15.2)

h(g) = hgh--1.

Similarly, for each x o E A, 7r 1 (A, x 0) acts on the n-th relative homotopy group n7,(X, A, x 0), n> 1, as a group of operators. As a consequence of these operations, let us consider two given homotopic maps f, g : X Y. Let ht : X --> Y, (0 < t < 1), be a homotopy such that ho f and h l = g. Choose a point Xo G X and denote yo = f() and Yi g(x 0). Define a path a:1- Y by taking a(t) = ht(x0), (t e I),

16. fi -SIMPLE SPACES

131

then c(0) = yo and a(1) = y i . According to § 5, the maps / and g induce homornorphisms

1* : grn(X, xo) -÷nn (Y, yo), g* :nn (X, x 0) ->grn(Y , y 1) for each n > 1. On the other hand, the path o- determines an isomorphism an : nn( Y , yi) -> an( Y, yo). Proposition 15.3.4 =

Let a e a„(X, x0) and choose a map 56 : (in, din) -› (X, xo) which represents cc. Define a homotopy v t : In ---> Y, (0 < t < 1), by taking yt = 495 for every t e ./. Then vo represents /* (a) and v l represents g* (a). Proof.

Since vt(a/n) — a(t) for each t E I, it follows that /* (a) = ag(a). I Corollary 15.4. 11/, g: X .-> Y are

homotopic maps such that f(x 0) ---- yo — g(xo), then there exists an element w en i (Y , yo) such that /* — log * .

As another consequence of (15.3), we have the following Proposition 15.5. 17 /

then

: X ---> Y is a homotopy equivalence and if / (x o) — yo, f * : nn (X , x 0) -÷(Y, yo)

is an isomorphism for every n > O.

/ is a homotopy equivalence, there exists a map g: Y ---> X such that g/ and /g are - homotopic to the identity maps. Let x1 — g(y0). Then g induces g* :nn (Y, yo) --->yrn (X, x 1). Proof. Since

Let ht : X --> X, (0 < t < 1), be a homotopy such that ho = g/ and h l is the identity map on X. Define a path o' : I --> X by cr(t) = ht (x0) for every t e I. Then, by (15.3), we have Since an is an isomorphism, this implies that /* is a monomorphism and g* is an epimorphism. Since g is also a homotopy equivalence, it follows that g* is also a monomorphism. Hence g* is an isomorphism and so is /* = Glern . I Similarly, if f: (X, A) --> (Y, B) is a homotopy equivalence and if / (xo) — yo, then I* :nn (X, A, xo) --->nn (Y , B, yo) is an isomorphism for every n> 1.

16. n Simple spaces -

A local system of groups { Gx } in a space X is said to be simple, if the homomorphism o4 depends only on the initial point G (0 ) and the terminal point a(1) of the path o': I ---> X. Let W be a group which acts on a group G. We shall say that W acts simply on G if wg . g for every w e W and g G G.

IV. HOMOTOPY GROUPS

132

The proofs of the following two propositions are straightforward and hence are left to the reader. local system of groups {G x } in a space X is simple i ff , for every xo G X, n1 (X, x o) acts simply on Gx0 . Proposition 16.1. A

system of groups { Gx } in a pathwis e connected space X is simple iff there exists a point xo c X such that 7c1 (X , x 0) acts simply on Gx0 . Proposition 16.2. A local

Corollary 16.3. A local

system of groups { G x ) in a simply connected space

X is always simple.

Let n> 0 be any given integer. A space X is said to be n-simple if the local system {n n (X, x 0)1 x o c X} of the n-th homotopy groups in X is simple. The following assertions are immediate consequences of the definition and (16.1)-(16.3). space X is n-simple iff, for every xo c X, ni (X, xo) acts simply on nn (X, x 0). Proposition 16.4. A

pathwise connected space X is n-simple iff there exists a point xo G X such that ni (X , x 0) acts simply on n„(X , x 0). Proposition 16.5. A

Corollary 16.6. A

simply connected space is n-simple for every n> O.

Corollary 16.7. A

pathwise connected space X is n-simple if gr„(X) . O.

Corollary 16.8. A

pathwise connected space X is 1-simple 19

7( 1 (X)

is

commutative.

Thus, the m-sphere Sm is n-simple for every m > 0 and n > O. Now let us consider the unit n-sphere Sn and a given point s o c S. The geometrical meaning of n-simplicity is given by the following space X is n-simple if, for every point, xo c X and any two maps /, g : Sn--> X with f (so) — x o — g(s 0), / _-_-_, g implies / _.-.-, g rel so . Theorem 16.9. A

Proof. Assume that X is n-simple. Then, by (16.4), 7( 1( X , x0) acts simply on 7c.(X, x0). Since /._,-,-,_ g, there exists a homotopy ht: Sn --> X, (0 < t < 1), such that ho — / and h = g. According to the remark given in the paragraph which precedes (2.1), the maps / and g represent elements a and p of respectively. Define a path o: / --)- X by taking a(t) = hi (s0) for each t c I. an element w of ni (X , x 0). By § 14 and Since a(0) . xo --= a(1), a represents _ ; §15, it is easy to see that oc = w,8. Since 7c 1 (X, xo) acts simply on (X, x0), we have wfi = 13. Hence oc = fi. This proves that / n_--,_ g rel so . Next, let us assume that the condition is satisfied. Let w E gr 1 (X, xo) and choose a loop a which represents w. Let a be any element of nv,(X, x 0) represented by a map f: Sn -÷ X with / (so) ,---- xo. Then the element woc of

16. fl-SIMPLE SPACES

133

an (X, x0) is represented by a map g : Sn --> X with g(so) = xo and satisfying / g. By our condition, this implies that / g rel so. Hence wcc = cc. By (16.4), X is n-simple. I As a consequence of (16.9), let us prove the following Proposition 16.10. Every

pathwise connected topologic' al group is n-simple

for every n> 0.

Let X be a pathwise connected topological group and xo its neutral element. Let I, g : Sn --> X be any two homotopic maps such that f(s0) xo g(s0). Then these exists a homotopy ht Sn X, (0 < t < 1), with ho and h1 = g. Define a homotopy k t : Sn X, (0 < t < 1), by taking Proof.

k t (s) = [ht (s0)] -3-- [ht (s)],

Sn, t

I).

Then we have k0 = /, k i = g and k t (so) = xo for each t r. Hence f g so . By the sufficiency proof of (16.9), 1 (X, x0) operates simply on rel 7C 1 (X, x0). By (16.2) this implies that X is n-simple. This proposition (16.10) can be generalized to the H-spaces as defined in (HI; § 11). See Ex. G. The usefulness of n-simplicity is that, for a pathwise connected n-simple space X, the abstract homotopy group gr(X) as defined in § 14 has a natural geometrical meaning as follows. Let us define 7r,,(X) to be the set of all homotopy classes of the maps of Sn into X. In other words, 7cn (X) is the set of all path-components of the mapping space = Xs n .

Choose an arbitrary basic point xo G X and consider the subspace W of 0 which consists of the maps of (sa, so) into (X, x0). Then the path-components of Iff can be considered as the elements of gr„(X, x0). Hence the inclusion map W c 0 induces a transformation X :nn (X, x 0) --+an (X). X is pathwise connected and n-simple, then X sends an (X, x 0) onto nn (X) in a one-to-one fashion. Lemma 16.11. If

Let cc E(X) and choose a map / : Sn -÷ X which represents cc. Since X is pathwise connected, there is a path o: I -÷ X such that o(0) =xo and o-(1) =x l = Aso). By the method used in the construction of c in § 14, one can show that there is a homotopy ft : Sn X, (0 7(n-1(A, x 0) is defined by

a

x0)• z`o) Finally, let f: (X, A, x 0) —> (Y, B, yo) be a given map. Then, f induces a derived map f' (X', A', 0) (Y', B', y'0). Let f' I (A', x' 0). Then

define

=

q*

, x' 0) --->nn _1 (B% y'0).

f* = /-*

Verify the seven axions of § 11 for the system H above.

7C, , * }

constructed

B. The equivalence theorem

Consider two given homotopy systems H = {7(,

a, * },

{7(' together with their natural equivalence h = {h7b ): H F..5 H' constructed

in § 12. By an admissible transformation k {k n }: H --> H', we mean for each triplet (X, A, x 0) and each integer n> 0 a transformation A, x0) --->nr.(X, A, xo)

satisfying the conditions: (AT!) For each triplet (X, A, xo) and each integer n > 0, we have the O'kn, where, in case of n — 1, k0 denotes commutativity relation 14,4 the identity map. (AT2) For each map f: (X, A, x0),—> (Y, B, yo) and each integer n > 0, we have the commutativity relation k n t* = kfj By (El), (E3) and (E4) of § 12, every equivalence between H and H' is an admissible transformation. Conversely, prove the following .

136

IV. HOMOTOPY GROUPS

Equivalence Theorem. Every admissible transformation le is an equivalence between H and H'. In fact, it coincides with the natural equivalence h, that is to say, kn hn for every n > 0. This shows that the natural equivalence h ={ hn,} is the only possible

equivalence between any two homotopy systems. Furthermore, in order to construct geometrically the natural equivalence between two homotopy systems given by geometric definitions, it suffices to establish an admissible transformation by means of some natural geometric method. C. Properties of the homotopy system

The uniqueness theorem (12.1) implies that the set of seven properties in §§ 6-10 is equivalent to the set of seven axioms in § 11. The following is an outline of deducing these seven basic properties together with other properties of the homotopy system right from the seven axioms. Since the properties I, II, III, V, VII are stated exactly the same as the corresponding axioms, it remains to prove the properties IV and VI. 1. If f: (X, A, x 0) --> (Y ,B, y0) is a homotopy equivalence, then, for each n > 0, f* sends A, x 0) onto 7c7,( Y, B, yo) in a one-to-one fashion. 0 for every n > O. 2. If X is contractible to the point xo, thenn„(X, x 0) 3. For any given triplet (X, A, x0), consider the derived projection p : (X', A', x' 0) --> (X, A, x 0). Then in the diagram

nn (X , A , x 0) .(2t 7rn(X' , A' ,

a 2., )

both p* and a are one-to-one and onto and hence we obtain a natural correspondence

z

ap;i :nn (X, A, x 0)

, x' 0)

which sends nn (X , A, x o) onto nn_ 1 (A', x'0) in a one-to-one fashion. 4. For every triplet (X, A, xo) and every n > 0,7cn (X, A, x 0) is non-empty. Furthermore, one can uniquely define a neutral element of nn (X, A, x 0) in such a way that X, a and the induced transformations send the neutral element into a neutral element. 5. In the homotopy sequence • • • .121,.. nn +1(X , A, x0) nn (A , x0) nn (X, x0) grn (X , A, x0) JL• • A, x0) -> 7c0(A xo) no(X, zo) of (X, A, x0) every set has a specific neutral element. By the kernel of a transformation in this sequence, we mean the inverse image of the neutral element. Then prove simultaneously the following two theorems: The exactness theorem. The homotopy sequence of any triplet is exact,' that is to say, the kernel of every transformation in the sequence coincides with the image of the preceding transformation. The fibering theorem. if f : X Y is a fibering, A = f --1-(B) and x 0 e f --1 (y 0 ), then the induced transformation 14, carries 7rn (X, A, x0) onto nn (Y B, yo) in a one-to-one fashion for every n > O. These two theorems cover the properties IV and VI respectively. •

n1(X ,

137

EXERCISES

D. The role of the basic point in the relative homotopy groups

Consider a given space X, a given subspace A of X, and two given points xo, x, connected by a path 0- : /

A , a(0)

xo , a(1) =

For each n > 0, define a transformation : nn (X , A, x l ) , A, xo) as follows. Let a enn (X, A, x 1). Choose a representative map (In , In-1 , in-1) (X, A, for a. Pull the image of njn-1- retreating along the path a back to xo with the image of In being dragged in such a way that the image of In -1 is always in A. The map obtained after this homotopy represents an element p of 7c„(X , A, xo) which depends only on a and a. Then, we define an (a) = 13. Give the details of this geometrical construction as in § 14, and prove the following assertions: 1. For every n > 2, an is a homomorphism. 2. For every n > 0, an depends only on the class of the path a. 3. If a is the degenerate path a(I) = x0 in A, then an is the identity transformation on nn (X , A, x0) for each n > O. 4. If ci , T are consecutive paths in A, i.e. ci(I) = T(0), then (crT)„ anrn for each n > O. 5. For every n > 0, an carries nn(X,A,x 1) onto grn (X,A,x 0) in a one-toone fashion. Hence an is an isomorphism for every n > 2.

6. Each rectangle of the following ladder is commutative • • • .2, nn (A, x 0 ) 2:t, nn (X , x 0)

nn (X , A, xo) ig„

1 • • . 2, nn (A , x l) 22!_>.

grn (X,

x l)

>• 741 -1(Ar

an

nn (X , A, x 1) 2, grn _ 1 (A, x 1) It, • • •

7. For any triplet (X, A,x 0),7c,(X, A, x0) is a crossed. [ 1 (A, xo), a ] -module.

By this, we mean that the following two conditions are satisfied for every w in n j (A, xo) and a, fl in 7V 2 (X, A, x0): (i) (w oc) =

(i1 ) ( 0a)/3 ==afla-1 . Hence 7.4,7c 2 (X, xo) is contained in the center of 7c 2 (X, A, x0) and i* n l (A, x 0) acts as a group of operators on f *7c2 (X , x 0). See [Hi, pp. 39-41]. The significance of the assertion (5) is as follows. If A is pathwise connected and n > 2, then all the groups Tc.(X, A, x0) for various basic points xo are isomorphic. Hence, as an abstract group, an (X, A, x0) does not depend on the basic point xo and may be denoted simply by nn (X , A). This abstract group 7c„(X, A) will be called the n-th (abstract) relative homotopy group of X modulo A. For example, we have mn (En, Sn -1) = 0, (m < n), 7Cn(En , Sn -i ) =-- Z, 0, (m > 2), 7c4(E3, S2) = Z. 7rm (E2 S')

IV. HOMOTOPY GROUPS

138

Next, consider the spaces of paths W o ------ [X ; A, xo], W i

[X ; A,x 1]

as well as the degenerate paths wo E Wo and w 1 G W t. Then a given path : I A which connects xo to x 1 induces a map : W 1 Wo defined as follows: for each w e W 1 , (2.0) G Wo is the path defined by

[(w)](t)

if 0 < t < if 1, then [cc, i3] is the element 13a — cc of nrn(X, x0). 3. If m> 1, then the assignment cc -> [at, p] for a given 15' e zn(X, xo) defines a homomorphism

p* : 7cm (X, xo) -->7tmi_n _1 (X, x0). 4. If m + n> 2, then, for every cc egm (X, x o) and i51 enn (X, x o) we have [/3, cc] =" (— 1 ) mn [OC) M•

5. If a:I--->X is a path joining xo to x i , then, for every cc enm (X, xJ ) and p enn (X, x1), we have

am+n-1[X, /3 ] [CM( a) , an(18)1 6. If 0 : (X, xo) ---> (Y, yo) is a map, then, for every a en.(X, x o) and p Enn(X, x0), we have px, p] ,_, [0* ( a) , ,t,* (p)] . 7. For any cc enin (X, x0), 13 Enn (X, x0), r EN(X, x0), the following Jacobi identity holds:

(— 1)m[[, P], y] + (— 1 ) 1un [[18, 5 ] , cc] + (— 1 )"[[Y, a], 18 ] = O. Whitehead products may be also defined between relative homotopy groups and between gm (X, A, xo) and nn (A, xo). See [Hu 11] and [Blakers and Massey 2]. G. Homotopy groups of H-spaces

Let X be a given H-space as defined in (III; § 11) and let xo be a homotopy unit of X. Then the group operation in 7r,,(X , x 0) is closely related to the multiplication in X as follows.

Iv, HOMOTOPY GROUPS

1 40

Let a, /3 be arbitrarily given elements of n92,(X , x 0) with n> O. Choose any representative maps

f , g:

(X, x0)

am)

for a and 13 respectively. By means of the multiplication in X, we may define a map h: (I n , aIn) -÷ (X, xo) by taking

h(t)

(t) • g(t), t e In.

Prove that h represents the element a fi of 7rn (X, x0). Furthermore, if X is a topological group and xo is the neutral element of X, then we may define a map k: aPi) -> (X, x0) by taking

k(t) = f (t) • [g(t)] -1 , t e /92'• Prove that k represents the element cc — p of 7r„(X, x0). Next, let a en.(X , x 0) and 13 Enn (X, x0). Choose representative maps

: (P4, 9/m) ÷ (X, x0), g (In, aln) -

(X, x0)

for a and 13 and define a map h: In4+71 -> X by taking

h(s, t) Prove that

h

f (s) • g(t), s e

te

I aim-Pa represents [oc, fi]. This implies that [ac, fi]

O.

Hence, it follows from the assertion (2) of Ex. F that every pathwise connected H-space is n-simple for every n> ; in particular, we deduce again that (X, x0) is abelian. H. Semi-simplicial complexes

A. semi- si mplicial complex K is a collection of elements { a } called cells together with two functions. The first function assigns to each cell a an integer ni > 0 called the dimension of a, m = dim (o-); we then say that a is an m-cell. The second function assigns to each rn-cell a, (m > 0), of K and each integer i, (0 . Hn (Y , B)

X0, we have an isomorphism

j * H(X) H n (X xo)

and hence we obtain a homomorphism hn =

:nn (X, x 0) H1 (X)

which will be called the natural homomorphism of 7c.(X, x0) into Hn (X). Clearly, h i coincides with the homomorphism h* of (II; § 6). Since a : H n 1-1(E n+1) S n) "". -1/ I I n(S n), it follows that = 19$211- is a generato r of the free cyclic group ii n (..Sn). If we identify the boundary Sn --1- of En to a single point so, we obtain an n-sphere sn and a point so . Hence there is a relative homeomorphism p (En , Sit-1) (Sn , so ) such that A(4n) = j(), where

p* : Hn(En, S n-1) rr:,) H n(Sn, so), i * : H(S)

lin(Sn, so)

are the isomorphisms induced by the map p and the inclusion map j : Sn (Sn, so).

Let oc enn (X, x0) and pick a map 0 : (En, SI") --> (X, x 0) which represents a. Then there exists a unique map v: (Sa, so) (X, xo) such that 0 0. We may consider a as represented by v. As a map of Sn into X, v induces a homomorphism

: H(n) -> Hn (X).

Then clearly we have h(oc) v * (rin). This may be used as the definition of the natural homomorphism hn.

V. THE CALCULATION OF 1101VIOTOPY GROUPS

148

Then, the following proposition is obvious. Proposition 4.3.

commutative:

For any triplet (X, A, x 0), the following rectangle is

a

nn.,_ 1 (X, A, x 0)

17,,

Hn +i(X, A)

,

nn (A , x 0

)

Ilan

a

Now consider the following homotopy-homology ladder:

• - 1•_!__,. zi, ±1 (X , A, x0) _A_.). gt,,(A , xo)

i. ,

nr,(X, x 0) .±,_, an (X, A, x0) ..__, h,

• • •

• ••

Pen

4' o . . . J±—).. Hn .,.. 1 (X , A) . 9___> H n (A) 2:1', H n (X) ._11._>. H n (X , A) _,

By (4.3), the rectangle on the left is commutative. By (4.2), the middle rectangle and the one on the right are also commutative. Hence the whole ladder is commutative. The remainder of this section is devoted to the proof of the Hurewicz theorem for polyhedra. Generalizations will be given in the exercises at the end of the chapter and also in a later chapter. In (il; § 9), we defined the notion of the n-connected spaces. In terms of homotopy groups, one can easily prove that, for a given integer n > 0, a space X is n-connected if it is pathwise connected and 2-t m (X) — 0 for every m < n. Theorem 4 4.

(Hurewicz theorem). If X is an (n — 1)-connected finite

simplicical complex with n> 1, then the natural hornomorphism hn is an isomorphism.

Proof. If X is the n-sphere Sn, then the theorem is given already by the

Hopf theorem in (II; § 8). By (3.2), it follows easily that the theorem holds for the case X — Sn.../S. Then, by means of (3.1) and finite induction, one can easily prove the theorem for the case where X is the one-point union of a finite number of n-spheres. Next, assume that dim X < n. If we identify the (n — 1)-dimensional skeleton Xn -.1 of X to a single point yo, we obtain a quotient space Y with a natural projection

p : (x-, xn-1) -÷ ( Y, yo).

If Y is different from yo, then it is obviously homeomorphic to the one-point union of a finite number of n-spheres with Yo as the common point. Pick a vertex x0 E Xn -1 . Since X is (n — 1)-connected, Xn -1 must be contractible to the point xo in X. By an application of the homotopy extension property, it follows that there exists a homotopy ft : X --)- X, (0 < t Xn joining xo to x i . Then xn+i clearly sends the element irs +1 (a) of an +1 (X Xn, x 0) into the generator s of Cn+1 (X). Since s is arbitrary, this proves that xn+1 is an epimorphism. By (IV; 3.4), we have nn (X , Xn, x 0) 0. On the other hand, we have Hn(X, Xn) = 0. Then, in the homotopy-homology ladder of (X, Xn, x 0), we have the following diagram nn,+1(X, Xn,

x0)

gin (Xn,

Ihn Hn +i (X, Xn)

x0) 1:L_>.

gn (X,

x0)

0

„

Hn (Xn)

H(X) 1±._> 0

where xn+1 is an epimorphism and fen, is an isomorphism. These imply that 'in is an isomorphism. In fact, let p E Hn(X). Since i* carries Fin (Xn) onto Hn (X), there exists an element y E H(X 1 ) such that i(y) = fi. Let oc i* Irn,l(y) ezn (X, x0). Then, we have h(cc)

i*lenie;1(y) = 1* (y) = 13.

150

V. THE CALCULATION Or HOMOTOPY GROUPS

Hence h is an epimorphism. On the other hand, let a E grn(X, x 0) be any element such that Iin(a) = O. Choose an element /3 Enn (Xn, x0) with 4 (i9 ) — a. Since i*kn (13) = hi(fi) = hn (a) 0, it follows from the exactness that there exists a y E lin+1(X, Xn) with 0(y) = k n (P). Since xn+i is an epimorphism, there is a (5 E nn„(X, Xn, xo) with xn-1-1( (5 ) = y. Then, we have

1:0(6)

* k ii-lkn0(6) = *ki-ilann+,(6)

= i*k:l a(7) i*k;,-1 kn(f3) =i(fl) ocHence a = 40(ó) = O. This proves that hn is also a monomorphism. So, the theorem is proved for the case dim X = n + 1. Note that the argument used in this paragraph is a special case of the "five" lemma, [E-S; p. 16]. Finally, let X be any (n — I)-connected finite simplicial complex. Consider the (n 1)-dimensional skeleton Xn+1 of X and the following diagram Irn (xn-f-i , x0) I. 7(n(X, x0)

I

'

Hn (Xn+1) i # H(X) where the natural homomorphism len is proved to be an isomorphism. By (IV; 3.4), it follows that i* is an isomorphism. On the other hand, i4t is also an isomorphism. Hence, hn = i 1j-k ni*-1 is an isomorphism. I

5. Direct sum theorems In the present section, we shall derive three useful consequences of the exactness of the homotopy sequence. Proposition 5.1.

If A is a retract of X and xo c A, then

an(X, x 0) arn(A x 0) + nw,(X , A, xo) for every n > 2 and i* :nn (A, x 0) --->7rn(X, x 0) is a monomorphism for every n > I. Proof. Let r: XD be a retraction. Since ri is the identity map on A, it follows that r*i* is the identity automorphism on nn(A , x0) for every n > I. This implies that i* is a monomorphism and r* is an epimorphism for every n > 1. If n > 2, then 2r1 (X, x 0) is abelian. Hence it follows from ri* = I that X, xo) decomposes into the direct sum

K, J = Image i* , K = Kernel r • grn(X, x0) = Since i* is a monomorphism, we have J 7(71(A , x 0). Further, by the exactness of the homotopy sequence of (X, A, x 0), it is easy to deduce that * : nn (X, x 0) -(X, A, x0) is an epimorphism for every n > 2. By exactness, the kernel of i* is J and so i* sends K isomorphically onto7N(X,A,x 0). Hence K 2rn (X, A, xo).

5.

DIRECT SUM THEOREMS

151

Consequently, if A is a retract of X, then n 2 (X,A, x 0) must be abelian. Proposition 5.2.

If X is de formable into A relative to a point x o e A, then

x 0)

nn,(X, x 0) + nn+i(X, A, xo)

for every n > 2 and i* :7En (A, x o) nn (X , x 0) is an epimorphism for every n >I. Proof. According to the hypothesis, there exists a homotopy ht : X (0 (A, x0) by taking h(x) = hl (x) for each x e X. Since ih hl ho rel xo, it follows that i* h* is the identity automorphism on grn (X, x 0). Hence h* is a monomorphism and is an epimorphism for every n > I. If n > 2, then nn (A, xo) is abelian. Hence it follows from i* h* = 1 that nn (A, x 0) decomposes into the direct sum nn (A, x0) =

K, J = Image h * , K

Kernel i* .

Since h* is a monomorphism, we have J nn (X, x 0). On the other hand, since i* is an epimorphism for every n > 1, it follows from the exactness that d : 7r,, +1 (X, A, x0) ---> nn(A , x 0) is a monomorphism. Hence

K = Kernel i *= Image d

A, x0). I

For the case n 1, n i (A, x 0) is an extension of gr2 (X, A, xo) by 7r1 (X, z o) if X is deformable into A relative to xo. Proposition 5.3.

If A is contractible in X relative to a point x o e A, then

7rn (X, A, xo)

xo)

7rn ,(A , x o)

for every n > 3 and i* sends 7c.(A , x 0) into the neutral element of 7c.(X, x 0) for every n > 1. Proof. By hypothesis, there exists a homotopy ht : A ---> X, (0 < t < I), x o and ht (x o) xo for each t e I. This implies that such that ho = i , hi (A) 0 for every n > 1. Now let n > 2. By the exactness of the homotopy sequence, i* implies that j* is a monomorphism and d is an epimorphism. This proves that grn,(X, A, x0) is an extension of 7c?,(X, x0) by nn _ i (A, x0) . By means of the contraction ht, we define a homomorphism h* : --->an (X ,A,x 0) for each n >2 as follows. Let oc e 7rn_ 1 (A,x 0) be represented by a map f: _1) -÷ (A, x0). Define a map g: (In, In -1 , jn -1) -÷ (X, A, x0) by taking Ow • • , tn) = latn i(ii, • • • , tn-i)-

/, Then h* (a) is defined to be the element represented by g. Since g I In . Hence h* it follows that Oh* is the identity automorphism on 7C n is a monomorphism for every n > 2.

152

V. THE CALCULATION OF HOMOTOPY GROUPS

If n > 3, then 7c,a(X , A, x 0) is abelian. Hence, dh* — 1 implies that X, A, x 0) decomposes into the direct sum Kernel d. mi (X, A , x 0) =J K, J — Image h* , K 1 (A , On the other hand, we have Since h* is a monomorphism, J

K = Kernel O = Image f * -,...y7c.(X, x 0) since f* is a monomorphism. I For the case n 2, the last argument of the preceding proof breaks since n 2 (X,A, x 0) is in general non-abelian. However, it can be proved that 7c 2 (X, A, x0) is isomorphic to the direct product of 7c 2(X, x0) and 7c 1(A, x0). The assumed relativity with respect to xo in (5.2) and (5.3) is convenient but not essential; for proofs without this assumption see [Hu 1]. 6. Homotopy groups of fiber spaces Let E be a fiber space over a base space B with projection p : E B. Choose a basic point b0 E B such that the fiber F =25 --1(b0) is not empty. CalIF the basic fiber and choose a basic point eo eF.Thus, we obtain a triplet (E, F, e9). (B, b0) defines a map Since p(F) = bo, the projection p : (E, e0) q: (E, F, e0) -› (B, bo) and p = qf , where j: (E, e0) c (E, F, e0) denotes the inclusion map. According to (IV, § 9), q* sends nn,(E, F, eo) onto nn (B , b o) in a one-to-one fashion for each n > I. Let

d*

e0), (n > 1).

:nn(B, b 0)

Since p* q* f *, we can construct from the homotopy sequence of the triplet (E,F,e0) an. exact sequence

••

7Cn„(B b0)

.

, e 0)

- • ±1•,. 1(B, b

A.

nn( E , e 0) 121, nn (B , b0) eo)

7r0 (E,

e 0)

which is called the homotopy sequence of the fibering p : E ---> B based at eo . Proposition 6.1'. If

the basic fiber F is totally pathwise disconnected, then :

nn (E e0)

grn (B,b 0), n > 2,

and p* is a monomorphism if n 1. Proof. If F is totally pathwise disconnected, then 7rn (F, eo) = 0 for every n > 1. Hence the proposition is an immediate consequence of the exactness of the homotopy sequence of the fibering 25: E B. I

fibering p E B admits a cross-section Z: B E, then, for every 1)0 e B and e0 = X(b0) G F 25 --1 (b0 ), we have Proposition 6.2. If the

nn (E , e 0) pr.,/ nn (B , b o) ± grn(17, e0) or each n > 2 and p* is an epimorphism for every n > 1.

6.

HOMOTOPY GROUPS OF FIBER SPACES

153

Since px is the identity map on B, it follows that p* x* is the identity automorphism on ni1 (B,b 0). Hence, ; is a monomorphism and p* is an epimorphism for every n > 1. If n > 2, then E, e0) is abelian. Then, p*x* 1 implies that 7c,i (E, e0) decomposes into the direct sum Proof.

K, J Image X * , K — Kernel p* • Since X,R is a monomorphism, we have b0). On the other hand, since p* is an epimorphism for every n > 1, it follows from an exactness argument that i* is a monomorphism for every n > 1. Hence e0)

K

Kernel

p* = Image i* tr.,/ grn(F,e0),I

If n > 1, then (6.2) is a generalization of (2.1). As immediate consequence of (5.1)-(5.3), we have the following three propositions concerning the homotopy sequence of a fibering p : E B. Proposition 6.3.

If F is a retract of E, then

, 1)0)

nn(E , e0)

nn,(F , e 0)

for every n > 2 and 75 4 is an epimorphism for every n > 1. Proposition 6.4. If E is de formable into F, then ,

nn (F , e0) or every n

2 and

Proposition 63.

e0)

p* = o for every

7C91 +1 (B,

n

b 0)

I.

If F is contractible in E, then

an(B,b0) sYinn(E,eo) an--1(F, eo) for every n > 2 and p* is a monomorphism for every n > 1. As an application, let us consider the Hopf fiberings

p : s2m-1

(m = 2, 4, 8).

In each of these fibering, the fiber F is an (m — 1)-sphere Sm -4 and is contractible in S2m -1. Hence, by (6.5), we have (6.6) 7Cn( 5 n ) atn (S 2m --l) nn _1(5m-1) for every m = 2, 4, 8 and every n > 2. When m — 2, this gives (6.7) gr1 (S 2) 7(72,(S3) n >3. When m 4 or 8, we have (S4) 7(11(5 8)

nn j(S 3) 7r9 -1(S 7) ,

7 7 (S 4)

7c15(S 8)

Z Z

(2 2. Let b0 denote the identified point of so and 1. By (II; Ex. A), we get i (B , bo)

Z

with a generator w represented by the exponential map p : / B of (II; § 2). In the product space Sn x R of Sn and the real line R, let us consider the subspace

E

(Sn

x Z) U (so x R),

where Z denotes the subspace of R consisting of all integers. This space E is pathwise connected and grtn (E) = 0 for every m < n. In particular, since n> 1, E is simply connected and hence n-simple. The n-th homotopy group grn (E) as interpreted in (IV; § 16) is a free abelian group with a countable set { cc J IEZ} of free generators, where ai is represented by the map fi Sn ->- E which is defined as follows: Ms) = (s, i) ES91 X

Z= E,

(s e Sn).

