Homotopy Theory
PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks
Edited by EILENBERG PAULA. SMITHand SAMUEL
Columbia University, New York
I: ARNOLDSOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) 11: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 111: HERBERT BUSEMANN AND PAULKELLY.Projective Geometry and Projective Metrics. 1953 IV: STEFAN BERGMAN AND M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 V: RALPHPHILIPBOAS, JR. Entire Functions. 1954 VI: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 VII: CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 VIII: SZETSENHu. Homotopy Theory. 1959 IX: A. OSTROWSKI. Solution of Equations and Systems of Equations. 1960 X: J. DIEUDONNE. Foundations of Modern Analysis. 1960 Curvature and Homology. 1962 XI: S. I. GOLDBERG. XII: SIGURDURHELGASON. Differential Geometry and Symmetric Spaces. 1962 XIII. T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963 XIV: SHREERAM ABHYANKAR. Local Analytic Geometry. 1964 XV: RICHARD L. BISHOP AND RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 XVI: STEVEN A. GAAL.Point Set Topology. 1964 I n prepnratioii JOSELUISM A s s E R A A N D J U A N JORGE SCHAFFER. Linear Differential Equations and Function Spaces.
Homotopy Theory
SZETSEN HU Wayne State University, Detroit, Michigan
1959
@
ACADEMIC PRESS New York and London
COPYRIGHfl 1959 BY
ACADEMIC PRESS INC.
ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS
ACADEMIC PRESS INC. 111 FIFTHAVENUE NEWYORK,NEWYORK10003
United Kingdom Edition Published by ACADEMIC PRESS I N C . (LONDON) LTD. BERKELEY SQUARE HOUSE, LONDON W. 1
Library of Congress Catalog Card Number: 5911526
First Printing, 1959 Second Printing, 1962 Third Printing, 1965
PRINTED IN THE UNITED STATES OF AMERICA
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 11.The fiber spaces, which are of fundamental importance, are defined and studied in Chapter 111. Homotopy groups are constructed and axiomatized in Chapter IV while theelementary techniquesof computation are given in ChapterV. ChapterVI 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 V
vi
PREFACE
concerning the material; the references are not intended to be a historical record of mathematical discovery. (Thesereferences 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 (11; 7.1), where I1 stands for Chapter I1 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 settheoretic notations have been adopted in the text; namely, is used to denote the empty set and A\B the settheoretic difference usually denoted by AB. On the other hand, the symbol I indicates the end of a proof and the abbreviation iff 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 a t Tulane University and from the Air Force Office of Scientific Research while at Wayne State University. SZETSENHu
Wayne State University,Detroit, Michigan
Contents
.
PREFACE
.
.
.
.
.
.
.
.
.
.
.
LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS
.
.
.
. xii
.
V
1 1
CHAPTER I MAIN PROBLEM AND PRELIMINARY NOTIONS 1. Introduction . . . . . . . . . . . . . . . . . . . 2 The extension problem . . . . . . . 3 . The method of algebraic topology 4. The retraction problem . . . . . . . . . 5 Combined maps . . . . . . . . . . 6 . Topological identification . . . . . . . . . . . . . . . . 7 The adjunction space . 8. Homotopy problem and classification problem . . . . . . . . . . 9 . The homotopy extension property 10. Relative homotopy . . . . . . . . . . 1 1 Homotopy equivalences . . . . . . . . . 12 The mapping cylinder . . . . . . . . . 13 A generalization of the extension problem . . . . . . . . . . . . 14 The partial mapping cylinder . . . . . . . 15 The deformation problem . . . . . . . . . . 16. The lifting problem . . . . . . . . 17 . The most general problem . Exercises . . . . . . . . . . . . .
3 5 7 8 9 11 13 15 17 18 20 21 22 24 25 25
CHAPTER I1. SOME SPECIAL CASES OF THE MAIN PROBLEMS 1. Introduction . . . . . . . . . . . . . . . . . 2 . The exponential map p I R + S1 . . . . . . 3. Classification of the maps S' + S1 . 4. The fundamental group . . . . . . . . . . . . . . . . . 5 . Simply connected spaces . . . . . 6 . Relation between n , ( X . xo) and H,( X ) 7. The Bruschlinsky group . . . . . . . . . 8. The Hopf theorems . . . . . . . . . . . . . . . . . . 9 The Hurewicz theorem . Exercises . . . . . . . . . . . .
35 35 35 37 39 42 44 47 52 56 57
CHAPTER I11. FIBER SPACES . 1. Introduction . . . . 2 . Covering homotopy property
61 61 61
.
. .
. . . . .
.
vii
. .
.
. .
.
. .
.
. .
.
. .
.
. .
.
. .
.
1
...
CONTENTS
Vlll
3. Definition of fiber space . . . . . 4 . Bundle spaces . . . . . . . 5 . Hopf fiberings of spheres . . . . 6 Algebraically trivial maps X + S2 . . 7. Liftings and crosssections . . . . 8. Fiber maps and induced fiber spaces . . 9 . 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 %Simplespaces . . . . Exercises . . . . . . .
. .
. .
.
.
.
.
.
.
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . .
. . . .
. . . .
. . . .
62 65 66 68 69 71 73 78 79 82 83 85 86 89 93 97
.
107
. .
. . .
.
. .
. . .
.
. .
. . .
.
. .
. . .
. . .
. . .
. . .
. . .
. . .
. 107 . 107 . 110
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
.
. . . . . . .
.
. . . . . . .
.
. .
. . . . .
.
. .
. . . . .
.
. .
. . . . .
. 112
. .
. . . . .
CHAPTER V . T H E CALCULATION O F HOMOTOPY GROUPS . 1. Introduction . . . . . . . . . . . 2 . Homotopy groups of the product of two spaces . . . 3. The onepoint union of two spaces . . . . . . 4 . The natural homomorphisms from homotopy groups to homology groups . . . . . . . . . . . 5. Direct sum theorems . . . . . . . . . 6. Homotopy groups of fiber spaces . . . . . . .
113 114 115 117 118 119 119 121 123 125 129 131 135 143 143 143 145 146 150 152
ix
CONTENTS
7. Homotopy groups of covering spaces 8. The nconnective fiberings . . 9 . The homotopy sequence of a triple 10 The homotopy groups of a triad . . . 11 Freudenthal's suspension Exercises . . . . . . .
. .
.
CHAPTER VI OBSTRUCTION THEORY 1. Introduction . . . . . 2 The extension index . . . 3. The obstruction c n f l (g) . . . . . 4. The difference cochain . 5 Eilenberg's extension theorem . 6 . T h e obstruction sets for extension 7. The homotopy problem . . . 8. The obstruction an(/.g; ht) . . 9 The group Rn(K.L ; f) . . . 10.The obstruction sets for homotopy 11. The general homotopy theorem . 12. The classification problem . . 13. The primary obstructions . . 14 Primary extension theorems . . 15. Primary homotopy theorems . 16. Primary classification theorems . 17. T h e characteristic element of Y . Exercises . . . . . . .
. .
.
.
.
.
.
.
. 154
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.
155 159 160 162 164
. . .
. . .
. . .
. . .
. . .
. . .
. .
. .
. .
. .
. .
. .
.
.
.
.
.
.
.
.
.
.
175 175 175 176 178 180 181 182 183 184 185 186 187 188 190 191 191 193 193
.
. .
.
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
. .
.
.
.
.
. .
. .
. .
. .
. .
. .
CHAPTER VII . COHOMOTOPY GROUPS . 1. Introduction . . . . . . 2 . The cohomotopy set n m ( X . A ) . . 3. The induced transformations . . 4 . The coboundary operator . . . 5. The group operation in n m ( X . A ) . . 6 . The cohomotopy sequence of a triple . . . . 7. An important lemma . 8 . The statement (6) . . . . . 9 . The statement (5) . . . . . 10. Higher cohomotopy groups . . . 11 Relations with cohomology groups . 12. Relations with homotopy groups . . . . . . . . . Exercises .
. .
. .
. .
. .
. .
.
.
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . .
. . .
. . .
. .
.
.
. .
. .
. .
. .
205 205 . 205 . 206 . 208 . 209 . 214 . 216 . 219 . 220 . 222 . 222 . 224 . 226
. .
. .
. 229 . 229
.
.
.
.
CHAPTER VIII . EXACT COUPLES AND SPECTRAL SEQUENCES . . . . . . . . 1. Introduction . . . . . . . .
.
. .
X
CONTENTS
. . .
2 Differential groups . . . . . 3 Graded and bigraded groups . . . 4 Exact couples . . . . . . 5. Bigraded exact couples . . . . 6. Regular couples . . . . . . 7 . The graded groups R(%)and S(V) 8. The fundamental exact sequence . . . . 9. Mappings of exact couples . . . 10 Filtered differential groups . 11. Filtered graded differential groups . 12. Mappings of filtered graded dgroups . Exercises . . . . . . . .
.
. .
. .
. .
. .
.
.
.
. . .
.
.
.
.
.
.
.
.
.
.
.
. .
. .
. .
. .
. . .
. . .
. . .
. . .
.
.
.
.
. .
229 231 . 232 . 234 . 236 . 238 . 240 . 242 . 244 . 245 . 248 . 249
CHAPTER I X . THE SPECTRALSEQUENCE OFA FIBER SPACE 1 Introduction . . . . . . . . . . . 2 . Cubical singular homology theory . . . . . . 3. A filtration in the group of singular chains in a fiber space . 4 The associated exact couple . . . . . . . . 5 The derived couple . . . . . . . . . . 6. Homology with arbitrary coefficients . . . . . . 7. The spectral homology sequence . . . . . . . 8 Proof of Lemma A . . . . . . . . . . 9. Proof of Lemma B . . . . . . . . . . 10. Proof of Lemmas C and D . . . . . . . . . . . . . . . 11. The PoincarC polynomials . 12. Gysin’s exact sequences . . . . . . . . . 13. Wang’s exact sequences . . . . . . . . . . . . . . . . 14. Truncated exact sequences . 15 The spectral sequence of a regular covering space . . . . . . . . . . 16. A theorem of P. A . Smith . 17 Influence of the fundamental group on homology and cohomology groups . . . . . . . . . . . 18 Finite groups operating freely on S r . . . . . . Exercises . . . . . . . . . . . . .
.
. .
.
. . .
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 nconnective fiber spaces . . . 8 The generalized Hurewicz theorem
.
.
.
. . . . . . .
.
.
. . . . . . .
.
.
. . . . . . .
.
.
259 259 259 262 263 266 269 271 272 274 275 277 280 282 284 285 287 288 290 292
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 X I HOMOTOPY GROUPS O F 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 odddimensional spheres . . . . . . . . . . . . 8. The iterated suspension . . . . . . . . . . . . . . 9 The pprimary components of 7rm(S3) . 10. Pseudoprojective spaces . . . . . . . . 11. Stiefel manifolds . . . . . . . . . . 12. Finiteness of higher homotopy groups of evendimensional spheres . . . . . . . . . . . . 13. The pprimary components of homotopy groups of evendimensional spheres . . . . . . . . . 14. The Hopf invariant . . . . . . . . . . 15. The groups n n + l ( S n )and nn+2(Sn). . . . . . . . . . . . . . . . 16 The groups nn+3(Sn) . . . . . . . . . 17. The groups nn+4(Sn) 18. The groupsnn,,.(Sn), 5 < Y < 15 . . . . . . . Exercises . . . . . . . . . . . . .
31 1 311 311 313 314 315 316
.
.
.
BIBLIOGRAPHY INDEX
.
.
.
317 318 319 321 323 325 325 326 328 329 330 332 333
.
.
.
.
.
.
.
.
.
.
. 337
.
.
.
.
.
.
.
.
.
.
.
343
List of Special Symbols and Abbreviations The symbols and abbreviations listed below are followed by a brief statement of their meaning and by the number of the page on which they first appear. Symbols and abbreviations which are universall'y used in most branches of mathematics, such as E, U, n, w , >, sup, inf, etc., are not listed. End of proof, 3 Settheoretic difference, 5 Is homotopic to, 1I Empty set, 17 Tensor product, 261 Boundary of, 47 Coboundary of, 50 Closed unit interval [0,1], 2 ncube, 3 Euclidean nspace, 4 Unit nsphere, 4 Unit nsimplex, 7 nDimensional homology group of, 3 %Dimensionalcohomology group of, 5 Fundamental group of, 40 nth homotopy group of, 109 nth cohomotopy group of, 205 Group of integers, 109 Group of integers mod p, 281 The set X consists of a single element, 13 Restriction of the map f on A, 1 Space of all paths f:I f Y such that f ( 0 ) ~ and A f ( l ) ~ B78 , Space of all maps X  t Y,12 Onepoint union of X and Y,145 ACHEP ACHP AHEP ANR AR BP CHEP CHP
Absolute covering homotopy extension property, 62 Absolute covering homotopy property, 62 Absolute homotopy extension property, 13 Absolute neighborhood retract, 26 Absolute retract, 26 Bundle property, 65 Covering homotopy extension property, 62 Covering homotopy property, 24 xii
LIST O F S P E C I A L S Y M B O L S A N D A B B R E V I A T I O N S
Coker deg dim HEP
iff
Im Int Ker LPLP NHEP PCHEP PCHP PLP re1 SSP Tor
Cokernel of, 298 Degree of, 37 Dimension of, 49 Homotopy extension property, 13 If and only if, 2 Image of, 215 Interior of, 8 Kernel of, 215 Local path lifting property, 98 Neighborhood homotopy extension property, 30 Polyhedral covering homotopy extension property, 62 Polyhedral covering homotopy property, 62 Path lifting property, 82 Relative to, 17 Slicing structure property, 97 Torsion product of, 270
...