For a proof of this result, see Ex. C at the end of the chapter. Define a map q: E B by taking q(s, t)

P (t)

Sn, t e Z), (s = so, t R),

where p : R --> S 1 denotes the exponential map of (II; § 2). One can easily verify that E is a covering space over B relative to q. Hence, gr(B, bo) for each m with 1 <m n, where eo E E and p(e0)

bo , and hence nn + r(B, bo)

Hn+1(E).

156


Thus, the identity map on B is a 0-connective fibering and the universal covering over B, if it exists, is a I-connective fibering. If p:E-÷13 is an n-connective fibering, then E is said to be an n-connective fiber space over B. The main objective of the present section is to construct an n-connective fiber space over a given pathwise connected space B.

B is a given space, 1)0 a given point in B, and n a given positive integer, then there exists a space X which satisfies the following three Lemma 8.1. If

conditions: (i) X contains B as a closed subspace. (ii) nn (X , b o) O. (iii) The inclusion map i: BŒ X induces an isomorphism

i*

, bo)

7r..m (X , b o)

for every m satisfying 0 < m n. Proof. Let A be a set of elements of grn (B , 1)0) which generates nn(B , b o) and consider A as a space with discrete topology. Let En+1 denote the (n 1)-cell bounded by the unit n-sphere Sn in the euclidean (n + 1)space. Let W En+1 x A, C Sn x A W. Let so = (1, 0, • • • , 0) e Sn. For each a e A, choose a map gŒ: (Sn, so) which represents a. Then, define a map g : C B by taking

(-13 )bo)

g(s , a) = ga(s), (s, a) EC. Let X denote the adjunction space obtained by adjoining W to B by means of the partial map g :C -± B. It remains to verify the conditions (i)—(iii). Since C is closed in W, (i) is obvious from the definition of adjunction space in (I; § 7). To verify (ii) and (iii), let us first prove that

nr,i(X B, b o) = 0, 0 < m < n. For this purpose, let be an arbitrary element of nm (X , B, b o) and pick a map -÷ (X, B, bo) gm , which represents Since m

A n+j

in the euclidean

(X, xo).

Since f sends the leading vertex of Sn onto xo , f represents an element [f] of nn (X, x0). On the other hand, let : Li n

(i

0 , • • *, n + 1 ) ,

denote the simplicial map defined in [E-S; p. 185]. Then, the composed map f

er: (An, DLL)

(X, x0)

represents an element [fen of nn (X , x0) for every i = 0, 1,• • -, n + 1. Prove the following Homotopy addition theorem. For any map : (Sn, Kn -4) -÷ (X, x0),

we alway have

n > 2,

n+1

[f] = E(—

[/e].

For the exceptional case, nl (X, x o) is not necessarily abelian. However, for any map f: (Si-, K°) (X, x0), the following relation is obvious: rfi = [fe e'] •[Poi] • [fe1 1 ] -1-

In the remainder of this exercise, we shall give an analog for the maps of cubi c boundaries. Let n > 0 and denote by Kn -4 the (n — 1)-dimensional skeleton of the boundary n-sphere ain -o- of the (n 1)-cube /91+ 1, i.e., K" consists of all points (t11 . • - , tn+1) of In+1 such that 4(1 —1 1) 0 for at least two indices i. Consider any given map : (din+1, Kr")

-->

(X, x0)•

Since / sends the point (0, • • • , 0) of 0/91+1 into xo, it represents an element [f] of n(X, x o). On the other hand, for each i = 1,• • • , n 1, let ni and 6 denote the homeomorphism of In into a/n-F1 defined by

n i(ti, • • • , tn,) '1(ti , • • •

tn)

(t1, • • • , ti.._ 1 (ID .

,

0,

• *, 4-1, 1 ,

ti,.

tn),

ti, • • , tn)•

Then, the composed maps hp and gi of (In,aIn) into (X, xo) represent elements [hp] and [/6] of 2rn (X, x0) respectively. Then, prove that n+i [f] =E (—

([gi]

Unin,

n > 2.

For the exceptional case, 1 (X, xo) is non-abelian in general. However, for an arbitrary map f: (012, K°) -÷ (X, x0), the following relation holds: [n

[fi] • [g- 2] • [ g- 11 -'• [M2]-1.

166


B. The relative homotopy addition theorem

For any triplet (X, A, xo) and any integer n > 2, the elements of the group nn (X , A, x0) can be considered as the homotopy classes of the maps of (An, OA n, y o) into (X, A, xo), whete yo denotes the leading vertex of /in. Let n > 2 and denote by Kn -1- the (n — I)-dimensional skeleton of the boundary n-sphere Sn of 4 n+1 . Consider any map

f: (Sn, K92-1-, yo) ---> (X, A, x0). - If we compose f with e n : Zi n ---> A n+i, we obtain a map fee of (A n , aA n) into (X, A) for each i = 0, 1, • • -, n + 1. If i 0, fein maps the leading vertex yo of ii n into xo and hence it represents an element [fee] of nn (X, A, xo). Set x l — f (v 1). Then, Pon sends y0 into x1 and so it represents an element [fee] of (X, A, x 1). Let u: I -÷ A denote the path joining xo to x1 defined by a(t) — f(1 —1,t, 0, • • • , 0),

tel..

By (IV; Ex. D), a induces an isomorphism an : nn (X, A, x i) ‘,-;,, nii (X , A , xo). Prove the following Relative homotopy addition theorem. For any map f: (Sn, K -1, y 0) -> (X, A, z 0), n > 2, we always have

i*U i — an Ueon]

:f :( — 1) i = + :E

ueelj,

where [I] is the element of n n (X, x o) represented by / as a map of (Sn, v o) into (X, x o) and i* is the h,omomorphism induced by the inclusion map j: (X, x 0) c (X, A, x 0). State and prove the analogous theorem for the maps of (0/nd 1 , Kn -4, yo) into (X, A, xo). C. The Hurewicz *theorem

Let X be pathwise connected space, A be a pathwise connected subspace of X, and xo be a given basic point in A. The pair (X, A) is m-connected if A, xo)-,-- 0 for every n satisfying 1 2 and (X, A) is (n — 1)-connected, then the natural homomorphism x n : 7rn (X , A, xo) -+ Hn (X, A)

in (4.1) is an epimorphism and its kernel is the subgroup generated by the elements of the fOYM OC — WOC, where OL E nn (X , A, x0) and w E 1 (A, x 0). For the special case n = 2, prove that the kernel of x2 contains the cornmutatOr subgroup of 7( 2 (X, A, xo).

167

EXERCISES

As corollaries of the theorem given above, we have the following propositions: 1. If (X, A) is (n — 1)-connected and n-simple for a given n > 2, then

: nn (X , A, xo) Re, Hn (X , A). 2. If it

> 2 and X is (n

1)-connected, then hn :7rn (X , x 0)

Hn (X).

The last proposition generalizes (4.4). As an illustrative example, let us consider the space E = (Sn X Z) U (so X R) of § 7. Then, E is (n, — I)connected and hence nn(E) H(E). This implies that nu (E) is a free abelian group with a countably infinite set of free generators. Finally, prove 3. If n > 2 and X is (n 1)-connected, then hn+1 is an epimorphism. [G. W. Whitehead 1]. D. The Whitehead theorem

Let (X, A, xo) be as assumed at the beginning of the previous exercise. By the aid of the Hurewicz theorem and the following commutative exact ladder

• -.--->,7Cn (A,x 0) 71k-->nn (X ,X0)Lt -->ilin(X ,A ,x0)- 11›7t n_ 1 (A ,x 0)11 --).•• • -A-.7r 1 (X ,A ,x 0) '9 >0

ih„

h i, Y

•• •

Ij-->Hn (A) i4--->1172,(X) 14--->lln (X,A)A--->H n _ 1 (A)

--)-• • • --1-4--->H i (X,A),

prove the following assertions: 1. Let m> 0 be a given integer. If i * is an isomorphism for every n < ni, then so is it-. If i* is an isomorphism for every n < m and is an epimorphism for n — nt, then so is i . 2. Assume that both X and A are simply connected and m > 0 is a given integer. If i# is an isomorphism for every n < m and is an epimorphism for n m, then so is i* . Furthermore, if i4 is an isomorphism also for n = nt, then the kernel of i* in 7Cn(A, ,C 0) is contained in that of kn. Next, let X and Y be pathwise connected spaces. Consider a given map : (X, xo) -÷ (Y, yo) and the induced homomorphisms f * :nn(X, x 0) --->-7rn (Y , yo), 1*: Il(X) ---> H71( 1.7 ).

Using the mapping cylinder M1, and (I; 12.1 and 12.2), prove the following assertions: 3. Let m> 0 be a given integer. If /* is an isomorphism for every n 0 is a given integer. If A* is an isomorphism for every n < In and is an epimorphism for

168


n m, then so is f* . Furthermore, if /It is an isomorphism also for n = m, then the kernel of f* in 7cn (X, x0) is contained in that of hn :an (X, x 0) Hn (X). (Y, yo) such Note that the condition that there exists a map f: (X, x0) that f* is an isomorphism for each n > 0 is much stronger than the condition that grn (X, x 0) zn (Y, yo) for every n> O. In fact, there are pathwise connected spaces which have isomorphic homotopy groups but some of whose homology groups are different [Wang 1]. E. Homotopy groups of adjunction spaces

Let X be a given space and xo E X a given point. Consider an ipdexed

family

: (Sn, so)

(X, x0), u E M.

Give M the discrete topology and define a map f : Sn X M -› X

by taking f (s, tt) 1n (s) for every s E.Sn and tu, E M. Let Y denote the adjunction space obtained by adjoining En-o- x M to X by means of the map f, (I; § 7). The inclusion map i induces the homomorphisms 7cm (X, x0) --> nm,(Y, x0),

ni > O.

Prove the following assertions: 1. i* is an isomorphism for every m < n. 2. For m = n, i* is an epimorphism and its kernel in 7r,,,(X, x0) is the subgroup generated by the elements wan , where w E n 1 (X, x0) and an is the element of(X, xo) represented by the given map fi., for all e M. [J. H. C. Whitehead 1, p. 281]. 3. If 0 for every tt E M, the i* is a monomorphism for m = n 1 and its image in an+i (Y, xo) is a direct summand. The complementary summand is isomorphic to the relative homotopy group +i( Y, X, 'c o) which is a free abelian group with free generators wbn , where w E7C 1 (X, x0) and b E grn+I(Y, X , xo) is the element represented by the map : (En+1, Sn, ,C0) -÷ (Y, X, xo)

defined by WO p(t, ,a) where i) : En+1 x M Y denotes the natural projection. [J. H. C. Whitehead 1, p. 285]. F. Spaces of homotopy type (a, n)

Let n be a given group and n > 0 a given integer. If n> 1, we assume that is abelian. A pathwise connected space X is said to be of the homotopy type (gr,

provided Thus, by (8.4), if

7T(X) Pr.,/ 7V , 7rm (X) = 0 5:

if m L n.

E 13 is an n-connective fibering over an (n, — 1)-

169

EXERCISES

connected space B, then the fiber F — p-1(b0) over any bo E B is of the homotopy type (grn (B), n — 1). Construct a space X of the homotopy type (gr, n) as follows. If n = I, then ar can be represented as the quotient group of a free group F over a normal subgroup R of F. If n> 1, then gr is abelian and can be represented as the quotient group of a free abelian group F over a subgroup R of F. Hence

Let M denote the set of free generators of F and let bo be a space consisting of a single point bo . Give M the discrete topology and adjoin En x M to bo by means of the map g : S n-1 X M -->b o . Let A denote the adjunction space obtained in this way. Then prove that 7cn (A) — F, grm (A) — 0

if m < 11,,

For each reRc nn (A), choose a map /r: (91, so) ---> (A, 1)0) which represents r. Let B denote the space constructed by means of the family { fr 1 Y e R} as in Ex. E. Then we have FIR — gr, gr. ni (B) -, 0 if m < n. Then, by (8.2), there exists a pathwise connected space X which contains B as a closed subspace and grm (X) — 0

if m 0 n.

This completes the construction. Finally, prove that if X is a space of the homotopy type (n, n) with n> 1, then the space of loops A(X) at a point xo E X is a space of the homotopy type ( 77 , n — 1). G. The realizability theorems

By means of the spaces of homotopy type (7 r, n), prove the following Realizability Theorem. Let ori 1 71. 2, • • •

42,

7 7

•••

be a sequence of groups. All groups except possibly the first one are abelian and ni operates on 7C n for every n > 2. Then there exists a pathwise connected space X and a point x, E X such that the following three conditions are satisfied, [J. H. C. Whitehead 5] and [Hu 10] : (1) There exists, for each n > 0, an isomorphism lin : 7cn (X, xo)

(2) For any w E ni (X, x0) and

Ot E 7Cn( X,

x0), n >2, we have

h(wa) ,----. h 1 (w)h(oc).

(3) For every a Egrm (X , x0) and p E(x, x 0), m > 2, n > 2, the Whitehead product [oc,

p . ]

O.

170


By constructing a cone over X, deduce an analogous realizability theorem for relative homotopy groups. H. Topological realization of semi simplicial complexes -

For a given semi-simplicial complex K as defined in (IV; Ex. H), we can construct a space 1 K I associated with K as follows. To every integer nt > 0 and each m-cell a of K, let us associate an open m-simplex I a I, called the open m-cell corresponding to a, which is defined to be the topological product I al = a x hit (Am), hit (A 0)

of a as a single point and the interior of the unit m-simplex zl.. Then we define the closed m-cell Cl la I to be the set

C/Ial — U r 2, the n-th barycentric subdivision, defined in the obvious way, of a semi-simplicial complex K is a simplicial complex with locally ordered vertices. Hence, every semi-simplicial complex is triangulable. I. The projection a) : S(X) X

For any given space X, there is a natural projection co of the singular complex S(X) onto X as follows. For an arbitrary point p of the space S(X), let o- denote the unique open cell of S(X) which contains p. Then, a is a singular simplex X, m a: dim (a) and the characteristic map Za sends In Am homeomorphically onto the open cell a of S(X). The natural projection to : S(X) -÷ X is defined by taking

(0(15)= ax,--1(p), p E y C s(x). Prove the following assertions: 1. The projection to S(X) -± X is a map of S(X) onto X. For every subspace A of X, co carries the subcomplex S(A) onto A. 2. The space X is pathwise connected iff S(X) is connected. Next, let (X, A, x0) be any given triplet and let po denote the vertex of S(X) such that co( 50 ) x0. Thus we obtain a triplet (S(X), S(A), p0), and a map : (s(x), S(A), po) -÷ (X, A, x0) defined by the natural projection to. Prove the following important proposition, [Giever 1 and j. H. C. Whitehead 7]. 3. The induced transformations co * of to are one-to-one correspondences, namely, 7r,m (X , A, x0), co* : nm (S(X),S(A), po) co * :2-cm (S(X), po)

am (X, x 0),

co* :am (S(A), po)

am(A, x0).

The significance of this is that, in computing the homotopy groups of a space X, we may assume without loss of generality that X is triangulable and hence locally contractible. In fact, we may replace X by S(X). J. Induced cellular maps

A cellular transformation T K 1 -± K2 of a semi-simplicial complex K 1 into another such complex K2 is a function which assigns to each m-cell of K 1 an m-cell T = T(6) of K 2 in such a fashion that

r(1)= T(0(0),

0, 1 , • • , ni).

172

V. THE CALCULATION OF

HOMOTOPY GROUPS

Prove that a cellular transformation T: K 1 ---> K2 induces a unique map IT: K1 -> K 2 which carries o- E K 1 barycentrically on r = T(ci) E K 2 . A map /: K 1 ---> K2 is said to be cellular if it carries Ken into K 2 m for every m = 0, l , • • • . Thus, the map fg-, : K 1 --» K 2 induced by T is cellular. Consider a given map 0 : K --- > X of a semi-simplicial complex K into a space X. induces a cellular transformation

To: K --->S(X) as follows: For each m-cell o- e K, 1- — T(o) is defined to be the singular m-simplex

r — OX, 7 : A m ---> X

where 4 denotes the characteristic map of a. Verify that r(i) = To(o-(i)) for every i = 0, • • -, m. This cellular transformation To induces a cellular map 04* : K --->S(X) which will be called the induced cellular map of 0. Prove that we = 0. As an application of this, let X = K and take 0 to be the identity map. Then 04* maps K homeomorphically and barycentrically onto a subcomplex of S(K). Hence, we may consider K as a subcomplex of its singular complex S(K). Prove that K is a deformation retract of S(K) and that the cellular transformation To defines a chain equivalence. The second assertion implies that the homology and cohomology groups of a semi-simplicial complex K are topological invariants. Next, let X, Y be spaces and consider a given map j: X --->- Y. f induces a cellular transformation Tf : S(X) ->S(Y) defined by

r — Tf(a) = fcr : A m ---> Y

for eVery singular m-simplex a E S(X). This cellular transformation Tf induces a cellular map fit : S(X) .--›- S(Y) which will be called the induced cellular map of f. Obviously, RI- is also the induced map of 0 = /co : S(X) ---> Y. Verify that, in the following diagram, we always have /co,

S(X) lw X

f 4t

---> S(Y) la'

f

--->

Y

This implies that, in studying the induced homomorphisms f * of a given map f: X -> Y on the homotopy groups, we may always assume that X, Y are simplicial complexes and that f is simplicial. K. Admissible subcomplexes of S(X)

Assume that X is pathwise connected space and that a fixed point x0 of X has been selected as basic point.

EXERCISES

173

Two singular m-simplexes a and I- in X are said to be compatible if their faces coincide, that is to say, a(i) — TM for each i — 0, - • • , m. Hence a and 'V are compatible if o- 1 Sm --1- . T iSn", Sm-1- = Ott, Further, two compatible singular m-simplex a and T are said to be equivalent if there exists a homotopy ht : 4 . -->- X, (0 < t 1, then one can deduce similar results by using local coefficients. See Ex. A at the end of the chapter. In the present section, let us consider a given map g :Rn -› Y. This map g determines an (n 1)-cochain cn9g) of K with coefficients in the homotopy group 'r(Y) as follows. Let a be any (n 1)-cell of K. Then the set-theoretic boundary Oa of a is an oriented n-sphere. Since Oac Kn, the partial map ga g I Oa determines an element [go.] of g„( Y). Then c+1(g) is defined by taking [0+1 (g)](0. ) == [gc] E7n,(17)

3.

THE OBSTRUCTION Cn+1

(g)

177

for every (n ± 1)-cell a of K. This (n + 1)-cochain cn+ 1(g) of K is called the obstruction of the map g. Lemma 3.1. The obstruction cn+1 (g) is a relative (n + 1)-cocycle of K modulo L; in symbols, c+1 (g) E Zn+1 (K , L;n n (Y)).

Proof. Let us first prove that cn+i(g) is in Cn+1 (K, L; nn(Y)). For this purpose, let o- be any (n + 1)-cell of L. Since g is defined on Cia C L, gG, has an extension over City and hence [g,] is the zero element of (Y). This proves that cn+1 (g) is a cochain of K modulo L. Next, let us prove that cn+1 (g) is a cocycle. For this purpose, let a be any (n + 2)-cell of K. It suffices to show [6cn-0-(g)](a) = O. Let B denote the subcomplex aa of K and Bn the n-dimensional skeleton of B. Then we have the homomorphisms

Cn+1 (B) --2—> Z, (B) = Zn (Bn) = IIn (Bn) -nn (Y), where h denotes the natural homomorphism and k * is induced by the partial map k = g I B. If n > 1, then Bn is (n — 1)-connected and hence h is an isomorphism according to the Hurewicz theorem in (V; 4%4). If n = 1, then h is an epimorphism and the kernel of h is contained in that of k * since (Y) is abelian. Hence, in either case, we obtain a well-defined homomorphism Zn (B) -->nn (Y).

Since C. 1 (B) is a free abelian group, the kernel Zn (B) of 0 : C(B) -÷ C 1 (B) is a direct summand of C(B). Therefore, the homomorphism sb has an extension d: Cn(B) -->nn(Y).

For every (n + 1)-cell T in B, the element fen±i(g)](-r) is represented by the partial map k I dx. Therefore, it follows that [0+1(0] (r) _ k * h _1 (a.r. ) _ d(.r) . This implies that —

[c " + 1 (g) ] (a a) ----- d (Ha) --= 0 .

Hence cn+1 (g) e Zn+1 (K, L;nn(Y)). 1 Because of (3.1), cn-0-(g) is usually called the obstruction cocycle of g. The following two lemmas are obvious. Lemma 3.2. The map g : kn -÷ Y has an extension over kn+1 if cn -o-(g) —

O.

go, gl : Kfl .4- Y are homotopic maps, then cn+1(g0) = cn+1 (g1). Now let (K', L') be another cellular pair and ç6: (K', L') ---> (K, L) be a proper cellular map, [S; p.161]. For a given map g: Kn ---> Y, we obtain a composed map g' = g5b : K'n ---> Y. Since sb is proper and cellular, it induces a unique cochain homomorphism, [S: p. 161], Y)). sb4t : C 1 +1 (K, L;n n (Y)) ---> Cn+1 (K% L'; Lemma 3.3. If

178

VI. OBSTRUCTION THEORY

Lemma 3.4. cn -0-(g')

To prove this, one must show that the two sides have the same value on an arbitrary (n 1)-cell a of K'. For this purpose, one should consider the minimal carrier of a and pick an (n 1)-cell from this minimal carrier; then the lemma is a consequence of a number of trivial commutativity relations. The details of the proof are left to reader, {S; p. 168]. 4. The difference cochain

In the present section, we are concerned with two given maps En --> Y which are homotopic on fcn -i; we shall see that the difference of the obstruction cocycles of go and g1 is a coboundary. For this purpose, consider go,

a homotopy

ht :Kn -1 Y, (0 < < 1),

such that ho — go I En --1 and h1 g1 Regard the closed unit interval I as a cell complex composed of two 0-cells 0 and 1 and the 1-cell I, where 60 = — I and 61 — I. Then the topological product

J—KxI

is also a cell complex. We shall denote by jn the n-dimensional skeleton of J and use the notation

jn = (L x I) U

in

(K n X 0) U

X U (K n X 1).

Define a map F : In ---> Y by taking

17(x, t)

(x 6 R, t = 0), t E .r), (x e (X eKn, t . 1).

go(x) , ht(x), gi(X),

Then, according to § 3, this map F determines an obstruction cocycle cit .-0-(F) of the complex J modulo L x I with coefficients in nn ( Y). It follows from the definition of F that 0-0-(F) coincides with c1+1 (g 0 ) x 0 on K X 0 and with cn+i(gi) x 1 on K x 1. Let M denote the subcomplex (K x 0) U (L x I) U(1‹ x l)of J=Kx I. Then it follows that

0-0-(g0) x 0 — cn+1 (g1) x 1 is a cochain of J modulo M with coefficients in nn (Y). Since Cl -÷ ci x ris a 1 — 1 correspondence between the n-cells of K\ L and the (n 1)-cells of J \M, it defines an isomorphism cn+1-(F)

k : Cn(IC, L; nit ( Y) )

Cn +1(J, M; an( 17 )). Hence there is a unique cochain dn(go, g1 h) in Cn(K, L; nn,(Y)) such that ;

(i)

kdn(go, g1 ht) ;

on+1{ c-g(F)

c1li-i(g0)

x0

0+1(g1) >< 1

1.

4.

THE DIFFERENCE COCHAIN

179

This cochain dn (go, gl, ht) will be called the deformation cochain. In particular, g1 I En-1 and ht (x) = go (x) for every x E En -1 and every t, if go Rn-1 we abbreviate dn(go, g 1; ht) by dli(go , g1) and call it the difference cochain of go and gl. The following lemma is an immediate consequence of (3.2).

The homotopy h t : En --1 Y has an extension ht* : rfn go and hl* = g, if dn(g o, g1 ; ht) (0 < t IT is n-extensible over K iff On 4-1 (f) is non-

empty.

L -> Y is (n + 1)-extensible over K iff On +1 (f) contains the zero element of 1-1 92-0-(K, L;n n (Y)). Lemma 6.5. The map f:

By recurrent application of these two lemmas, one can easily prove the following Proposition 6.6.

If Y is r-simple and Hr+1(K, L ; nr(y)) , 0

for every r satisfying n -Y over K implies its m-extensibility over K. In particular, if K \L is of dimension not exceeding In, then the hypothesis of (6.6) implies that a map f: L ---> Y has an extension over K iff it is it-extensible over K. Hence, we have the following Corollary 6.7.

If Y is r-simple and lir+1 (K, L; nr(Y)) — 0

for every

Y

> 1, then every map f: L -> Y has an extension over K. 7. The homotopy problem

Let us study two given maps

f:K->-Y, g:K->Y

agreeing on L, that is to say, such that fIL=gIL. As described in (I; § 8), the homotopy problem (relative to L) is to determine whether or not f and g are homotopie relative to L, in other words, whether or not there exists a homotopy ht : K --›- Y, (0 < t < 1), such that ho ---= f, h 1 = g, ht i L.-- f f L for each / El. The most important special case is that L = o. Then the problem is to determine whether or not two given maps f, g : K -± Y are homotopic. Since the homotopy problem is a special case of the extension problem, the obstruction method can be applied. Let us begin by defining the notion of n-homotopy as follows. The maps f and g are said to be n-homotopic relative to L if f 1 En and g 1 En are homotopic relative to L. If f and g are homotopic relative to L, then they are obviously n-homotopic relative to L. Since Y is assumed to be pathwise connected, the following proposition is obvious. Proposition 7.1. Every pair of maps f,

O-homotopic relative to L.

g : K -> Y with f IL =g1L are

8.

THE OBSTRUCTION

dn(f,g;ht)

183

The least upper bound of the set of integers n such that f and g are n-homotopic relative to L is called the homotopy index of the pair (f, g) relative to L. The following two propositions can be proved as in § 2. Proposition 7.2. If fc_—_ f' and g nt-2 g' relative to L, then the pair (f', g') has the same homotopy index relative to L as the pair ( f, g). Proposition 7.3. If (K, L) is a simPlicial pair, then the homotopy index of any pair of maps f, g: K --4- Y relative to L is a topological invariant.

8. The obstruction dn(f,g;ht) Throughout the present section, we are concerned with two given maps f,g:K-÷Y, fiL=g1L

which are (n — 1)-homotopic relative to L. Let ht : Er" -4 Y , (0 < t < 1),

be a given homotopy such that ho . f 1 En--1, h l = g I En -1 , and ht 1 L -= f 1 L for every t e I. Since f and g are defined on En, the construction in § 4 gives a deformation cochain Cln (f , g; ht), which will be called the obstruction of the homotopy ht in connection with the pair (f, g). Lemma 8.1. The obstruction dn(/, g; ht) is a relative n-cocycle of K modulo L; in symbols, dn(f, g; ht) e Zn (K, L; an(Y)).

Proof. Since f and g are defined on Hence, by (4.2) we have

1

, it follows that c+1 (f) = 0 ---,---

e n-Fifkg) -, .

6dn(f, g; ht) --= cn-o-(/) — cni -1 (g) = O. I Lemma 8.2. The homotopy h t has an extension h t* : Kn --> Y, (0 < t < 1), such that he —f I En and h1 * -- —- g I En iff dn(/ , g; h t) = O.

Proof. Consider the pair ( J, M), where j = K x I and M = (IC x 0) u (L x I) U (K x I). Let jn = M U jn '-. (K x 0) U (Kn-1 X I) U (K x 1). Define a map F :

In --> .17 by taking f (x), F(x, t) -=.- { ht(x),

(x e IC, t = 0) , (x ek'n --1, t G I), (xeK,t = I).

g(x) , Then F determines an obstruction cocycle cn+1 (F) of the complex j modulo M. By the formula (i) of § 4, we obtain

(i)

kdn(f, g; ht) — (— On+10+1(F).

Since the homotopy ht has an extension ht* iff the map F has an extension


1 84

F* : In+1 -4. Y, the lemma is an immediate consequence of (3.2) and the preceding formula (i) since k is an isomorphism. The cocycle do(f, g; ht) represents an obstruction cohomology class ôn(f,, g; ht) E Hn(K , L;n n (Y)).

Analogous to (5.1), we have the following Eilenberg's homotopy theorem. Theorem 8.3.

5n(/, g; ht) — O if there exists a homotopy h t*

Y,

(0 < t 1. Then k t represents an element a of R7,1(K L; f). By (4.5), we have

= M(f, g; ht) — (5n(1, g; h't). Since $n(a) E Pi, this proves that O(/, g) is contained in the coset bn(f , g; ht) + jfn.

On the other hand, let a be any element of Rn(K , L; f). Then a is represented by a homotopy kt and ,n (a) (59/(f, f; k t). Define a homotopy k 2t if t < and h' = h21 _, if h' t : K'n -1 ---> Y, (0 Y. By (6.6), one deduces that h has an extension g : K -4- Y. Then it is clear that fIL and cony, g) = a. I If L is empty, then the preceding lemma can be formulated as follows.

Y is r-simple and Hr+1 (K; ,(Y)) = 0 for every Y satisfying n Y such that x fl(/) (x. Now let / : K --> Y be a given map and let us consider the totality W of the maps g:K-->Y such that fIL gr.L. Since nni (Y) =0 for every m < n, there is only one (n 1)-homotopy class of the maps W relative to L. We are going to enumerate the n-homotopy classes of the maps W Lemma 16.2. If

relative to L. is r-sini ple and Elr-fi(K, L; r( Y)) = 0 for each 7 satisfying n < r < dim (K\ L), then the n-homotopy classes relative to L of the maps W are in a one-to-one correspondence with the elements of the group Hn(K, L;n n (Y)). The correspondence is given by the assignment g ---> , g). Theorem 16.3. If Y

Proof. By (13.3), we have

Qfn(K, L;n, z (Y))

Hn(K, L, nn (Y)).

On the other hand, (16.1) implies that every element of Qf n(K, L;n n (Y)) is f-admissible. Hence, we have

API

=

I-In(K L; nn (Y)).

Then, (16.3) follows from (12.2). The second assertion is obvious. I If L is empty, then (16.3) can be conveniently restated as follows. is r-simple and fIr-f-1 (K;70(Y)) = 0 for each r satisfying n Y are in a one-to-one correspondence with the elements of the group Hn.(K; nn (Y)). The correspondence is determined by the assignment f Corollary 16.4. If Y

If K is of diménsion n, then we have the following corollary which includes the Hopf classification theorem (II; § 8) as a special case. See § 17.

homotopy classes of the maps f: K ---> Y are in a one-toone correspondence with the elements of the group lin(K;nn (Y)). The Correspondence is determined by the assignment f ---> The following generalization of (16.5) is an immediate consequence of (11.2) and (16.3). Corollary 16.5. The

Theorem 16.6 If

Y is r-simple and

L; grr(Y)) = 0 --- lir(K L; r(Y)) for every r satisfying n < r < dim (K \L), then the homotopy classes of the maps W relative to L are in a one-to-one correspondence with the elements of

EXERCISES

193

the group Hn(K, L; n( Y)). The correspondence is determined by the assignment g con (f , g). In particular, the hypothesis of (16.6) is satisfied if nr(Y) n Y dim (K \ L).