Xlll
This Page Intentionally Left Blank
CHAPTER I M A I N PROBLEM A N D P R E L I M I N A R Y NOTIONS
1. Introduction There is a general type of topological problem which will be called the extension problem. One of the principal objectives of the book is to show that this problem is fundamental in topology. It will be shown that many theorems of topology and most of its applications in other fields of mathematics are solutions of special cases of the extension problem. The objective of the first chapter is to formulate the extension problem precisely, and to study the problem in its most general terms. It will be shown that various other problems are fully equivalent to extension problems. We shall begin with a restricted form of the extension problem, and one of its special cases, namely, the retraction problem. The solution is shown to depend only on the homotopy class of the map involved, and then only on the homotopy types of the spaces. These considerations lead naturally to a moregeneral type of problem. The latter is then shown to be reducible to the simplest type of problem, namely, the retraction problem. Finally, dual problems of deformation and lifting (finding a crosssection) are discussed in an analogous fashion together with their interrelations with the extension problem. Thus it will appear that underlying these various questions is one central question, namely, the extension problem.
2. The extension problem By a map, or mapping, f :X .+ Y of a space X into a space Y , we mean a singlevalued continuous function from X to Y . The space X is called the domain off or the antiimage of f ; and the space Y is called the range of f . We shall not recall the definition and the elementary properties of continuous functions, since these can be found in any textbook on general topology, for example, [K; pp. 8488]. On the other hand, we assume that the reader is familiar with the popular notions and notations concerning maps such as given in [ES] . Let f : X .+ Y be a map and A a subspace of X.Then f defines a unique map g : A + Y such that g(x) = f(x)for each x E A . This map g is called the restriction of f to A or the partial map of f on A and is denoted by
g = f IA;
f will be called an extension of g over X . 1
2
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
If h : A c X denotes the inclusion m a p defined by h ( a ) = a E X for each a E A , then the relation g = f I A is equivalent to the commutativity relation / h = g in the diagram: A L + 1’
x The (restricted) extension problem is concerned with whether or not a given map g : A + Y defined on a given subspace A of a space X has an extension over X . When X , A , Y and g are given in some reasonably effective manner, the problem is to find an effective procedure for deciding whether g has an extension over X,and finding one when it exists. As will be seen, solutions have been given in numerous special cases; these are quite varied in nature, and the situations in which they apply are very restricted. As yet there is no reasonably complete theory. Actually it is convenient to concentrate on a broader problem which will be stated in 5 9, but several important ideas arise in connection with this problem. Let us begin by considering a few simple examples. 1. Let X be a given space and A be a subspace of X which consists of two points x, and x l . Let Y be a 0sphere, say the boundary sphere of the closed unit interval I.Consider the map g: A + Y defined by g(xo) = 0 and g(xl) = 1. Then g has an extension over X iff x,, x1 lie in different quasicomponents of X,[ES; p. 2541. Hereafter, the symbol “iff” will stand for “if and only if”. 2. Let X = I and A be the boundary sphere of I . Let Y be any given space, and g : A + Y a given map. Then g has an extension over X iff g(O), g( 1) lie in a compact, connected and locally connected subspace of Y satisfying the second countability axiom. 3. Let A be the union of two disjoint closed subspaces B, C of a normal space X , let Y = I , and let g : A + Y denote the map defined by
g(B) = 0, g(C)
=
1.
Then, by Urysohn’s lemma [L,; p. 271, g has an extension over X . 4. Let A be a closed subspace of a normal space X , let Y = I, and let g : A + Y denote any map. Then, by Tietze’s extension theorem [L,; p. 281, g has an extension over X . See Ex.D a t the end of the chapter. In the last example given above, the space Y = I 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. T H E M E T H O D
OF ALGEBRAIC TOPOLOGY
3
Proof. Let { Y , 1 p E M } be a collection of solid spaces and Y = P,Y, denote the topological product of this collection, [L,; p. 101. 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 p,,: Y + Y,, u , E M , the natural projection of Y onto Y,, and set g, = p,,g:A + Y,,.
Since Y,, is solid, g,, has an extension f,: X + Y p ,Define a map f :X+ Y by taking P p f W = f,(4, ( x E X).
It is obvious that f is an extension of g over X.I Since the closed unit interval Z is solid as noted above, it follows from (2.1) that any compact parallelotope, [L,; p. 191, is solid; in particular, the ncube I n and the Hilbert cube I" are solid. Their homeomorphs are likewise solid, hence the ncell and the nsimplex 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
L
Y
x of maps as described in the preceding section. In any homology theory satistying the EilenbergSteenrod axioms [ES; pp. 10121, the maps f, g, h induce for each m the homomorphisms f,, g,, h, indicated in the following diagram :
According to Axiom 2, [ES; p. 111, the relation fh mutativity relation f* h, = g*
=
g implies the com
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
+ : H m ( X )+ H m ( Y )
4
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
such that the commutativity relation $h, = g, holds. Thus, the existence of the homomorphism $ 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 f Y fails to have an extension over X . For example, let us take X to be the unit ncell En of the ndimensional euclidean space R n , A = Y to be the boundary (n1)sphere Snl of En, and g : A + Y to be the identity map on 9  l . We shall prove the following Proposition 3.1.
For each n 2 1, the identity map g:Snl fSnl has no
extension over En. Proof. Assume that there is some extension f :En + 9  l 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  1, then we have
Since g is the identity map on Snl, it follows from Axiom 1 that g, is the identity automorphism of Hm(Snl). Since Hm(Snl) # 0, this implies that g, # 0. On the other hand, since H,(En) = 0, the inclusion map h:Snl c En induces h, = 0. Thus, we obtain f,h, = 0 and g, # 0. This contradicts the relation f,h, = g., In fact, the derived algebraic problem has no solution. It remains to dispose of the case n = 1. In this case, Snl 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 f (En) = 9  l . This is a contradiction. I Note. The case n
1 of (3.1) can also be proved from the derived algebraic problem provided that one uses the reduced homology groups, [ES; pp. 1819]. As an important application of (3.1), we shall give the following =
Theorem 3.2. (The Brouwer FixedPoint 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 . Proof. Assume that f : En + En is free of fixed points. Then, we may define a map r :En 9  l as follows. Let x E En. Since f has no fixed points, we have f ( x ) # x . Draw the line from f ( x ) to x and produce until it intersects 9  1 a t a point r ( x ) . One verifies that the assignment x + r(x) defines a continuous function Y:En f 9  1 . If x E Sn1, it is obvious from the construction that r ( x ) = x . Hence r is an extension of the identity map on 9  l . This contradicts (3.1). I This application of (3.1)shows that the negative nature of a “nonexistence” f
4. T H E R E T R A C T I O N P R O B L E M
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 3A. According to (3.1), the boundary (n1)sphere 9  l of En is not a retract of E n . 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 9  1 is a retract of X.In fact, a retraction Y : X Sn1 ~ is given by Y(X) =
(5,. . . 5) ,
1x1
for every point x = ( x l , . . . , xn) of X = En \ 0, where I x 1 denotes the distance between 0 and x . The same formula also gives a retraction Y of Rn \ 0 onto 9  l . For another example of retracts, let us consider the topological product X = A x B. Pick a point b, 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, b,) for each a E A . This having been observed, it becomes clear that the natural projection X = A x B + A gives a retraction of X onto A . I n particular, a meridian of the torus T , = 9 x S1is a retract of T,. 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 proposition 4.1. A is a retract of X has a n extension over X .
io,for a n y space Y ,every m a p g : A + Y
Proof. If A is a retract of X with a retraction r : X 3 A , then g r : 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 Y : 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 * : H , ( A ) + H,(X), i * : H m ( X )+ H m ( A ) .
6
I. M A I N P R O B L E M A N D P R E L I M I X A R T N O T I O N S
The derived algebraic problem is to determine whether or not there exist homomorphisms
4 : H m ( X )+ H m ( A ) , y : H m [ A )+ Hm(X) such that +i* and i*y are the identities on H m ( A )and Hm(A) respectively. The existence of the homomorphisms 4 and y is a necessary condition for A to be a retract of X . In fact, if r : X 2 A is a retraction, then ri is the identity map on A and hence4 = r* and y = r* are solutions of the derived algebraic problem. Furthermore, since r,i, and i*r* are the identities, it follows that i,, r* are monomorphosms, that r*, i* are epimorphisms, and that H m ( X ) , H m ( X ) decompose into the following direct sums:
H,(X) = Image i, H m ( X ) = Kernel i*
+ Kernelr,, + Imager*.
If the coefficient group of the cohomology theory is a ring, then the cohomology groups H m ( X ) , m = 0, 1, * * , constitute a ring H * ( X ) with the cup product as multiplication. The inclusion i:A c X and the retraction r : X 3 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 r* is a monomorphism, i* is an epimorphism, and H * ( X ) decomposes into the direct sum
H * ( X ) = Kerneli*
+ Imager*
where Kernel i* is an ideal and Image r* 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 r with 0 < r < n. Then A is not a retract of X . To prove this fact, let us assume that there is a retraction r : X 2 A . Then we obtain a ring monomorphism r * : H * ( A )+ H * ( X ) of the cohomology rings with integral coefficients. Let a and E denote generators of the free cyclic groups H 2 ( A )and H2(X) respectively. Since r* is a monomorphism, there is a nonzero integer k such that r*(a) = k t . Since n > r , we have a n = 0. Since r* preserves multiplication, we obtain kntn
This contradicts the fact that
H" (S) .
=
En
.*(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
Proposition 4.2. I f ( X ,A ) is a (finitely) triangulable pair, [ES; p. 601, then the closed subspace L = ( X x 0) U ( A x I )
of the product space M
=
X x I is a retract of M .
Proof. First, let us prove the special case where X is the unit nsimplex An of the euclidean (n + 1)space, [ES; p. 551, and A is the boundary (n1)sphere of A , which is empty if n = 0. Then a retraction r : M 3 L can be constructed geometrically as follows. Since I R, it follows that M is a subspace of An x R. Then we define r to be the central projection of M onto L from the point (c, 2) of A n x R, where c denotes the centroid of A,.
This proves the special case. For a finitely triangulable pair ( X ,A ) , we may assume that X is a finite simplicia1 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 9 10. Besides, (4.2) can also be generalized to some nontriangulable pairs; see Ex.0 a t 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 , I p E M } 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,, p E M , is the whole space X . Let D,, = X,, il X , for each pair of indices p, v in M . For any given map g : X f Y of X into a space Y , the partial maps g, = g I X , are welldefined and satisfy the relation g, I D,, = g , I DPv for every pair of indices p,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 p E M , there is given a map f,:X, + Y such that I Dpv = f v I D,v
f,
for each pair of indices p and v in M . Then we may define a function f :X + Y by taking We are concerned with the problem whether or not f is continuous. Proposition 5.1. If M is finite and all the subspaces X,, p E M , are closed in X , then the combined function f is continuous.
Proof. Let F be any closed set in Y . Then it follows from the continuity of f,, p E M , that f;'(F) is a closed set of X,. Since X, is closed in X, this
8
1. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
implies that f;l(F) is a closed set of X . Since M is finite,
is a closed set of X . Hence f is continuous. I A frequent application of this proposition is as follows. Let f,g:X x I f Y be two given maps such that f (x,l) = g(x,O) for every x E X . Then we may define a function h : X x I + Y by taking:
( X E X , O < t
(if x E K , ) , (if x E t E K , \A).