0 whenever

17. The characteristic element of Y Throughout this last section of the chapter, we assume that Y is a finite cell complex in addition to the conditions assumed at the beginning of § 13. Then, according to § 13, the identity map t Y Y determined a unique characteristic element n n( t) = co n(t, 0) c Hn(Y :nn(Y)), where 0 denotes the constant map 0(Y) = yo, This will be called the characteristic element of Y and will be denoted by fl( Y). For example, if Y is the n-sphere, then(Y) Z. It can be easily verified that the characteristic element nn( Y) is a generator of the free cyclic group Hn(Y) which depends on the given orientation of Y. Now, consider an arbitrary map f: L -± Y. This map f induces a homo-

morphism

: Hn(Y; an ( Y) )

Hn(L ; nn ( Y)).

The following lemma can be proved by means of (4.6) and simplicial approximation. Lemma 17.1. ;OW

This lemma justifies the assertions of §§ 14-16 that (14.2), (15.3) and (16.5) include the Hopf theorems given in (II; § 8).

EXERCISES A. Minor generalizations of the theory.

I. Make use of the characteristic maps Za in (V; Ex, H) and establish the obstruction theory for the case where K is any topologically realized semisimplicial complex instead of a finite cell complex and L is a subcomplex of K. 2. Study the notion of a CW-comp/ex, [J. H. C. Whitehead 4], and generalize the obstruction theory to the case where K is any CW-complex and L is a subcomplex of K. 3. By the use of local coefficients, construct the obstruction theory for the case where Y is not necessarily n-simple.

B. The fundamental group of a semi simplicial complex. -

Let K be a given connected semi-simplicial complex and vo a given vertex of K. Then, as in (II; Ex. A) for finite simplicial complexes, the fundamental group n1 (K, vo) may be given in terms of generators and relations as follows. A broken line joining a vertex of K to another is a path which consists of

194


a finite number of edges of K. Since K is connected, yo can be joined to any vertex 7) of K by a broken line p(v). Suppose that these broken lines fl(v) have been chosen and that /3 (v o) consists of only a single point y o. To each edge e of K, let us consider the loop

2 e = 10 (e(0))* e [Me(1))14 which represents a element ge of (K, u0). Prove that the set { ge e E K J. generates n i (K, u 0). It remains to find the relations among the generators { ge }. For each edge e e K, the loop A e consists of a finite number of edges of K; more precisely, die e a, ea 2 ... ea n 1

2

where oci = ± 1 for each i = 1, 2, • • • , n. Then, deduce the relation (Re)

ge = g g • On the other hand, for each 2-cell a of K, verify the following relation (Ra)

ga(o ,') ga(1,2) = gc10,2)•

Now prove that n1 (K, y o) can be defined as the group with generators { and relations { R e ) and Ra }. By means of this expression of ni(K , vo), establish the following results: I. Condition for 2 - extensibility. Let L be a connected subcomplex of K (L, v o) (Y yo) be a map into a pathwise connected containing vo and space Y. Then / and the inclusion map i:Lc K induce the homomorphisms 1* ni(L, yo) --1(Y, Yo), i* ni(L, ni(K, vo)Prove that the map / is 2-extensible over K if there exists a homomorphism h:n1 (K,u 0) -->n/ (Y, yo) such that 1 * = hi* and that, for any such homomorphism h, there exists an extension g : K2 --> Y of f such that g* h. 2. Condition for 1 - hornotopy. Let us consider any two maps /, g (K, vo) , yo). They induce the homomorphisms

g*

1 (Y, y0).

:n1 (K,

Prove that / and g are 1-homotopic iff there exists an element such that oc (K, u 0). g* (oc) f * (oc)- c

e 7t 1 ( Y, yo)

,

For an indication of the proof, see [Hu, 7]. C. Generalizing the obstruction theory by using

Z ech cohomology theory

The obstruction theory studied in this chapter is formulated for finite cell complexes so as to avoid the complications involved in limiting processes. One can extend the theory to more general spaces by using äech cohomology theory as follows. Assume that X is a compact Hausdorff space, A a closed subspace of X, and Y a given ANR. For every finite open covering oc a l , • • • , ar of X, the nerve Ka of oc is a finite simplicial complex. The non-void sets A 11 ai,

EXERCISES

195

(i — 1, • , r) , form an open covering of A whose nerve L a is a subcomplex Of Ka. With the aid of the bridge theorems in (II; Ex. B), we can formulate an obstruction theory as follows. A map f : A --> Y is said to be n-extensible over X if it has a bridge OE and a bridge map ya : L a -* Y which is n-extensible over Ka in the sense of § 2. Prove the following assertion which gives set-theoretic meaning to the extensibility defined above. 1. If X is metrizable with dim (X \A) Y is n-extensible over S(X) in the sense of Ex. Al. Prove that a map / : A -÷ y is n-extensible over X if, for every map q: (K, L) -± (X, A) of a semisimplicial pair (K, L) into (X, A), the map /56 : L ---> Y is n-extensible over K. Then, prove that, if (X, A) is a semi-simplicial pair, the definition of n-extensibility defined here is equivalent to that of Ex. Al. Pick x o E 261 and yo — 1(x0) e Y. Then the map / : A -÷ Y and the inclusion I : Ac X induce the following homomorphisms

ft : ni (A, x 0) —>ni (Y , yo), i* :7(10, X o) -÷ Z1(X , x0). Assume X, A, Y to be pathwise connected and prove that / is 2-extensible over X iff there exists a homomorphism h :ni (X, x 0) -->ni(Y, yo) such that /* — hi* . [Hu 7; p. 177]. Assume Y to be pathwise connected and n-simple with nn — nn( IT, yo). Define for any given map f: A --> Y the (n + 1)-dimensional obstruction set On+1 (/) in the singular cohomology group Iln+1(X, A; nn) in obvious way, and prove that / is (n + 1)-extensible over X if On+1 (f) contains the zero element of the group 11914-1 (X, A ; n.). Deduce further results on the extension problem. Now, let us turn to the homotopy problem. Two maps f,g:X -->Y are said to be n-homotopic if the maps /co, gco : S(X) --> Y are n-homotopic in the sense of Ex. Al. Prove that /, g are n-homotopic if, for every map q5: K --> X of a semi-simplicial complex K into X, the maps /96, g56 are n-homotopic. Then, prove that, if X is a semi-simplicial complex, the definition of n-homotopy given here is equivalent to that of Ex. Al. Pick xo E X and yo e Y and assume 1(4 — yo = g(x 0). Then, the maps /, g: X —> Y induce the following homomorphisms

1* , g* :n i (X, xo) -* 1(Y, Yo). Assume X, Y to be pathwise connected and prove that f,g:X—>Y are 1-homotopic iff there exists an element en i (Y, yo) such that

g* a) (

. N - f* (a) - ,

a e ni (X, x 0).

Assume Y to be pathwise connected and n-simple with 7r1i, = n.„(Y, yo). Define for a given pair of maps f, g: X -->- Y the n-dimensional obstruction set On(f, , g) in the singular cohomology group Hn(X ;niz) in the obvious way, and prove that f,g:X-->Y are n-hornotopic iff On(f,g) contains the zero

EXERCISES

197

element of Hit (X; 7c.). Deduce further results on the homotopy problem and the classification problem. E. The obstruction theory of deformation

Throughout this exercise, let (Y, B) be a given pair and (K, L) a given finite cellular pair. A map f: (K, L) --> (Y, B) is said to be deformable into B if there exists a homotopy ft : (K, L) ---> (Y, B), (0 < t - (Y, B) is said to be n-deformable into B if the partial map f I (En, L) is deformable into B. A map f is said to n-normal if f (En --1) c B. Hence f is (n — 1)-deformable into B ill it is homotopic (relative to L) to an n-normal map. Hereafter, let us assume that both Y and B are pathwise connected and pick yo e B. Then it is obvious that every map f: (K, L) --> (Y, B) has to be 0-deformable into B. Prove that every mkp f: (K, L) --> (Y, B) is 1-deformable into B provided that 7c1 (Y, B, yo) — O. The latter is equivalent to the condition that the induced homomorphism i* : ni(B, Yo) -. 1 (Y, y0 )

is an epimorphism. Next, let n > 1 and assume that (Y, B) is n-simple in the sense of (IV; Ex. E). Then the n-th relative homotopy group 7rn -.- 7c.(Y , B) is abelian and may be used as coefficient group. For each n-normal map f: (K, L) --> (Y, B), define an n-cochain cn(f) of K with coefficients in 7C n in the obvious way. Prove the following assertions, [Hu 9; p. 194] : I. c(f) is a cocycle of K modulo L. 2. c(f) = 0 iff there exists a homotopY ft : K -÷ Y, (0 < t Y , (0 (Y, B) is (n — 1)-deformable into B iff On(f) is non-empty. 5. The map I: (K, L) --> (Y, B) is n-deformable into B iff On(f) contains the zero element of Hn(K, L; 7c.). 6. Every map f: (K, L) --> (Y, B) is deformable into B if n i ( Y, B, yo) --= 0 and if, for each n> 1, (Y, B) is n-simple and Iln(K, L;n, i) --- 0, where an — an (Y, B) .

198


7. Assume that K and L are pathwise connected and xo E L. Then, L is a deformation retract of K if 7c,a (K, L; x 0 ) — 0 for every n > I. Develop this obstruction theory of deformation as far as possible and then generalize this theory as in Exs. A, C and D. In particular, deduce a proof of the following assertion, [Hu 9; p. 216]. 8. If Y and B are ANR's and if B is closed, then the following statements are equivalent: (i) B is a deformation retract of Y. (ii) There exists a homotopy ht : (Y, B) ---> (Y, B), (0 < t 0, consider the set On of all ordered sets (xo, • • - , x.) of n 1 elements of 7r called the ii -sets of 7V . Two n-sets (x0, • - • , xn) and (y0, • - - , yn) of 7V are said to be equivalent if there xxi for each i O,» n. The n-sets exists an element x of 7( such that yi On of 7-c are thus divided into disjoint equivalence classes, called the n-cells of 7r.. Tin> 0, the i-th face of the n-cell cr = [xo ,- ••,x.] of 7V , 0 <j < n, is defined to be the following (n — 1)-cell Let

7V

00) = [zo, • • '

xi+i , " xn].

200


Verify the condition (SSC) of (IV; Ex. H). Hence the n-cells of r, (n 0, 1, • • • ), constitute a semi-simplicial complex K(z), called the complex of 7r. 2. The non-homogeneous definition. For each n > 0, consider the set of all n- tu ples (x 1, x) of elements xi e n. If n > 0, the i-th face of the n-tuple ci — (x1, • - • ,xn), O 0, the i-th face of cc, O - Ilni-1(17 , X) °4 H(X) i4*, 14(Y) --.-. where g, h, k are natural homomorphisms, the spherical subgroup Eit(X) ----.. Im(h) of Ji(X) coincides with /m(94t) = Ker(ilt). 5. The induced homomorphism )

Hm(X) .,j1t Hm(Y) R-f, Hm(7c) is an isomorphism for in < n and is a monomorphism for m = n. 6. The subgroup An(X) = /m(i4) of H(X) consists of all elements which annihilate En (X). Summarizing these assertions, we obtain the classical theorem: If 7c i (X) •-.-z, 7( and 7Cm(X) = 0 whenever 1 < m < n, then

Hm (X)r..-ii Hm (n), Hm(X),-P..,, Hm(n), m < n, .1-1n (X)1E n (X) 'AY, Hr), An(X)

Hm(; G) and Hin(n; G) Let G be a given n-group; that is to say, G is an abelian group on which n acts as left operators. A 7c-group G is said to be n-free with ( ga } as a K. Computation of


202

a-basis if the elements g!.,, for all G a and g, e f g,, } form a basis of G. A homomorphism f : A . -)- B of a-groups is called a gr-hornomorphism it it commutes with the operators. A n-group G is said to be a-infective if, for every a-group B and every a-subgroup A of B, every n-homomorphism f : A -›- G has a a-extension g : B -* G. For any n-group G, we shall denote by 1(G) the subgroup of G which consists of the elements g e G such that eg = g for all en, and denote by J(G) the quotient group of G over its subgroup L(G) generated by the elements g — g for all $ G a and g e G. Prove the following assertions: 1. .1/0(n; G) j,r(G), H°(gr;G),-,-; I ,t (G). 2. If G is n-free, then H.(gr; G) = 0 for each m> 0; if G is n-injective, then Hm(a; G) - 0 for each m > O. 3. If a = 1, then H.(gr; G) - 0 - Hllqa; G) for each in > 0; if n is a free (non-abelian) group, then Hm (; G) = 0 = Ho ( c; G) for each m > 2. 4. If a is a free abelian group with r generators and if 7c operates simply

on G, then

lint(7c; G) = 0 - Hn't(a; G),

for m > r,

Hm (gr; G)r-p. G(L) -,-..4 Jim fry; G),

for m < Y,

where 0 denotes the direct sum G + • • • + G of f terms of G. Now let gr be a cyclic group of finite order r and G a 7c-group. Let be a generator of 7C and define a a-homomorphism z1 :G--->G by taking /1(g) = g ± 4 + • • - + e. -- i-g for each g e G. Denote the image of II by 111„(G) and the kernel of Ll by N(G). Then M(G) Œ 1(G) and 1,„(G) Œ 1V,r(G). 5. If n is a cyclic group of finite order r, then for each p > 0 we have H2p, 1 (a; G)

I„(G)1111„(G),,-.), H2P+2(7c; G),

H22, +. 2 (a; G) ,,-,,e N(G)/L(G) F..,, H2P44 (7c; G).

In particular, H2p+2(2.0 , H 22,+2 (70 , 0 , H2p+i(n) .

6. Universal coefficient theorem. If n operates simply on G, then Hm(; G) ;,.•.,/ Hm (a) 0 G ± Tor (Hm ,(7c), G), Hm(; G) ,-,-2, Horn (H,,,(a), G) + Ext (H0 _,(7c), G). 7. Kiinneth relations. For the direct product a x r of two groups 7C and r,

we have

Ho(n X -r) = E Hp(a) 0 Hq (c) + E Toi (Ii 23(7 ) , 14(r)). p-F q =m-i P+q=nt

The assertions 4-7 allow for a complete computation of the groups H.,,,,(a; G) and Hm(a; G) for a finitely generated abelian group 7C which operates simply on G.

EXERCISES

203

K(gr, n) Let gr be a given abelian group written additively and n a positive integer. Define a semi-simplicial complex K(7c, n) as follows. An ni -cell a of K(gr, n) is an ft-cocycle of the unit. m-simplex Am with coefficients in gr. To define the i-th face or(i) of a, 0 < j I and i < j, verify the condition (SSC). Thus we obtain a semi-simplicial complex which is called the Eilenberg-MacLane complex K(7c, n), [Eilenberg-MacLane 1]. Prove that K(n, 1) is essentially K(n) and that K(gr, n) is a space of the homotopy type (gr, n).

a(X) Let n > 2 be a given integer and X an (n 1)-connected space. Study the influence of gr = gr(X) upon the structure of homology and cohomology groups of X as follows. According to (V; 8.2), X can be imbedded in a space Y of the homotopy type (gr, n) such that the inclusion map 1:Xa Y induces an isomorphism i* :t(X) nn (Y). Let q> n be a given integer such that gr.(X) = 0 whenever n q. 2. 11.(Y , X) = 0 for nt (Y, B) is a relative homeomorphism, that is to say, f maps X \A homeomorphically onto Y \B, and if X is a compact space, A is non-empty, Y is a regular Hausdorff space and B is closed, then the induced transformation f* sends nm(Y , B) onto am(X , A) in a one-to-one fashion. Theorem 3.2.

Proof. Since A is non-empty, so is B. It follows that we may identify A and B to single points q4 and qB respectively. Let $ : (X, A) --> (X A , qA), 7] : (Y, B) --->- (Y11, q B ) denote the natural projections of identification. One can easily verify that f induces a one-to-one map g: (X4, qA) --> (Y B, qB)

of (X A, qA) onto (Y11, 0) such that Tif = a. Since X is compact, so is XilSince Y is a regular Hausdorff space and B is closed, it follows that YB is a Hausdorff space. Hence, g is a homeomorphism. By a property of contravariant functors, we have f*Ti * = $*g*. Since g is a homeomorphism, it is obvious that g* is a one-to-one correspondence. On the other hand, $* and 21* are one-to-one correspondences according to (3.1). Therefore, f* — $*g*27* -1 sendsam(Y, B) onto nm(X , A) in a one-to-one fashion. I The auxiliary conditions in (3.2) are essential. In fact, we have the following counter examples. First, let Y denote the unit m-cell of the euclidean m-space and B its boundary (m — 1)-sphere. Let yo , Yi E B where yo 0 3/ 1. Let X=--- Y \ yo

208

VII. COHOMOTOPY GROUPS

and A B \ yo . Then clearly there is a deformation retraction of (X, A) into the pair (Yi' Yi). Therefore 7Cm(X , A) = 0 although 7CM (3{ B) c Z. This shows that the induced transformation i* : 7rm(Y, B) --> 70 (X , A) of the inclusion map i (X, A) c (Y, B) , which is a relative homeomorphism, cannot be one-to-one. In this example, X fails to be compact. 1)-cell of the euclidean (m 1)-space Next, let Y denote the unit (m and X the boundary m-sphere. Let A consist of a single point of X and let B (Y \ X) U A. Then nm( X , A) Z while nm(Y , B) 0. This shows that the induced transformation i* : am(Y , B) -3-7rm(X, A) of the inclusion map i: (X, A) c (Y, B) , which is a relative homeomorphism, cannot be onto. In this example, B fails to be closed. Corollary 3.3.

(The excision theorem). Let (X, A) be a pair where X is a

compact Hausdorff space and A is a closed subspace of X. If V is any open set of X contained in A, then the induced transformation e* of the inclusion map

e : (X \V , A \V) c (X, A) carries nm(X , A) onto vxm(X \V , A\V) in a one-to-one fashion.

if Since e is a relative homeomorphism, (3.3) follows from A \V is non-empty. If A \V is empty, then A = V and hence A is both open and closed in X. Therefore, every map b: X \ A —> Sm has a unique extension p : (X, A) --->- (Sm, so). This proves that e* is a one-to-one correspondence. I Proof.

(3.3) that X is a compact Hausdorff space can be removed if we assume that the closure of V is contained in the interior of A. The verification is left to the reader. Remark. The condition in

4. The coboundary operator In the present section, we are going to construct a coboundary operator for cohomotopy sets which will be analogous to the coboundary operator

for cohomology groups. Unfortunately, this coboundary operator (5 is not defined for a completely arbitrary pair (X, A). Since, in the construction of 6, one has to use some form of Tietze's extension theorem, it is necessary to assume a normality condition on the pair (X, A). By a binormal pair (X, A), we mean a binormal space X, (I; § 9), and a closed subspace A of X. Then, by definition, both X and X X I are normal spaces. In particular, (X, A) is a binormal pair if X is a semi-simplicial complex and A is a subcomplex of X, or more generally, if X is a para and A is a closed subspace of X. Throughout thecompatHusdrfe present section, we assume that (X, A) is an arbitrarily given binormal pair. 1)-sphere obtained by joining Sm to two points, Let Sm 11 denote the (m the north pole and the south pole. Denote by E:+1 and Eln+1 the north and the south hemispheres respectively. Then we have Sm+1 E7+1 U E 1!+1, so e sin E' n Ent+1.

4.

THE COBOUNDARY OPERATOR

209

Consider the following diagram

n(X , A; Sm+1, so)

n(A: Sm)

H

Tc(X, A; E7+1 , Sm) IL(X, A; 5 n1+1, a 1) where the transformations are natural, namely, a is defined by restriction while /3 and y are induced by inclusions. Since Er', Em + .1 are contractible to the point so and (X, A) is a binormal pair, it is straightforward to verify that both OL and y are one-to-one correspondences. Thus, we obtain a transformation gen(A) 7.0 n 4-1(x A) , which is called the coboundary operator. Geometrically, the coboundary operator 6 can be determined as follows. Let e E nm(A). Choose a map 95 : A -›- Sm which represents e. Since Er I- is contractible, q has an extension

04* : (X , A) -± (EV -1, Sm). Then, 4 4* represents oc-1 (e). Let

$t (Er', Sm)

(Sm+1, ET,±1.),

0
n. For any given pair (Y, yo) consisting of a space Y and a point yo e Y, consider the set n(X , A; Y, yo) of all homotopy classes of the maps (X, A) Y', y' 0) induces a transformation (Y yo). Then every map f: (Y, yo)

1* gr(X, A; Y, yo) --->-n(X, A; Y', y' 0) by means of composition. In particular, let us consider a triplet (Y, B, yo) where Y and B are pathwise connected. Then the inclusion map 1: (B, yo) (Y, yo) induces a transformation 7( (X,

A; B, yo) --->- 7t(X, A; Y, yo).

We recall that (Y, B) is said to be n-connected if 7rq(Y , B) = 0 for every q < n. Then the following lemma can be easily established by the obstruction method described in (VI; Ex. E).

5.1. If (X, A) is n-coconnected and (Y B) is n-connected, then i,R sends n(X, A; B, yo) onto n(X, A; Y, y o) in a one-to-one fashion. Throughout the remainder of this section, let m be any given positive integer and assume that (X, A) is (2m 1)-coconnected. Under this assumption, we shall define a group operation in m(X, A). For this purpose, Lemma

let us consider the triplet

yo — (so, so). Then, by (IV; 3.4), (Y, B) is (2m — 1)-connected and hence i* is a one-toone correspondence according to (5.1). s Next, consider the map j : (B, yo) (Sm, so) defined by j(s, so) i(so, s) for every s in Sm. Then j induces a transformation j* : (X, A; B, yo) -->nm(X, A). Y---- Sm x Sm, B

Sm V Sm,

Finally, we shall define a transformation

h* : 7Cm (X, A) x nm(X , A) -->n(X, A; Y, y o) as follows. Let (cc, #) be any pair of elements in 7(In ( X , A). Choose represen(5m, so) for oc, /3 respectively. Then the map tative maps 0, v : (X, A) çb X p: (X, A) (Y, yo) represents an element of n(X , A; Y, yo) which obviously depends only on the ordered pair (cco3). We define h* (cc, ,e) to be this element. Composing these three transformations, we obtain a transformation

k=

Am(X, A) x am(X, A) --÷7rm(X , A).

5.

THE GROUP OPERATION IN

nrn(X, A)

211

For any pair (a, /3) of elements in 9X, A) , the element k(a f3) of nm (X , A) will be called the sum of ac, /3 and denoted by cc + p. a (2m —1)-coconnected cellular pair, then nm(X, A) forms an abelian group with the addition a + 13 as its group operation and 0 as its group-theoretic neutral element. If a EZ 121 (X, A) is represented by a map : (X, A) -->- (Sm, so), then its negative — a is represented by the composed map rO, where : (Sm, So) (Sm, So) is an arbitrary map of degree —1. Theorem 5.2. If (X, A) is

(a) Commutativity. Let us consider the homeomorphism : (Y, y o) (Y, yo) defined by t) (t, s) for every pair of points (s, t) in Sm. Since maps B onto itself, it defines a homeomorphism : (B, yo) --> (B, yo). Obviously we have in = i and j j. Therefore, the following diagram is commutative: Proof.

, A; B, yo)

3.E- (X , A; Y, yo)

m( X, A) n(X, A; Y, yo)

z(X,A;B,3 1 0)

Let a, fi be any two elements of 7rm(X, A). It follows from the definition of h* that * h * (a , 13) = h * (p, oc) Hence

fl + a = /** 4 1h* (fl, oc) = j* ii, 14h* (a, 13)

= i*N4, h*(cc, 13) = i*4111*(cc, 18) (b) Associativity. Consider the triplet (Z, C, 2'0), where Z

Sm x Sm x Sm,

C

= oc

Sm V Sm V Sm, zo = (s

+

P. so).

Then, by (I; Ex. S), it is easy to verify that (Z C) is (2rn — 1)-connected ( Z, z o) induces a oneand hence, by (5.1), the inclusion map (C, 2'0) to-one transformation : n(X, A; C , zo) -->-n(X , A;

, zo).

so) denote the map defined by (if 5 2 = so 53 ), S1, (if s i = so = so), 2,(s 1, s 2' = s2 , (if s i = so = s 2 ). s3 Then A induces a transformation Let A.: (C,

2'0) ---> ( Sm,

A * : 7c(X , A; C, z o) .-)-vxm(X , A).

Now, let oc, fl, y be any three elements of nm(X, A) and choose representative maps sb, 7/), Z : (X, A)

(Sm, so)

for a, 16, y respectively. Then the map 95 x x X represents an element


212

y) of n(X, A; Z, z o) which depends only on the triple (a, 13, y). itt.(a, Thus we obtain an element A, * 1,u* (cc, p, y) of nm (X , A). We are going to prove that (cc + 18) + y = A 1 (cc, /3, y = cc + (r3 ± Y). (Y, yo) denote the map defined by For this purpose, let p: (Z, z p(s i , s 2 , .33) = (.31, s 2). Since p sends C onto B, it defines a map 0 (C, z o) (B, yo). Then the rectangle n(X, A; Z, z o)

n(X,A,C,z o) 1 °.

n(X, A ; Y , y o)

A; B, y o)

is commutative. Furthermore, by the definition of p, it is clear that PO*(cc , 06, y) = h*(cc , Choose a map : (X, A) -> (C, z o) representing the element %;,-, 1 /2* (a, fi, y). Then A/represents the element Aoci;ly,R (a, fi, y) and j0/ represents the element s3. cx + p. Let co : (Z zo) (Sm, so) denote the map defined by co(s l , 5 3 , s3) Thén the map (o/ clearly represents y. Since / sends X into C, it is easy to see that jef x co/ carries (X, A) into (B, yo) and that j(jOf X (o/) This implies that (cc + 13) ± y = Aidç1itt*(0E, 13, y) Similarly, we can prove cc + (13 + y) = A*x,-,-1/2* (cc, /3, y). This completes the proof of associativity. (c) Existence of neutral element. Let a be any element of m(X, A) and choose a representive map 56 : (X, A) -> (Sm, so) of a. Let : (X, A) -> (Sm, so) so. Then x sends (X, A) into (B, y o) denote the constant map (X) and clearly /(0 x = 0. This implies that cc -I- O = cc for every oc e nm(X , A). (d) Existence of negatives. Let cc E 2-Cm (X , A) be represented by ç : (X, A) -> (Sm, so) be any map of degree — I. Then p f. (Sm, so) and let r : (Sm, so) represents an element fi E 71m (X, A) which depends only on cc. We are going to prove that a + 9 = O. Without loss of generality, we may assume that r is defined by

" xm--1, xm) = (xo, " xm-i,

xm)•

Let ET and Er denote the hemi-spheres of sin defined by xm > 0 and xm (Sm, s o), 0 < t (E + 74 +1, Sm). Composing with the relative homeomorphism $, of § 4, we obtain a map

$ 1g4 : (X, A) -> (5m 41 , so). Since gII I A = g, /It represents the element 6(13) according to § 4. We have to prove that f and /4 are homotopic relative to A. Since 1h— /4 I K relative to B. g4 j K h, we can easily see that f j K f#: 1 K U A relative Since 1(A) = s o 14(A), this implies that fIKUA to A. The cones over the simplexes of A constitute a triangulation of X. In this special triangulation of X, we have K U A = X 2r4 U A. Hence f and /4 are 2m-homotopic relative to A. Since A is 2m-coconnected and X is /ft


218

contractible to a point, it follows that (X, A) is (2m + 1)-coconnected. Hence, by (VI; 11.2), / and /* are homotopic relative to A. Thus we have proved the lemma for the special case that either dim A < 2m — 1 or m -= 1. Next, let us prove the lemma with the condition that m> 1 andA is simply connected. 1)In this case, we shall first prove that A is dominated by its (2m dimensional skeleton B. Since 2m — I > 3 and A is simply connected, so is B. Hence the pair (A, B) is n-simple for every n > 2m. Consider the inclusion map e: (A, Am -2) c (A, B),

and apply the obstruction method described in (VI; Ex. E) for deforming e into B. Since Hn (A) — 0 Hn A 2M- 2) =

for every n > 2m, it follows that e is deformable into B with A*71 -2 held fixed. In particular, there is a map : A -› B such that the composition ij with the inclusion map 1:Bc A is homotopic to identity map on A. This proves that B dominates A. Since X and K are the cones over A and B respectively, the maps i

extend to maps

(K , B) : (X , A) : (K , B) (X , A) , A the obvious way so that i in is homotopic to the identity map on (X, A). A

This gives a commutative diagram i*

nm(A)

, nm(B)

* 70(A )

6 A

am-f-1 (X , A)

•*

_>

am4-1-(K , B)

am -f-1 (X , A).

Given oc e04-1 (X, A) , then iA*(oc) enm 4-1 (K, B). Since dim B < 2m — 1, we have prove that sends 7Em (B) onto m-f-1 (K, B). Hence there exists an element y e am(B) such that (y) iA* (a) . Let fi = j *(y) e nm(A). Then we have (5 68) = 6i * (Y) = Ai"(y)

Thus we have proved the lemma also for the case that m> 1 and A is simply connected. Finally, let us prove the lemma for the general case m> 1. Let A' C X denote the union of A and the cone over its 1-skeleton. Then A' is simply connected and 2m-coconnected. Consider the inclusion maps k : A C A' and A) C (X, A'). We obtain the commutative rectangle :

am(A.')

nm(A)

16 am+1-(X,

A') !AL* nm 4-1-(X , A).

8.

THE STATEMENT

(6)

219

Since dim (A' \A) 2, it follows that kA* is an epimorphism. Let X' denote the cone over A'. Then (X, A') is homotopically equivalent to (X', A'). Hence 8 sends 70(A') onto 7rm -1-1 (X, A') since A' is simply connected and 2m-coconnected. Then it follows from the commutativity of the rectangle that 6 carries 7rm(A) onto 70+ 1 (X, A). I

8. The statement (6) A triple (X, A, B) is said to be (finitely) triangulable if there exists a finite triangulation of X such that A and B are both subcomplexes.

1/ (X, A, B) is a triangulable triple such that (A, B) is 2mcoconnected, then the statement (6) of § 6 holds. More explicitly in the sequence nin(A B) 71271+1 (X, A) _it). nnti-i(X , B) Lemma 8.1.

the image of 6* contains the kernel of j*.

Proof. We shall be dealing with subsets of X x I and will use the following abridged notation. If Y is a subset of X, we define Yo = Y x 0,

YX

Y, Y x 1.