Now we shall complete the construction of the sequence { fn } by induction. Let m > 2 and assume that fm1 : Km1 +S1has already been constructed. Let 8 be an arbitrary msimplex in X \A, and vo be a vertex
50
11. S O M E S P E C I A L C A S E S O F T H E MA I N P R O B L E M S
of 0. Since m > 2, the ( m  1)sphere80 is simply connected by (5.2).Then, according to (5.3), there is a unique map j e : a0 + R such that ie(vo)
=
0,
Pie = f m  1 180
where p : R +S1 denotes the exponential map in 3 2. Since R is solid, the map iehas an extension ke : 0 + R. Then we define fm by setting (if x E Kml), (if X E 0 E Km \A).
imi(x),
f m f x )=
pke(x),
This completes the construction of the sequence { fn }. Since fn I Kn1 = fn,, we obtain a map f : X +S1 such that f I Kn = fn. In particular, /(KO) = 1 and f I K , = f l . By the construction of f l , this implies that cl( f ) = z. Hence, k* is an epimorphism. To prove that k* is a monomorphism, let GC be any element of n l ( X ,A ) with h*(a) = 0. Pick a representative map f : (X, A ) + (Sl, 1) of a which sends all vertices of X to 1. Then the cocycle c1( f ) is a coboundary of X modulo A and hence there is a cochain c0 E CO(X,A ) such that cl( f ) = 6co. For each n = 1,2;. dim X + 1, let Jn denote the subspace of X x I defined by Jn = (X x 0) U (Kn1 x I ) U (X x 1). a ,
To prove that f is homotopic to the constant map O(X) = 1 relative to A , let us construct a sequence of maps
F n : Jn+S1, n
=
1,2;*.,dimX
+1
as follows. To construct F,, let us take for each vertex v in X \A 5, : I +S1such that
L(0) = 1
=
tAl), deg (E,)
=
a loop
c0(~).
Then we define F , : J 1 +S1 by taking
F,(x,4
=
1
1,
W), f (4,
(if x E A or if t (if x = v ) , (if t = 1).
=
0),
Next, let us construct the map F , : J , + S1. Let a = v,,v,, be any 1simplex in X\A. Then the partial map F , I d(a x I ) on the boundary 8(a x I ) of a x I is a loop in S1 of degree
 co(vo) = (dco) (a) [c'(f)l (a)= 0. Hence F , I d(a x I) has an extension q,, : a x I +S1.We define F , by the co(vl)  [c'(f)I
(0)
Now we shall complete the construction of the sequence { Fn } by induction. Let m > 2 and assume that Fm, : Jml + S1 has already been
7.
51
T H E BRUSCHLINSKY GROUP
constructed. Let t be any (m  1)simplex in X\ A . Since m > 2, the (m 1)sphere a(t x I ) is simply connected. Hence, it follows just as before, that F,#(t x I ) has an extension 5,: t x I ,S1. Then we define F,: J , S1by setting
.
This completes the construction of the sequence { Fn }. Since Fn I Jn1 = Fnl, we obtain a map F : X x I +S1 such that F I Jn = Fn. In particular, we have F ( x , 0) = 1, F ( x , 1) = f ( x ) , F ( a ,t ) = 1 for every x E X , a E A and t E I . This implies that f is homotopic to the constant map O(X) = 1 relative to A . Hence a = 0 and h* is a monomorphism. I Since H l ( X , A ) is effectively computable, the theorem (7.1) solves the classification problem for the maps ( X ,A ) + (9,1). In particular, if we take A = 0 , 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 f , g : ( X ,A )
ifl I*(&) = g*(1).
+
(9,1) are homotopic (relative to A )
This corollary is an immediate consequence of (7.1). In particular, let us take A = a. The inclusion map j : S1c (9,1) induces an isomorphism j* : Hl(S1, 1) M H1(S1)and x = j * ( i ) is the generator of the free cyclic group H1(S1)determined by the counterclockwise orientation of S1. Then (7.2) gives the following corollary as a special case. Corollary 7.3. Two maps
f, g : X +S1 are homotopic it f
* ( x ) = g*(x).
Corollary 7.4. A map f : A +S1 can be extended over X iff the element f * ( x ) of H 1 ( A )is contained in the image of the homomorphism i* : H 1 ( X )+ H 1 ( A )induced by the inclusion map i : A c X . In fact, if a is an element of H 1 ( X )such that i*(a)= f * ( x ) , then f has an extensiong : X + S1wilhg*(x) = a. 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 + S1 such that (jk)* ( L ) = a.Then'we have
( j k i ) * ( i ) = i*(jk)*(t) = i*(a)= f * ( x ) = f * j * ( i ) =
(jf)*(i).
By (7.2), this implies that jki N j f and hence ki N f . According to the homotopy extension property, there exists an extension g : X + S1 of f such that g = k . Then we have g*(x) = k * ( x ) = k*j*(i) = ( j h ) * ( i ) = a. I Following the definition of triangulable pairs in [ES; p.601, we have assumed above that X is a finite simplicia1 complex. However, the proof of
52
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
(7.1) is so arranged that it extends to the case that X is an infinite simplicia1 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 a t 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 S' can be generalized to higher spheres. For this purpose, let us first consider some preliminaries. For every n > 1, let Sn denote the boundary nsphere of the unit (n + 1)simplex A = An+,. For each m = 0, l;.., n 1, let A ( m ) denote the mth (ndimensional) face of A . Then Sn is the union of the nsimplexes d ('4, A W , ,
[email protected]+'). The ndimensional cocycle $ of Sn defined by$ (A(O))= 1 and + ( A ( m ) ) = 0, m = 1; .,n + 1, represents a generator x of the free cyclic cohomology group Hn(Sn). For any map f : X +S", the element f*(x)of P ( X ) depends only on the homotopy class [ f ] off and will be called the degree of the map f or of the class [ f ] . In particular, if X = 9, then the element f * ( x ) of Hn(Sn) determines a unique integer deglf) such that f * ( x ) = deg(f)x.In this case, it is this integer deg(f)which is traditionally known as the degree of the map f : Sn +Sn. If n = 1, this definition of deg(f) is obviously equivalent to the one given in 5 3. Let vo denote the leading vertex of Sn. For each q = 0, 1; n + 1, let IW denote the union of all A @ ) with m # q. Then the inclusion maps
+



5 : Sn c (9, ?lo), ?jQ : (Sn, vo) induce isomorphisms sions. Let
t*,
a ,
(9, F(Q))
on the cohomology groups of positive dimen
& = f*l
(4
J
AQ = ?jg*'(
I).
Then L and & are generators of the free cyclic groups Hn(Sn, vo) and Hn(Sn, I'M)) respectively. Next, let us consider the unit nsimplex dn and its boundary (n 1)sphere ad,. The ncocycle y of An given by y(dn) = 1 represents a generator ,u of the free cyclic group Hn(& ad,). For any map f : ( A n ,ad,) + (Sn, vo), the element f * ( ~determines ) a unique integer deg(f)such that /*(L) = deg(f)* p This integer deg(f) will be called the degree of the map f. For each q = 0, 1; * *, n + 1, let cQ: (An,ad,) + (Sn, D O ) ) denote the map defined by the orderpreserving onetoone assignment of vertices of An into A W . Then it is not difficult to verify that (8.1)
&* (3.9) = (
1Pp
8. T H E
53
HOPF THEOREMS
Now, let f : S n  + S n be a map which sends the ( n  1)dimensional skeleton of S n into vo. Then, by (8.1) and a theorem in cohomology theory, [ES; p. 37, Theorem 14.6~1,one can easily deduce the following relation n+i
deg(f) =
(8.2)
(q=o
1)g
deg(ftA.
Finally, let us prove the following Lemma 8.3. For every integer m, there exists a m a p fm : (An, adn)+(Sn, vo) with deg(fm) = m. Proof. If m = 0, then the constant map fo(An) = vo is obviously of degree 0. Next, assume m > 0. Take a sufficiently fine simplicial subdivision K of An so that we may pick m mutually disjoint closed nsimplexes u1; * ,a m contained in the interior of An. Let the vertices uij of at be ordered in such a way that the orientation of at =
(utu ~
2 , *' * ,
utn+J
agrees with that of An. We now define fm to be the unique simplicial map of K into Sn which sends utj into the vertex vj of S n for each i = 1,. * , m and each j = 1, ,n + 1 and sends every other vertex of K into vo. Then i t is easily seen that fm(aAn) = vo and deg(f,) = m.Finally, let 2 denote a linear homeomorphism of An which interchanges a pair of vertices of An and leaves other vertices fixed. Then deg(fmR) =  m. I An organization of the Hopf theorems is as follows.


Theorem H". (Homotopy). If a m a p f : S n +Sn has degree deg(f) = 0 , then f i s homotopic to the constant map O(Sn) = vo. Theorem En. (Extension). Let (X,A) be atriangulablepairwith dim(X\A) + S n be a given map. If there exists a n element a E Hn(X) such that i*(u)= f*(x), where i : A c X , then f admits a n extension g : X +. Sn such that g*(x) = u.
< n + 1 and f : A
Theorem C". (Classification).If X is a triangulable space with dim X < n, then the assignment f f*(x) sets up a onetoone correspondence between the homotopy classes of the maps f : X + S n and the elements of the cohomology group Hn(X). f
The converse of En is trivial: if f admits an extension g : X + S n , then there exists an element u E Hn(X) such that i*(u)= f*(x). In fact, we have u
=
g*(x).
Since H' is a special case of (3.2), these theorems can be established inductively by proving H" + En, En + H"+l, En * C" for every n > I . Also, it worth noting that Hnis a special case of C", C1is a special case of (7. l), and El is a special case of (7.4).
54
11. SOME S P E C I A L C A S E S O F T H E M A I N
PROBLEMS
Proof of Hn En. Let X be a simplicia1 complex and A a subcomplex of X. We may assume that f sends the (n  1)dimensionalskeleton of A into the point wo of Sn, for otherwise we may replace f by a homotopic map which satisfies this condition by applying the method of simplicial approximation and observing that every proper subspace of S n is contractible in Sn. Under these ass,umptions,we may define an ncochain cn(f) of A as follows. Let u be any nsimplex of A and 1, :An + u denote the linear homeomorphism which preserves the order of vertices. Then f& is a map of (An, ad,) into ( 9 , zio) and hence we may define the cochain cn(f) by taking = deg(fL7) “)l(u) for every nsimplex (I 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 cohomology class f*(x) E Hn(A). Assume that a is an element of Hn(X) such that ;*(a)= f*(x). Then it follows that there exists an ncocycle zn of X which represents ci and satisfies the relation zn (4 = [cn(Pl(4 for every nsimplex u of A. Let K denote the union of A and the ndimensional skeleton of X . We shall define a map k : K + S n as follows. Let u be any nsimplex of K which is not in A. By (8.3),there is a map /u
: (An,
a&)
+
(Sn,vo)
with deg(f,) = zn(u). Let A, : An + u denote the linear homeomorphism which preserves the order of vertices. Then we define k by setting
[
k ( x ) = f(X)J
(4,
(if x € A ) , (if ~ E U E K \ A ) .
Finally, we are going to construct an extension g : X + S n of k as follows. Let t be any (n 1)simplex of X \A and 1, :An+, + t the linear homeomorphism which preserves the order of vertices. Denote k, = k& 19.By (8.2), we have n deg (k,) = X ( l)(zn(t(O) = zn(at) = dzn(t) = 0.
+
i0
According to Theorem H”,this implies that k, is homotopic to the constant map 0. Then it follows from the homotopy extension property that k, has an extension g, :An+, + 9.Then we define g : X +S n by taking
Since g is obviously an extension off, it remains to verify that g*(x) = a. According to the construction of g, it can be seen that g maps the (n  1)dimensional skeleton of X into vo and that cn(g) = 2”. Since cn(g) represents g*(x) and 2%represents a, this implies g*(x) = a. I
8. T H E HOPF
THEOREMS
55
Proof of E n * Hn+'. Let f : Sn+l+ Sn+l be a given map with deg(f) = 0. We are going to prove that f 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 Sn+l into 1 triangulation J of Sn+l. Pick an (n 1)simplex (T = (u,, u l ; * , Hn+l) of J such that the cocycle defined by +(u) = 1 and+(t) = 0 for every (n 1)simplext of J other than (I represents the generator x of Hfi+l(Sn+l). Let M denote the subcomplex of K consisting of the closed ( n + 1)simplexes of K which are mapped into (T 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 + I)simplex of J" none of whose vertices is also a vertex of J . Under these assumptions, we have H9(M) = 0 for every q > 0. The simplicial map f : K + J induces an (n 1)cocycle y = I#(+) of K which represents the element

+
+
+
+
f * ( x ) = deg(f).x = 0 of Hn+l(K).Hence there exists an ncochain c of K such that y = dc. The simplicial inclusion map 9 : M c K induces the following two cochains of M c1 = Y1 = P#(w) = P#f#(+) = (fP)#(+).