Let /: (X, A, B) X 1 U A 0 U B1, X1U B1) denote the map defined by f (x) = (x, 0) for every x E X. Let /1 = (A B), iz=li (X A)According to § 3 and § 4, commutativity holds in the following diagram: (5*

7cm(A, B)

1 nm(X 1 U A ° U B1, X 1 U Bi) /1*

nm+1-(X, A) — 1* orm+1-(X , B) *

12* nm+1 (X1, X 1 U A ° U B

Since f, is essentially the excision map obtained by excising (B B0) U X, from X, U A o U B1 , it follows from (3.3) that f l* is one-to-one and onto. By the definition of homotopy, one can easily see that 1m (/ 2*) Ker (i*). By the commutativity relation 6*1 1 * 1 2 *(5 1 *, it suffices to prove that the coboundary operator 6 1* is onto. Let g: (X 1 U AI, X 1 U A o U B1) (X1, X, U A o U BI) denote the inclusion map. Since X 1 U Ai is a deformation retract of XI according to (I; 10.1), it is easy to see that the induced transformation g* is one-to-one and onto. By the commutativity of the triangle 7CM(X 3 U A o U B a

, X 1 U Bi)

0,4-1 (Xi, X 1 U A o U Bi) g*/

U AI, X 1 U A o U it remains to prove that the coboundary operator 6 2 * is onto. 7Cm+1 (2 1


220

Identify B to a point q and X 1 to a point r, and let h: (X 1 U Al, X, U A o U Bi, X 1 U B7) —> (AB, AB U

4, .'4)

denote the identification map, where if Y is a subspace of Ao = A, Y denotes the join of Y with the point r, and AB denotes A 0 with Bo identified to a point q. Let X,U Br), h, = h 1 (X,U AI, X„U A o U B1).

h/ — hi (X,U A o U

According to § 3 and § 4, the following rectangle is commutative: nm()C i

*

U A o U /31, X, U

1

111* 7CM(AB

7rm+1(X 1 U AI, X, U A o U 134 1h2*

U

15 3*

'14)

--->- 7Cm+1 (AB, B

U

A

Since h, and h 2 are obviously relative homeomorphisms, it follows from (3.2) that both h1 * and h 2* are one-to-one and onto. Hence, it remains to prove that 63* is onto. Consider the inclusion map k: (AB, AB, 0) and let .k, = k 1 (A B , o) and k 2 k

(1413, AB U 4)

1(4B , AR).

Then, we obtain a commuta-

tive rectangle: 7rm(A B U

4,4) '3%

m+1 (A 13, AR

U q)

lh 2* 7Cm(AB)

4*

nm+1 (A B, A

B) •

Let a E7Cm (AB) be an arbitrary element represented by 0 : A B ---> Sm. Since Sm is pathwiseA connected, we may assume that 0(q) = so. Then 0 has a unique extension 0 : (A B U 4, 4' ) -÷ (sin, so). Hence /a l* is onto. Since A B is a strong deformation retract of A B U cl , it follows that k 2 * is one-to-one and onto. Since (A, B) is 2m-coconnected, so is AR. Hence, by (7.1), 6+4 * is onto. This implies that 63* = 4 -1-154*k 1 * is also onto. I

9. The statement (5) If (X, A, B) is a triangulable triple such that (X, A) is 2mcoconnected, then the statement (5) of § 6 holds. More explicitly, in the sequence Lemma 9.1.

7(74 (X,

B)

am(A, B)

m 41- (X, A),

the image of i* contains the kernel of cr Proof. Let a enm(A, B) be an arbitrary element with 6*(a) = O. Let : (A, B) (Sm, so) be a map which represents a. We are going to extend 0 throughout X by means of stepwise inductive construction over the ndimensional skeletons In = Xi/ U A of X.

9.

THE STATEMENT (5)

221

It is obvious that 95 can be extended over Xm. Assume that n is an integer with m 2m, then the following part of the cohomotopy sequence is exact: am(X,A) 7cm(X,B) nm(A ,B)

---> 7CM-1-1 (X , A) f---> 704-1 (X ,B) • • •

(X, A) is a triangulable pair such that H(A) = O = Hn(X , A) for every n > 2m, then the following part of the cohomotopy sequence is exact: i* — 7tm(A) 7(M + 1 (X A) 12:-->- 7CM +1 (X) i* nm(X , A) ± Corollary 9.3. If

:

W. Higher cohomotopy groups Theorem 10.1.1/

(X, A) is an m-coconnected cellular pair,,thennm(X , A) = 0.

Proof. Since Sm is (nt 1)-connected, every two maps 51), v : (X, A) (Sm , so) are (m — 1)-homotopic relative to A. Then, by (VI; 11.3), sb and v are hornotopic relative to A. I —

-

The significance of this theorem is that, for a (finite) cellular triple (X, A, B), the cohomotopy sequence of (X, A, B) ends with a term O. Corollary 10.2.11X

is an m coconnected triangulable space, then 7Cm(X) = O. -

In particular, the corollary includes the obvious special case : If X consists of a single point, then nm(X) = 0 for every ni > 0; on the other hand, 70(X) clearly consists of two elements.

11. Relations with cohomology groups In (V; § 4), we constructed a natural homomorphism hm from the homotopy group grm (X, A) into the homology group Hm (X, A) over integral coefficients. In the present section, we shall define a similar operation hm : nm(X , A) ÷ Hm(X , A;n m (Sm , s0)), ni> 0, -

which will be a homomorphism if (X, A) is an (2m — 1)-coconnected cellular pair. Since 7rm(SM, so) is free cyclic and the identity map on Sm represents a generator, it can be identified with the group Z of integers in a natural way. Hence we may denote the cohomology group Hm(X , A ; grm (Sm , so)) simply by Hm(X, A). In order to construct the natural homomorphism hm, let us denote by X. the characteristic element of the cohomology group Hm(Sm, so) as defined in (VI; § 17). This characteristic element Xm can also be defined by the natural homomorphism hm :7(m (SM, so) .Hm (Sm , so) as follows. By Hurewicz's theorem, (V; §4), hm is an isomorphism and hence the inverse hm-1 is a well-1 defined homomorphism of Hm (Sm, so) into am (Sm, so) = Z. Therefore, hm determines a generator of Hm(Sm, so) which can be easily proved to be Xm.

II. RELATIONS WITH COHOMOLOGY GROUPS

223

Now let us define the natural operation hm as follows. Let a enm(X, A) be an element represented by a map s6 : (X, A) (Sm, s o). st. induces a

homomorphism

56* : ,Hm(Sm, so) --> fim(X, A).

If we use singular cohomology, than 0* depends only on a according to the homotopy axiom. In this case, we define hm() 0*(Xm). This completes the definition of the natural operation hm. Proposition 11.1. If

(X, A) is a (2m — 1)-coconnected cellular pair, then

hm is a homomorphism. Proof. Let pi : Sm x Sm .÷ Sm, (i = 1, 2), denote the projections defined Sm denote the map defined in § 5; by pi(y i , y 2 ) yi; let f: Sm V Sm and consider the inclusion map k : Sm V Sm Sm x Sm. These maps induce

the homomorphisms pi* : Hm(sm, so)

Hm(sm, so)

x sm,

Hm(sm

-÷ Hm(sm V

(so ,

sin, (so, s0)),

: Hm(Sm x Sm, (so, so)) Hm(Sm V Sm, (s o, so)). It is easy to see that j*(Xm) ep,*(x.) + k*15 2 *(X m). Now let a, p E nm (X , A) be represented by 95, p: (X, A) (Sm , so) respectively. By (5.1), sb X ip is homotopic to a map g: (X, A) -÷ (Sm V Sin, p 2kg relative to A. (so , so)) relative to A. Then we have st p, kg and v Hence

hm (oc) + hm (/9)

0 * (xm) + V* (Xm) g*k*p 1*(X m) g*k*p 2*(X m ) g *i*(x.) .

On the other hand, since fg : (X, A) -÷ (Sm, so) represents a we have hm(ot p) .„ (7. g)* (Zm ) , g* i* (X m) This implies that hm(a Proposition 11.2.

is commutative:

# by definition,

hm(13). I

#) = hm(x)

For any map f: (X, A) –› (Y, B), the following rectangle gm(X, A)

P

am(Y , B) hm

/en

Y

Hm(X, A)

P

Hm(Y, B).

Proof. Let a Enm(Y, B) be represented by *(a) is represented by Of and hence

4: (Y, B) (Sm,

hint * * = (01) * ( 7m) = /*0*(Xm) = i*hm(x).

so )• Then

VIL COHOMOTOPY GROUPS

224

Proposition 11.3. For any binormal pair (X, A), the following rectangle is

commutative:

nm(A)

I

6

nm+1

kin

hm -

Hm (A)

(X, A)

6

—>. Hin-F1 (X, A).

Proof. Consider the map i of § 4. Then we have Hm(Sm, so ) 6*

Hin+1 (Em + +4, Sin) . cY1 9 (B).

One can easily verify that the operation ,T is a covariant functor from gd to itself, [E-S; p. 1 1 1 ]. Two mappings /, g : A -- B in gd are said to be homoto pic (notation: f g) if there is a homomorphism : A B such that g

The homomorphism is called a homotopy (notation: :/ g). One can (g). easily prove that f rze. g implies'A' (/) Let an exact sequence (2.1) in gd be given. We shall define a mapping (C) ---> A9(A)

as follows. Let oc e(C) and choose x eff (C) which represents oc. There is some y E B with g(y) = x. Since gd(y) = d(gy) = 0, there is a zEA with (z) = d(y). Since /c1(z) d(/z) = 0, d(z) = 0 and z represents a 13 e Ye (A) which depends only on cc. Then 0 is defined by d(oc) = /3. Since the differential operators on. Yf (C) and A°(A) are trivial, 0 is a mapping in gd. Thus we obtain a triangle in gd: '(A).

` rt° (1)

Xt.° (B) /.‘(g)

(C)

3.

GRADED AND BIGRADED GROUPS

231

One can verify that this triangle is exact in the sense that the kernel of each homomorphism is precisely the image of the preceding.

3. Graded and bigraded groups An abelian group A is said to be graded, or to have a graded structure, if there is prescribed to each integer n, (positive, zero, or negative), a subgroup A n of A such that A can be written as a (weak) direct sum

A = E A R. The elements of the subgroup A. are said to be homogeneous of degree n. Similarly, an abelian group A is said to be bigraded, or to have a bigraded structure, if there is prescribed to each (ordered) pair (m, n) of integers a subgroup A m ,. of A such that A can be written as a direct sum A =E

nz,n

The elements of the subgroup A m , n are said to be homogeneous of degree

n) When dealing with graded or bigraded groups, only a certain limited class of homomorphisms are of interest, namely, the homogeneous holmmorphisms. If A E B E Bm , R

tn,n

m,ft

are bigraded groups, then a homomorphisrn f : A geneous of degree (p, q) if RAM,n) C BM n+q

B is said to be homo-

for every pair (m, n). With bigraded groups as objects and homogeneous homomorphisms as mappings, we obtain a category gb called the category of bigraded groups. Similarly, one can define the category gg of graded groups. If f :A-4-B is a mapping in gb of degree (0,0) such that A c B and f (a) = a for each a E A, then A is called a subgroup of the bigraded group B. Let C = BIA. Then one can verify that C is isomorphic to the direct sum of the groups Bm ,.IA m ,.. We agree to identify these naturally isomorphic groups. Then C is bigraded and the projection g:B-÷C is a mapping in gb of degree (0, 0). With this bigraded structure, C is called the quotient bigraded group. Note that both the kernel and the image of a mapping in gb are subgroups of the corresponding bigraded groups. One can easily formulate the analogous concepts for graded groups. In algebraic topology, we have to deal with graded (or bigraded) groups with a differential operator which is homogeneous. In this case, the derived group is also graded (or bigraded). For example, let us consider the group C(X) of all singular chains in a space X with integral coefficients as defined in [E-S; p. 187]. C(X) has a natural graded structure

C(X) =

Cn(X).

232

VIII. EXACT COUPLES AND SPECTRAL SEQUENCES

where C(X) is the group of n-chains if n > 0 and CU (X) — 0 if n < O. The boundary homomorphism 0 : C,,,(X) --> C 1 (X) extends to a differential operator d = 0 : C(X) --> C(X) which is homogeneous of degree — 1. Hence the kernel Z(X) of d and the image B(X) of d are subgroups of the graded d-group C(X) with graded structures

Z(X) = E Zn (X), B(X) — E B.(X). n

n

Furthermore, the derived group II(X) of C(X) has a graded structure H(X).., t, 11.(X),

where H(X) is the n-dimensional singular homology group of X if n > 0 and H(X) = 0 for n

which consists of two abelian groups D and E, and three homomorphisms i : D ---> D, i : D -> E, k : E ---> D

such that the following triangle is exact: D

i

,\

--> D

1\

/ E

X

There is an operation which assigns to an exact couple t9 another exact couple

(‘' — < D', E'; i', j', k' >

called the derived couple of W constructed as follows. Define an endom orphi sm d : E -> E by d = fk. Since ki = 0 by exactness, it follows that dd = f kf k — f(k1)k — O. This implies that d is a differential operator on E. Let D' = i(D), E'

then D' is a subgroup of D and E' is the derived group of the d-group E. Since D' c D, i(D`) c D'. We may define an endomorphism D' --> D' by i' = I I D'. Since one can easily verify that k[Y(E)1= D', k[R(E)1 = 0, k induces a homornorphism le ; E, , D,.

EXACT couPLEs

233

Let xED' and choose a yED with i(y) = x. Then f(y) is in T(E) and the coset of i(y) mod M(E) does not depend on the choice of y. Denote this coset by jr(x). The assignment x f'(x) defines a homomorphism

: D' -÷ E'. This completes the construction of The verification that i', h' form an exact triangle is straightforward and is left to the reader. This process of derivation can be applied to W' to obtain a second derived , and so on. In this way, we obtain a sequence of exact couples couple

n = 1, 2,• • •, defined inductively by (el

7,

7n

= ( Wn-ly

n > 1.

} has two important properties. Firstly, the groups Du The sequence { form a decreasing sequence D

D' D D2 • • •

Dn+1 D • • •

with in : Dn —> Dn defined by the restriction of i to D. The intersection of the groups Dn is denoted by Dc . Secondly, it has been shown above that the endomorphism

dn juku : En

En

is a differential operator on En and En-H- is the derived group of En with respect to dn. Hence we obtain a sequence of differential groups E = El, E2, ••, En, • • such that En+1 'clf (En) for each n > O. This will be called the spectral sequence associated with the exact couple W. In the spectral sequence { En } there exist natural homomorphisms

: f(En) --> En+1 defined by assigning to each element of Y(En) its coset modulo (En). Thus hn is an epimorphism of a subgroup of En onto En+1. We can define an epimorphism hp, of a subgroup of En onto EP by the formula hn+p - 0'724-P - 2

hn.

Then hn.1 The precise domain 212, of definition of 4, can be defined inductively as follows :

{ a E 2t-1 1 h 1 (a) E

(En+P-1) }, p> 1.

1, 2,— • , then, Let En denote the intersection of all subgroups 2, p for each a e En, ht(a) is defined for all values of p. Define jin : E,n to be the restriction of hn to En. Then the sequence of groups { En } and constitutes a direct sequence of groups in the usual homomorphisms { }

234


sense, [H—W; p. 132]. The limit group .E°° of this direct sequence of groups will be called the limit group of the spectral sequence { En }, In particular, if there is an integer r > 1 such that cln = 0 for each n > Y, then (En) = En, h. : En En+1

= h. for each n > r. This implies that and therefore En = En and Er. Returning to the general case again, we can express the limit group E' in terms of the original exact couple W as follows. Consider the epimorphisms i(n) inin-1. • : D Dn+lc D. Since in j I Dn for each n > 1, i(n) is actually the n-fold iteration of the homomorphism I D -9- D. Let D"

n

Im [i (n) ],

Dn

U Key [1(10],

D°

where 1m [1(n) .] and Ker [i(n)] denote the image and the kernel of i(n) respectively. Then it is easy to verify that EG° In the following chapters as well as in the exercises at the end of this chapter, we shall give detailed accounts of some of the important exact couples. For the moment, let us be content with a simple illustrative example. Consider a (finitely) filtered space, i.e., a space X furnished with a finite increasing sequence of subspaces

(0)

C Xo c

X1

c •••c

X , = X.

Using total singular homology groups over a given coefficient group G, we define D = E H(X p), E E H (X2, Xp_1). Then the homology sequence of the pairs (X2 X2 couple WH(0) = ,,

„.. 1

) give rise to an exact

called the homology exact couple of the filtered space X. 5. Bigraded exact couples In the exact couples W < D, E,i,j,k> which we shall deal with in the sequel, the groups D, E are usually bigraded and the homomorphisms , k are homogeneous. In this case, in the successive derived couples Wn of W, the groups Dn, En are also bigraded, i.e.,

Dn = E Dn En = E E" PO'

P

' q

Pig Pa

and the homomorphisms in, in, kn, and dn inkn are homogeneous. The

5.131GRADED EXACT COUPLES

2 35

elements of £;; g and EL are said to be homogeneous of degree (p, q). We shall call p the' primary degree, q the complementary degree, and p q the total degree.

It is easy to verify the following relations about the degree of homogeneity: deg (in) = deg (i),

deg (kn)

deg (in) = deg (f)

deg (k),

(n — 1) [deg (i)].

Hence the degree of the differential operator dn, is given by the formula: deg (du) = deg (j) ± deg (k)

(n —1) [deg (I)].

The following two special classes of bigraded exact couples are important. A bigraded exact couple < D, E; ti, j, k 7. will be called a 0-cou75/e if i is of degree (1, — 1), i is of degree (0,0) and k is of degree (— 1, 0). In this case, the degrees of in, f, k, and dn are listed as follows:

(1) in is of degree (1, — 1) ; (2) in is of degree (— n + 1, n — 1) ; (3) kn is of degree (— 1, 0) ; 1). (4) PI is of degree (— n, n

Similarly, ee' will be called a b-couple if i is of degree (— 1, I), j is of degree (0, 0), and k is of degree (1, 0). Then the degrees of in , in , kn, and dn are listed as follows: (61) in is of degree (— 1, 1); (62) in is of degree (n — 1, — n + 1) ; (63) kn is of degree (1, 0) ; n 1). (M) dn is of degree (n,

< D, E;i,j,k> is a quite elaborate structure; it can be developed into a "lattice-like" diagram as follows: A 0-couple

I • •.

i

D p +1

>

Ep , q+1

h

E

q+1

h

i

•

h

np +1, q

Ep+i, q

...

h

DP+21q- I

—P+2, q--3

h

Dp , Q,

) Ep,

h

•Dp

h

--> DP+1,q- 1

k --> EP+11q- 1 —>

p

• •

236


The steps from upper left to lower right are exact sequences, for example, h •••

j

r

1Jp _1, q + 1 -÷ p q

k pq

Dp -1, q

i

D ,

is an exact sequence. Similarly, one can develop the derived couples Wn, n = 2, 3, • • , into analogous diagram. It is more or less evident that the notions of 0-couple and 6-couple are dual to one another. Thus, let = < D, E, 1, j, k> be any given bigraded exact couple. We shall construct another bigraded exact couple (C* — by merely reindexing the groups < D*, E*; 1*, f*, k* >, called the dual of Dp,q and Ep, g, as follows: Dp* , q Then we obtain D* — D and E* = E; therefore, we may take i* j* f, and k* k. Their degrees are given by — deg (k). — deg (1), deg (f*)= — deg (f), deg (k*) deg (1*) Hence the dual of a 0-couple is a 6-couple and vice versa. For example, the homology exact couple WH(0) of § 4 is a 0-couple Indeed, the groups D and E are bigraded, namely, Ep,q Hp +q(Xp, Xp-1); Dp,q = Hp,q(X p), and the homomorphisms i,f,k are obviously homogeneous of degree (11,— 1), (0, 0), (— 1,0) respectively.

6. Regular couples A 0-couple = < D, E ;1, f, k> is said to be regular provided: (R1) Dp,q = 0 if p < O; Ep,q 0 if q < 0. (R2) (R01) and the exactness of Dp, q

imply that (R3)

Ep,q =

0 if p

DP - 1, q

4,48

< 0. Hence,

EL = 0 = EL,

for any positive integer n, we have

if p

< 0 or q < O.

1). If n> p, each element of E7, q is a cycle since dn is of degree (— n, n If n> q + 1, no non-zero element ' of EL can be a boundary under dn. Hence (R4) EnM+1 • • • = E' if n> max (p, q 1), P4

Since each ir is of degree (1, — 1), we have D'P4 = in-1 in- 2 • • • il (Dp-n-f-i,q+n-i)• (R5) This and (R01) imply D3 = 0 D' if n > p + 1. (R6) P4

6. REGULAR COUPLES

Consider the homomorphismi : Dp, q D

237

q . (R02) and the exactness of

Dpq D19÷1;q-1' EP+1)q imply that i is an epimorphism if q < O and an isomorphism if q < O. Thus we obtain a sequence Do

,

m

7-N

Dm ,

•••

Dm-f-i, —1 R-id

Lint+ 2,- 2

"

16 /1

-ilm-f-r,—r

"

where 1: Dm, 0 -› D +11 is an epimorphism. This suggests defining a graded group Ye(V) E.Yem,(V), Ye rn(W) = Then D„n, +,,_, is a subgroup of Y4 (') and (R05) gives a homomorphism )1,2), q : Dp,q

where

9 p+q(V)

(q > 0),

4, q iq +1 iq •

is actually the (q 1)-fold iteration of the homoD. Denote the image of 4, q by

morphism 1: D

q(DP,q) q(W) 144:1-2q If p < O then Jfp , q (V) = 0 since Dp , q = 0. If q = 0, then Ye p , q(W) Yepfg e) since 4 1 , reduces to the epimorphism i. Hence, for each m > 0, we obtain a finite decreasing sequence: (

(R07) dfm (V)

o (W)

Let -n > max

(p, q + 2)

k"

E113 -1-n -hq -n-f-2

.m-h 1(W) D • • 'D Aao,m(W) D =e-i,m+i(W) =

O.

and consider the diagram:

in

in

nn

q--n-f-2 -->

1

gn

D11) -1-n —1,

kn

nn --> -1-/p

--1,q

cc :e7)---1,q+i(W) -1-> Ye2),q(W) The first row is exact since it is a part of Vn. Since i maps Dpi. q+,,_r isomorphically onto Dp ,q+r+1,,i for each Y > 1, we obtain two natural isomorphisms a and /3. y denotes the inclusion and the rectangle is commutative. Since n is-greater than max (p, q 4- 2), we get — 0, E;56,.

ET3 , q ,

q

0

according to (R 1 3), (R04), and (R06). Hence we have (R08 )`'el),q(W)PrP-1,q-Ei(W)

Ec ° ,q.

In other words,crep, q (W) is an extension of q + 1 (V) by E73,q. Similarly, a 6-couple = < D, E; i, j, k> is said. to be regular provided: (R81)

0 if q < O; Ep , q = 0 if p < O.(R62) Dp,q

By methods dual to those used above, one can prove the following assertions. Ep ,q = 0 = E ;c , q if 75 < 0 or q max (p, q + 1). .. . v (n l k—p-f-n—i,q—n-f-1)•

= Dr9 , q if n > q + 1.

D; , q =

The homomorphism i Dp , q Dp - i , q+1 is an isomorphism if we obtain a sequence 0 ---> D 122,

D 0. m

•

Dm

LI

m-f -1

•

p — r,

< 0. Thus m-f-r

This suggests defining a graded group

Ye(?) = Yern (?), nz

Ye rn (?) = Darn D!,, rn+1 .

Let Ye p , q (?) denote the subgroup D 6,. of Yep+q (?). Then, for each m > 0, we obtain a finite decreasing sequence:

(R67) Yern (?) =Yearn (?)

l , rn -i (?) D • •D Ye rn, o (?)

Yern+1 ,_1(?) =

which satisfies the relations: (R68 )crep,q(W)/A 91)+],q-i(W) `^-'d Epc q• Let us consider again the a-couple 'H(Ji) of § 5. (Ra 1) is obviously true while (Ra2) is false in general. However, in case X is a finite simplicial complex and XD is a subcomplex of X containing the 75-dimensional skeleton of X, then (R02) is also satisfied. Hence, in this case, WH(0) is a regular a-couple. If, in particular, X p is the p-dimensional skeleton of X, then E, q = 0 for each q 0 and E, 0 = Hp (Xp, Xp _.1) is the group of 75-chains of X over G. One can verify that the differential operator cl: E, 0 --> E 1 , 0 is precisely the boundary operator on 75-chains. Hence we obtain

O. If n > 2, dn = 0 since it increases the complementary degree q. Hence we obtain H (X) = E2 = E ,-- • • • = E. Hp (X), E , q = 0 if q

7. The graded groups R ( ) and S() Let = < D, E; ti, j, k> be a regular a-couple. Define a graded group R(W)= R q (%) , R q (W) = E 0 , q .

Since dn is of degree (— n,n— 1), each element of 4, 2 is a cycle. Thus we obtain epimorphisms Rq (W)

E oj• q --+ E q

-> • • • Cal-1 —)-

Er; = EN .

Let x denote the composition. By (R08) of § 6, we have E,q rd e q(W) œ q(W) • Denote the composed monomorphism by t. Then we have

(Ra9)

R q (W)

Ecoc: q

APq (W) .

7.

THE GRADED GROUPS

R( ) (

AND

SM

2 39

On the other hand, define a graded group

Sp(V), S p (V) — 4 0.

S(W) —

If n > 2, no non-zero element of obtain the monomorphisms Ec° P10 '" P, 0 —EP÷'

E921,, 0

EPP10 '"-i

›

can be a boundary under da. Thus we 62 >.Eit , 0 = Sp ( W) . P, 0 -

• . • - 13->E3

Let t denote the composition. By (Ra8) of § 6, we have

ELI

l( W) =" A9 P(W)/ A9P - 1,

Let x denote the natural epimorphism. Since

tal)(W) = N+1,--1,

we have

/2 : ,;le° p (V) -÷ S p (W) . One

can verify commutativity in the triangle

Ye p (V)

(R010)

Sp(W)

i2

\ / E;,o

Thus / 2 is factored into the composition of an epimorphism and a monomorphism. If W is a regular 6-couple, then we define the graded groups 1?(W) and S(W) exactly as above. Since dn is of degree (n, — n + 1), no non-zero element of E:s can be a boundary under dn and every element of E75, 0 with n > 2 is a cycle. Hence, one may establish the epimorphisms x and the monomorphisms t in an analogous way as above with the roles of R(W) and S(W) interchanged. Furthermore, since z/eq (V) — D0, q and R07) E 0, q, we have

i:

'cfeq ((f)

---›- Rq (V). Thus we get

(R69)

S(

) .--'t ---).

E 0

and a commutative triangle

(R6I0)

Rq (V)

\ / EP°, q

might define R q (r) to be E 02 , q instead of E0 , q. Then all the results in this section stand as they are except that the i in (R610) should be replaced by /2. Our choice of E0 , q is based on the fact that it gives a natural expression for Rq (W) if W' is defined by a filtered graded differential group. See § 11 below. Note. One

240


8. The fundamental exact sequence = < D, E; i, j, k> be a regular 0-couple and > 0 an integer. Let Under certain conditions on the bigraded group Ev, we shall obtain a useful exact sequence which will be referred to as the fundamental exact sequence. Theorem 8.1. If E2 has only two rows which might be non-trivial, more precisely, if there are two integers a E2aon-a

671,(W)

tpm

77 2

Ea2on-i-a

Xyn

cytem

(

...

Proof. The hypothesis implies that E 9 = 0 for each n > 2 and hence a and p b. Then (R07) and (RN) of § 6 give rise to an E;g = 0 if p

exact sequence

0 --->

EZ),n.1,

O.

Since drt is of degree (— n, — 1) and a 2. Hence we obtain epimorphisms Eaa ,m_a

Ea3,m-a

Ean+z =

--> M3 • •

where n — max (a, m — a + 1). Similarly, if n > 2, no non-zero element of ET, ifi_b can be a boundary under du and we obtain monomorphisms Ea° b,ns-b = En+1 b,m-b

T,n

tn-i Ebon-b

—>" 'botz-b

where n -= max (b, ni — b +1) . Thus we obtain an exact sequence d'rni(W) YLn.-› E g,m-b'

, n. To determine the kernel of O m, we have to study that of gr, r = 2, The kernel of e consists of those elements of Eram.a which are boundaries under dr. There are only two terms of Er of total degree nt + I which might be non-trivial, namely, Ear,. +Fa and ET,„, +1.b. The elements of the first are cycles; and dr maps the second into Eram_a iff Y = b — a. Hence Om is a monomorphism if b — a = 1. Now let r b — a > 2; we have an exact sequence xr r ar

> a,m-a

Erb,m+1-b

Furthermore, Ear,„,_,,— Eaam.„ and Eartni.a E a`zon.a. A similar argument shows that Efkm-J-1-8 Eg m1 . Thus we obtain an exact sequence . b

E2b,m+1-b XM-1-1

)

aon-a

0m

where Z. + , =Oifb—a ---- 1 and X. +1 = drifb—a = r > 2. By similar methods, one can prove the exactness of the sequence in(W)

Vm — >

Xm —>

E2a,m-1-a*

The fundamental exact sequence is obtained by putting the various parts together. 1

8.

THE FUNDAMENTAL EXACT SEQUENCE

241

It was proved above that 2tm+1 — 0 if b — a — I. Hence we have the following Corollary 8.2.

If in the hypothesis of (8.1), b — a

I, then, for each ni,

we have an exact sequence I'm(') Pm

0

Henceze m (V) is an extension of Ea'

0.

E12)

by Egon.b.

Analogously, one can prove the following

E2 has only two columns which might be non-trivial, more precisely, if there are two integers a 1197n( W )

Corollary 8.4. If,

n- -->

-a,a

) A 9m—i(W)

2 -1-b,b Em

Em 2

"

in the hypothesis of (8.3), b — a = 1, then, for each m,

we have an exact sequence

0

Em2 .b,b

Vm --> El-ara ---> O.

Hence 'cffm (?) is an extension of EL. 6,1, by El2n_a,a • Although the preceding two theorems cover most of the applications, it is sometimes necessary to deal with fundamental exact sequences obtained under weaker conditions. The two-term condition. Let tt, y be integers such that A 1. We shall say that V satisfies the two-term condition { A, p.; y } if the bigraded group Ev has the following properties. For each integer in such that A <m < E;, q = 0 if p q = m and (p, q) is different from two given pairs (am , bm) and (cm , dm), where

bm = m = cm dm ,

am < cm .

Moreover, we require that the following two conditions should also be fulfilled:

(1)

E;,q = 0 if p

(2)

Epv

g = m —1, p < am — v, and

o if p + q =

A < ni cm v, and Â

m
• • • ---> Eva m ,b 7n ,

Corollary 8.6.

,

—>r

')

--->

Ecvnt ,c17n,

••

-->

Ecv da

Let y >1 and { am } be a sequence of integers such that am < am ..1 +

242


for each m > 0. I/ Eti,,q --, 0 for every pair (p, q) such that p p am , then Ye m (W) E av,n,b„,, bm m — am . Proof. Put cm

q = m and

am + 1 and dm = bm — 1. Then V satisfies the two-term

condition { 0, tt; w } for every u> O. Since Fey no dm = 0 for each m > 0, the fundamental exact sequence implies the conclusion of this corollary. I If V is a regular 6-coup 1 e, then the two-term condition is stated by precisely the same words. The fundamental exact sequence in (8.5) becomes

Eavt,,bt,

••

Eav

'inf.

Ecvm ,d,„

••

Eve,1,4

and similarly in (8.1) and (8.3). Regular 0 - and 6-couples are considered further in Ex. B at the end of the chapter.

9. Mappings of exact couples Let Wr mapping

< Dr, Er; 4, ir, kr>, r = 1, 2, be two exact couples. By a (sb, V)

we mean a pair of homomorphisms : D 1 D 2 and v E 1 E 2 such that Oil = izO, Vii = 120 , Oki — k2V• If dr = frkr, then yci i dep and v is a mapping in the sense of § 2. ip induces a homomorphism vi : -4- 4 Since 0(D1) = 0i 1 (D 1) = i 20(D 1) c i2(D 2) = we may define a hom.omorphism : D1' 1;1 by 0' = I D. It can be readily verified that the pair (0', v') of homomorphisms constitutes a mapping of the derived couples. This mapping (0', : V2 is called the derived mapping of (56, v). Thus, the set of all exact couples and their mappings forms a category, and the operation of derivation is a covariant functor. If we iterate this process, we obtain a sequence of successive derived mappings (0n, r): _÷ W21i, (n = 1, 2, • • •). In particular, the homomorphisms p2: E 1n,E 2n, (n = 1, 2, • -), of the spectral sequences commute with the differential operators dn, and vn+1 is the induced homomorphism of vn. If the exact couples Vi and W2 are bigraded, then only a limited class of mappings are of interest, namely, the homogeneous mappings. The mapping (0, v) is said to be homogeneous of degree (p, q) if both 0 and v are homogeneous of the same degree (p, q). By reindexing one of the given exact couples, we may always assume that (0, v) is of degree (0, 0); in this case,

9.

MAPPINGS OF EXACT COUPLES

2 43

we say that (.0, ip) preserves degree. Then each of the derived mappings (# n 1 vn) also preserves degree. Now let Wi, W2 be regular d-couples, and (0, v) : WI --> W2 a mapping which preserves degree. Then the homomorphism 412 defines a homomorphism

(0, V) * :.Ye(Vi) -atiV2) which carries 'cfen,(Vi) into en.,(V 2 ) and :16(W 1) into :A°2,, a(V 2). Furthermore, y and ze define homomorphisms

R(Vi ) -). R(V 2), S(V 1) --> S(V2). These obviously commute with the epimorphisms x and the rnonomorphisms t in § 7.

9.1. (O., v)* is an isomorphism if either of the following two equivalent conditions is satisfied: (i) 02 : D12 F....,,, D 22 ; (ii) v2 : E 1 2 gt:i, E 22. Proposition

Proof. It is obvious that (i) implies that (95, v)* is an isomorphism. That (i) (ii) is an immediate consequence of the "five" lemma, [E-S; p. 16]. Finally, that (ii) = (i) follows from an easy induction on the primary degree by using (ROI) and the "five" lemma. I Note. (9.1) remains true if we replace the superscripts 2 in (i) and (ii) by any positive integer n. In fact, it can be easily seen from § 6, that (i) implies that (0, v)* is an isomorphism while the equivalence of (i) and (ii) is proved exactly as above. Obviously, similar results can be obtained for regular 6-couples. Now, let us go back to the general case where Vi, W2 are any two exact couples and consider two mappings

(0, V), (0', r) : They are said to be homotopic (notation: (0, v).„‘--.-. (0-, -r)) if there is a homomorphism : El --> E 2 such that

0.(x) —95(x) = kgi i (x),

T(Y) — V(y) ---- .c/i(Y) + dg(Y) for every x E D 1 and y c E l . Then one can easily prove the following obvious but important

9.2. If the mappings (st, y) and (o-, -r) are homotopic, then the derived mappings (4/ , y') and (o-', -r`) are equal. Proposition

Hence, (On, vn) -..- (01t, rn) for every n > 2. In particular, if W i, W2 are regular 0-couples or regular 6-couples and if (95, v), (o-, r) preserve the degree, then (0, y) n-z (a, -r) implies that (95, ip)* = (0-, r)*.

244


10. Filtered differential groups Let A be a d-group. By an increasing filtration in A, we mean an increasing sequence of subgroups { AP } of the d-group A, p ranging over all integers, with their union equal to A; in symbols, we have U pAP

— A, AP

C

AP+1, d(AP)

Œ

AP.

If an increasing filtration { AP } is given in A, A will be called an increasing filtered d-group. Let A be an increasing filtered d-group. For each a e A, the greatest lower bound of the integers p such that a E AP is called the weight of a and is denoted by w(a). The following properties are obvious:

w (a — b) < max [w (a) , w (b)] , w(da) < w(a). Conversely, if there is given a function w defined on a d-group A with integral values (including — co) which has the properties given above, we can define an increasing filtration { AP } by taking

AP '=: {aEAlw(a)

- R(V) in (R010) of § 7 reduces to the derived homornorphism of the natural projection / : A -->A 0. There is a natural monomorphism

248


which preserves degree and commutes with d. The derived homomorphism

A9(g)

Ye(A)

is given by the composition ot in (R69) of § 7. As an example of regular 0-complexes and 6-complexes, let us consider a filtered space Xr X D X_ i X 0 c Xi • • (0 ) where X is a finite simplicial complex and X p is a subcomplex of X. Assume that X p contains all p-simplexes of X. Then, for any given coefficient group G, the group A = C(X) of chains form a regular 0-complex with AP

=

C(X p), A p Cp (X).

One can easily see that V(A) is precisely the exact couple Wi(t) of § 4. Hence %'H(Ï) is also a regular 0-couple in this case. Similarly, the group of cochains of X over G forms a regular 6-complex.

12. Mappings of filtered graded d groups -

A -> B Let A and B be two regular 0-complexes. A homomorphism is said to be a mapping if it commutes with d and preserves both degree and filtration, i.e. B., /(A1') BP. = di, (A.)

The set of all regular 0-complexes and their mappings constitutes a category which is a subcategory of „Fi. Now let f : A -> B be a mapping in X' s. According to § 10, / induces a mapping (B) : (A) — (Of,

x'a

in the sense of § 9. On the other hand, we have a derived homomorphism Ye(f) : if(A) -> The following proposition is a corollary of (9.1) and its note. Proposition 12.1. For any mapping /:A

B in X .,9, Yt°(f) is an iso-

morphism if there exists an integer n > 1 such that either of the following two equivalent conditions is satisfied:

(i)

:

alt

(A )

D(13); (ii) 1p7 :

(A)

En(B) .

Similarly, one can define the mappings of regular 6-complexes and the category lça of regular 6-complexes and their mappings. The preceding proposition is also true for any mapping in Yfd.

EXERCISES

2 49

EXERCISES A. Direct construction of the spectral sequence of a filtered differential group

Let A be a filtered d-group with an increasing filtration { AP the following notations:

} .

Introduce

x e AP I d(x) — 1,

Zp

f x E AP I d(x) e AP -n }, (n > 0),

Znp'

B; = d(Z),

(n > 0).

Verify the following inclusions and equalities: Zp

p +1,

c 131,c • • • e B; c= B;+1 Œ • • •

d(A P) =

Zp c•• c Z;

c•••=4ŒZ

ZnP-1 = ZnP

(1

BP 1

n AP-1Œ

P

AP -1

,--

AP

,

Zn4-1 P Bnp-1.

Then define the following quotient groups: D;Zp _n+11$75,1-, +1

,

(

n• >

1\ ,

E; Z;I(Z7) =1 + BF-1), (n > 1) , and the graded groups

Dn E D" En E En. P P The inclusions Zp , Zp ,+1 and By)':4 /373-_-. 1 define a homomorphism in : Dn Dn which is homogeneous of degree I. The inclusions Zp , 4. 1 31 _,I+1 and Bri,-, +1 (Zi + /37,-L 1 ) define a homomorphism j1 : Dn -÷Z1 En which is homogeneous of degree — (n — 1). Finally, since d(Z;) c d(Zr.1) = Br_I-z , and d(B7) --') 0, the differential operator d induces a homomorphism kn : En -÷ Dn which is homogeneous of degree — I. Prove that

I . 1 Ep,q ='. 0 if p < 2 or q < 0 or p + q < 2. 4. %7 (K , y) is a regular 8-couple in the sense of § 6. 5. If yo and y i are any two vertices of K, and if a: I --» K' is a path joining vc, to v l , then induces a natural isomorphic mapping

W'() --- ((h, ipz) : W(K, y l)-,-...-, W(K, y o) in the sense of § 9. Furthermore, if two paths , n : I -4 Kl - from yo to u 1 are homotopic in K with endpoints fixed, then W()— W(27). These facts can be concisely expressed by saying that the set of homotopy exact couples ? (K , v) for various vertices y of K constitutes a local system, of exact couples in K. The same remarks hold for the successive derived couples Wn(K, Yl. 6. Let K, L be connected cellular complexes, y, w be vertices of K, L respectively, and /: (K, y) -÷ (L, w) be a cellular map. Then f induces a mapping WU) --- (Of, w) : W(K, y) ---> W(L, w). Let X' denote the category in which the objects are all pairs (K, y) of a cellular complex K with a vertex y E K and the mappings are all cellular maps. Let wa denote the category of all regular 8-couples and their mappings. Then the operation (K, y) -± W(K, y) and f -÷ W (/ ) is a covariant functor from .1(' to Wa. 7. Let k be a connected covering complex over K with projection

co :

k

---> K and i) be a vertex of

fc-- with NCO

.(co) : W(i?, '77; )

= v. Then

W (K , u).

The significance of this result is that there is no essential lack of generality if we assume that K is simply connected when discussing the properties of W(K, y) and its derived couples. 8. The derived couple W2 (K, u) . < D2, E2 ; i2, j2, k2 > is an invariant of the homotopy type of K. 9. D g — 0 if p < 3 or p --i- q < 2.

EXERCISES

25 3

np +q (K , v) if g < 0 and p + q> 1. 11. Ep2 07 =0 if p < 2 or q < O. 12. If K is simply connected, then 4,0 is isomorphic to the homology group H p (K) of the complex K for each p > 2. 13. If K is r-connected, then DP, q. = 0 for p < r + 1 and EP, q = 0 for p < r . 14.The exact couple W2 (K, y) contains the following exact sequence of J. H. C. Whitehead:

10.

• • - it.,. DP , 0 2-> -D13 +1 ,

-1

.±-+ EP , 0 _12.--> DP _1, 0 2_). • • • .

If K is simply connected, then, by 10 and 12, there are natural isomorphisms , -. -; 7rp(K , y), E,0

H p(K)

for each p > 2. Hence we obtain the following exact sequence for a simply connected K: • • • L -> DP, 0 .2_'_>.

7Cp(K,

V) 2--> Hp (K) L'2-> DP , 0 2H • • • . - 1

>

As a consequence of this and 12, we obtain the result that, if n > 2 and K is (n — 1)-connected, then i2 : 7C29(K , y) -› H p (K) is an isomorphism for p < n and an epimorphism for p _, n ± I. This includes essentially the Hurewicz theorem. 15. Since ce = ? (K , y) is a regular 0-couple we may apply the results of § 6 and § 7. Thus we obtain that '.-/ fp(W)ri.-'., 7(2)(K , y), Rp(V) =

0, Sp (V) ,-,,, Hp (K)

for every p > 2 and that these groups are all trivial whenever (R010) of § 7 gives a commutative triangle for each p > 2: rtp (K , v)

12

)

p < 2. Then,

Hp (K)

\,/ r )0

This implies that ETpo is isomorphic to the image of (K, v) in Hp (K) under /2. 16.The inclusion KP K induces homomorphisms il.p , q : gTp +q(KP , y) --->-7723 +q(K , y).

Let us denote the image of 2p , q by 7Cp,q(K , V). Then Y fp,q(W) r eti -

p > 2 and p + q > 2; = 0 if p < 2 or p ± q < 2.

np , q(K , y)

Ye p , q (V)

if

Hence (R07) of § 6 gives a finite sequence nt,,(K, y) = am , o (K , y) D

7Cm-i, 1 (K,

v) D • - • D n 1 , m -1 (K . u) -- 0

for every in > 2. Furthermore (R08) of § 6 implies that 7(23 , q (K , V)12rp

- I

, q+1 (K , v) F.2, E.;Z q .

254


If q = 0, the triangle in 15 shows that 1 (K, u) is the kernel of the horn om o rphi SRI / 2. 17. Generalize the foregoing results to semi-simplicial complexes or, more generally, to CW-complexes. 18. Finally, let X be a pathwise connected space and xo a given point in X. Consider the singular complex K = 5(X). Then K is a connected semisimplicial complex and xo determines a vertex y of K. W(K, u) is a homotopy invariant of the pair (X, x0) and will be called the homotopy exact couple of X at xo, denoted by W(X, x0). By means of the relations 7Cin(K , V) F:5

Xm(X ,

X),

Hm(K)

Ilm(X),

deduce results concerning 7Cm (X xo) and Hm (X) analogous to those given above for the pair (K, v). The usefulness of the homotopy exact couple and its associated spectral sequence is limited by the fact that we know very little about the groups E, q• F. The cohomotopy exact couple

Let K be a finite cellular complex of dimension n. Let Y denote the least integer greater than En + 1). Then we have the following exact cohomotopy sequence of the triple (K, KP+1 , KP): grf (K,R -P+1) 2:–› • • •–›rn (K,KP+1 ) ±> m (K,KP) no +1(K JO +1)

am

(KP-1-1,KP)

where i and j are homomorphisms induced by the inclusions and k denotes the cob oundary operator. Define two bigraded groups D and E as follows: Dp 9 q

709 -1 0' (K, 1 r, q

r;

q > Y, r q

1,

The homomorphisms i, I, k in the cohomotopy sequence may be extended in an obvious way to define homomorphisms î, I, k of the triangle

which are homogeneous of degrees (— 1, 1), (0, 0) and (1, 0) respectively. Prove that the triangle is exact. Thus we obtain a 6-couple W*(K) which is not necessarily regular in the sense of § 6; it is called the cohomotopy exact couple of K, [Massey 1] and [Peterson 1].

255

EXERCISES

Study *(K) and its derived couples as in the previous exercise. In particular, the exact couple V2 (K) contains the following exact sequence - - • -I' --> -1, 1 2 2--> 14,1 Prove that, for each p > Y, we have = FP(K), .13$ _2, 2 7rP(K) , 4 _1 , 1

• HP (K),

where FP(K) denotes the image of the homomorphism i 2vP (K , KP) -÷ aP(K, KP -'). Hence the preceding exact sequence becomes > FP -.4. nP (K) t> HP (K) ••• FP4-1 • Fr where h denotes the natural homomorphism defined in (VI; § 11). Prove that Pp = 0 for all p > n. This implies that h is an isomorphism for p = n and is an epimorphism for p n — 1. Thus we obtain the Hopf theorems as corollaries. G. The Gamma functor of abelian groups

A function f : A B of an abelian group A into another B is said to be

quadratic if f (a + b + c) — f (b + c) — f (c + a) — f (a

for all elements a,b,cinA. Let Of : A x A Of (a , b)

b) +1(a) + f (b) + f (c)

0

B denote the function definedby

f (a + b) — f (a) — f (b)

for every pair of elements in A; in other words, Of is the deviation of f from a homomorphism. Prove that f is quadratic iff Of is bilinear. Hence every homomorphism is quadratic. Also prove that compositions of quadratic functions with homomorphisms are quadratic. A quadratic function f : A -÷ B is said to be homogeneous if f (—.- a) =f (a) for every a e A. Prove that every quadratic function is the sum of a homomorphism and a homogeneous quadratic function. Define an abelian group P(A) as follows. Let F (A) denote the free abelian group generated by a set of generators in a 1 — 1 correspondence with the elements a e A. Let R(A) denote the subgroup of F(A) generated by the elements of the forms w (a) — w(— a) ,

w(a + b + c) — w (b + c) — w(c + a) — w(a + b) + w (a) + w(b) + w(c)

for all elements a,b,c in A, where w(a) denotes the generator of F(A) which corresponds to the element a e A. Then the group P(A) is defined by r(A) = F (A)/R(A). For each a e A, let Lai denote the element w (a) + R(A) in r(A). Prove that the assignment a [a] gives a homogeneous quadratic function y : A -÷ TO).

256

VIII,

EXACT COUPLES AND SPECTRAL SEQUENCES

If /: A -->B is any given homogeneous quadratic function, prove that there exists a homomorphism h : r(A) ---> B such that hy f. Next, let f : A ---> B be a homomorphism. Prove that the assignment [a] -÷ [fa] gives a homomorphism P(f) : r(A) called the homomorphism induced by /. Verify that the assignments IV) define a covariant functor r from the category of A -4- r(A) and abelian groups and homomorphisms into itself and hence f : A i B implies

P(f) :P(A) r(B). Prove the following elementary properties, [J. H. C. Whitehead 7] : 1. If A is additive group of rationals, then f(A) A. 2. If every element in A is of finite order and also divisible by its order, then r(A) = O. In particular, if A is the group of rationals mod 1, then f(A) = 3. If A is a free abelian group with a free basis { ai } indexed by an ordered set i , then P(A) is the free abelian group freely generated by the elements y(ai), v (ai, ak) with j < k. 4. If A is a finite cyclic group of order h and if a is a generator of A, then F(A) is a cyclic group of order h or 2h, according as h is odd or even, and is generated by y(a). 5. If A is the direct sum E A p of its subgroups { 4 } indexed by an ordered set { p }, then r(A) = E r(Ap)

E A g A y. 9 A defined in § 11. Let 4,---- A° n A. If p 0, then g maps A into 0 and hence defines a homomorphism

h AplAY,--->

257

EXERCISES

If p > 1, then h commutes with d and hence induces a derived homo-

morphism

h* :.;fp (AIA°) ---> .0' 23 (J), (p

> 2).

Prove that the images of g* lf(g) and h* in Ye p (A) are g* [Aep(A)] = Erol, h* [.;fp(AIA°)] = 4 0 . Next, consider h* together with the boundary homomorphism a: .16,(A)

(AIA°)

.16, _ 1(A°) ,

(p > 2).

Denote by J the image of h* , K the kernel of h* , L the image of a, and M the kernel of a. Let x e j. Choose y e '.09p (AIA°) with h(y) x, and consider a(y). It is an element of z092,1(A°) which, when y varies, describes a coset mod a(K). Hence we obtain a homomorphism T : J ---> Yfp_ i(A°)10(K) called the transgression. 1 is a quotient group As the image of h* , we have J---- 4, 0. By § 7, E of Yf p _1 (A°). Denote the projection by x. Prove that the rectangle :492) (AIA °)

L

Yep - 1 (A°)

dP -÷ El)

is commutative and a(K) is the kernel of x. Hence n induces an isomorphism Ee, p —1, (p > 2). After this identification, the transgression T reduces to the differential operator dP : E ,0 For an arbitrary 0-couple, we may define the

transgression to be this

differential operator dP. Analogous to the transgression, define a homomorphism : L --> ,Ye p(A)111 * (M) called the suspension. If rp (A) = O .Yf p _1 (A), then ô becomes an isomorphism and the suspension reduces to h*O -1 : fp _1 (A°) ---> In this case, the transgression dP T is an isomorphism and in the diagram :Yep(Â)

fep-10°)

4

EP P,o

258


the image of E coincides with the image of the monomorphism t and the kernel of E is precisely the kernel of the epimorphism x. Furthermore, prove that the following relation holds: E—

Establish analogous results for a regular a-complex A. I. Properties of Steenrod squares

Let (X, A) denote any pair consisting of a space X and a subspace A of X, and Z 2 denote the cyclic group of order 2. Reproduce the definition of the Steenrod square operations Sqi : Hn(X, A; Z 2) -)- Hn+i(X, A; Z 2), i > 0 , and prove the following properties of these operations, [Steenrod 2 and Cartan 1] : 1. Sqi 01* = f* 0 SO for any map / : (X, A) -÷ (Y , B). 2. Sql 0 6 — 6 0 SO, where 6 denotes the coboundary operator in the cohomology sequence. 3. Sqi(oc u 13) — . E Sqi(oc) u Sqlc( 13). 4. Sqi(oc) — 0 if dim (a) < 1. 5. Sqi(a) = a, u oc if dim (oc) = 1. 6. Se(oc) — oc. 7. Sql- coincides with the coboundary operator induced by the exact sequence 0 --> Z 2 --> Z4 ---> Z2 --> 0

and hence we have an exact sequence Sq 1 • • • --->Hn(X,A; Z 4)---> lin(X,A, Z 2)— ---- Hn+1 (X , A ;Z2)--)• Hni-1 (X , A ;Z4)-÷• • •

CHAPTER IX THE SPECTRAL SEQUENCE OF A FIBER SPACE

1. Introduction We turn now to the study of the relations among the homology and cohomology groups of the various spaces of a fibering. As already indicated, the principal tool is the machinery of spectral sequences developed in the preceding chapter. Since we shall follow Serre, we use the singular groups; our principal hypotheses will be that the fibering have the covering homotopy property and that the fiber be pathwise connected. It is convenient to use cubes rather than simplexes in defining the singular groups, and hence we begin with an outline of this construction of the singular complex of a space. The associated exact couple of a fibering is then introduced, and the term E 2 of this exact couple is computed ( §§ 3-10; the main result appears in §§ 5-6). A variety of applications follow. Relations among the Poincaré polynomials of the fiber space, base space, and fiber are deduced in § 11; and several exact sequences, including those of Gysin and Wang, are constructed in the next three sections. In the final sections of this chapter ( §§ 15-18), regular covering spaces are treated: there is a difficulty, namely that the fiber fails to be connected, but this is avoided by the introduction of certain auxiliary fiberings. A spectral sequence due to H. Cartan is constructed and used to deduce certain facts about the action of finite groups on a sphere and the celebrated results on the determination of certain homology and cohomology groups in terms of the fundamental group. 2. Cubical singular homology theory In the traditional simplicial singular homology theory, [E-S; pp. 185-211], the unit n-simplex A n is used as the anti-image in defining the n-chains of a space. However, to study the spectral sequence of a fiber space, it will be more convenient to use the cubical definition in which the n-cube In plays the role of A. In the present section, we shall give a sketch of the cubical theory. As to the equivalence of the cubical theory and the simplicial theory, a proof is given in [Eilenberg and MacLane 2]. By a singular n- cube in a space X, we mean a map u: X. If n 0, then u is interpreted as a single point in X. If n > 0, we define the lower and upper /aces Aeu and Ailu of u to be the singular (n —1)-cubes given by 259

260

IX. THE SPECTRAL SEQUENCE OF A FIBER SPACE

(248u) (tp . • • , tn--1)

u(11,• • • , ti-i, e, lb' •

for every i = 1, 2, - • , n, e = 0, 1, and (4, • • • , 4_1) we have 2.;;

tn-i)

E rn -4 •

Then, for i < j,

= 2.7_1 )4

where s and n may be either 0 or 1. Define Q(X) to be the free abelian group generated by all singular n-cubes in X if n > 0 and Q(X) = 0 if n O. Hence, we have to "normalize" { Qn (X), 0 J. For each singular (n 1)-cube u in X, n> 0, we define a singular n-cube Du in X by taking (Du) (t 1 ,• • , tn) 14 (4," • ,191-1). A singular n-cube y in X is said to be degenerate if y — Du for some u. In other words, 7, is degenerate iff it does not depend on the last coordinate tn of the point (t1, • • - , tn) in /n. The degenerate singular n-cubes in X, n> 0, generate a subgroup D(X) of Qn (X). Since it follows that 0Du =

AieD = D2 (— 1)i (24 1Du

,

C(X) of degree — 1. We shall call C(X) the group of all normalized cubical singular chains in X. For any subspace X0 of X, the group C(X 0) can be considered as a subgroup of the group Cn (X). The quotient group

Cn(X, X0) . C(X)/C(X 0) is obviously isomorphic to the free abelian group generated by the nondegenerate singular n-cubes in X not contained in Xo . Since a carries C(X) into C I (X) and C(X 0) into Cn _ 1(X0), it induces a boundary homomorphism --)-Cn _1 (X, X 0). Since aa — 0, we obtain a chain complex { Cn (X, X0), a } and hence the relative homology groups II (X, X0 ; G) and the relative cohomology groups Hn(X, Xo ; G) over an arbitrarily given abelian group G. Furthermore, if G -----{G x 1xEX} denotes a local system of abelian groups in X, then one can define the groups H.(X, X 0 , G) and Hn(X, X 0 ; G) with local coefficients in G in a way similar to that in the traditional singular homology theory. See [Steenrod 1] and [Eilenberg 3]. The direct sum C(X, X0) of the groups Cn (X, X0) is a graded differential group, called the group of all normalized cubical singular chains in X modulo Xo. The direct sum H(X, X o ; G) of the groups Hn (X, X o ; G) will be called the total singular homology grout of X.modulo X 0 Over G, and the direct sum H*(X, X o ; G) of the groups I/9/(X, Xo ; G) will be called the total singular cohomology group of X modulo X 0 over G. For the important special case that both X and X0 are pathwise connected, pick a point xo of X as basic point and assume xo e X0 if X0 is non-empty. Just as in the traditional singular theory, it may be proved that we may consider only the singular cubes with all vertices at xo . Let G . { Gz i x E X } denote a local system of abelian groups in X. Since all vertices of the singular cubes are at xo, the coefficients of the chains and the cochains with local coefficients in G are all in the group Gx0 on which the fundamental group al (X, x0) operates. For this particular case, the homology groups with local coefficients can be simply defined as follows. Let G denote an abelian group on which al (X, x 0) acts as left operators. For each n, consider the group

Cn(X, X0 ; G) . Cn (X, X 0) 0 G,

262

THE SPECTRAL SEQUENCE OF A FIBER SPACE

where C,i (X, X0) denotes the group of normalized cubical singular n-chains in X modulo X0 (defined by the n-cubes with all vertices at x 0). Define a boundary homomorphism : C,i (X, Xo ; G) --> Ca...1 (X , X 0 ; G) as follows. Consider a generator u 0 g of the group C.(X, X0) 0 G, where u : In X is a nondegenerate singular cube not in X0 with all vertices at X xo and g is an element of G. For each i = I, • - , n, we define a loop MU by taking aiu(t) = u(t i , • - • , 4,) :

where t = t and — ii0 for every j i. The loop o•iu represents an element [mu] of 1 (X, xo). Then a is defined by

a(u

g)

E ( — 1)/{ Ailu [criu]g —

J=1

Ai°u 0 g }.

It can be easily verified that ad = O. Hence we obtain a chain complex C.(X, X0 ; G), 1. The n-th homology group of this chain complex is defined to be the n-dimensional homology grout 11.(X, X 0 ; G) of X modulo X 0 with local coefficients in G. Analogously, one may define the n-dimensional cohomology grout lin(X, X o ; G) of X modulo X 0 with local coefficients in G. In the sequel, we shall need a slight refinement of the notion of the degeneracy of a singular cube. A singular cube u: In X is said to be of degeneracy q iff there exists a non-degenerate singular cube : /n -q X such that u = D v , where Dg denotes the q-fold iteration of the operation D. Hence the degeneracy of a non-degenerate cube is 0, and degeneracy of u : In X is not less than q if

a

u(ti, for every point (t i ,

u(t i ,• • , tn _q, 0, • • , 0)

• , in)

in) of I.

3. A filtration-in the group of singular chains in a fiber space Throughout §§ 3-14, let co: X -->- B be a given fibering as defined in (III; § 3). In other words, X is a fiber space over a base space B with co as projection. Pick a point x0 e X and let b0 = co(x0) e B. The subspace

F

co _1(bo)

of X will be called simply the fiber. We assume that both B and F are pathwise connected. As an immediate consequence of the existence of covering paths, X is also pathwise connected. Hence, according to a remark of § 2, we may consider only the singular cubes with all vertices at x o or 1)0 when dealing with the cubical singular theory. We assume once for all that every singular cube considered in the sequal is of this limited kind.

4 . THE ASSOCIATED EXACT COUPLE

263

A singular cube u : In -± X is said to be of weight p (in notation : w(u) = p) if the singular cube wu : In --> B is of degeneracy n — p ; thus u is of weight p if u(t i , • • , tn) remains within a fixed fiber as tp+1 , • • • , t,, vary, but moves from fiber to fiber as t2, varies. For each singular cube u in X, the weight w(u) satisfies the condition

0 < w(u) < dim (u).

(3.1)

It is also straightforward to verify the following two relations: (3.2) (3.3)

w(2u) < w(u) if 1 < 1 < w(u) and e----- 0, 1 ; w(u) = w(u) if w(u)
p, it follows from (A3) of § 4 that Ai 6M(u, v5) — M(u, A4v5), (8 = 0, 1).

Since / is a cycle, the expression r

E

q 1.11

(

—

1)icej(Âi 1vi — )4°7)5)

j----1 i----1

is equal to a linear combination of degenerate singular cubes in F. Hence, if we neglect the degenerate terms, we can write az in the following form: r

aZ =

P

E E (— 1)ia5 { 241M(u, v.') — APM(u, u7) 1. i = 1 i=1

268


According to (3.2), Oz is a cycle in CP-1. Projecting to the quotient group 6P-1, az represents a cycle of 0 -1 and hence an element y of E1 ,q. By the definition of d, we have y = dX -1 (u h).

Hence the element X(y) == XdX- 1 (u h) is represented by the cycle X

E E (— 1)ict5 { B p _..1)41- M(u, yi)

1=1

Fp ._. 1 1 41- M(u, v5) —

Bp _ 1 24°M(u, y5) 0 Fp _ i Ai° M(u, y5) }

of Cp _ 1 (B, B0) C(F) under dF. We have to consider the singular cubes B9_12i8 M(U, 7/5) and Fp_agli€ M(u, yi) for i < 5 The first is evidently Aisu; the second is defined by

M(u, vf) (0, • , 0, E, 0, • • • , 0, t i , • • , k),

(F2,1 /146 M(u, vi)) (ti , • • • ,k)

where e is at the i-th place. Obviously we have F2,1 2i°M(u, x = E (— 1)i {

where

gi

o gi — ),?//t

y5) = yi;

hence

},

denotes the cycle M(u, y5)

E

of C(F).

To determine the homology class of gi, let us construct for each q-dimensional singular cube y in F a (q 1)-dimensional singular cube Dv in X defined by (Dv) (t, t 1 ,- - , tq) = M(u, y) (0, • • , 0, t, 0, • • , 0, t1, • • • , te),

where t is at the i-th place. Since (03v) (t, t 1 , • • • , tq)

(op) (t),

B denotes the path defined as in § 2, one can easily verify where aiu : that the assignment y Dv determines a deformation D : C(F) C(X)

covering the path

criu.

Since

(Dyj) (0, t 1, • • • , tq) = yi(t i, • •

(Dv))

(1, ti, • • • , tq)

tq),

F --] Ai l M (1/t , 5) (t ,• • • , t q),

it follows that gi represents the element [aiujh of Hq(F). Therefore, we obtain 7

XdX -1 (1/1

h) = E (— 1)i { AP-u [aiu]h

Ait'u 0 h } =dB(u 0 h).