m),
The relation y = 6c in K implies that y 1 = dc, in M . Next, consider the leading nface a(") = (ul,* * , u n + l ) of the (n 1)simplex(T and let 4, denote the ncochain of J defined by+,(d0)) = 1 and+,(t) = 0 for other nsimplexes t of J . Then the simplicial map fp : M + J induces an ncochain c, = = (fp)#(+,) of M . Since+ is the coboundary of+, on the subcomplex u of J and fp maps M into (T, it follows that y 1 = dc,. Then d(c,  c,) = y1 y1 =O and hence c1  c, is an ncocycle of M . Since H n ( M ) = 0, this implies that c1  c, is a coboundary of M . Let M n denote the ndimensional skeleton of the complex M . Then f defines a simplicial map X of Mn into the nsphere au. By the construction given above, it is clear that c, is a cocycle of Mn and represents the degree of the map X . Since c1c, is a coboundary, the degree of X is also represented by the cocycle c1 of Mn. Let N = (K \ M ) U M n and consider the inclusion map q : N c K . Since dc(t) = y ( t ) = 0 for any (n + 1)simplex t which is not in M , it follows that cz = q#(c) is a cocycle of the complex N . Since c2 is an extension of c,, we may apply Theorem En to conclude that X has an extension p : N +da. Define a map g : Sn+l+ Sn+l by taking

+
56
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
Since f maps N into the space W
=
(J\ o) U do which is solid, the maps
f I N and p are homotopic in W modulo Mn. This implies f II g. Next, since the image of g is contained in u, it follows that g is homotopic to a constant
map. Since Sn+l is pathwise connected, this implies that f
II
g
II
0. I
X be a simplicia1complex with dim X < n. First, let f S n such that f*(x)= u. Let A denote the (n 1)dimensional skeleton of X and i : A c X the inclusion map. Then we have P ( A ) = 0 and i*(a)= 0. Hence, by E", the constant map k(A) = oo has an extension f : X +Sn such that f * ( x ) = u. Next, assume that f, g : X +Sn are two maps such that f*(x) = g * ( x ) . We are going to prove f =g. For this purpose, let us study some elementary properties of the product space X x I . Obviously, X x I is triangulable and dim(X x I\ < r z + 1. Let p : X x I + X and q r : X  f X x I, [i =0,1), denote the maps defined by Proof of En 3 C". Let
a be any element of P ( X ); we shall find a map f : X
P ( x , t) = x , q o ( 4 = ( x , O), q1(4 = ( x , 1). Since $9, is the identity map on X , we obtain qe*fi*(u) = u for every u E Hn(X). Next, consider the subspaces x , = x x o , x , = x x 1, A = X , U X , of X x I . Then, according to an elementary theorem [ES; p. 331, Hn(A) is the direct sum of Hn(X,) and Hn(X,). Hence the elements of Hn(A) can be represented by the pairs (p, y ) of elements of Hn(X). Let r : A C X x I denote the inclusion map. Then we have r*p*(a) = (u,a). Define a map k : A +Sn by taking h(x,t ) =
f (x)J
dx),
(if x E X and t = 0), ( i f x E X a n d t = 1).
Then, h*(x) = (u,a) E Hn(A). According to En, there exists an extension H : X x I +Sn such that H * ( x ) = qo*(u).This proves f N g. I Thus we have proved the celebrated Hopf theorems H",En,C" for every n = 1,2; * In particular, if we take X = Sn in Theorem C", we obtain the following a .
Corollary 8.4. The homotoPy classes of the maps f : Sn + S n are in a onetoone correspondence with the integers. The correspondence i s given by the assignment f + deg ( f ) . Finally, the remarks given at the end of fj 7 are also true in the present circumstances.
9. The Hurewicz theorem The dual of the Hopf classification theorem is the Hurewicz theorem, the statement of which is the main purpose of the present section. To this aim, let us define the important notion of nconnected spaces.
9.
57
THE HUREWICZ THEOREM
Let n > 0 be a given integer. A space X is said to be nconnected if, for every triangulable space T of dimension < n, any two maps f , g : T + X are homotopic. So, X is 0connected iff it is pathwise connected, and X is 1connected iff it is simply connected. Using the extension property, one can easily prove that an ( n  1)connected space X is nconnected iff every map f : Sn + X is homotopic to a constant map. Now, let X be a given space and consider the maps f :Sn+X,
+
where Sn denotes the boundary nsphere of the unit (n 1)simplex = A,,,. Since we have fully studied the case n = 1 in 3 4 and 3 6 , we will assume that n > 2. The fundamental ncycle
d
n+1
2 =
(
l)tA(O
i=o
of Sn represents a generator i of the free cyclic group Hn(S”). For each map f : Sn + X , the element f , ( i ) of H n ( X ) depends only on the homotopy class of f and will be called the degree of f . If X = Sn, then one can easily verify that f , ( i ) = deg ( f ) 1 , where deg(f ) denotes the integer defined in 3 8. Theorem 9.1. (Hurewicz Theorem). If X i s a n ( n  1)connected space with n > 2 , then the assignment f + f , ( i ) sets up a onetoone correspondence between the homotopy classes of the maps f : Sn + X and the elements of the singular homology group H n ( X ) .
As an immediate consequence, we have the following important Corollary 9.2.
for every m
=
2;
A simply connected space X i s nconnected ifl H m ( X ) = 0 * , n.
In all existing proofs of the Hurewicz theorem, one must use the group structure and a few elementary properties of the homotopy group. In the sequel, we shall prove a much more gencral theorem. See (V; $4) and (X ;$ 8 ) .
EXERCISES A. The fundamental group of a connected simplicial complex
Let X be a connected simplicial complex. Denote the vertices of X by vo, v,; . * , vm. A broken line joining a vertex to another is a path which consists of a finite number of 1simplexes. Since X is connected, we can join vo to of by a broken line 11. Suppose that these broken lines &, ( i = 0,l; * m),have been chosen such that 1, consists of only a single point vo. Let Acl denote the reverse of At. To each oriented 1simplex vrvj of X , consider the loop a ,
58
11. S O M E S P E C I A L C A S E S O F T H E
M A I N PROBLEMS
which represents an element au E ~ , ( Xuo). , Prove that these elements ail form a system of generators of n,(X, uo) with the set of relations described as follows. First, for the 1simplex v j q , we have trivial relation
ajr
=
(afj1l.
Second, for each 1simplex ucvj, the loop f i j is a product of the Isimplexes up,, of X, say
Then we have the relation
ftl = pu
ail
("€Vq).
= Pij (a€,,),
where Pi, (at,,) is the product obtained by replacing Z'CV,, with the corresponding element a h in n,(X, uo). Third, for each 2simplex u = u t u p k of X, we have the relation atjawkr = 1. This gives an effective method for computing the fundamental group nl ( X ) of X. For examples, prove the following assertions: 1. The fundamental group n l ( X ) depends only on the 2dimensional skeleton X2 of X . More precisely, the inclusion map i : X2 c X induces an isomorphism i, :n,(X2, uo) M n,(X, vo). 2. The fundamental group of a closed orientable surface of genus g is the abstract group generated by 2g elements a$,/$, (a' = 1; ,g), with a single relation

Hence the fundamental group of the (2dimensional) torus is the free abelian group with two generators. If g > 1, then the fundamental group is nonabelian. 3. The fundamental group of a closed nonorientable surface of genus g is the abstract group generated by g elements a,, * * , agwith a single relation

alala2a2**
[email protected],
=
1.
Hence the fundamental group of the projective plane is the cyclic group of order two. If g > 1, then the group is nonabelian. 4. If a connected simplicial complex X is the union of two connected subcomplexes A and B with connected intersection D = A rl B, then nl(X) is the quotient group of the free product n,(A) On,(B),[ST; p, 301, obtained by identifying, for each element 6 € n , ( D ) , the element t,(d) E n , ( A ) with the element q*(6)E ~ , ( Bwhere ), t : D c A and q : D c B denote the inclusion maps. In particular, if D is simply connected, then we have
n d X ) = n , ( A ) On,(B). 5. If X is a connected 1dimensional simplicial complex, then n l ( X ) is a free group. In particular, if X is the space which consists of two circles intersecting at a single point, thenn,(X) is the free group on two generators.
EXERCISES
59
B. The bridge theorems
Assume that (1) X is a normal space, (2) Y is a connected ANR, and (3) either X or Y is compact. Let f : X + Y be a given map and a be a finite open covering of X . A map g, : N u + Y of the geometric nerve N , of a into Y will be called a bridge map for f if g,h, N f for every canonical map ha : X + N,, [L,; p. 401. If such a bridge map g, exists, a is said to be a bridge for the given map f . Prove the following assertions, [Hu 51 : 1. Every map f : X + Y has a bridge. 2. Every refinement of a bridge is also a bridge. 3. If a, ,!Iare two bridges for a given map f : X + Y , where X is compact, and if g, : N, + Y ,gp : Np + Y are bridge maps, then there exists a common refinement y of a and tf? such that gapya N gppyp, where pya : N , + N u and pyp : N , f N p are arbitrary simplicia1 projections. Study the analogous assertions for a map f : X + Y of a paracompact Hausdorff space X into a connected ANR Y by considering the locally finite open coverings of X . C. Maps of compact spaces into spheres
The bridge theorems in Ex. B furnish us with a link between a map on a compact Hausdorf space and maps of simplicia1 complexes. Therefore, if one uses the tech cohomology groups and the notion of dimension both defined by means of finite open coverings, one can extend the results in fj 7 and fj 8 to compact Hausdorff spaces. Establish the following theorems: 1. Hopf extension theorem. Let ( X ,A ) be a compact Hausdorff pair with dim X Q n + 1, u a generator of the free cyclic group Hn(Sn),and f : A + Sn a given map. If there exists an element a E H n ( X ) such that i*(a)= f * ( u ) , where i : A c X , then there exists an extension g : X + Sn such that g*(u) = a. 2. Hopf classification theorem. If X is a compact Hausdorff space with dim X < n and if u is a generator of Hn(Sn),then the assignment f + f *(u) sets up a onetoone correspondence between the homotopy classes of the maps f : X +Sn and the elements of the Cech cohomology group H n ( X ) . In the case h = 1, the conditions on the dimension of X may be removed. Furthermore, the assignment f + f * ( u ) ,in this case, is an isomorphism h* :nyx) M H'(X).
For further generalizations to noncompact spaces, see [Dowker 11. D. The degree of a suspended maps
Consider the nsphere Sn as the equator of an ( n + 1)sphere Sn+I with north hemisphere E";tl and south hemisphere E?+l. By Tietze's extension
60
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
theorem, every map f : Sn + Sn has an extension f * : Sn+l+ Sn+l such that f *(E:+l) c E:+l and f *(En_+')c EE+l, which is called a suspended m a p of f. Prove 1. deg(f*) = d e g ( f ) . 2. (8.3) can be obtained inductively by the construction of suspended maps of a map f : S1f S1of degree m.
CHAPTER I l l
F I B E R SPACES 1. Introduction The concept of fiber space is crucial in homotopy theory: it usually appears in the application of homotopy theory to geometric problems; it is a powerful weapon in the computation of the homotopy groups of various spaces; and it plays a key role in the axiomatization of homotopy theory. Thus, before turning to the discussion of the homotopy groups themselves, it is appropriate to develop certain properties of fiber spaces in some detail. Already we have seen one important example, namely the exponential map p : R +S1 of Chapter 11: in the language of the present chapter, we should say that i? is a fiber space (indeed a covering space) over S1with projection p . As to the usefulness of this particular example, recall that upon it was based the classification of maps of a space X into S1 of $ 7 of Chapter 11. Historically, there were a number of examples, and definitions of fiber spaces were abstracted from these in a variety of ways; but in each case it was possible to prove a socalled covering homotopy theorem. It remained for J.P. Serre in 1950 to single this theorem out as the crucial property, and to base on it his study of the singular homology of fiber spaces. The influence of this study on homotopy theory has been profound, and it now seems quite clear that his is the proper definition; hence we adopt it and consider some of its immediate consequences in $$ 23. On the other hand, the classical examples of fiber spaces belong to a much more narrow class, in that all of them have a local product strzcctzcre. This concept, together with an important example, the Hopf fiberings of spheres, is considered in $5 46. After considering certain mappings of fiber spaces ($6 78) we develop (for later use, especially in Chapter IV) certain properties of spaces of paths ( $ 5 914). In particular, it will appear that they are fiber spaces in the sense of Serre mentioned above, but they do not have a local product structure. Finally ($9 1517) we study the case of discrete fibers and in particular the classical covering spaces.
2. Covering homotopy property Throughout the present section, we shall consider a given map P:E+B of a space E called the total space into a space B called the base space. 61
62
111. F I B E R S P A C E S
Let X beagivenspace,f : X +. B agivenmap, andft : X + B , (0 Q t Q I ) , a given homotopy of f . A map f* : X +. E is said to cover f [relative to p ) if $f* = f . Similarly, a homotopy ft* : X + E , (0 Q t Q l),off* is said to cover the homotopy f t (relative to p ) if pft* = f t for each t e 1 ; ft* is called a cuvering homotopy of f t . The map p : E +. B is said to have the covering homotopy property (abbreviated CHP) for the space X if, for every map f* : X f E and every homotopy f t : X +. B, (0 < t Q I ) , of the map f = pf* : X f B, there exists a homotopy ft* : X + E , (0 Q t < l ) , of f* which covers the homotopy f t . The map p : E + B is said to have the absolute.covering homotopy property (abbreviated ACHP) if it has the CHP for every space X . The map p : E +. B is said to have the polyhedral covering homotopy property (abbreviated PCHP) if it has the CHP for every triangulable space X . The map p : E +B is said to have the covering homotopy extension property (abbreviated CHEP) for the space X relative to a given subspace A of X if, for every map f* : X +. E and every homotopy f t : X +. B, (0 Q t Q l ) , of the mapf=pf*:X+B,everyhomotopygt*:A+E,(O Q t 0, let Xm denote the mdimensional skeleton of X . Let Km denote A U Xm. By successive application of (iii), one can construct for each integer m = 0, 1,2; a homotopy htm : Km + E , (0 Q t < l), such that f

hr
= f*
hy I A
=
I Km, gt*,
= ft
hY+' I Km
I Km, =
hr.