Hence we have XdX ---1 dB. I As an immediate consequence of the lemma, we have the following

6. HOMOLOGY WITH ARBITRARY COEFFICIENTS Theorem 5.2. The isomorphism X induces for each

an isomorphism

269

pair (p, q) of integers

xp , q E,q Hp (B , Bo ; Hq (F))

of ET,,q onto the p-dimensional singular homology group of B modulo B o with local coefficients in Hq (F).

The isomorphisms xp , q reveal an important fact, namely that the bigraded group E2 is the same for all fiber spaces having the same base space B, the same subspace B o of B, the same fiber F, and the same operations of n1 (B) on 1/(F). In particular, if al (B) operates simply on H(F), the bigraded group E2 is the same as that of the product space B x F. The deviation from the product structure is bound up in the deviation of the differential operator d2 from O.

6. Homology with arbitrary coefficients In the preceding two sections, we restricted ourselves to integral coefficients for the sake of simplicity. However, the results can be easily generalized to arbitrary coefficients as follows. Let G be an abelian group and C denote the regular 0-complex C(X, X0) of the previous sections. Consider the group

A ----CoG of normalized singular chains of X mod X 0 with coefficients in G. Then A is a graded group with A m = Cm coG

and a differential operator 0 of degree — 1. Define an increasing filtration { AP } by taking AP CP ® G. Thus, A becomes a regular 0-complex. We shall study the associated exact couple W(A) < D, E;1,j,k> and its derived couple < D2, E2 ; 2 j2 , k2 >. Since CP is a direct summand of C, it follows that

AP = AP/AP -1 = (,‘P 0 G. Next, let us consider the group

KP = Cp (B , B0) C (F)

G.

Define a differential operator cip on KP by

dp(b

/

g)

(-1)Pb ® Of ® g

for every generator bofog of KP . Then

Yep ,q (KP) = Cp(B, B 0) 0 Hq (F ; G),

IX. THE SPECTRAL SEQUENCE OF

270

A

FIBER SPACE

where Hq (F ; G) denotes the g-dimensional singular homology group of F with coefficients in G. KP The homomorphism tp of § 4 defines in this case a mapping v: AP and hence induces the homomorphisms Xp , q : E

--->Cp (B, B0) Hq (F ; G).

The lemmas of § 4 imply the following Theorem 6.1. Xp,gis

an isomorphism of E

Cp(B, B 0) Hq(F; G).

By the lemmas C and D in § 5, 2-(1 (B 1'0) acts on H(F ; G) as a group of left operators. Let dB denote the boundary operator in C(B,B 0) ø H(F; G) as chains with local coefficients in H(F ; G). Then, one can easily verify that (5.1) is also true in this general case. Hence we have the following Theorem 6.2.

The isomorphism X induces for each pair

isomorphism

(p, q) of integers

an

4 1, Hp (B, Bo ; Hq (F ; G))

Of E2.1) ,q onto the p-dimensional singular homology group of B modulo Bo with local coefficients in Hq (F ; G).

An important special case of (6.2) is that A = C(X) G. Then Bo is empty and xp , q is an isomorphism of 4,q onto the homology group Hp (B; Hq (F ; G)) with local coefficients in Hq(F ; G). Analogous results hold for cohomology with arbitrary coefficients. See Ex. A at the end of the chapter. In most of the applications, we shall deal with the case where al (B, b o) operates simply on the homology and cohomology groups of the fiber F. In particular, this is the case if B is simply connected or if X is a fiber bundle over B with a pathwise connected structural group, [Serre I ; p. 445]. Now assume that the coefficient group G is either the additive group of integers or a field. Then the isomorphism xp , q in (6.2) is actually an isomorphism of G-modules. If 1 (B, bo) operates simply on Hq(F ; G), then the group Hp (B, Bo ; .111 (F , G)) in (6.2) reduces to the singular homology group with coefficients in Hq (F ; G) in the usual sense. Hence, by the Universal Coefficient Theorem, [E—S; p. 161], we obtain the following Theorem 6.3. F:l a

Hp

(

B

4/7‘ 1(B, 60) operates simply on Hq (F; G), then -

Bo ; G) ®0 14(F; G)

T or G(II(B, B o ; G), Hq (F ; G)).

The torsion product TorG has the important property that TorG(L, M) if L or M is a free G-module, [E—S; p. 134]. Hence we have the following Corollary 6.4. 1/ Hp_ i (B,

Bo ; G) or Hq (F ; G) is a free G-module, then Hp (B ,

0 ; G) 0 G -Iar(F ; G) .

Note that the hypothesis of (6.4) is always true if G is a field.

7.

THE SPECTRAL HOMOLOGY SEQUENCE

271

7. The spectral homology sequence Let us consider the regular 0-complex

A =C0 G, C C(X, X 0) of the preceding section. As in (VIII; § 4), we denote by

Wn(A) = < Du, En ; in, in, kn > the successive derived couples of the associated exact couple W(A) WI(A) of A. Then, En is a bigraded differential group with cln inle as differential operator and is the n-th term of the associated spectral sequence

{E

n = 1, 2, • • • }

of W(A), which will be called the spectral homology sequence of the fiber space X modulo X0 over the coefficient group G. Since A is a regular 0-complex, we may apply the results of (VIII ; §§10-1 1). In particular, :Ye(?) = e(A) = H (X , X0 ; G). Then H(X, X 0 ; G) is filtered with Ec° as its associated graded group; more precisely,

Hm (X, X0 ; G) =

, 0(W) e

—1,1(8) D ' ' D Aeo , m(W)

A9P1 q( W WP — 1, q4-1( W )

m +I (V) 0,

EZq.

Hereafter, we s11all use the notation Hp , q (X, X 0 G) Zep e q(W). For further studies in this section, we have to specify whether or not X0 is empty. Let us first consider the case that X0 is empty. According to (VIII; § 1 1) , Rq(W) = X° q (A°). A singular cube u in X is of weight w(u) 0 if and only if the image of wu is a single point. Since all vertices of u are assumed at x0, this single point must be b0 and hence u is in F. Therefore, A° . C(F) 0 G. Then it follows that ;

Rq(W) = Hq (F ; G). Next, Sp (W) = AP (A) with Ap Ep , 0. However, by means of Xp , 0 , we may identify Ep, 0 with C(B) H 0 (F; G). Since F is path.wise connected, H 0 (F ; G) may be identified with G on which 7r1 (B) operates trivially. Hence C(B) 0 G and Sp (W) = Hp (B ; G). Then, we may apply the results of (VIII; § 7) and obtain the following two commutative triangles

Hq (F ; G) x\sk EO,q

> 14(X, G)

Hp (X ; G)

G)

Ec° I/ P,0 where 0* , co* are induced by the inclusion 0 : F X and the projection w: X —> B, the ye's are epimorphisms, and the t's are monomorphisms. It is

272


an immediate consequence of these triangles that Ho , q(X; G) is the image of 0 * and Hp _1, 1 (X; G) is the kernel of co * . It remains to study the case that X0 is not empty. Then F c X o. One can easily see that A° = 0 and hence Eon.q = = Eac°4 ; in particular, Rq(W) = O.

On the other hand, we have Sp (W) = Hp (B, B o ; G). Thus, we still have a non-trivial commutative triangle

) Hp (B, Bo ; G)

1-/(X, X • G)

E' _AO The kernel of co* is Hp _i ,i(X, X0 ; G) and the image of co* is isomorphic to EL. Similarly, one can study the spectral cohomology sequence. See Ex. B.

8. Proof of Lemma A We shall prove the lemma by induction on the integer q. First, assume that q = O. Then the singular cube y reduces to the point x0 ; and the problem reduces to constructing, for each singular cube u: B, a singular cube z M(u, zr) : X such that (oz = u. Let Vo denote the leading vertex (0, • , 0) of /P. Since Vo is a strong deformation retract of /P, we may apply (III; 3.1) and obtain a map y: I ---> X such that coy = u and y(V0) xo. Let Va denote the various vertices of IP Since u(V a) u, it follows that y(Ta) is a point of F. Since F bo and coy is pdthwise connected, there exists a path cra : / --> F such that (I G (0 ) y(V a) and aa (1) xo . Let Q denote the subspace of /P which consists of all vertices of /P. Define a homotopy ft : /23 ---> B, (0 < < 1), and a homotopy gt : Q ---> X, (0 - X such that cov — f and v I Q —g. Since Q contains all vertices of P, y is a singular cube with all vertices xo . Let Du = v and the conditions (B1)—(B7) are obvious. Next let q > 0 and assume that we have constructed Du for every u with w(u) < 5 and dim(u) < 5 + q. Now let u be a singular cube in X with w(u) < p and dim(u) — p -1- q. To construct y — Dru, let us dispose of the special case where u is degenerate. Then u does not depend on the last variable and 2,1,° +6,u -- 4,.gu. Define y = Du by taking y(4, - • • , tp +0, +1) — DrA°15 ,124 (4, • • • , tp+a). By means of the hypotheses of induction, one can easily verify the conditions (BI)—(B7).

IO. PROOF OF LEMMAS C AND D

2 75

It remains to construct y = Du for a non-degenerate u with w(u) < ,5 and dim(u) = p q. For this purpose, let P 1-23 x/x/q-, Q = (IP x 0 x Ig) U (IP x 1 x Ig) U (V o x I x Ig) (IP x I x 01- ), where Vo denotes the leading vertex of /25' and OP the set-theoretic boundary of /q. Define two maps f : P B and g : Q X by taking

f (t1 , • - • , tp, t, s1,- • • , sq) = Bpu(t 1,- • • ,t23). g(t i ,• • • , tp , 0, si ,• • • , sq) = u(11 ,• • • , tp, s1 ,• • • , sq), g(t i ,• • • , tp , 1, s l ,• • • , sq) = M(Bpu,Fpu) (t 1 ,- • • , tp , s 1,• • , sq), g(0,- • • , 0, t, s 1 ,— • , s q) = u(0,- • , 0, si.,• • • , sq) = Fpu(s i ,- • • , sq), g(t i ,- • , tp , 1, sp • • • , si _i,

8,

Si,*

.13pA23,+i tt (t p • • • , tp, t, S i ,

••,

Sq_ i )

••',

Since Q is a strong deformation retract of P and cog = IQ, there exists a map y : P -4- X such that coy = f and y Q = g. Since Q contains all vertices of P, y is a singular cube with all vertices at xo. Let y Du and the conditions (B1) through (B7) can be easily verified. This completes the proof.

10. Proof of Lemmas C and D To prove Lemma C, we consider a as a singular 1-cube in B. For each singular n-cube y in F, we take

Day . 111(c, y) given by Lemma A. Then Dav is a singular (n 1)-cube in X. The assignment y Day defines a homogeneous homomorphism

Da : C(F) -±C(X) of degree 1. The conditions (A1)—(A4) imply the conditions (C1)—(C4). This proves Lemma C. To prove Lemma D, let a, T be two loops in B representing the same element of 7c1 (B, b o). Then there exists a singular 2-cube u : I2 B such that

u(0, t) = 1)0 = u(1, I),

14(1, 0) = a(t), u(t, 1)

for each t e I. Let Da and D, be deformations of C(F) covering the paths a and T respectively. For each singular n-cube y in F, we shall construct a singular (n 2)-cube Qv in X satisfying the conditions: (Dl) ze)(Qv) < 2. (D2) B 2Qy = u. (D3) ,,,.°Qv(t, t 1,'.., tn) = v(ti ,• • • , ta). (D4) 01 20Qv Day; 2 21Qv = Dry. (E q, ; i • • n). (D5) QA.tv= (D6) If y is degenerate, so is Qv. —

276

IX. THE SPECTRAL SEQ UENCE OF A FIBER SPACE

The cube Qv will be constructed by induction on the dimension n of v. First assume n = O. Then y reduces to the point xo. Consider the subspace

A = (I x 0) U (I x 1) U (0 x I) of 12 and define a map f : A --> X by taking f (t, 0) --. 131v(t), f (1,1) . D rv(t),

AO, t) ---- aco

for every t E I. Since cof —urA and since A is a strong deformation retract of 12, f has an extension g:P-->-X covering u. We define Qv to be the singular 2-cube g. The conditions (D1)-(D6) are obviously satisfied. Next assume that n > 0 and that Qv has been constructed for every singular cube y in F of dimension less than n. Let y be a singular cube in F of dimension n; we are going to construct Qv. If y is degenerate, we set

Q v(s, t, t1 , • . • ,tn.) = Q24/° v(s, t, t 1 ,.. • , If y is non-degenerate, let us consider the subspace

A = (I x 0 x In) U (I x 1 x In) U (0 x I x In) U (I x I x aln) of In+2 =' 1 2 X In and define two maps f : A -4- X and st, : In÷2 ---> B by setting

C s, t, t i , • • • , tn)

= u(s, t),

i(s, 0, t1,• • • , tn) — D ov(s, t i ,• • • , ta),

f (s, 1, t i ,• • • , tn) — Di-v(s, t 1 ,• • • , tn), /(0,1,1,,••• , tn) = v(t i ,• • • , tn), / (s, 1, I D • - - , ti _3 , 8, ti ,• • • , tn _1)----- Nisv(s, I, ti, • • • ,

Since cof — 9!. IA and A is a strong deformation retract of In+2, f has an extension g: In+2 -* X covering 0. We define Qv to be the (n ± 2)-cube g. The verification of the conditions (Dl)-(D6) is left to the reader. The inductive construction of Qv is complete. Now consider the endomorphisms J0 , J, : C(F) --->C(F)

defined by the deformations Da, D, respectively. Let K denote the homogeneous endomorphism of C(F) of degree I defined by (Ku) (1,11 ,- • • , In) — (Qv) (1, t, t 1 ,• - - , In) for each singular cube y : In -). F. Then, one can easily verify that My + Kay . jTv — j f,v. Hence, J„ and IT are chain homotopic. This completes the proof of Lemma D. Remark. If X is a bundle space over B with respect to the projection co : X ---> B, then the proofs of the lemmas A-D can be simplified as follows.

II. THE POINCARÉ POLYNOMIALS

277

For a given singular cube u: IP B, one considers the bundle space U over IP induced by u. By a theorem of Feldbau [S; p. 53], U is equivalent to the product space IP x F and hence the constructions can be easily carried out in U instead of X.

11. The Poincaré polynomials Throughout this section, let G be a field. We shall consider vector spaces M over G graded by the spaces M p ; the dimension of Mp is called the p-th Betti number of M and is denoted by Rp(M). Then M is of finite dimension iff the Betti numbers { R(M) } are all finite and only a finite number of them are different from zero. We assume once for all that, unless there is a statement to the contrary, a given graded vector space M is of finite dimension over G and Mp 0 if p < O. For such an M, we define the Poincaré polynomial /p(M) and the Euler characteristic X(M) by

/p(M)

Rp (M)tP, X(M)

E (— 1)PRp (M).

A few elementary properties of the Poincaré polynomials are listed as follows. Proofs are left to the reader. (1) If L is a subspace of M and N = MIL, then 'ON) = OM) (2) For any two graded vector spaces M and N, we have Tp(11/ oGN) --(3) If M is a graded vector space with a linear differential operator d : M > M of degree 1, then we have ip (. r(m)) v (m) (1+1)Vd(M)). For any two polynomials f and g, the symbol f < g will mean that g — f is a polynomial with non-negative coefficients. (4) Let M be the same as in (3). If L is a subspace of the graded vector space M such that L n d(M) = 0, then N = L n Z(M) is isomorphic to the subspace of .09(M) represented by the cycles N and ip(L) — ON) < hp(d(M)). (5) Let M be the same as in (3). If L is a subspace of Z(M) and N denotes the subspace of f(M) represented by the cycles L, then we have v(L) — V (N ) < V(d(M)). Let (Y, Yo) be a pair of a space Y and a subspace Yo of Y. If H(Y, Yo ; G) is of finite dimension, then its Betti numbers, its Poincaré polynomial, and its Euler characteristic will be defined to be those of the pair (Y, Yo) over G denoted by R( Y, Yo ; G), /p( Y, Yo ; G), and X(Y, Yo ; G) respectively. Now let us consider the spectral homology sequence of § 7. Assume that , bo) oprates simply on H(17 ; G) and that H (B , B0 ; G) and H (F ; G) are of finite dimension. Then we have the following two theorems:

Theorem

Rm (X , X 0 ; G) -
cc > 0), (F; G) = 1 + cit + c212 + • - • + cytY,

(y > 0).

We shall also consider the following four polynomials:

P = ba+2ta+2 + • • • + bptfl , Q = bat' +,..+ bp _20-2, U = cif ± c2t2 + • • • + crtv , V = I + cit + • • • ± cy _1tv -1. TIM is a vector space over G bigraded by the subspaces M, q, then M can be graded by means of the total degree p + q and its Poincaré polynomial 'OM) is defined in terms of this grading. Now let LI = E dn(En). n>2

Then A is a bigraded vector space over G of finite dimension. The general theorem mentioned above can be stated as follows.

11(B, B 0 ; G) and li(F; G) are of finite dimension, then so is H(X, X 0 ; G) and its Poincaré poly nomial is given by Theorem 11.3. I/

y(X, X 0 ; G) = v(B, B o ; G)Ip(F ; G) —(I + t)v(A). Furthermore, the polynomial y(A) satisfies the following inequalities: 'OA) . . . • ip(Mni- 1)

v

moo

(dn(En))

for each n > 2 and. hence P(M2) tp(E 2) PV. Hence we deduce tp(A) < VAPV .

Next, let us consider, for each n > 2, the subspace Nn E

E EL + EEZ 0

of E. No non-zero element of Nn can be a boundary under dn and hence

we have the monomorphisms N " -->••• -->

By (4), we have

Nn ---> • • --> N3 ---> N2.

v(Nn) — v(Nn±') < hp(dn(En))

for each n > 2 and hence we obtain

V(N2) v(E 2)

QU.

Hence we obtain ip(4) < QU. Corollary 11.4.

Rp +y(X, X 0 ; G) =

Rp +y _1 (X, X o ; G) =

B o ; G) Ry (F ; G).

B o ; G) Rx_ 1 (F ; G)

Rp_ i (B, B o ;G)RA(F,G).

As an application of (11.3), let us study the fiber spaces where the base space and the fiber are homology spheres over G. For this purpose, let us assume that

v(B; G) = I + 0,

(F; G)

I + lq

with p > 1 and q > O. Leaving aside the critical case q p I, the Poincaré polynomial of X over G is completely determined by B and F Proposition 11.5.

v(X; G) = (1 + 123) (1 + tq) if q

op

1.

280


Proof. P tP, Q = 1, U = tq, V = 1. According (11.3), we have v(X; G) = (1 + tP) (1 + ta') — (1 + t)v(4) with v(d) < t23 ' and v(z1) < j. Since q p — 1, the latter conditions on O. I y(Z1) imply that v(A) For the critical case q p— I, the conditions on y(4) imply that either v(Z1) — 0 or v(d) = ti". Hence we obtain the following Proposition 11.6.

p

If q

1, then tp(X G) is either (1 + tP) (1 + t2)-1)

or I + t2P A consequence of (11.5) and (11.6) is that the only possibility for a homology n-sphere X to be a fiber space over a simply connected homology p-sphere B with a homology q-sphere F as fiber is the critical case q = p — I and n = 2p — 1, s; p. 145]. Examples for p = 2, 4, 8 are the Hopf fiberings, [

(III; § 5). 12. Gysin's exact sequences Let G denote either the additive group of integers or a field. Assume that the fiber F is a homology r-sphere for some Y > 1 and that 7C1 (B, bo) operates simply on the total homology group I- (F ; G) and the total cohomology group H*(F ; G). Then we have the following Theorem 12.1.

There is an exact sequence `- ' Hm (X, X0 ; G) 22---4

• • • 2-1-* 11m , i (B,B o ;G) ••• Ilm (B,B 0 ;G)

called Gysin's homology sequence, where co * is induced by the projection (B, B o). co : (X , X 0)

Proof. Since F is a homology r-sphere, it follows that E2 contains only two columns which might be non-trivial, namely Hp (B,B 0 :G), E 7

Hp (B,B o ,G)

00

Hr(FG)

Hp(B,B 0 ;G).

Applying (VIII; 8.3) and the commutative triangle at the end of § 7, we obtain the theorem. I Similarly, one can establish the following Theorem 12.2.

There is an exact sequence

•• Hm(B , B 0 ; G) ... HM4 1 (B , B o; G)

H211(X , X 0 ; G)

Hm -r(B , B0 ; G) —

called Gysin's coh,ornology sequence, where co* is induced by the projection w: (X, X 0) -÷ (B,130).

0*(1) e Ilr+1.(B; G). Then it can 56*(x)=xUs=skix

If Bo = o, then H°(B ; G)

be proved that

G. Let s

12. GYSIN'S EXACT SEQUENCES

281

for every x c Hm -r(B; G) and that 2s -,-- 0 if r is even. Since this result will be used only in the present section, the proof is omitted. See [Serre I; p. 470]. As an application of Gysin's sequences, let us consider the fiberings of spheres by spheres and study the structure of the integral cohomology ring of the base space. For this purpose, assume that X itself is a homology n-sphere for some n > r and consider the cohomology ring H*(B). By the exactness of Gysin's cohomology sequence, we deduce that the homo-

morphism

95* : Hm -r-1(B)

Hm(B)

is a monomorphism if ni = 0 or ni = n, is an epimorphism if ni = I or n ± 1, and is an isomorphism for other values of nt. It follows immediately that

Hm(B) Z, m 0 mod (r + 1), 0 n depends on the relation between rt and r. If n = p(r + 1) + q, 0 < q < r, then it is easy to verify that the cohomology ring H*(B) is generated by three elements 1 e H°(13) Z, s = 0*(1) e Hr+1 (13), and t H(B) with co*(t) as a generator of H(X) Z. More precisely, the cohomology group H*(B) is free abelian and has a free basis

{ I, s, s2,• • • ,t,st,s 2t,• • • 1,

(12.3)

where juxtaposition denotes cup product. If

n = p(r ± 1) + r, we have to consider the exact sequence -

Hm(B)

> H9 X)

HP(r+i)(B) 4* > H -'-'(B) -- --- 0,

where we have H(X) Z and HP(r+1)(B) Z. Since H(B) is isomorphic with a subgroup of Hn(X), it follows that either H(B) Z or H(B) = O. If H(B) Z, it follows that v* O and hence co* and 95* are both isomorphisms. In this case, H*(B) is free abelian and has (12.3) as a free basis. If H(B) = 0, then it follows that H+1 (B) is a cyclic group of finite order k > 1. In this case, the groups Hm(B), m > n, are given by

0 mod (r + 1), Hm(B) = 0, ni 0 mod (r + 1). Hm(B)

Zk,

Hence H* (13) has a system of generators

(12.4)

{ 1, s, s 2 ,•

},

where si is free if i p.

282


13. Wang's exact sequences Let G denote either the additive group of integers or a field and assume that the base space B is a simply connected homology r-sphere over G for some r > 2. First, let us consider the case Bo = 0. Theorem 13.1.

•••


Hm (F ; G)

Hmi (F ; G)

Hm _r(F ; G) P "---±-> • • -

-›- H.(X ; G)

called Wang's homology sequence, where 0* is induced by the inclusion 0 :Fc X. Proof. Since B is a homology y-sphere, it follows that E2 contains only two rows which might be non-trivial, namely

Hp (B ; G)

OG

Hq (F ; G),

Hq (F ; G)

(f)

a, r).

Applying (VII; 8.1) and one of the commutative triangles in § 7, we obtain the theorem. I Similarly, one can establish the following Theorem 13.2.

•••

Hm (X ; G)


Hm (F ; G)

Hm+1 (X , G) 2-7—> • -

Hm -r-1-1 (F ; G)

called Wang's cohomology sequence, where 0* is induced by the inclusion 0:F X. The homomorphisms p* define an endomorphism

p* : H*(F; G)

H*(F ; G).

According to Leray, this endomorphism p* is a derivation if r is odd and an anti-derivation if r is even, that is to say, p*(x U y) = p* (x) U y + (

—

1)(r+1-)P

x U p*(y)

for each x e HP(F; G) and y e (F; G). The proof of this result is left to the reader. [Serre 1;33. 471]. Next, consider the case Bo = bo and hence X0 = F. SiAce B is a homology r-sphere, Hr(B, 1)0 ; G) G and ,H,n (B, b 0 ; G) = 0 for m r. Then E2 and E* 2 contain only one row which might be non-trivial, namely

Hq(F ; G), E ; H(1 ; G). Hence, by (VIII; 8.1) we obtain the following Theorem 13.3.

For every integer m, we have

H.(X, F; G) Hni _r(F ; G), Hm(X, F; G)

Hm -r(F ; G).

As an application of these theorems, let us consider the fiber space

X = [B; B, bo]

13.

283

WANG'S EXACT SEQUENCES

over B with the initial projection co : X ---> B as defined in (HI; § 10). Then we have F [B; b0, 1)0]. Since n1(B) = 0, it follows from (IV; § 2) that F is pathwise connected. Therefore, we may apply (13.1) to this case. Since X is contractible, we obtain

Hm (F) Hm-r +i (F)

for every Irk >1. Together with Ho (F) Z, this implies the following theorem of Morse. Theorem 13.4. If

[B; b0, b0], then

B is a simply connected homology r-sphere and F =

Hm (F) Z i/rn 0 mod (r — 1), Hm (F) = 0 if rnE1--7. 0 mod (r — 1).

For another application, let us consider the sphere bundles over spheres. Assume that the fiber F is a homology s-sphere for some s > 1 and consider the integral homology groups Hm (X). Assume that s > 2. Then, by the exactness of Wang's homology sequence, we deduce that the induced homomorphism 0* : Hm (F) ---> .11,,(X) is an epimorphism if m Y I or m = r s —1, is a monomorphism if ni = Y or m = r s, and is an isomorphism for other values of ni. Therefore, it remains to compute Hm (X) for the four critical values r —1, r, r s 1, r s of m. First, let us compute lir _ i (X). If 1' —I S, then it follows that

Hr_ i (X) = 0 * [Hr _ 1 (F)] = 0 s, we define a numerical invariant since F is a homology s-sphere. If r — 1 k of this fibering as follows. Consider the homomorphism p* : 110 (F) Hs(F).

If p* = 0, then k is defined to be zero; otherwise, the quotient group Hs(F) f p* [110 (F)] is a finite cyclic group and k is defined to be the order of this finite cyclic group. Hence, if r — I = s, we have Hr_1 (X)

if k = 0, if k O.

{ Z' Zk,

Next, let us compute Hr(X). From the exact sequence 0 P* -÷ Hr(F)

Hr(X)

it follows that Hr(X)

(

Z Z

' Z,

Ho (F) if Y 0 S, if r = s.

0,

284


Then, let us compute Hr-f-s--, (X). Since r>1, we have 14+8-1(F) - O.

Hence, we obtain

Hr+s, (X) - P[Hr-f-s-i(F)] = O.

Finally, let us compute Hr +8(X) . In the following exact sequence Hr .8(1- ) ±3."---> '- I Ir +s(X) --T-I --)- H s(F) P* >

we have IIr +8(F) = 0 = .Hr8_ 1 (F) and hence Hr„.8(X)

A.-;

Hs(F) R.,., Z.

For the remaining case s - I, the induced homomorphism

0* : Hm (F) -÷ 11m (X)

is an epimorphism if m --,-, r — 1, is a monomorphism if m - r + 1, and is an isomorphism if m < 1' — 1 or m > r + I . Therefore, it suffices to compute the group Hm (X) for three critical values of m, namely, m = Y — 1, r, and Y + I. The computation is similar to the case s > I and hence is left to the reader. Note that the same results can be obtained by using Gysin's homology sequence instead of Wang's homology sequence. 14. Truncated exact sequences

,6.

Let G denote either the additive group of integers or a field. Assume that X0 --, EJ and that n i (B, 1)0) operates simply on H (F ;G) .

1/ Hm (B , G) = 0 /or 0 < m < f and Hm (F; G) - 0 for 0 < In < q, then we have an exact sequence Theorem 14.1.

a m (B ; G)-21->1I m_i (F ; G) ° -41m (X ; G) ---2-+H Hp+.q _ 1 (F ; G) 0 * > . • • • —7:-.4-Hm (F ; G)--If -> • --

H 2 (B ; G) ---2 - ---> H 1 (F ; G)f .'' --> Hi (X ; G)--f-. 2.'- -> Hi (B ; G) —T -> 0

where 0 * and a)* are induced by the inclusion 0 : F C X and the projection co : X ---> B, and T is called the transgression.

Proof. According to (6.3), we have H (B ; G)

OG

Hi(F; G) ± TorG[Hi_1 (B; G), Hi(F; G)].

Therefore, El i 0 if i.---A 0, i 0, and i±j 0, then to * : Hm (X; G) Hm (B; G) for each nt > 0. Corollary 14.2.

Proof. Apply (14.1) with p -,- 1 and q = co. 1 If Hm (B; G) = 0 for each m > 0, then 0* : Hm (F; G) Rd Hm (X; G) for each nt > 0. Corollary 14.3.

Proof. Apply (14.1) with p ,--- co and q = 1. 1 Hm (X) = 0 for each m > 0 and Hm (B; G) = 0 for 0 < In < I), then we have T:Hm (B; G) R.-1 Hm _ 1 (F; G) for 2 <m < 2p— 2. Corollary 14.4. If

Proof. Applying (14.1) with q .- 1, we find that Hm (F; G) = 0 for 0 Hq (B; G) )

where co * is induced by the projection co :X ---› B, tt* = 461 , the n's are epimorphisms, and the t's are monomorphisms; p* will be called the natural homomorphism. As immediate consequences, Ye o, q (() is the image of co * and 1 , 1 () is the kernel of /2* . Similarly, we have the cohomology exact couple W* = < D, E*; i*, f* , k* , > and a spectral cohomology sequence { E*n } of the regular covering space X over B. In particular,

(15.3)

E*2 -,-.,, H*(P; H*(X; G)) = H*(7r; H*(X ; G))

and c'ff(W*) = H*(B; G) is filtered with E*°° as its associated graded group. Furthermore, , Rq(w*) = Hq(X; G), S p (W*).=.- Hy(yr; G) and we have two commutative triangles

Hq(B ; G) w* -÷ Hq(X; G)

E*0c;

1123 0; G) - - -(B ; G) \x\ \L

if" Pop with properties analogous to those in the homology case. If G is a commutative associative ring with unity element, then (15.3) is a ring isomorphism.

16. A theorem of P. A. Smith In the present and the following two sections, we shall give a few of the applications of the spectral sequences obtained in § 15.

If a discrete group n acts freely on a locally contractible acyclic space X of finite dimension, then n has no element of finite order other Theorem 16.1.

than 1.

1. Proof. Assume, on the contrary, that n has an element e of order r The subgroup of yr generated by is a cyclic group of order Y and acts freely on X. Thus we may assume that gr itself is a cyclic group of order r. Let B--..=- X hIn denote the orbit space. Then X is a regular covering space of B with yr as quotient group. This implies that B is locally contractible and finite dimensional. Let G = Z be the additive group of integers and let n operate simply on G. Consider the spectral homology sequence of § 15. Since X is acyclic, it follows that Ç0 E 2 0 s.,-, H P(n) and E2P a — 0 if q O. Then, by (VIII; 8.3), we obtain Hp (B) ,-,-..,, Hp (yr), (p > 0).