Then the required homotopy ft* is defined by taking ft* I Km = h r for each m = 0, 1,2; * * (iv) + (v). Since A is a strong deformation retract of X , there exists a homotopy ht : X + X , (0 Q t < I), such that ho is a retraction of X onto A , h, is the identity map on X , and ht (a) = a for every a E A and t E I. Define a map f # :X +E and a homotopy ft : X + B , (0 Q t < l), by taking f # = g*ho and ft = fht for each t E I. Then fo = pf#. Since ft(a) = f ( a ) for each a E A and t E I , we may define a partial covering homotopygt* : A +E , (0 Q t < l), of f # by setting gt* = g* for every t E I. According to (iv), gt* has an extension ft* : X + E , (0 Q t Q I), such that Pft* = f t for every t €1. Let f* = f l * . Then we have f* 1 A = g* and pf* = f l = fh, = f. (v) + fi). This implication follows inmmediately from the fact that X x 0 is a strong deformation retract of X x I. I As noted a t the end of 3 2, the product space E = B x D is obviously a fiber space over B relative to the natural projection p : E f B .
4. B U N D L E S P A C E S
65
4. Bundle spaces A map p : E +. B is said to have the bundle property (abbreviated BP) if there exists a space D such that, for each b E B, there is an open neighborhood U of b in B together with a homeomorphism +,y:
U x D+pl(U)
of U x D onto pl(U) satisfying the condition
(DF)
p + , y ( ~ d, ) = U , (U E
U , d E D).
In this case, the space E is called a bundle space over the base space B relative to the projection p : E +. B. The space D will be called a director space. The open sets U and the homeomorphisms +U will be called the decomposing neighborhoods and the decomposing functions respectively. As an immediate consequence of the definition, we have + ( E ) = B except in the trivial case where E is empty. The main idea of this definition is that a bundle space is a space with a local product structure over every point of the base space. In particular, the product space E = B x D is a bundle space over B relative to the natural projection fi : E + B with D as director space. In this definition of bundle spaces, we have essentially followed that of Ehresmann and Feldbau, [S; p. 181. For relations with coordinate bundles in the sense of Steenrod, see [S ; pp. 1820]. Many examples can be found in
[SI. Theorem 4.1. Every bundle space E over B relative to p E + B i s a fiber space over B relative to p . Since CHP is trivial for product spaces, the idea of the following proof is to reduce the construction to local ones where the local product structure is available. Proof. Let X be a (compact) triangulable space, f * : X + E a given map, and f l : X +B, (0 < t < l), any homotopy of the map f = pf * : X +. B. I t suffices to construct a homotopy f t * : X + E , (0 < t < l ) , of f * which covers ft. Pick a collection o = { U } of decomposing neighborhoods U which covers the base space B and define a map F : X x I +. B by taking
F(x,t)
= ft(x),
(x E
x,t E I ) .
The collection { F  l ( U ) I U E O } of open sets of X x I forms an open covering of X x I . Since X x I is compact, { F  I ( U ) I U E o } has a refinement of the form { W Ax I , }, where { W A} is a finite open covering of X and { I , } = { I , , * * , I f } is a finite sequence of open subintervals of I which covers I . We may assume that I , meets only I,,l and I,+l for each p = 2; * .,r  1. Choose numbers 0 = to < t , < . . * < tr = 1
66
111. F I B E R S P A C E S
so that t, is in the intersection I , n I,+,. We shall assume, inductively, that the covering homotopy ft* has already been defined for all t Q t, where p > O is a given integer less than 7. We proceed to extend ft* over the closed subinterval [t,, t p + l ] of I . Taking a sufficiently fine triangulation of X,we may assume that X is a simplicial complex such that every closed simplex of X is contained in some W Aof the finite open covering { W A }constructed above. Hence, for each closed simplex n, we may choose a decomposing neighborhood U, E W such that f t ( x ) E ua, ( X E 0 s t, Q 1 Q tp+l). Let yua : U, x D + D denote the natural projection. Let a be a vertex of X.Then ft* can be defined by taking
ftp*(a)l, ( f , Q t t,+J. Thus ft* is defined on the zerodimensional skeleton X o of X for each t E [t,, t,+l]. We shall assume that ft* has already beendefinedon the ( n 1)dimensional skeleton Xnl for each t E [t,, t,+J, where n > 0 is a given integer. We proceed to extend ft* over the ndimensional skeleton X n of X . Let a be any closed nsimplex of X . Then ft* has been defined on the boundary aa for every t E [t,, t,+l]. Let M = a x [t,, t,+l] and consider the subspace N = (a x t,) u (80x [t,, t,+,l) ft*(a) = +ua [ft(a),WU~+U,'
of M . According to (I;4.2), these exists a retraction p : M 2 N . Let O : N +E denote the map defined by O(x, t ) = f t * ( x ) . Then we may extend ft* over a by setting f t * W = +u, [ f t b ) , yua+u,' O p b , 01 for every x E a and t E [t,, t,+J. This completes the construction of ft*. 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 p : E + B has the BP, then it has the C H P for every paracompact Hausdorf space. For a proof of (4.2), see [Huebsch 11 and also [S; p. 501.
5. Hopf fiberings of spheres Among the early examples of bundle spaces were the three fiberings of spheres p : S2nl + S n , n = 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, S3 consists of the points (zl,z 2 ) in C2 such that Z 1 Z l + z2z2 = 1.
5. H O P F F I B E R I N G S O F S P H E R E S
67

Let S2 be represented as the complex projective line, that is to say, as pairs [z,, z2] of complex numbers, not both zero, with equivalence relation [z,, z2] [hl, h , ] where 1 # 0. Then the Hopf map p : S3+S2 is defined by p ( z l ,z 2 ) = [z,, z,] for each ( z l , z 2 ) E S ~The . continuity of p is obvious. Since any pair [z,, z z ] can be normalized by dividing by (zlZ1 + z,Z2)*, p maps S3 onto S2. To prove that S 3 is a bundle space over S2 relative to p , let us represent S1 as the set of all complex numbers 1 with I 1 I = 1. Consider the points a = [l, 01 and b = [0, 11 of S2 and the open sets
U
V = S2\b. Then U and V cover S2. Every point in U can be represented by a pair [z, 11. Hence we may define a map +v of U x S1 into S3 by taking = S2\a,
for each [z, 11 E U and each 1 E S1. One can easily verify that +v maps U x S1 homeomorphically onto p  l ( U ) and that p + ~ ( ud,) = zc for each u E U and d E D. Hence +u is a decomposing function. Similarly, we can construct a decomposing function This completes the proof that S 3 is a bundle space over S2 relative to the Hopf map p . If ( z l , z 2 ) E S3, then one verifies immediately that the fiber p  l [ z l , z2] consists of all the points (h,, h2) with 1 ~ 9 Hence . the fibers are just great circles of S3. In this way the 3sphere is decomposed into a family of great circles with the 2sphere as a quotient space. The Hopf fiberings : S7 +S4 and p : S15 + SS 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. 1051 101. In these fiberings, the fibers are 3spheres and 7spheres respectively. The Hopf maps are all essential; in fact, this is a consequenceof the following
+".
Proposition 5.1. I f a sphere Sn is a fiber space over a base space B which contains more than one point, then the projection p : Sn + B is an essential map. Proof. Assume that p were inessential. Then there exists a homotopy ht : Sn + B (0 Q t Q 1) such that h, = p and Izl(S")is a single point b, of B . Let i denote the identity map on Sn; then we have pi = p . According to the CHP, there exists a homotopy kt : Sn + Sn, (0 < t Q 1),such that k , = i and pkt = ht for every t E I . Now, k , maps Sn into the fiber pl(b,) which is a proper subset of Sn since B contains more than one point. Hence, according to ( I ; 3 8), k is inessential. This implies that the identity map i is homotopic to a constant map. This is impossible by ( I ; Q 8). I I t is interesting to observe that the Hopf maps are algebraically trivial, that is to say, their induced homomorphisms on the homology groups and the cohomology groups are all trivial. Historically, these Hopf maps were the first examples of essential maps which were algebraically trivial. The
68
111. F I B E R S P A C E S
existence of these maps shows that induced homomorphisms are not by themselves sufficient to classify all maps.
6. Algebraically trivial maps X f S2 In the present section, we shall show that, by the aid of the Hopf map p : S3 + S2, one can solve the classification problem of the algebraically trivial maps f : X + S2of a 3dimensional triangulable space X into S2. Let X be any given triangulable space. For an arbitrary map F : X + S3, i t is obvious that the composed map f = pF : X +S2 is algebraically trivial and that the homotopy class o ff depends only on that of F. Proposition 6.1. For any given triangulable space X , the assignment F + f = pF sets up a onetoone correspondence between the homotopy classes of the maps F : X + S3 and those of the algebraically trivial maps f : X + S2. As an immediate consequence of (6.1) and the Hopf classification theorem C3 in (I1 ; fj 8), we have the following Theorem 6.2. T h e homotopy classes of the algebraically trivial maps f : X + S2 of a 3dimensional triangulable space X into the 2sphere S2 aye in a onetoone co~respondencewith the elements of the integral cohomology group H3(X).For any u E H 3 ( X ) ,the homotopy class which corresponds to u contains the m a p f = PF : X +S2, where F : X +S3 i s a m a p with a as its degree. In particular, if X = S3, then every map f : X +S2 is algebraically trivial. Hence, we have the following Corollary 6.3. T h e homotopy classes of the maps f : S3 + S2 are in a onetoone corresfiondence with the integers. For a n y integer n, the homotopy class which corresponds to n i s represented by the composition f = pF : S3+ S2 of the Hopf m a p p : S3 + S2 and a m a p F : S3 + S3 with deg ( F ) = n. The proof of (6.1) consists of the following two lemmas. Lemma 6.4. If a m a p f : X +S2 i s algebraically trivial, then there exists a m a p F : X + S 3 such that PF = f . Proof. Since S3 is a bundle space over S2 relative to the Hopf map p ;S3+ S2, we may choose a collection w = { U } of decomposing neighborhoods U which covers 9. Taking a sufficiently fine triangulation of X , we may assume that X is a simplicia1 complex such that the given map f carries every closed simplex u of X into some decomposing neighborhood U , E w . Let Xm denote the mdimensional skeleton of X , and let f m = f I Xm. For each m = 2,3; * * , we shall construct a map Fm : Xm + S3 such that pFm = f m , Fm+i I Xm = Fm. First, let us construct F,. Since the given map f is algebraically trivial, so is f 2 . According to the Hopf classification theorem C2 of (11; fj 8), f , is homotopic to a constant map which can be lifted. Hence it follows from the CHP that there is a map F , : X 2 + S3 such that p F , = f 2. Next, assume that n > 2 and that Fm has been constructed for every
7. L I F T I N G S
A N D CROSSSECTIONS
69
m < n. Let u be an arbitrary closed nsimplex of X and choose a decomposing neighborhood U , E w which contains f (a).Let 4, : u, x s1+ p  l ( U,), yo : u, x s1+ S' denote the decomposing function and the natural projection respectively. Define a map [, : du + S1 by taking 5,(x ) = yo&lFn 1 ( x ) for every x E da. By (11; fj 7), [, has an extension 7, : u + S1. Then we define a map Fn : X n + S3by taking (if x E Xnl), Fn I(%), Fn(x) = (if x E u E X n ) . 4 J f n ( 4 ,~ d x1 ) Obviously we have PFn = f n and Fn I Xnl = F n l. This completes the inductive construction of the maps Fm, m = 2,3; . Finally, define a map F : X + S3by taking F I X m = Fm for every rn > 2. Then we have PF = f . I Lemma 6.5. If F , G : X + S3aye two maps such that pF N PG, then F N G. Proof. Since pF N 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 qq = 1. Then the fiber which contains the quaternion 1 is a subgroup S1 of S3 and the other fibers are cosets of S1 in S3.In fact, in the usual representation q = x1 x2i + x 3 j x4ij = x1 x2i ( x , x.,i)j = z1 zzj where z , = x1 + x 2 i and z2 = x 3 + x4i, the multiplication is based on the rules j 2 =  1 and z j = $5. Then S1is the subgroup defined by z2 = 0 and the right cosets of S1 are the fibers of the Hopf map p : S3+ S2. Define a map H : X +S3by taking H ( x ) = F ( x )* [G(x)]' for every x E X . Since pF = pG, F ( x ) and G ( x ) are contained in a coset of S1 and hence H carries X into a proper subspace S1 of S3.Then it follows that there exists a homotopy Ht : X +S3, (0 < t < l ) , such that H , = H and H , ( X ) = 1. Define a homotopy J t : X +S3, (0 < t < l ) , by taking It(%)= H t ( 4 .G(x) for each x E X and t E I . Then we have J , = F and J 1 = G. Hence F N G. I Since the proof of (6.4) and (6.5) is based on special properties of S1 andS3, there are no analogous results for the maps X + S4 and X + SS.
{

+
+ +
+
+ +
7. Liftings and crosssections Let E be a fiber space over a base space B with projection p : E + B and let f : X + B be a map of a space X into B. By a lifting of f in E , we mean
70
111. F I B E R S P A C E S
a map g : X + E such that p g = f . As a special case of (I; 4 16), the lifting problem for the given map f is to determine whether or not f has a lifting in E . For example, if p : S3 + S2is the Hopf map and X is a triangulable, space, then (6.4) and (6.5) solve the lifting problem for any map f : X + S2, namely, f has a lifting in S3 iff f is algebraically trivial. If p : E + B has the CHP for X , then the lifting problem for a map f :X + B is equivalent to a broadened lifting problem, namely, to find a map g : X + E such that p g = f . According to the definition of fiber spaces, this is always the case if X is triangulable. By (4.2), this is also true if E is a bundle space over B relative to p and X is a paracompact Hausdorff space. If X is a subspace of B and f : X B is the inclusion map, then the notion of a lifting for f reduces to that of a crosssection over X . A crosssection in E over a subspace X of B is a map x : X E such that
.
$x(x)
=
x,
.
(XEX).
Thus, a map x : X + E is a crosssection iff the image x ( x ) of an arbitrary point x E X is contained in the fiber over x. I t is easy to verify that every crosssection x : X + E maps X homeomorphically onto x ( X ) with p I x ( X ) as its inverse. Therefore, a crosssection 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 q5u: U x D+P'(U), y u : U x D + D , then, for any point e in Pl(U), there is a crosssection xe : U + E given by
xe(u) = $u(u, 4, d = YU#JU'(~) €or each U E U . If u = p ( e ) , then xe(u)= e. Thus, in bundle spaces, local crosssections always exist. However, the existence of a global crosssection, i.e. a crosssection over the whole base space B, is a rather strong condition on the structure of the fiber space. In fact, if a global crosssection x : B + E exists, then, in any homology theory satisfying the EilenbergSteenrod axioms, the projection p : E + B and the crosssection x : B + E induce for each m the homomorphisms 9,: Hm(E)+ H m ( B ) , x * : Hm(B)+ H m ( E ) . Since px is the identity on B , p,x, must be the identity on Hm(B).I t follows that x + is a monomorphism, that p , is an epimorphism, and that Hm(E) decomposes into the direct sum Hm(E) = Kernel p , + Image x * . An immediate consequence of this necessary condition is that the Hopf fiberings in 4 5 do not have global crosssections. Dually, one can deduce necessary conditions for the existence of a global crosssection in terms of cohomology : Hm(E) = Imagep* + Kernel%*.
8. F I B E R
MAPS A N D I N D U C E D F I B E R SPACES
7'
These conditions (both those on the groups Hm(E) and those on the groups Hm(E))resemble those for retracts. This is no accident : if x is a crosssection, then the image of x is a homeomorph of B and is a retract of E . Let x : X + E be a given crosssection. Corresponding to the extension problem of maps in (I ; 5 2), we have the extension Problem of crosssections to determine if x can be extended over the whole base space, that is to say, whether OT not there exists a crosssection x* : B + E such that x* 1 X = x . This extension problem of crosssections is a generalization of that of maps in (I; 3 2). Indeed, if E = B x D where D is a given space, then E is a bundle space over B with projection fi : E + B defined by p ( b , d ) = b. For a given map f : X + D on a subspace X of B, we have a crosssection x f : X +E defined by xf (4= ( x , f (4) 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; tj 8) is the classification problem of crosssections. Let K denote the set of all crosssections x : B + E . Introduce an equivalence relation in K as follows: For any two crosssections f , g E K , f g iff there exists a homotopy ht : B + E , (0 < t < l ) , such that h, = f, h , = g, and ht E K for every t E I . Then the classification problem of crosssections is to enumerate the classes of K divided by this equivalence relation . An argument similar to that used above for the extension problem shows that this classification problem of crosssections is a generalization of that of maps in ( I ; 5 8).


8. Fiber maps and induced fiber spaces Let p : E +B 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 iff, for every point b in B, there exists a point b' in B' such that F carries p'(b) into p'l(b'). Now let F : E + E' be a given fiber map. Then F induces a function f : B + B' defined by f ( 4 = P'FPW
for every b E B. For any arbitrary set U in B', we have
f'(U)
=
PF'P'l(V).
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+E'
72
111. F I B E R S P A C E S
Some special cases of fiber maps are important. First, let us take p' : E' +.B' to be the trivial fibering over B, that is to say, B' = B, E' = B, and p' is the identity map. In this case, the projection : E +. B 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 = p'F : X + B' as induced map. This suggests the following extension of the lifting problem in 3 7. Let : E +. B and 9' : E' + B' be two fiberings. By a lifting of a given map f : B +. 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' iff P'F = fp. 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 crosssections, the answer to this problem is not always affirmative. However, for a given fibering p' : E' B' and a given map f : B + B' of a given space B into B', we can construct a fibering : 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 = { (b, e') E B x E' I f ( b ) = p'(e') } and let p : E +. B denote the natural projection defined by p ( b , e') = b. Let F : E + E denote the map defined by F ( b , el) = e'. We are going to prove that p : E + B is a fibering and that F is a lifting of f . By the preceding construction, we have f p = p'F. Hence it remains to prove that p : E + B is a fibering. Let : X +. E be a given map of a triangulable space X into E and ht : X +. B , (0 < t Q l ) , be a homotopy of the map y = +p. Let 6 = F+ : X +. E', kt = fht : X +. B', (0 Q t Q 1). Then kt is a homotopy of the map k , = f h , = fp+ = pfF+ = p ' l Since p' : E' + B' is a fibering, there exists a homotopy kt* : X +. E', (0 < t < l ) , of 6which covers kt. Define a homotopy ht* : X + E , (0 Q t < I ) , by taking ht*(x) = (ht(x),k t * ( x ) ) for every x E X and t E I . This definition of ht* is justified by the relation f k t * = kt = fht. Since
+
f
+
ko*(x) = ( h O ( X ) > ko*(x)) = F + ( x ) ) = 9(x) for every x E X , ht* is a homotopy of +. Since ht* obviously covers ht, this completes the proof that p : E + B is a fibering and F is a lifting off. The fibering p : E + B constructed above is said to be induced by f ; the lifting F : E +. E' off will also be said to be induced by f . Note that, for each b E B, F maps the fiber p  l ( b ) homeomorphically onto the fiber @'l(b'), 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. (
P
W
8
9. M A P P I N G
SPACES
73
Finally, if x' : B' + E' is a crosssection, then the map x : B + E defined by x(b) = (b, x ' f ( b ) ) is also a crosssection. We shall call x the induced crosssection 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'l(B) and p with p' I B in an obvious way. In this case, the induced fibering p : E + B will be called the restriction of p' : E' + B' on B. Hence we have the following
p
Proposition 8.1. If E i s a fiber space over a base space B with projection : E + B and if A i s a n y subspace of B, then pl(A) i s a fiber space over A
with p 1 pI(A) as projection.
9. Mapping spaces Let
X and Y be arbitrarily given spaces and denote by
a= Y X
the totality of maps of X into Y . There are various ways of topologizing $2, but we will be concerned only with the compactopen topology, [Fox 31. I t is also called the ktopology, [Arens 21, and the topology of compact convergence, [B; I111 and [K]. 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 ) = { I f ( K )c W 1. M ( K , W ) will be called a subbasic set of $2 if K is compact and W is open. The compactopen topology of D is defined by selecting as a subbasis for the open sets of IR the totality of the subbasic sets M ( K , W )of 52. According to the usual definition of a subbasis, every subbasic set is open in D and every open set of f2 is the union of a collection of the finite intersections of subbasic sets. Throughout the present book, mapping spaces are understood to be topologized by their compactopen topologies, unless otherwise stated. Lemma 9.1. I f X i s a Hausdor8 space and { U } i s a subbasis for the open sets of Y , then the totality of the sets M ( K , U ) , for K a compact subset of X and U E { U }, constitutes a subbasis for the compactopen topology of a [Jackson 21. Proof. It suffices to show that if K is a compact subset of X and W an open subset of Y , and if f E M ( K , W ) , then there exist compact subsets K,; . * , Km of X and members U,; . * , Um in { U } such that
Let X say U f ;
E
n M(Km, Urn)c M ( K , W ) . / E M ( K , , V,) n K . Since f ( x ) E W , there are a finite number of sets in { U }, * , U& such that f(X)EU:n
n U$,C
w.
I l l . FIBER SPACES
74
Since f is continuous, there is a neighborhood G, of x in X such that As a compact Hausdorff space, K is regular. So there is an open neighborhood H , of x in K such that the closure K , = H , is contained in G,. The collection { H , I x E K } is an open covering of the compact space K , and hence there are a finite number of points in K , say x1; x,, such that K = Hzl U * * * U HZ,. a ,
In the subscripts and superscripts involved above, we shall simply replace x j b y j , j = l;*.,q. Now the sets K,; ., K, are compact. Moreover,

Hence we have
a
ni
Suppose that g e l 2 is contained in the set on the righthand side of the preceding formula. If x E K , then xis in some H f and hence is in Kj. Therefore Thus g E M (K, W ) ,and so a
ni
Next, there is a natural function w:QxX+Y defined by w (f , x ) = f ( x ) for each f E Q and each x E X . w will be called the evaluation of the mapping space Q. Proposition 9.2. If w of Q i s continuous.
A’ i s a locally compact regular s+ace, then the evaluation
Proof. Let f E 9,x E X, and an open set W of Y which contains f ( x ) be arbitrarily given. Since f is continuous, the inverse image f l( 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 is compact and is contained in f 1(W).Then U = M ( 7 , W ) is a subbasic open set of i2 which contains f . U = M ( V , W ) implies that w maps U x V into W . This proves the continuity of w . 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 f2 which maps X into the single point y . The assignment y + j ( y ) defines a function
j:Y+Q
9. M A P P I N G
75
SPACES
called the natural injection of Y into B.It can be easily verified that j maps Y homeomorphically onto a subspace j ( Y )of B.Furthermore, if Y is a Hausdorff space, then j ( Y ) is closed in a. For any given point a E X , let pa:.R+
Y
denote the function defined by p,( f ) = f (a) for each f E B.Clearly, p, is a map and sends the subspace j ( Y ) of B onto Y . We shall call p, the projection of Sa onto Y determined by the given point a E X . From the definitions given above, we observe that p,j is the identity map on Y and i p , is a retraction of 9 onto its subspace j ( Y ) .Hence we have the following Proposition 9.3. The natural injection i maps Y homeomorphically onto a retract i(Y ) of B. Next, let us consider three given spaces T , X , Y and the mapping spaces Q = yx, @ = y x x * , y = Q T .
For each map taking
+ :X
x T
+
[O(+)(t)l(x)
Y in @, define a function 8(#) : T +B by = +(x,
O(+) is said to be associated with 9.
4,
Proposition 9.4. The associated function
( J E Tx~ , X).
8(+) of
+
E
@ is continuous.
Proof. Let y = 8(+) and let U = M ( K , W )be any subbasic open set in B. I t suffices to prove that yl( U ) is an open set of T. Let to be any given point in y  l ( U ) . By definition, we have
K x t0c+1(W) The continuity of implies that +'(W) is an open set of X x T . Hence #l(W) is the union of a collection of open sets of the form G, x H,, where C, and H , are open sets of X and T respectively. Since K is compact, K x to i. contained in the union of a finite number of these open sets, say
+
GI x HI, Gz x H z , . . . , Gn x Hn with to E Ht for each i
=
1,2;
* *,
n. Then
H=H,nH,n **nH, is an open set of T containing to and is contained in y  l ( V ) . Therefore, yl( V )is an open set of T. I As a consequence of (9.4) the assignment + 8(+) defines a function
e:o+y; 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 i s a Hausdor8 space, then 8 :@ + Y is continuous.
76
111. F I B E R S P A C E S
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
W L , M ( K , W)l
=
{y
EY
I y ( L )= M ( K , W) 1
form a subbasis for Y, 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 8 that
81 { M [ L , M ( K , w) 3 1 = M ( K x L, w). Since K x L is compact, M ( K x L, W )is open
[email protected] { M [ L , M ( K , W ) ]} is a subbasis of Y, it follows that 8 is continuous. I It is obvious from the definition that 8 carries @ into Y in a onetoone fashion. In general, 8 is not onto. However, we have the following Proposition 9.6. T h e evaluation o of Q i s continuous i f , for every space T , the function 8 : @ + Y i s onto.