288


Since B is locally contractible, Hp (B) = 0 if p > dim B. However, Hp(oz) if p is odd. See (VI; Ex. K5). This is a contradiction. I 17. Influence of the fundamental group on homology and cohomology groups Let B denote a pathwise connected space. Since the singular complex S(B) has the same homotopy and homology structure as B, (V; Ex. I), we may assume without loss of generality that B is locally contractible. Then the universal covering space X of B is defined and the fundamental group ai ( B) operates freely on X. Theorem 17.1. nnp(B) =

0 for 1 < p

- Hr(B; G) is .ro ,r(W). Since E°° contains only two terms of total degree Y which might be different from zero, namely, E 7 and Efo, it follows that A 92,.. 1 , 1 (W) 90 ,r(W). Hence we obtain the exact sequence (a). By considering the spectral cohomology sequence of § 15, one can also establish the exactness of the sequence (b). In the proof of (17.2), we did not make full use of the two-term condition r 1, r; 2 J. Indeed, we actually have longer exact sequences given in the following Theorem

17.3. /17rp (B) = 0 for 1
Hr(a) --> 0,

(iii)

. ‹___P___ ' * Hr(B) ...--- 2, then the cohomology algebra H*(F ; K) is isomorphic to an exterior algebra generated by an element of odd degree n — 1. [Serre I, p. 492]. 4. If K is a field of characteristic zero and the cohomology algebra

2 95

EXERCISES

H* (B; K) is isomorphic to an exterior algebra generated by an element of odd degree n, then the cohomology algebra H*(F ; K) is isomorphic to a polynomial algebra generated by an element of odd degree n-1. [Serre 1; p. 489]. 5. If K is a field of characteristic zero and the cohomology algebra H*(B; K) is isomorphic to an exterior algebra generated by an element of even degree n > 2, then the cohomology algebra H*(F; K) is isomorphic to the tensor product of an exterior algebra generated by an element of degree n — 1 and a polynomial algebra generated by an element of degiee 2(n — 1). [Serre 1; p.489]. 6. If K is a field of characteristic p 0 and .Tin6 (B; K)P.:,, Hm(Sn; K) for every rn < p(n — 1) ± 1, where n > 3 is an odd integer, then the subspace of the cohomology algebra H* (F; K) formed by the elements of degree not exceeding p(n — 1) admits a homogeneous basis which consists of the elements { I, y, y2, • • -, yp -i, z }, where [Serre 1; p. 494]

O. 7. If K is a field of characteristic p 0 and the subspace of the cohomology algebra H*(B; K) formed by the elements of degree not exceeding pq, where q > 2 is an even integer, admits a homogeneous basis which consists of the elements { 1, y, y2, — • , yP -1 , z }, where deg (y) q, deg (z) pq, yP 0, then the subspace of H*(F; K) formed by the elements of degree not exceeding pq —2 admits a homogeneous basis which consists of the elements { 1, u, y 1, where [Serre I; p. 495] deg (u) — q —1, deg (v) — pq 2. deg (y)

—

n

—

1, deg (z)

—

—

p(n

—

1), yP

—

—

—

—

G. The cohomology algebra of (Z, n)

Let Z be the free cyclic group, Prove that the cohomology algebra H*(Z) over the ring of integers is isomorphic to the exterior algebra generated by an element of degree 1 and the cohomology algebra H*(Z, 2) over the ring of integers is isomorphic to the polynomial algebra generated by an element of degree 2. Next, let K be a field of characteristic zero. By means of the relations 1 and 2 in Ex. F, prove that the cohomology algebra H*(Z, n; K) over K is isomorphic to an exterior algebra generated by an element of degree n if n is odd and is isomorphic to a polynomial algebra generated by an element of degree n if n is even. H. The cohomology algebra of A

(S")

Consider the sphere Sn of dimension n > 2 and study the cohomology algebra of the space nn = A(Sn) of loops in Sil with so as basic point. Let T , [Sn; Sn , so]. Then T is a contractible fiber space over Sn with projection co : T ÷ Sn and fiber On = co-'(s0). From (13.2), deduce an isomorphism -

-

p* : Hm(Qn)

sd.,

Ilm-n+1(S2n)

296


for each m; p* is a derivation or an anti-derivation according as n is odd or even. Thus, a homogeneous basis { ei }, i = 0 1 1, 2, • • • , of H*(f27) is given by p* -1 (ei_ 1 ), i

e0 ----- 1, e i

1, 2, • • • ,

with dim (ei) = i(n — 1). Prove the following

Theorem. The multiplicative structure of the integral cohomology algebra H*(S2n) is given by the rule epeq cp, qep+q, where cp , q is an integer given as follows: (1) II n is odd, then (p + q) Cry)- fy

(ii)

n is even, then

p! q !

•

p and q are odd; otherwise ± q)12]1 [p/2] [qi12]!

0 if both

c

C 13 ,g—

where [x] denotes the largest integer not exceeding x. Then prove I. If n is odd, then (e l)P (p!)e p . If n is even, then (e 1) 2 0, (e 2)P e ve, (p!)8 2p, and e 1e 2p 2. If n is even, then H*(S271) is isomorphic to the tensor product of the algebras H* (SI") and H* (Q2n-1). 3. Let K be a field of characteristic zero. If n is odd, then H*(Qn ; K) is isomorphic to a polynomial algebra generated by an element of degree n 1. If n is even, then H*(S211; K) is isomorphic to the tensor product of an exterior algebra generated by an element of degree n — I and a polynominal algebra generated by an element of degree 2(n — 1). 4. Let K be a field of characteristic p O. If n is odd, then H*(Qn ; K) is isomorphic to a polynomial algebra with infinitely many generators h, (i = 0, 1,• •), modulo the ideal generated by (h)23, (1 = 0, 1, • • • ), being of degree pi(n — 1). 5. The cohomology algebra H*(Sn-1 x p2n-1) is isomorphic to the tensor product H*(Sn -1) I. The connective fiber spaces of S3

For each n > 3, let X n denote an n-connective fiber space over S3. Verify the following results on the coliomology algebra H*(X n ; Z 2) due to [Serre 3] : I. For the dimensions Ker(f)-->. A —I--->- B -÷Coker(f)--> O.

4.

THE ?-NOTIONS ON ABELIAN GROUPS

299

Furthermore, any pair of homomorphisms

A ---(--> B --> C gives rise to a natural exact sequence 5(1, g):

0 ----> Ker(f) --> Ker(d) ----> Ker(g) --->- Coker(f) -4- Coker(d) -> Coker(g) --> O.

In the applications of class theory, the groups in a class? are usually to be neglected in a certain sense. Thus we are led to the following terminology: Let ? be a given class. A group A is said to be (C--null if A c W. Let / :A-4- B be a homomorphism. Then / is said to be a W-pnonomorphism if Ker(f) E?, a W-epimorphism if Coker(f) EW, and a ?-isomorphism if it is both a Vmonomorphism and a W-epimorphism. If a W-isomorphism / : A --)-B exists, then A is said to be ?-isomorphic to B. If ? is the class 0, these notions coincide with the corresponding classical notions; and . it is more or less obvious that, for an arbitrary class?, these notions have the same formal properties as the classical notions. The detailed statement of these facts is deferred to Ex. B at the end of the chapter. Two abelian groups A and B are said to be ce-equivalent if there exists an abelian group L with two ?-isomorphisms / :L-->A and g : L ---> B.

abelian groups A and B are W-equivalent if there exists an abelian group M and two ?-isomorphisms h ; A - - > M and k : B -->- M. Proposition 4.1. Two

Proof. Sufficiency. Let L denote the subgroup of the direct sum A +B consisting of the elements (a, b) such that h(a) --= k(b). Define homomorphisms / : L----> A and g : L-> Bby f(a,b) = a and g(a, b) , b. Then one can verify that / and g are ?-isomorphisms. Necessity. Let A and B be ?-equivalent. Then there exists an abelian group L together with two ?-isomorphisms / : L -› A and g : L ----> B. Let M denote the quotient group of the direct sum A + B over the subgroup consisting of the elements (Al), g(1)) for all lei,. Let p : A + B ----> M denote the natural projection. Define homomorphisms h .A--->M and k:B---)-M by h(a) -,- p (a, 0) and k(b) . p(o, b). Then one can verify that h and k are ?-isomorphisms. I

The relation of being ?-equivalent is obviously reflexive and symmetric.

It

is also transitive. To verify this, assume that A, B and B, C are both ?-equivalent. By definition and (4.1), there are two abelian groups L, M and four ?-isomorphisms

f:L-->-A,g:1,-)-B, h:B-)-M,k:C-->M. Since kg : L --). M is a?-isomorphism, it follows that A and C are ?-equivalent. This proves the transitivity and the relation of being ?-equivalent is an equivalence relation.

300

X. CLASSES OF ABELIAN GROUPS

S. Perfectness and Completeness The classes of abelian groups with which we will deal in the sequel usually satisfy some further conditions described as follows. A class ? is said to be perfect if A c? implies that Hm (A) E? for every m > O. ? is said to be complete if A G? implies that A 0 B E? and Tor(A, B) e %2 for every B. ? is said to be weakly complete if A e? and B e? imply that A ®Bece and Tor(A, B) E?.? is said to strongly complete if every finite or infinite direct sum of groups in ? is also in?. Every complete class is obviously weakly complete, and it can be verified that every strongly complete class is complete and perfect. See Ex. C at the end of the chapter. The usefulness of these completeness conditions can be illustrated by the

following Proposition 5.1. If ? is a complete class, X a

pathwise connected space,

X 0 a subspace of X, and G = { Gx 1 xeX} a local system of groups in X with each Gx e W ' , then Hm (X, X o ; G) is in? for each nt > O. Proof. Pick xo e X as in (IX; § 2) and denote Gx0 also by G. Then Hm (X,X 0; G) is isomorphic with a quotient group of some subgroup of Cni (X,X 0)0G

which is in?. Hence Hm (X ,X 0; G) is in?. I As to the examples of classes (1)—(6) in § 2, one can verify that (I), (2), (6) are strongly complete, and that (3), (4), (5) are prefect and weakly complete but not complete. The classes in § 3 are all strongly complete; in particular, the class .922 of the p-primary groups is strongly complete. For an example of a class which is perfect and complete but not strongly complete, see Ex. A2, at the end of the chapter. In the sequel, we will deal with two different kinds of applications. For the first kind, it suffices to assume that the class ? involved is weakly complete (and, sometimes, perfect). For the second kind, we have to assume that ? is complete. However, this difference is not of much practical importance. In fact, -the homotopy and homology groups considered in the applications are usually finitely generated; and if ? is a given class and Wf is the class consisting of those abelian groups all of whose finitely generated subgroups are in ?, then Wf is strongly complete and Wf

(la,.

--...%7

(1 ,(21f.

6. Applications of Classes to Fiber Spaces

Let us go back to the notation of (IX; § 3) and assume that 7c 1 (B, b o) operates simply on the homology and cohomology groups of the fiber F. Unless otherwise stated, the coefficient group G is the group of integers and hence is omitted from the notations.

6.

APPLICATIONS OF CLASSES TO FIBER SPACES

301

Let be any given class of abelian groups. Let us consider the spectral homology sequence of (IX; § 7). Lemma 6.1.

If, for some n > 1, EL

then E;,°,q E

Since EV-q-. 1 is the quotient group of a subgroup of EL, it is in W. Then it follows by finite induction that KZ, is in W. I Proof.

If , for a given pair (p, q), Erj EW whenever i+f=p+q and i < p, then Ilp, q(X, X 0) eV. In particular, if EZi E V whenever i f = p q, then Hp+q(X, X0) E W. Lemma 6.2.

The lemma follows from the fact that Hi,i(X, X0) is an extension of Hi_1,1 +1 (X, X0) by E. and that 1/0,,,(X, X0) E. I Proof.

If V is a weakly complete class and if, for some integer Y> 0, Hm (B, B 0) E W and Hm (F) EV whenever O <m < Y, then we have Hm (X, X0) G V whenever 0 < m < Y. Proof. According to the above lemmas, it suffices to show that 4q e whenever 0 < + q < Y. By (IX; 6 .3), EL F.:: Hp (B, B 0) 0 Hq(F) Tor(Hp _1 (B, B 0), Hq(F)). If p> 1, q> 0, and 0 < p + q H p (X) H p (X , F) we have H(X) G morphism. I

Hp _1(F) Hp _1 (X) • • •

and H_1 (X) G W. This implies that

If V is a complete class, Hm (X) E Hm (B) G ï whenever 0 < m < 5 for some given integer whenever 0 0, and > 0, then Hm(F) G

Hm (F) Hm+i (X , F) Hn _f_ 1 (B , 1)0) m+i (B) define a -equivalence of Hm (F) and Hm±i(B) whenever p — I O.

is a weakly complete class, 111 (B) = 0, and two of the spaces X B, F are -acyclic, then so is the third. Theorem 6.9.1/ V

Proof. If the two spaces are B and F, then it follows from (6.3) with r = co and Bo = D that X is V -acyclic. If the two spaces are X and B, then it follows from (6.7) with p = co that F is V-acyclic . If the two spaces are X and F, then we shall prove Hp (B) e V by induction on p. The case p I is trivial. Assume 5> 1 and Hm (B) E if 1 <m < p. I By (6.7), Hp (B) is '-equivalent to H. 1 (F) and hence Hp (B) As an application of these results, let us consider the special case where X = [B; B, b 0] and co : X -± B is the initial projection. In this case, the fiber F becomes the space A(B) = [B; b p , 1)0] of all loops in B with 1)0 as basic point. Assume that B is simply connected and hence A(B) is pathwise connected. Therefore, we may apply the results of this section. Since [B; B, b0] is contractible, the following theorem is an immediate consequence of (6.7) and (6.8).

is a class and Hm (B) GW' whenever O < m < p, then Hm (A(B)) is -equivalent to Ilm , i (B) for the following values of is weakly complete. (i) O < m 1, then A(B) is a space of the homotopy type (n, n 1). Thus, we may apply (6.10) to this special case and obtain Theorem 6.10. // V

Hm (n, n — I)

n), 0 < 1, and a weakly

complete class W, the following two statements are equivalent: (i) II:m (7r) E W for each m > O. (ii) Hm (7r, n) E W for each m > O. 7. Applications to n connective fiber spaces -

Let B be a pathwise connected space and bo a given point in B. According to (V; § 8), we may construct inductively a sequence of spaces (B, n), n = 0, 1, 2, • • , and a sequence of maps

fin

:

(B , n) -4- (B n — 1) , n — 1, 2, 3,- •,

as follows. Let (B, 0) = B. For each n > 0, let (B, n) be an n-connective fiber space over (B, n — 1) with projection fin. This system { (B, n) , fin } will be referred to as a connective system of the space B. If B is locally pathwise connected and semi-locally simply connected, then of course we may take (B, 1) to be the universal covering space over B with /ill as the projection. Now let { (B fl.} be any connective system of B. Since (B, n) is an n-connective fiber space over an (n 1)-connected space (B, n — 1) with fin as projection, it follows that the fiber of this fibering is a space F. of the homotopy type (7r.(B), n— 1). Next, let

con = fiifi 2 "

fln:

(B,n) -B,

n

I , 2, • • ,

then it is clear that (B, n) is an n-connective fiber space over B with con as projection. Applying (6.9) and (6.11) to the fibering fin , we obtain the following Proposition 7.1. If

W is a perfect and weakly complete class, n> 1, and

71n (B) e W, then the following two statements are equivalent: (i) (B, n) is W -acyclic. (ii) (B, n — 1) is W -acyclic.

co to the fibering fin, we obtain the following

Applying (6.6) with q Proposition 7.2.

If W is a perfect and complete class,

then

> 1, and 7E.(B) E W,

m (B , n) .HM(B , n — 1) is a W-isomorphism for each m > O. *:

Finally, if B is (n — 1)-connected, then we may take (B, ni) = B for every m 1, and 5> n is an integer such that Hm (B) e W whenever n 0,

we obtain (A)* — 1?-1 (187)4ti. 8. The generalized Hurewicz theorem

Let be a perfect and weakly complete class. If X is a simply connected space and n > 2 is an integer such that n m (X) e whenever 1 2 is an integer such that Hm (X) e V whenever 1 1).

Hence, gri ( Y0)

0 and 7cm ( Y0) E W for 2 <m A 2 -/-L--> A 3 ±-> A4 ->' A 5 Assume that there exist two homomorphisms gl : A 2 -÷ A 1 , and g4 : A 5 --->A 4 such that the endomorphisms / 1 g 1 and /4g4 are '-isomorphisms. Define a homomorphism h: A 3 + A 5 --)-A 4 by taking h(x,y) — / 3 (x) + g5(y). Prove that h is a ''-isomorphism. F. The Products of C-Equivalent Groups

Let W be a complete class. Prove that, if A and B are 'C-equivalent respectively to A' and B', then A 0 B and Tor(A,B) are W-equivalent respectively to A' 0 B' and Tor(A' , B'). G. C-Exact Sequences

Let W be a class and A, B two subgroups of an abelian group G. We say that A and B are ce-equal if the inclusion homomorphisms A n B --- > A and A n B --> B are ?-isomorphisms. Replacing equality by W-equality, one can define the notion of a W-exact sequence. 1. Establish the elementary properties of W-exact sequences as in [E—S; p. 50]. 2. Generalize the results in (VIII; § 8) to obtain various fundamental W-exact sequences. H. On Induced Homomorphisms

Let X, Y be simply connected spaces, f: X -4- Y a map such that /* : 7 r2 (X) --> n 2 (Y) is an epimorphism. Assume that the homology groups are finitely generated.

310

X. CLASSES OF ABELIAN GROUPS

1. Let eF denote the class of all finite abelian groups, ,F the class of all torsion groups, and G a field of characteristic zero. Prove that the following four statements are equivalent: (a) /* : Hm (X) -->- .11.(17) is an g--isomorphism for In < n and is an g--epimorphism for In -- n. (b) 14t : Hm (X) ---). - H m (Y) is a F-isomorphism for in < n and is a .r-epirnorphism for In — n. (c) /4 : II.(X; G) --->- Hm (Y ; G) is an isomorphism for m Sn+1 defined by co(x) = x(1) for every path x e X and with fiber W = Let U and V denote the north and the south hemispheres of Sn+1 respectively and let ,K0 0) -1(Sn) , X tt 0) -1(U) , Xv co _1(v) . We are going to define a map

x : (U x W, Sn x W) -> (X u , X0) which will be called the canonical map. For each point b E U, let y(b) E Xu denote the path joining so to u and then u to b by geodesic arcs. The assignment b ---> y(b) defines a cross-section y: U X. Then x is defined by x(b, i) = /y(b) for each b E U and f e W, where / y(b) denotes the product of the paths f and y(b). As a consequence of the construction, we have cox(b, Lemma 3.1.

b, (b E

e W)

The canonical map x is a homotopy equivalence.

Proof. Let 2: (X e , X 0) ->

x W, Sn X W) be the map defined by

2(x) = (co(x), x • [yco(x)] -1 ),

(x

E Xu).

where 1yco(x)] -1 denotes the reverse of the path yco(x). Then we have x• fyco(x)] -1)

2, (I'Y(b))

(x• [yco(x)] -1).yco(x),

( 1), [17(b)1'[Y(v)] -1).

Hence, n2 and An are both homotopic to the identity maps. I Next, let us define a map :S x W W by taking ,u(b, f) / • i(b) for each b E Sn and fE W, where i : S1Z ---> W denotes the imbedding in § 2. Lemma 31.

y =

The map ,u is homotopic in X v to the map y : S X W X 0

I Sn

definby

X W.

Proof. Intuitively, a homotopy of itt to y is accomplished by "unwinding" the path i(b) to half its original length. More precisely, define a homotopy : Sn -> Xv , (0 < < 1), by taking [ht(b)](s)

ri(x)]

s

(

1

t

), (b E Sn, S E

t

/).

3 14

XI. HOMOTOPY GROUPS OF SPHERES

Then ho = i andh1 = y. Define a homotopy k t : Sn x W X v , (0 0 an isomorphism

p* : H2,(Sn) Hq (W)

Let in follows.

lin÷q(W).

n q + 1. The construction of p * will be made in six steps as

Step 1. Since the south hemisphere V of Sn-o- is contractible to the point so,

an application of the covering homotopy theorem proves that W is a strong deformation retract of X. Hence the inclusion map induces an isomorphism

: Hm(X, W) Re, Hrn,(X, Xv).

Step 2. The inclusion map induces a homomorphisrn :

Let

Hm (xu, X 0) ---> Hm (X,

Du = Sn+1 \y, Dv = Sn+1 \u, D =Du n Dv ; Yu = co -1 (Du) , Yv = co -"(D), Y = Yu

n Y.

Since Yu and Yv are open sets whose union is X, the excision theorem holds and hence the inclusion map induces an isomorphism

Hm ( Yu , Y)

117,1(X, YV) •

Since, U, V, Sn are strong deformation retracts of Du , Dv , D respectively, an application of the covering homotopy theorem proves that Xu, Xv, X0 are strong deformation retracts of Yu , Yv , Y respectively. Hence n is an isomorphism. Step 3. Since the canonical map x is a homotopy equivalence, it induces an isomorphism ff n,(Xu, X0). Ilm (U x W, Sn X W) Step 4. By the Kiinneth theorem, we get an isomorphism

: Hu +i (U, Sn) Hq (W) H in(U X W,S X W).

Step 5. Since X is contractible, we have an isomorphism :

W) Ilm-i(W).

Step 6. Since U is contractible, we have an isomorphism Hu,i(U, Sn)

H(S).

Taking tensor products, we obtain an isomorphism

: .1-4 4. 1 ( U, Sn)

I/6 ( W)

Hu(Sn) H q (W).

5.

RELATION BETWEEN

p * AND

315

Composing these steps, we get an isomorphism

p*

--1-Tpe*0 -1 : H(S) 14(W) Hn +q (W).

This isomorphism p * is the same as the homomorphism p* in Wang's exact sequence (IX; 13.1) for the fibering to : X Sn41-. See also [Wang 21. 5. Relation between p* and 14,v The space W of loops has a continuous multiplication

M:Wx W.-÷W defined in (III; § 11). The total homology group OD

H (W) = H m (W) 112-O

becomes a ring under the Pontrfagin multiplication defined as follows. Let oc E Hp (W) and /3 G 14(W). By the Kiinneth theorem, a and 13 determine a unique element cc X fl of .Hp+q (W X W). The map M induces a homomorphism M * lim (W X W) ---> .H m (W) for every m. Then the Pontrlagin product of a and 16 is defined to be the element ocfl = 11/*(tx X fl) E 1-11)+4. (W). Proposition 5.1.

For every a. E H(S) and 13 e I4(W), we always have P*(oc

fl)

where i# : J-1.(Sn) Hn (W) denotes the homomorphism induced by the imbedding i : Sn W of § 2.

Proof. Consider the diagram

lim (U x W, Sn X W)

Hm (Xtt, X0) -IL Hm (X,

v)

Hm (X , W)

a 10

Rin _1 (Sn

X W) .

Hin-i (X0)

10

Hm-i(W) where o- and 7' are induced by inclusion maps and the homomorphisms a are boundary operators. The rectangules are all commutative and hence (1) By (3.2), we have (2)

xv * = Crit * .

By the Kiinneth theorem, we have an isomorphism X: H(S) 14(W),--zed Hn,q(Sn X W)


316

and a commutative rectangle Hn+q,i (U X W, Su x W)

Hn+i (U, S ) 0 14(W)

Hn ,q (S21 X W).

H(Sn) O Hq (W)

Hence we obtain

00 -1 .

X

(3)

Using (1), (2) and (3), we deduce p* =

= 140-1 = p*X.

==

Then it follows from the definition of la that P*(0c fi) = ra*Z(.7. 0 ,8) = fi /4(0c).

In particular, if q = 0, then Ho(W) is a free cyclic group generated by the element e represented by wo as a 0-cycle of W. For each oc e H(S), we have i44_(oc)

—

e • i 44.(a) = p* (cic

e) .

This proves Lemma 2.2. 6. The triad homotopy groups

Consider the space of paths By (IV; 3.1), we have

T

[W ; Sn, so].

cm (T) = nm , i (W , SI?)

for every m. Hence the homotopy sequence of the pair (W, an exact sequence • --> rc.(T) --> nIn,(Sn)

+i (S

1)

Sn)

gives rise to

n,(T) ---> • • .

This is essentially the suspension sequence of the triad (Sn+1 ; U, V), n. (T) being essentially" the triad homotopy group nm+2(Sli+1 ; U, V). See (V; §§ 10-11).

Because of this exact sequence, it is desirable to determine the triad homotopy groups nn,(T). The following lemma is an immediate consequence of the suspension theorem (2.1).

6.t nm (T) = 0 for every rit < 2n —2. To determine the higher homotopy groups of T, let us study the space of paths Q [W ; Sn, TV] Lemma

which is of the same homotopy type as Sn. Consider the projection co : Q ->W defined by co(a) = a(1) for every a E Q; then Q becomes a fiber space over W with fiber co - i(so) T.

7.

FINITENESS OF HIGHER HOMOTOPY GROUPS

Since Hm (W) O whenever 0 <m < n and .NT) -,0 <m H1(W) --->0 -

Since Q is of the same homotopy type as Sn, we have Hm+i (Q) 0 - Hm (Q) for every ni > n. This implies that Hm (T) Hm+1 (W) whenever n < ni < 3n —2. Hence, we deduce the following Lemma 6.2.

H 21 (T) c Z and Hm(T) -= 0 whenever 2n —1 < ni < 3n —2.

Choose a map f : S 271-1 T which represents a generator of the free cyclic group 7C 2 _ 1 (T) I/21, ( T). Then f induces an isomorphism

Then, by (6.1) and (6.2), it follows that fit : lim (S 2n -1) Hm(T) for every m Hin(X; K) Hm (F ; K) -› Hm --92-0-(F ;K) -÷ Hm+1 (X ; K)

-,


318

where p* is a derivation since n is odd. Since fin(X; K) — 0=.- Hu-1 (X; K), p* sends Hn -1 (F ; K) isomorphically onto ll"°(F ; K). Therefore, p*(oc) is a non-zero element of H°(F ; K)r:.-; K. Now let us prove that p* :

HP(92-1) (F ;

K) F.', H(7?-4 (n-1)(F ; K)

for every positive integer p. In fact, the vector space HP(n-1)(F ; K) over K admits oc29 as a basis. Since p* is a derivation, we have p * (kOCP) — pkp*(a)ccP -1, (k e K).

Hence p* is an isomorphism for every m > O. Then an exactness argument proves that .Fini(X; K) ,- 0 for every m> O. Since Hm (X) is finitely generated and K is of characteristic zero, this implies that Hm(X) is finite for every m > O. An application of the generalized Hurewicz theorem (X; 8.1) proves that 7Cm(X) is finite for every m. Thus, we have proved the following Theorem 7.1.

If Sn is an odd-dimensional sphere and m > n, then nn2,(Sn)

is finite.

8. The iterated suspension The natural imbedding Sn+1 c A(Sn4-2) of § 2 induces an imbedding i : A(Sn-o-) --->A 2(Sn+2).

Composing with the natural imbedding i:Sn--›-A(Sn+ 1), we obtain an imbedding k — fi : Sn -)-11_ 2 (Sn 4-2).

For each rn, k induces a homomorphism k * :n.(Sn, so) -÷7r,m (A 2 (Sn 4-2), so).

As in § 2, we have a natural isomorphism 1* : nm (Ap(sn+2) , so ) R.e nm+2 (Sn-1-2 , Proposition 8.1.

1 *k * is equal to the iterated suspension E2.

Proof. By § 2, there is an isomorphism :7rtn (A(s2H 3.) , so -

)

76

. +I (sn+i , so) .

Similarly, there are isomorphisms ,. p : 76.(A2(st14-2) , so ) s.. , jrnt+1 (A(Sn+2) , so) , y : nra+1 (A (Sn+2) , so) rrtz," n (sn+2, so ) . Then l* = yfl and k * = . The proposition is a consequence of the cornmutativity of the diagram: i* grm (Sn, so) ---> ani (A(Sn 4-1), so) l '—'--)- 7-cm (A2(Sn4-2) , so)

la am +1 (91+1

So)

? -->. 7Cm+ 1(11(Sn+2 ) , So)

I

Y

ani+2(sn+ 2 so). 1 ,

The following proposition is an immediate consequence of (8.1) and (2.1),

9.

THE P-PRIMARY COMPONENTS OF nnt(S 3)

319

homomorphism k * is an isomorphism if m < 2n 1 I. and is an epimorphism if m = 2n If we study the p-primary components instead of the whole homotopy groups, then we can deduce more detailed information from the iterated PrOposition 8.2. The

—

—

suspension E 2. n > 3 be an odd integer, p a prime number, and the class of all finite abelian groups of order prime to p. Then the iterated suspension E2 :m(S) „grm+2(914.2) Theorem 8.3. Let

-isomorphism if m < p(n + 1) — 3 and is a W-epimorphism if m P(n + 1 ) 3. Proof. According to (8.1), it suffices to prove the theorem for the homomorphisms k*:am(Sn) ___>. 7cm (A2(Sni-2) ) is a

indUced by the natural imbedding h : Sn A 2 (Sn+2). Let K be a field of characteri3tic p. Then k induces the homomorphisms k# : Hm(A 2 (Sn+2); K) Hm(Sn K). By Whitehead theorem (X; 10.1) and (X; Ex. H2), it suffices to show that klt is an isomorphism for every m 0 and hence q + 4p 8 < p(q — 1) — 3. This implies In 2 < (q — 1) — 3 whenever m 2n — 1.

13. THE P-PRIMARY COMPONENTS OF HOMOTOPY GROUPS

325

(7) If p is an odd prime number, then the 5-primary component of arm (V) is isomorphic to that of n m (S2n --1).

12. Finiteness of higher homotopy groups of even-dimensional spheres is an even-dimensional sphere and m is an integer such that m> n and m 2n 1, then nni(Sn) is finite and n21 (Sn) is isomorphic to the direct sum of Z and a finite group. Theorem 12.1. // Sn

—

Since 7r,n (S 2)r,-;_, 7c.(S 3) for every m> 2, the theorem is true for n . 2. Hence we may assume n >4 and apply the results of § 11. If ni> n and m 2n — 1, then both am (V) and mi (Sn') are finite. Therefore, the exactness of the sequence nm (V) -+7 m (S) -÷ n 2,_ 1 (Sn -1) implies that 7m(S) is finite. To study the critical case m . 2n — 1, let W denote the class of all finite abelian groups. It follows from the exact sequence that Proof.

w* : 7r271,-1( 17) -›- n2n-1(S n) is a W-isomorphism. This implies that n 2._1 (Sn) is isomorphic to the direct sum of Z and a finite group. I

13. The p-primary components of homotopy groups of even-dimensional spheres In the present section, we are concerned *ith the p-primary components of the homotopy groups nm (Sn) of an even-dimensional sphere S. Since nm (S 2) -,-.., nm (S3) for every m > 3, we may restrict ourselves to the case n _> 4 and apply the results of § 11. Consider the following part of the exact sequence appearing in § 11: amil.(S n) Ll* nni(Sn-1) '''-÷ nnt(V) a ' --' 7m(S) -cL'L' Using the homomorphism co * and the suspension /, we define a homomorphism r :7(T7) ± 7l m _ 1 (Sn --1) --) - 7r,(Sn) by setting Iloc, ig) — co* (a) ± /(16) for each a e 7rm (V) and ,8 en._1 (Sn A ). Lemma 13.1. 1/ W

denotes the class of all finite 2-primary groups, then P is a

'-isomorphism. According to (11.1), the suspension E followed by d* is the induced endomorphism k * on nni-i (S n-1) of a map k : Sn -1- --> Sn -1 of degree d — ± 2. By Ex. A6 at the end of the chapter, k * is a W-automorphism. Hence the lemma follows as a consequence of (X; Ex. E). 1 Proof.

denotes the class of all finite 2-primary groups, then the homotopy group am (Sn) of an even-dimensional sphere Sn is (g-isomorphic to the direct sum of grm (S 2n -1) and nni_1(Sn-1). Theorem 13.2. If W


326

Proof. Since yon (S 2) 7(.(S 3) for every m > 3, the theorem holds for n = 2. If n > 4, then (13.2) is a direct consequence of (13.1) and (11.2). I The importance of (12.1) and (13.2) is that the calculation of the homotopy groups of an even-dimensional sphere, except for their 2-primary components, reduces to that of the homotopy groups of odd-dimensional spheres. Precisely, we have the following Corollary 13.3.