Proof. Necessity. Let y E Y and consider the map X : X x T + Q x defined by X ( X , t ) = f y ( t ) ,x ) , (xE t E T).
X
x,
Let 4
= W X E @.
Since
[W(t)l(x) = 4(%4 = W X ( % t ) = o [ y ( t )XI * = [y(4l(x) for every 1 E T and x E X,we have 8(+) = y. Hence 8 is onto.
Suficiency. Assume that the condition holds. In particular, select T = 52 and take y E Y to be the identity map on Q. Then there is a map 4 E @ with O(4) = y . Since
4(% f ) = [ Y ( f ) I ( X = ) f(x) =4f4 , for every x E X and f E Q, the continuity of 4 implies that of o.I As an immediate consequence of (9.2) and (9.6), we have the following Corollary 9.7. If X is a locally compact regular space, then the function 8 : @ + Y sends @ onto Y in a onetoone fashion. Proposition 9.8. If X and T are Hausdorf spaces, then the association m a p
0 : @ + Y i s a homeomorphism of @ onto a subspace of Y [Jackson 21.
Proof. Since 8 is onetoone and continuous by (9.5), i t remains to prove that 81 is continuous on 0(@)c Y. For this purpose, it suffices to prove that, if J is a compact subspace of X x T and W is an open set in Y , then the image O [ M ( J ,W)] is an open set of 8 ( @ ) . Let y E 8 [ M (J , W)] be arbitrarily given. Choose a #IJ E M (J , W) with 8(4) = y . Let J X and J T denote the projections of J in X and T respectively. For each point z = ( x , t ) E J , choose an open neighborhood U, of x in J X and an open neighborhood V zof t in J T , such that
+(UZx Vz)=
w.
9.
MAPPING S P A C E S
77
As compact Hausdorff spaces, J X and JT are regular. Hence we may shrink U, and V, a little bit so that
L)=w,
x
+(&
where Kz denotes the closure of Uz in J X and L, that of V, in J T . The collection { (U, x V,) n J 1 z E J } is an open covering of the compact space J. Hence there is a finite number of points in J , say z1; ., z,,, such that J C ( U Zx~ VzJ U U (Uz, x Vz,).

For the subscripts in the notations of the various sets involved above, we shall simply replace zt by i, i = 1, ,n. Now the sets Kg and Lr, i = 1, * n, are compact. Moreover,
a ,
for each i
=
[y(Ldl (4= +(Kix Lt) = W l;.., n. Hence Y E
e(@)n {.;
1=1
W L t , M(&,
Wl 1.
Suppose that X E Y is contained in the set on the righthand side of the preceding formula. Since X E O(@), there is a 5 E @ with X = 8 ( 5 ) . If z = ( x , t ) E J , then z is contained in some Ui x Vt and hence in Kt x Li. Since X is in M [ L i , M ( K t , W ) ]and since x E Kr, t E Lt, we have 5(z) = t ( x ,t )
=
[X(t)l ( x ) E w.
This proves that t ( Jc) W . Therefore, 8 [ M ( J ,W ) ] Thus, . we obtain n
Y E ~ P )n {.n M L , S=l
t E M ( J , W ) and hence X
MW, w)i I =
=
8 ( [ )E
e w ( J ,wi.
This proves that 8 [ M ( J ,W ) ]is an open set of 8 ( @ ) .I As an immediate consequence of (9.7) and (9.8), we obtain the following Theorem 9.9. Let X and T be Hausdor8 spaces and Y any space. If X i s locally compact, then the association map 8 i s a homeomorphism of the mapping space @ = Y x x onto the mapping space Y = ( Y X ) T .
Hereafter, when the assumptions of (9.9) are satisfied, the two mapping spaces will be identified by means of the homeomorphism 8 . In symbols, we have y x x T = (Y")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 Q. Such a condition is given by the following Proposition 9.10. If X i s a locally compact regular space and contractible to a point a E X , then j ( Y ) i s a strong deformation retract of Q.
78
111. F I B E R S P A C E S
Proof. By (9.2) the evaluation o :9 x X + Y is continuous. Since X is contractible to the point a E X , there exists a map h : X xI+X such that h(x,0 ) = x , h(x, 1) = a, ( x E X ) ; h(a, t ) = a, (t E I ) . Define a map C# : X x 9 x I + Y by taking C#(x,f, t ) = o [ fh(x, , t ) l , ( X E X , f €9, 4. According to (9.4),the associated function =:
e(+) :JZ x
is continuous. Define a homotopy
Xt
I +Q
: 9 +J2, (0
< t < l ) , by setting
X d f ) =p(f,t) (fEQ,tEI). Then it is easily verified that X, is the identity map, X , fcr every f~ j(Y)and t E I . I Since it is easily verified that Xt[PYY)I = P 3 Y L we have proved the following
= jPa,
and &(f)
=f
(YEy , t E 4 ,
Proposition 9.1 1. If X is a locally compact regular space and contractible to a point a E X , then, for each y E Y , the subspace p;'(y) is contractible to the point j ( y ) . 10. The spaces of paths Let Y denote a given space. By a path in Y we mean a map a:I+Y of the unit interval I = [0, I ] into Y . The points a(0) and a(1) are called respectively the initial point and the terminal point of the path a, and are said to be connected by the path CT. 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 pathcomponents of Y . We shall denote by Cy the pathcomponent of Y which contains the point y E Y . Y is said to be pathwise connected if it consists of a single pathcomponent and hence every pair of points in Y can be connected by a path. The totality of paths in Y forms a space 9 = Y' with the compactopen topology defined in the previous section. This space 9 together with certain of its subspaces is of fundamental importance in homotopy theory as well as in the functional topology of Morse, [ M , and M J . 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 nonempty. For a given generalized triad ( Y ;A , B ) , let us denote by [ Y ;A , Bl
11. T H E S P A C E OF LOOPS
79
the subspace of l? which consists of the paths u in Y such that u(0)E A and 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 8 simply by 8,.It is called the space of paths in Y with a given terminal point y. A loop in Y is a path u : I + Y such that a(0) = a(1). The point o(0) = a( 1) will be called the basic point of the loop u. The set of all loops in Y forms a subspace A of B 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 B simply by A,. It is obviously a subspace of A and will be called the sface of loops in Y with a given basic point y. Clearly we have
a(1) E
A,
=A
n 8,.
The projections Po, p , : 8 + Y determined by the points 0, 1 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 Po and p , implies that the subspaces A, A,, 8,of 8 are all closed. Let 6, denote the degenerate loop &(I) = y. By (9.11), we have the following
6,. Intuitively, a contraction of 8, to the point 6, is obtained by pushing all paths of Q, along themselves simultaneously to the terminal point y. Proposition 10.1.8, is contractible to the point
11. The space of loops Let Y denote a given space and y E Y a given point. The space A , 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, as follows. For any pair of loops f , g E A,, the product f g E A , is the loop in Y defined by
f (24, ( f g ) ( t ) = g(*t  I),
I
(0 Q t < 41, (3 Q t Q 1).
Intuitively speaking, fg is the loop in Y which travels along the loop f with double speed while t is in the first half of I; then f g travels along the loop g with double speed while t is in the second half of I . The correspondence (1, g) +. ,~i(f, g) = fg defines a function /L
: A , x A , +Ag
called the multiplication function in A,. Proposition 11.1. The natural multiplication in A , is continuous; by this we mean that the multiplication function ,u is continuous.
80
111. FIBER SPACES
Proof. Let U = M ( K , W ) n A, be an arbitrary nonempty subbasic open set of A,. It suffices to show that the inverse image pl( U)is an open set of A, x A,. Let 4, y : I + I denote the maps defined by
+
+(t) = i t , y w = i ( t 1) for each t E I . Let A = +l(K) and B = yl(K) ; then A and B are compact subspaces of I. Consider the subbasic open sets F = M(A, n A,, G = M ( B , n A ,
w)
w)
of A,. For any pair of loops f , g E A,, it is easy to see that fg E M ( K , W )iff f E M ( A , W ) and g E M ( B , W ) . Hence pl(U) = F x G. This proves that pl(U) is open. I Corollary 11.2. Every loop f
EA,
determines two maps L f , Rf :A , + A y defined by Lf(g)= f g and R f ( g ) = gf for every g E A,. Proposition 11.3. I f SEA, denotes the degenerate loop & I ) = y , then L.3 and R6 are deformations of A,. I n fact, there exists a homotopy ht : A , +A,, (0 t l), such that h, = L a ,h, i s the identity m a p , and h,(d) = S for each t E I ; and analogously for Rd. Proof. According to (9.2), the evaluation (o : A, x I + Y is continuous. Hence we may define a map 4 :I x A, x I + Y
<
4. I In the preceding proof, we have also proved the following
82
111. F I B E R S P A C E S
Corollary 11.5. Under the hypothesis of (11.4),iftwo elements a,/? €zl ( X ,xo) are represented by the loops f , g : I + X at x,,, then the element a/?is represented by the functional product h : I + X of f , g defined by
h(t) = f (4 g(t)n tt E 4 . As we shall see later, this assertion also holds for the higher homotopy groups. Now let us come back to the space A , of loops and consider its pathcomponents. By (9.9), one can easily see that the pathcomponents of A , are exactly the equivalence classes of the loops of AV as defined in (11, 9 4). Hence we have the following Proposition 11.6. Under the natural multiplication of A,, the pathcomponents of A, furm a group which is essentially the ficndamental grouPz,(Y, y ) .
12. The path lifting property In the present section, we are going to define, for any given map p :E + B the path lifting property (abbreviated PLP) which was recently introduced by Hurewicz and Curtis, and to prove that it is equivalent to the absolute covering homotopy property (abbreviated ACHP) defined in 9 2. Roughly speaking, a map p : E + B is said to have the PLP if, for each e E E and each path f : I * B with f (0) = p ( e ) , there exists a path g : I + E such that g(0) = e, p g = f , and that g depends continuously on e and f . For a precise definition, let Z denote the subspace of the product space E x B' defined by z = (e, f ) E E x B ' I p ( e ) = /(o) 1. Define a map q:E'+Z by taking q(g) = (g(O),p g ) for each g :I f E in E'. Then p : E + B is said to have the PLP if there exists a map r:Z+E' such that qr is the identity map on Z.In this case, Y is clearly a homeomorphism of Z onto a retract of E'.
A map p : E + B has the PLP in it has the ACHP. Proof. PLP j. ACHP. Let g : X + E be any given map of a space X into E and f t : X  + B , (0 < t Q l ) , a given homotopy of the map f = p g . According to (9.4), the homotopy ft gives a map h : X + B' defined by [h(X)l(4 = f&), ( X E X , t E 4. Let k : X +Zdenote the map defined by Proposition 12.1.
k(x) = (g(x), h(x)). ( % E X ) . Then the composition rk is a map of X into E'. According to (9.9),we may define a homotopy gt : X + E , (0 < t < l), by taking g t ( 4 = [ ~ k ( x ) l ( t ) (, X E X , t E 1).
13. T H E F I B E R I N C
THEOREM FOR M A P P I N G SPACES

83
One can easily verify that go = g and pgt = f t for each t E I . This implies ACHP. ACHP PLP. Let q :2 + E denote the natural projection defined by q(e,f ) = e for each (e, f ) €2.Define a homotopy & :2 .+B , (0 < t < l), by taking Me,/)= f(t), ( ( ~ , ~ ) E Z , ~ E I ) .
Then we have to= h.According t o ACHP, there exists a homotopy qt :2 + E , (0 < t < l), such that qo = q and pqt = 66 for each t E I . According to (9.4), qt gives a map Y : Z + E' defined by
[r(z)l(t)= q&), ( z E 2,t E 4. Then it is easily verified that qr(z) = z for each Z E Z .Hence, p : E + B has the PLP. I Corollary 1 2 3 If a map over B relative to p .
p
:E
+B
has the PLP, then E is a fiber space
13. The fibering theorem for mapping spaces Let X be a locally compact ANR and A be a closed subspace of X which is also an ANR. On the other hand, let Y be an arbitrary space. Let us consider the mapping spaces E = Yx, B = Y A . There is a natural map
p:E+B defined by p ( f ) = f I A for every map f : X + Y . We are going to prove that E is a fiber space over B with p as projection; in fact, we have the following stronger result.
p : E + B has the PLP. Let us consider the subspace Z of E x B'
Theorem 13.1. The map
Proof. and the map q : Ex + 2 described in the preceding section and try to construct a map r :2 + E' such that qr is the identity map. Consider the closed subspace T = ( X x 0) U ( A x I ) of X x I . By (I; Ex. 0),T is a retract of X x I . Let p : X x I 1 T denote a retraction. Consider the mapping space YT.For each g : T + Y in YT,define two maps e : X f Y and f : I + Y" as follows:
e ( x ) = g(x, 0). ( x E X ) ;
[ f ( t ) l ( a= ) g(a, 4, ( ~ E ~ I, € 4 . The continuity of f follows from (9.4). Hence, the assignment g + 8(g) = (e, f ) determines a function 8 yT +z. By the methods used in establishing the exponential law of mapping spaces in 5 9, one can prove that 8 is a homeomorphism of YTonto Z. Using the inverse of 8, we can define a map r :Z + E' by taking { [r(z)I(t)1 (4= [8'(z)lp(x, t )
84
111. F I B E R S P A C E S
for every t E I and x E X . Since p is a retraction, it follows immediately that qr(z) = z for each z in Z. I Note. The condition that X is locally compact is inessential. In fact, the theorem holds without this condition; for (9.7) may be replaced by Ex. L at the end of the chapter.