If n is even and p an odd prime, then the p-primary com-

ponent of 7m(S) is isomorphic to the direct sum of those of ani(S 291-1) and

14. The Hopf invariant In order to strengthen (13.2), we propose to present Serre's version of the notion of Hopf invariant. Consider the n-sphere Sn and a given point so E sm. Let .§2n A(Sn) denote the space of loops in Sn with s o as basic point. Consider the natural iso-

morphism

:g2n-2(nn) n2n-i(S n)

and the natural homomorphism h gr2n - 2(Qn) -->H2n-On) Z.

Let f: S2n-1 -->Sn be a given map representing an element [f] e7 27,1 (Sn). The Hopf invariant of f is the integer 11(f) uniquely determined by hi-1 ([f]) = H(i)u2 ,

where u 2 denotes the generator p* 2 (1) of H2._2 (0n) determined by the homomorphisms p* in (IX; 13.1). For other definitions of Hopf invariant, see Ex. C at the end of the chapter. When n is odd, H(f) is always zero; when n is even, there exists a map f with H(f) = 2; if n 2, 4 or 8, then there exists a map f with H(f) = 1, namely the Hopf maps of (III; § 5). See [Hopf 2] and [S; p. 113]. Theorem 14.1. Let n be an even integer and f S 2n -1 -÷Sn

a map with Hof

invariant II(f) = I? 0 O. Let V denote the class of all finite abelian groups of order dividing some power of k. If Xf:

m .1(S n-1) + nm (S 2

) --)-nm (Sn)

is the homomorphism defined by Xf(a , P)

E(00

f(fi)

x enm-1( 9"),

e 7tnb(S 2n-1),

then Xf is a V-isomorphism for every m > 1.

Proof. The theorem is obvious if n = 2. Hence we assume n > 4. The map / defines a map g: S22n -1 nn in the obvious way and g induces a homomorphism g* : .H*(S2n) H*([22n-1)

14. THE HOPF INVARIANT

327

of the cohomology algebras with integral coefficients. According to (IX; Ex. H), 1-/*(S2n) admits a homogeneous basis { ai} with dim (ai) = i(n —1) for each i — 0, 1 , • • • such that ao = 1, (a1) 2 = 0, (a 2)P = (5I)a 2p, a 1ot 2p

a2pa1

a 2p, 1 .

Similarly, H*(S221z -1) admits a homogeneous basis {bi} with dim (bi) i(2n — 2) for each i = 0, 1,» such that 1)0 = 1, (b 1)P

Since H(/)

(N)b p .

k, it follows that g*(a2) = kb,. Then we get (p!)g*(a2p) = g*(a229) = kP(bi)P

and therefore g*(a 2p) = kPbp. Since obviously g*(a2p .+1 ) = 0, it follows that g* is completely determined. Now let i : Sa -1 --)..f2n denote the natural imbedding in § 2 and consider the induced homomorphism i* • H(Q) ->11*(Sn -1).

By (2.2), e = i* (al) is a generator of Hn --1 (Sn -1). By means of the multiplication in Qn, we may define a map ,95 : Sn -1 X s22n---1 „Du by setting 0(x, w) = i(x) g(w) for each x c Sn -1- and %V E S2 211-1 . By (IX; Ex. H), H*(Sn -1 x .02n -1) is naturally isomorphic to the tensor product H*(Sn -1) H*(D2n -1). This enables us to determine the induced homomorphism

as follows:

y51* : .H*(Qn) 11*(Sn -1 X Q27 1) 1954*(a 2p) 964 (a2P+i)

hP•I 0 bp,

0*(ai.)95#(a2p)

kPe

bp.

D271-1) is a monomorphism and its Hence, 0,4* Han (pn) wiz cokernel is finite of order equal to a power of k for every dimension m. By duality, this implies that 0.# : Hin,(Sn-1 >< syn.-1) Hm(Q) is a f-isomorphism for every m. An application of the Whitehead theorem shows that the induced homomorphism

4)* grnz(Sn-1

X 122n-1)

„nin (Qn)

is a ce-isomorphism for every In. The group 7Cm(.91-1 X Q2n -1) is isomorphic to the direct sum 7Cm (Sn -1) ani (D2n-1‘) ; on the other hand, nm(Q) n.+1 (Sn). Then, the construction of çb shows that O. * reduces to the homomorphism Z1. I I/ ,11(1) = ± 1, then ,Cf is an isomorphism and hence the suspension E :nm _1 (Sn -1) -->nm,(Sn) is a rnonomorphism for every ni> 1. Corollary 14.2.

Because of the existence of a map f with 11 (f) = 2, (14.1) implies (13.2).

XL HOMOTOPY GROUPS OF SPHERES

328-

15. The groups an+1 (S ") and an+2 (S")

Since an,(51) -= 0 for each m> 1 and 7r.(52) nm (S 3) for every m> 2, we may assume n > 3. Theorem 15.1.

n. +1(Sn) is cyclic of order 2 for every n > 3.

Proof. Let X denote a 3-connective fiber space over S3. Then, by (9.1), we have Z 2. n 4(S3) rk-,, TC 4 (x) c .ff 4 (X)

By the suspension theorem (2.1), we deduce • • • tr -' n +1(Sn) ''Jj • • • • n,5(54) Hence n 1 (S) Z 2 for every n > 3. 1 One constructs the generator of 1 n+1(S9i) as follows. Let us consider the Hopf map p : .s3 5 2 defined in (III; § 5). According to (V; § 6), p represents the generator of n3(52). Since the suspension E n3(52) ->r4 (S 3) is an epimorphism, the suspended map Ep : S4 --> S 3 represents the generator of 714(53). Then the generator of n(Sn) is represented by the (n 2)-times iterated suspension En -2P of the Hopf map p. 7r4(S3)

Theorem 15.2.

n,,, 2( 5n) is cyclic of order 2 for every n > 3.

Proof. Applying (10.1) with n -= 4, h

sequence

2, m 5, we obtain an exact

0 -)-Z 2 (23) Z 2 > 5 (P ) -> Tor(Z , Z 2) -> O.

Since Z20 Z 2 W, Z 2 and Tor(Z , Z 2) = 0, this implies r5 (P) Z 2. Then, by (10.2), the 2-primary component of 'r5 (S 3) is isomorphic to Z 2. Since the p-primary component of 2 r 5 (53) is 0 if p > 2 by (9.2), it follows that n 5 (53) Z2. Since the Hopf map 57 -*S is of Hopf invariant 1, we may apply (14.2) with n .= 4 and ni 6. Hence, (2.1) and (14.2) imply that E :n5 (S 3) 7c6 (S 4). Thus, n6 (S4) c Z 2. Finally, by (2.1). we deduce 7C 6(S 4 )

277(.55)

•••

n +2(Sn)

•••

Hence 7C n + 2 (S n ) Pe Z 2 for every n > 3. 1 To obtain the generator for 7rn + 2(S n) , let i :S4 -÷ .11 denote the imbedding given by the definition of P. Then, by the proof of (10.1), the generator of n 5 (.11) is represented by the composition of i and E275: 55 -->- 54 • Composing with the map X: P -÷S3 in § 10, we obtain a representative map Xi E2p for the generator of n 5 (S 3). This implies that Xi represents the generator of 7c4 (S 3) and hence is homotopic to Ep :s4 -3.53. Therefore, the generator of n 5 (.5 3) is represented by 9, Ei5 E275 s5 where 75 : S 3 52 denotes the Hopf map. Then it follows that the generator of 7rn + 2 (Sn) is represented by the (n 3)-times iterated suspension En --3q of q for every n > 3.

16. THE GROUPS nn4.3(Sn) Corollary 15.3. n 3 (S 2)

Z, 7C 4(S2)

Z 2, 5 (S2)

329

Z2.

By (V; § 6), the generators of these cyclic groups are represented by respectively the maps

P s3

P

EP S 4 ÷ S2 , P 0 EP 0 E 2P s5 s2 -

-

16. The groups r +3(Sn) Theorem 16.1.

8 (S3)

Z12 .

Proof, Applying (10.1) with n — 4, h

sequence

0

Z

—

2, m = 6, we obtain an exact

Z -± 6 (4 ---> T (2 2 , Z2) -÷ 0.

Since Z20 Z 2 Z 2 and T or(Z 2 , Z 2) c Z 2, 7C6(11 is isomorphic to an extension of Z 2 by Z 2 and hence has 4 elements. Hence, by (10.2), the 2-primary component of 7T6 (S3) has 4 elements. By (9.2), the 3-primary component of 7(6 (S3) is isomorphic to Z3 and the p5-primary component of 6 (S 3) is 0 for every prime > 3. It follows that gr6 (S3) has 12 elements and hence is isomorphic to either Z 12 or Z Z6. Suppose that n 6 (S 3) Z6 . Let X denote a 5-connective fiber space Z over S3, then H6 (X) g6 (X) .7r6 (S3) Z ± Z6, and it follows from the universal coefficient theorem [E-S; p. 161] that H6 (X; Z2 ) i Horn VI 03 (X) ; Z 2)

Z

Z2.

This contradicts to (IX; Ex. 1); hence, we conclude that 6(S 3) Z 12'2. I Examination of the first paragraph of the proof reveals that the composition of the maps SG --)- S 5 — z2P-> S4 Sa represents an element of 7c6 (.53) of order 2. A generator of 7C6 (S 3) is represented by the characteristic map : S 6 -9- S 3 of the fiber bundle sp(2) over S 6 with S(l) as fiber, [Borel and Serre 1; p. 442]. For the definition of the characteristic map, see [S; p. 97]. Corollary 16.2. n6 (52)

Z12 .

A generator of a6 (S 2) is represented by the composed map Theorem 16.3. 7r7 (S 4 )

Z

:56 52.

Z 12.

Proof. Let us denote by q; S 7 --- S 4 the Hopf map in (III; § 5). Since H(q) = 1, we may apply (14.2) with n = 4, na = 7, and / = q. Thus, we obtain an isomorphism zq 2.r 7(54) 6(S3) + 7r ( 5 7 )

Since 7c7 (S 7) Z and n 6 (S 3) Z 12 , this proves the theorem, I From the preceding proof, it follows that q represents the generator of


330

the free component Z of :r7 (S4) and the suspended map E$ : S7 --> S 4 represents an element of order 12 which generates the torsion component Z 12 of 7C7 (S4). Theorem 16.4.

r+3(S)

Z24 i/ n > 5 .

By the suspension theorem (2.1), E maps n7 (S 4) onto n8(S 5). According to Ex. D at the end of the chapter, the kernel of E is the free cyclic subgroup of n7 (S 4) generated by the Whitehead product [e, e], where e denotes the generator of 7c4 (S4) represented by the identity map on S. On the other hand, it follows from a theorem on characteristic maps, [S: p. 121 1 , that 2 [q] e E [$] [e, e] Proof.

where e =± I depends on the conventions of orientation. Hence, in 7C8(S 5), we have

E2 [$]

e2E

[q].

This implies that n8 (.35) is isomorphic to 224 with I [q] as a generator. Finally, by the suspension theorem (2.1), we deduce 7r9 (S 6)

7c8(S 5)

••

7C714.3(Sn) rr:-5 • • • .

for every n > 5. 1 Obviously, a generator of n 11+3 (Sn), n > 5, is represented by the (n — 4)times iterated suspension En -4q : Sn 4-3 Sn of the Hopf map q : S7 -÷ S4 .

Hence 741+3 (Sn)

Z24

17. The groups arn+4(S ) Theorem 17.1..7r7 (S3)

Z2.

By (9.2), the p-primary component of n7 (S 3) is 0 for every prime > 3. By (10.4), the 3-primary component of 7(7 (S 3) is also O. Hence 7(7(S3) is a 2-primary group. Next, consider a 6-connective fiber space X over S. Then 7c7 (S 3) H7 (X) and hence we have 7C7 (X) Proof.

Hom (7c7 (S 3), Z 2) r.s H7 (X; Z2).

By (IX; Ex. 14), H7(X; Z 2) Z 2 . This implies that 7(7 (S3) is isomorphic to a cyclic group Zq with q 24 , h > 1. If h> 1, every homomorphism of a7 (.53) into Z 2 can be factored into n7 (S 3)

4 -÷ Z2.

Then it follows from the exact sequence in (VIII; Ex. 17) that Sea = 0 for every element a c H7 (X ; Z 2). This contradicts (IX; Ex. 14). Hence It = 1 and

7(7 (S 3)

c

According to [Hilton 2; p. 549], the two maps 0 E475 S7 S 3, E P 0 q S 7 S3

17. THE GROUPS n224.4(S-4)

331

are both essential. Hence they are homotopic and represent the non-zero element of n7 (S 3). Corollary 17.2. gr7 (S 2)

Z2.

The non-zero element of

7(7 (5' 2)

is represented by the homotopic maps

PO OI4P:S7.-÷ S2, PO EPOq: 51 .-S2Theorem 17.3. Proof.

a8(S4) Z 2 Z2.

As in the proof of (16.3), we obtain an isomorphism :

7 (S3)

+ 768(.5 7)

a8 (S 4).

Since n7 (S3) Z 2 and n8(S7) Z 2 , the theorem is proved. 1 The group 7r8 (S4) is generated by two elements cc and /3 of order 2. cc is represented by the hornotopic maps (e 0 E4p)

st, E (E

p

s4

q) S

and le is represented by q (j) Egp : S8 S4. Theorem 17.4. n9(S5) Z 2 . Proof.

Consider the following part of the suspension sequence in § 6:

n8 (S 4) -÷ 9(S5) -÷ 7(7(T ) 7(8(S 5) -± 0 n7(-54) As mentioned in the proof of (16.4), the kernel of E :7c7 (S 4) -÷7c8(S 5) is a free cyclic group. By (6.3), 7C.7 (7" ) Z. It follows from the exactness of the sequence that s6 :n7 (T) --->-7r7 (S 4) is a monomorphism and hence : 7c8 (S 4) -->-vr9(S5) is an epimorphism. Since 7 8 (T) 7C8 (S 7) r,■". Z 2 , the kernel of E : n8 (S 4) -± 7C 9 (S 5) contains at most two elements. On the other hand, consider the element oc of 7(8 (5.4) represented by E (e C Vp) : S 8 -÷ S 4. In 7c9(S5), we have grs(T)

X(cc) E 2 [4:] 0 E6 [P] ( 82 E[q]) 0 (E 6 [P]) = (e:E[q]) 0 (2 D[P])

0Hence the kernel of E :7c8 (S4) -->- 7c9 (S 5) consists of exactly two elements, namely, 0 and a. This implies that n9 (S 5) fr.,/ Z 2 . I The non-zero element of 7( 9 (S 5) is E(p) represented by the map E(q0E 6 5) : S 9 S 5. On the other hand, the Whitehead product [e, e] of the generator e of (S 5) is also non-zero and hence [e, e] = E(p) [Serre 3; p. 230]. Theorem 17.5.7c. +4 (Sn) ---- 0

n > 6.

Proof. By the suspension theorem (2.1), E maps n9(S 5) onto n 10 (S8). According to the delicate suspension theorem in Ex. D at the end of the chapter, E [e, e] O. Hence we obtain n10(S 6) 0. Finally, by (2.1), we

deduce Hence

7V10(S)nu(S

grn +4(S n) =

7) r•`-'

0 for every n > 6. 1

• R-1, grn+4(Sn)

•••

332


18. The groups 7c,, + ,(S"), 5
9).

8. Z15

n10(S2) 7r11(S 3)rr

j-, Z2 Z2

gr 13 (S5) ,,,,-,

Z2

,-...5 Z24 + Z2 n15(S 7),,..--3, Z 2 + Z 2 +Z 2 7C 16 (S8) g--.d, Z2+ Z 2 +Z 2 + Z2 gr n (S9),s--,., Z 2 +Z2 +Z 3 a 14 (S 6)

Z 2 + Z2,

(n > 1 0) .

EXERCISES

333

EXERCISES A. The distributive laws

Let ot c nn,(Sn, so) and /3 c 7c.(X, x0). If cc and /3 are represented by the maps f: (Ent, Sm-i) --->- (Sa, so), g: (Sn, so) ---> (X,, xo) respectively, that the composed map gf represents an element y of 7c.(X, xo). Prove: 1.The element y depends only on the elements cc, /3 and will be called the composition 18 0 OL of oc and p. ' 2. The right distributive law. For a given p enn(X, x0), the assignment a -+j9 0 a defines a homomorphism. In fact, this is the induced homomorphism g* . 3. The left distributive law. If X is an H-space with xo as homotopy unit or if a is the suspension E(6) of some element 6 e n.,_ 1 (Sn ---1-, so), then the assignment )8 -- > p 0 a defines a homomorphism. {S; p. 122]. In particular, let (X, x o) ---- (Sn, so). Consider a map g: (Sn, so) -÷ (Sn, so) of degree d and. study its induced homomorphism g* : nm,(Sn, so) -÷7c.(Sn, so). Prove: 4. g * (a.) — da for every oc C nin(Sn, so) if n — 1, 3, 7 or if in n2n+I (Sn+1) is the subgroup of z2n-Fi(S n+1) consisting of the elements of Hopf invariant zero.

336


2. The kernel of E :7r27,..1 (S/j) --)-7r27,(Sn4-1) is the cyclic subgroup of ar21 (Sn) generated by the Whitehead product [e, e], where e denotes the generator of nn (Sn) represented by the identity map. If n is even, [e, e] has Hopf invariant 2 and therefore has infinite order. If n is odd, 2[e, e] = 0, and [e, e] . 0 iff there is an element of gr 2,„±1 (Sni-1) with Hopf invariant 1.

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243-327. 2. On adding relations to homotopy groups. Ann. of Math., 42 (1941), 409-428. 3. On the groups nr(1771,m) and sphere bundles. Proc. London Math. Soc. (2), 48 (1944), 243-291. Corrigendum, 49 (1947), 478-481. 4. Combinatorial homolopy I, II. Bull. Amer. Math. Soc., 55 (1949), 213-245, 453-496. 5. On the realizability of homotopy groups. Ann. of Math., 50 (1949), 261-263. 6. Note on suspension. Quart. J. Math., Oxford (2), 1 (1950), 9-22. 7. A certain exact sequence. Ann. of Math., 52 (1950), 51-110. 8. On the theory of obstructions. Ann. of Math., 54 (1951), 66-84. WOJDYSLAWSKI, 1. Raractes

M.

absolus et hyperespaces des continus. Fund. Math., 32 (1939), 184-192.

Index A Absolute neighborhood retract, 26, 29-32, 59, 184, 198 Absolute retract, 26, 29 Admissible transformation, 135-136

B Basic point, 125, 137 Betti number, 28, 277 Bigraded exact couple, 234-236 Bigraded group, 231 Borsuk extension theorem, 29 fibering theorem, 103 hornotopy extension theorem, 31 Boundary operator, 112, 159 Bridge theorems, 59, 195 Brouwer fixed-point theorem, 4 Bundle property of a map, 65, 99 Bundle space, 65

C Y-notions, 298 Y-acyclic, 303 Y-aspherical, 306 Y-epimorphism, 299 Y-equal, 309 f-equivalence, 299 Y-monomorphism, 299 Y-null, 299 Characteristic class, 88 Characteristic element of a complex, 193 of a map, 189 of a pair of maps, 186 Classes of abelian groups, 297-310 complete, 300, 308 perfect, 300, 308 strongly complete, 300 weakly complete, 300 Classification problem, 13, 16, 187, 198 Classification theorem of covering spaces, 96 Hopf, 53, 59 primary, 191

343

Closed surface, 28 fundamental group of, 58 Coboundary operator, 208,214 Cohomology algebra of a space, 294 of a space of loops, 295 of (Z, n), 295 Cohomology group cubical singular, 261-262 of (n, n) 199 Cohomotopy group, 205-228 Cohomotopy sequence, 2 14-2 16 set, 205, 213 Cohrntpy Complex contractible, 200 CW-, 193, 254 Eilenberg-MacLane, 203 d-, 246 6-, 246 semi-simplicial, 140-142 simplicial, 140 singular, 195 Cone over a space, 20, 28 partial, 22 Contraction, 12 Covering homotopy, 62 Covering homotopy extension property, 62 absolute, 62 polyhedral, 62, 63 Covering ho-motopy property, 24, 37, 62, 63, 70, 99 absolute, 62, 98 polyhedral, 62, 63, 99 Covering homotopy theorem, 62, 66, 314 Covering map property, 43 Covering path property, 36 Covering space, 89-97 generalized, 104 of a torus, 105 regular, 92 universal, 91, 96 Covering theorem, 91 Covering transformation, 93 Cross-section, 70, 73, 99, 313 Curve, 103

INDEX

344 D Deformation, 12 cochain, 179 homeomorph, 103 obstruction theory of, 197 problem, 22 retract, 16, 33 Degree base, 266 complementary, 235 fiber, 266 of a map, 12, 38, 52, 57, 60 of homogeneity, 231 primary, 235 total, 235 Derived triplet, 111 Difference cochain, 179 Differential group, 229-231 derived group of, 230 filtered, 239, 244 filtered graded, 245-248 Differential operator, 229, 232 Dimension, 266 Direct sum theorems, 150-152

Eilenberg extension theorem, 180 subcomplex, 45, 174 Eilenberg-MacLane complex, 203 Equivalence theorem for covering spaces, 91 for homotopy systems, 135 Euler characteristic, 277 Evaluation, 74, 76 Exact couple, 232 associated, 263-266 bigraded, 234-236 cohomology, 251 • cohornotopy, 254 derived couple of, 232, 266-269 homology, 234, 251 hornotopy, 251, 252 of a bundle space, 251 regular, 236-238 Exact sequence, 115 Gysin's, 280-281 natural, 299 truncated, 284 Wang's, 282- 284 Exactness property, 115 Exactness theorem, 136 Excision theorem, 208

Extension, 1 Extension index, 176 Extension problem, 1, 2, 15, 20, 71, 198 Extension property, 28-29 Extension theorem Borsuk, 29 Eilenberg, 180 Hopf, 53, 59 primary, 190

Fiber, 62, 262 Fiber map, 71, Fiber space, 62-106 homotopy groups of, 152-154 n-connective, 156, 304, 320 sliced, 97 spectral sequence of, 259-296, 300-304 Fibering, 62 induced, 72 n-connective, 155 of spheres, 66, 100 Fibering property, 118 Fibering theorem for homotopy groups, 136 for mapping spaces, 83-84 Filtered d-group, 244-245 associated graded group of, 244 exact couple associated with, 245 spectral sequence of, 249 Filtered graded d-group, 245-248 Filtraction, 244-246, 262 Five lemma, 309 Freudenthal's suspension, 162, 227, 311 Fundamental exact sequence, 240-242 Fundamental group, 39-47, 193 as a group of operators, 130 influence on homology and cohomology, 201-202, 288-290 Fundamental homotopy lemma, 186

G Gamma functor, 255-256 Graded group, 231, 237, 238-239 associated, 244 Grassmann manifold, 101 Group bi graded., 231 Bruschlinsky, 47-52 differential, 299-231 fundamental, 39-47, 57, 130, 193, 201, 288

INDEX graded, 231 homology, 44, 261

homotopy, 107-164

re-, 201 p-primary, 298 torsion, 297, 310 Gysin's exact sequence, 280-281 H Homology group, 44, 261 cubical singular, 261 of (n, n), 199 of a group, 199

Homotopy, 11 addition theorem, 164-166 class, 13, 16 connecting maps f and g, 11, 15 index, 183 invariant, 18 partial, 13 relative, 15 unit, 81 Homotopy equivalence, 117 Horno-topy extension property, 13-15,

30-34 absolute, 13, 31 covering, 62 neighborhood, 30 Hom.otopy extension theorem, 30-31 Homotopy group, 107-164 absolute, 107-110 abstract, 128, 137 of adjunction spaces, 168 of covering spaces, 154 of fiber spaces, 152-154 of H-spaces, 139 of spheres, 311-336 relative, 110-112 relations with cohornotopy groups, 224-226 triad, 160, 316 Homotopy property, 117 Homotopy sequence of a fibering, 152 of a triad, 160-161 of a triple, 159 of a triplet, 115, 120, 136 Homotopy system, 119 axioms of, 120 equivalence theorem for, 135 equivalent, 121 group structures for, 123 inductive construction of, 135

345

properties of, 136 uniqueness theorem for, 121 Homotopy theorem Hopf, 53 primary, 191 Homotopy type, 17, 198 Hopf classification theorem, 53, 59 extension theorem, 53, 59 fiberings of spheres, 66 homotopy theorem, 53 invariant, 326-327, 334-336 map, 66 theorems, 52-56, 59, 255

H-space, 81 Hurewicz theorem, 57, 148, 166, 253 generalized 305 relative, 306 I Induced transformation of a map on cohomotopy groups, 206, 213 on fundamental groups, 42 on homotopy groups, 113, 125

K Kan complex, 141-142 Klein bottle, 28 Kiinneth relations, 202 Kuratowski 's imbedding, 27

L Leray, 229, 249 Lifting, 24, 69, 72, 86 Lifting problem, 24, 70, 72 Local system of groups, 129 simple, 131 Loop, 36, 311 degenerate, 40 equivalent, 39 product of, 39, 79 representative, 40 reverse, 40 space of, 79 -82, 295

M Map, 1 algebraically trivial, 67-69 association, 75 canonical, 313

INDEX

346

cellular, 172 characteristic, 170, 333 combined, 7 deformable into a subset, 16, 22, 24, 197 derived, 114 essential, 16 exponential, 35 fiber, 71 homotopic, 11, 15 inclusion, 2 inessential, 16 n-extensible, 175, 194-196 n-hornotopic, 182, 194-196 n-normal, 197 null-homotopic, 16, 33 of surfaces, 106 of exact couples, 242-243 of filtered d-groups, 244 of torus into projective plane, 106 on topological products, 102 partial, 1 suspended, 60 Map excision theorem, 207 Mapping, 1 Mapping cylinder, 18 partial, 21, 31 Mapping space, 73-78, 101-102 evaluation of, 74, 76 exponential law for, 77 fibering theorem for, 83-84 induced map in, 85 subbasic sets of, 73 Maximal cycle theorem, 293 Möbius strip, 27

Natural correspondence, 119 Natural equivalence, 122 Natural homomorphism, 44, 48, 287, 305 Natural projection, 9, 21, 143

0 Obstruction, 175-204, 227 cocycle of a -map, 177, 195 cohomology class, 180, 184 of a homotopy, 183 of a map, 177 primary, 188-190 set, 181, 185 One-point union of two spaces, 145

Pair binorm al, 208 n-coconnected, 226 n-simple, 197 Path, 36 component, 78 equivalent, 41 space of, 78-79, 313, 316 Path lifting property, 82-83, 98-99 Poincaré group, 40 polynomial, 277-280 Primary component, 298, 319, 325

Realizability theorem, 169 Regular couple, 236-238 a-, 236 6-, 237 graded groups of, 238-239 two-term condition for, 250 Relative homeomorphism, 10 Relative homotopy, 15 addition theorem, 166 class, 16 Relative homotopy group, 110-112, 137 Relative n-cell, 11, 333 Restriction of a map, 1 Retract, 5, 25 absolute (AR), 26, 29 absolute neighborhood (ANR), 26, 29-32, 59, 194, 216 deformation, 16, 33 neighborhood, 26 strong deformation, 16, 32, 33 Retraction, 5 Retraction problem, 5

Semi-simplicial complex, 140-142 complete, 141 degeneracy operators in a, 141 fundamental group of a, 193 of a group, 199 topological realization of, 170 Singular complex, 45, 196 admissible subcomplex of, 172 first Eilenberg subcomplex of, 45, 174 Singular n-cube, 259 degenerate, 260 faces of, 259

INDEX

of degeneracy q, 262 weight of, 263 Singular simplex, 45, 196 Space adjunction, 9-11, 31 base, 61, 65 binormal, 14 bundle, 65, 286 connective system of, 304 contractible, 12 covering, 89-97 director, 65 dominating, 32 fiber, 62 filtered, 234, 248 generalized covering, 104 H-, 81 homogeneous, 99 homotopically equivalent, 17 locally contractible, 32 locally pathwise connected, 43 locally simply connected, 93 mapping, 73-78 n-coconnected, 210, 226 n-connected, 57, 148, 166, 210 n-connective, 155 n-simple, 131-134 of curves, 103 of homotopy type (a, n), 168, 198 orbit, 9, 200, 286 pathwise connected, 41, 78 pseudo-projective, 321-323 quotient, 9, 99 real projective, 321-323 regular covering, 92 semi-locally simply connected, 93 simply connected, 42 solid, 2, 26 topological sum of, 10 total, 61 totally pathwise disconnected, 89 universal covering, 91 Spectral cohomology sequence, 292 Spectral homology sequence, 271 Spectral sequence associated with exact couple, 233 limit group of, 234 of a fiber space, 259-285 of a regular covering space, 285-287 of filtered d-groups, 249 Sphere as homogenous space, 100 connective fiber space over, 296

347

finite groups operating freely 290-291 homotopy groups of, 311-336 Hopf's fiberings of, 66 Steenrod square, 258 Stiefel manifold, 100-101, 323-325 Suspension, 163, 257, 312, 335 iterated, 318 theorem, 311, 312, 335 Suspension sequence of a triad, 163

on,

Tietze's extension theorem, 26 Topological identification, 8, 27 Topology admissible, 102 compact-open, 73 identification, 9 of uniform convergence, 102 quotient, 9 weak, 200 Whitehead, 170 Torus, 28 Transgression, 256-258, 284, 293 Triad, 78, 160 generalized, 78 homotopy groups of, 160, 316 Triple, 214-216, 219-222 binormal, 214 cohomotopy sequence of, 214 Homotopy sequence of, 159 Triplet, 110 derived, 111 hornotopy sequence of, 115, 120, 136 Two-term condition, 241 Li Uniqueness theorem for homotopy, 121 Unit n-simplex, 7 Universal coefficient theorem, 202, 329 Universal covering space, 91

Wang exact sequence, 282-284, 320 isomorphism, 314 Weight, 244-245 Whitehead exact sequence, 253 product, 138-139, 330, 336 theorem, 167, 307 topology, 170 Wojdyslawski's theorem, 27

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