Now let { A , j p E M } and { Y , I p E M } be given families of subspaces of A and Y respectively, both indexed by the elements of a set M . Consider the subspace B' of B which consists of the maps g E B such that g(A,) c Y , for each p E M . Let E' = Pl(B'). Then E' is the subspace of E consisting of the maps f : X + Y such that f (A,) c Y , for every p E M . By (8.1) and (12.2), we have the following
p' : E' + B' i s a fibering. In fact, it is also obvious that p' : E' + B' has the PLP. So, this generalizes (13.1). For important examples, let us take X to be the unit interval I . First, take A to be the single point 1 of I . In this case, the space E is the space SZ of all paths in Y , and the space B can be identified with Y in an obvious way. Furthermore, the natural projection is essentially the terminal projection p , :SZ+ Y . Hence P I : 52+Y has the P L P by (13.1). Let 2 be a subspace of Y , then P,l(Z) = [ Y ;Y , Z ] . Corollary 13.1. T h e m a p
Therefore, by (13.2), we have the following Corollary 13.3. For any subspace Z of a given space Y , the space [ Y ;Y , Z ] i s a fiber space over Z relative to the terminal projection. Similarly, the space [ Y ;Z , Y ]i s a fiber space over Z relative to the initial projection.
Next, let X = I and take A to be the subspace of I which consists the boundary points 0 and 1 of I . Let M be an index set containing a single element p. Define A , and Y , to be the single points 1 E A and y E Y , where y is a given point. In this case, the space E' in (13.2) becomes the space 52, of all paths in Y with y as terminal point, and the space B' can be identified with the pathcomponent C, of Y in an obvious way. Furthermore, the natural projection p' is essentially the initial projection p,. Hence, we have the following important Corollary 13.4. The space i s a fiber space over Y relative to the initial projection. T h e fiber over the given point y E Y i s the space A , of all loops in Y with y as basic point.
Let Z be a subspace of Y . Then the following corollary is an immediate consequence of (13.4) and (8.1). Corollary 13.5. For every subspace 2 of a space Y and any given point y in Y , the space [ Y ; Z , y ]i s a fiber space over Z relative to the initial projection.
14.T H E I N D U C E D M A P S
I N MAPPING SPACES
85
14. The induced maps in mapping spaces Let X , Y ,Z be arbitrarily given spaces and : Y + Z a given map. For
+
each f E Yx,the composition +f is in Zx.The assignment f ++f defines a + x : Y X +zx function which is obviously continuous and will be called the induced map of on Y x into Zx.Let C denote the category composed of all spaces and all maps, [ES; p. 1 101. The operation Y + Y xand+ ++x, where X is a given space, defines a covariant functor of C into itself, [ES; p. 11 11. Furthermore, if we consider the natural injection and the projection p a determined by a given point a E X , the commutativity relatioiis
+
j+ = +xj, +pa = pa+x obviously hold in the following rectangles of maps: Y+Z4
4 Y+z
Throughout the remainder of the present section, we are concerned with the important special case that X = I . Consider a given map : Y + 2. Let y be a given point of Y and denote z = +(y) E 2. As above, we shall use the following notation:
+
YI:
Q, = [ Y ;Y , YI, A , = [Y; Y, Q, = [ Z ; Z,z], A , = [ Z ;z, ZI. Then the induced map obviously sends Q, into Q, and Av into A,. More
+'
generally, 4' maps [Y ; A , B] into [Z ; + ( A ) ,+(B)] for any given subspaces A and B of Y . According to 3 11, there are natural multiplications defined both in A, and in Az. I t follows from the definition that (14.1) +'(fg) = +'(f)+'(g) for any f E A , and g E A,. Since 4' is a map, it sends the pathcomponents of Av into those of A,. Hence, (14.1) shows that 4' induces a homomorphism of the group of pathcomponents of A , into that of A, which is the induced homomorphism +* of n,(Y , y) into nt,(Z,z). Now let Y , Z be given spaces, A C Y , B C Z given subspaces, and yo E A , z,, E B given points. Let u = [Y; Y , Yo], c = [ Y A;, yo], V = [Z ;2, z0], D = [Z ; B, z0]. and denote by uoE C and vo E D the degenerate loops u,(Z) = yo and vo(Z) = zo respectively. Let P : U  + Y , q:v+z denote the initial projections. We are going to establish a covering map theorem which will be important in the next chapter.
86
111. F I B E R S P A C E S
Theorem 14.2. If y : U + Z is a map swch that y(C) C B and d u o )= z,, then there exists a map y* : U+V such that qy* = yl y*(C) c D , and
v*(uo) = Vo. Proof. According to (9.2). the evaluation o:U x I + Y is continuous. Define a map a : I x I x U ,Y by setting a(s,t,f) = o ( f , s + t  s t ) , (SEI,tEI,fEU).
By (9.3),the associated function X : I x U
+U
defined by
W ,f ) I ( s ) = a(s,t , f ) , (t E I,f E u,s E 4, is continuous. Let 6 = yX : I x U +Z. By (9.3), the associated function y* : U + V defined by
w,
[y*(f)I(t) = f L ( t E I,f E U), is continuous. It remains to verify the relations qy* = p, y*(C) c D and Y*(”O) = .V, From the definition of X , one can easily see that X(0,f) = f for every fcs U. Hence we have qy*(f) = [Y*(f)l(O)= f ) =w(f) for every f E U. This proves that qy* = y. y*(C) c D is an immediate consequence of qy* = y and y(C)c D.To check y*(uo),= u, we first note that X ( t , u,) = u, for every 1 E I.Then we have l u X ( O 9
[y*(u,)I(t) = 6V, uo) = @(t, u,) = y ( 4 = 2,. for every t E I.This implies y*(uo)= u,. I If we removed the condition y*(uo) = uo 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 y* : U + V of the map y : U + 2.That y*(C) c D is obvious but the condition y*(uo)= u, does not necessarily hold. The lifting y* constructed in the proof of (14.2) may be called the canonical lifting of y .
15. Fiberings with discrete fibers Motivated by the results concerning the exponential map p : R +S1 in Chapter 11, 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. Lemma 15.1. Fm each path u : I + B joining b, to b1 and for each e, E pl(b0), there exists one and only one (covering) path a* : I + E such that o*(O) = e, and pa* = a. Proof. The path u may be considered as a homotopy of the partial map a I 0; hence, the existence of a* is an immediate consequence of the covering homotopy property. To prove the uniqueness, let u*, a# :I + E be any two paths in E such
15. F I B E R I N G S W I T H D I S C R E T E F I B E R S
87
that fiu* = u = #a# and u*(O) = e, = u#(O). Let s E I be arbitrarily given. It remains to show that u*(s) = a#(s). For this purpose, let us define a map g : I + E by taking u*(s  2st), (if 0 Q t Q i), g(t) = u#(2st  s ) , (if 4 < t < 1). Then the map f = fig : I + B has a homotopy fr : I + B , (0 < I Q l), defined by u(s 2st Brst), (if 0 < t < i), fr(4 = u(2rs 2st  s  2rst), (if 4 < t < 1). Since fr(0)= u(s) = fr( 1) for every r E I , it follows from (iv) of (3.1) that g has a homotopy gr : I f E , (0 < I < l ) , such that
(
[
+
+
gr(0) = u*(s), gr(1) = a#(s) for every Y E I . Since / , ( I ) = a(s), p g , = f l implies that the connected set g,(I) is contained in the fiber over u(s) which is discrete. Therefore, g,(I) must be a single point. This implies that a*(s) = u#(s). I We recall that two paths u , t : I B joining b, to b, are equivalent, u N t,if they are homotopic with the end points held fixed. The paths in B joining b, to b, are thus divided into disjoint equivalence classes which are actually the pathcomponents of the space [ B ; b,, b , ] . Pgr
= fr,
f
Lemma 12.2. The terminal point u*(l)of the covering path a* : I + E i n (15.1) depends only on e,E#l(b,) and the equivalence class of the path u:I+B. Proof. Assume that u, t : I + B are equivalent paths joining b, tb b, and that u*, t*:I  + E are the covering paths with common initial point e o E f i  l ( b o ) . It suffices to prove that u*(l) = t*(1). Since u N t,there exists a homotopy ht : I + B , (0 Q t < l), such that h, = u, h, = t and ht(0) = b,, ht(1) = b, for each t E I. According to (iv) of (3.1), there exists a homotopy ht* : I + E , (0 < t < l), such that ho* = u*, fiht* = ht, ht*(O) = e,, and ht*(1) = u*(l) for every t E I . Since h,*(O) = e, and fib,* = h, = t, it follows from the uniqueness part of (15.1) that h,* = T*. Hence we have t*(1) = h,*(l) = ~ * ( 1 )I.
Lemma 15.3. For each b, E B and each e, E fil(bo),the projection p : ( E , e,) +
(B, b,) induces a monomorfihism
p,
: n , ( E ,e,) + n , ( B , bob Proof. Assume that a E ~ , ( Ee,), and $*(a) = 1. Let g : I f E be a representative loop of a. Since the loop f = p g : I + B represents the element fi+(a)= 1, there exists a homotopy ft : I B , (0 < t < l), such that f,= f, / , ( I ) = b, and ft(0) = b, = ft(1) for each t~ I . According to (iv) of (3.1), there exists a homotopy gt : I + E , (0 < t < l), such that go = g, #gt = ft f
88
111. F I B E R S P A C E S
and g,(O) = eo = g,(l) for each t E I . Since f l ( I ) = b, and Pg, = f l , the connected set gl(l) is contained in the fiber pl(b0) which is discrete. Hence, g!(I) must be a single point. This implies that g , ( l ) = eo and a = 1. I Therefore, n,(E, e,) is isomorphic to a subgroup p,[nI(E, e,)] of n l ( B , b,) which obviously depends on the choice of e,. In general, there are no relations among these subgroups p , [ n l ( E ,e,)] for various choices of e, E pl(bo) unless they can be joined by paths in E. Hence, let us assume that E is pathwise connected. It follows, of course, that B must also be pathwise connected. Now let el be another point in pl(bo). Since E is pathwise connected, there exists a path a : I + E joining e, to el. According to (11; 4.1), u determines an isomorphism
u, :n l ( E ,el) M n l ( E , 8,). On the other hand, pa : I + B is a loop a t b, and hence represents an element w of n,(B, b,). By the definitions of p , and a,, one can easily see that +,a*(.) = w*p,(a)*wl for each a E n l ( E , el). This implies that p,[n,(E, e l ) ] is the transform ~  l . P * [ n I ( Ee0)l.w , of P*[n,(E,e0)I. Conversely, let w be an arbitrary element of n,(B, b,). Pick a representative loop t : I + B for w . By (15.1) there exists a path a : I + E such that a(0) = e, and pa = t. By (15.2), the point e, = a(1) in fil(b,) depends only on the element w . One can also easily see that el = e, iff w is in the subgroup fi,[nl(E, e,)]. Hence, we have proved the following Theorem 15.4. If E i s a pathwise connected fiber space over B relative to
p :E
B with discrete fibers, then, for each b, E B, the images p,[nc,(E,e,)] of the induced monomorphisms +
P, :n,(E, 8,) +n,(B, b,) for all e, E pl(b,) constitute a class of conjugate subgroups of n,(B, b,). Fwthermore, for a fixed e, E #1(b0), there i s a natural onetoone correspondence between the poivrts of pl(b,) and the right cosets. of fi, [nl(E,e,)] in ntl(B,b,) with e, corresponding to the subgvorrfi p,[n,(E, eo)]. This class X(E : b,) of conjugate subgroups { P,[nl(E, e,)] I e, E pl(b,) } of n , ( B , b,) will be called the characteristic class of E a t b,. Each group in X(E : b,) is isomorphic to the fundamental group n,(E) of E. X(E : b,) will consist of a single group iff p , [nl(E,e,)] is an invariant subgroup of n l ( B ,b,) for some and hence every e, E #l(b,). If X(E, b,) consists of a single group, then so does X(E, b,) for any other b, E B. In this case, we say that the fibering ( E ,p , B ) is regular. For a fixed e, E pl(bo), the natural onetoone correspondence is defined as follows: The right coset of p,[nl(E, e,)] in n,(B, b,) corresponding to el E pl(b,) is the one which contains the element w E ~ , ( Bb,), represented by the loop pa : I f B, where a : I + E is any path joining e, to el. Note. All the assertions as well as their proofs obviously hold for fiber
16.C O V E R I N G S P A C E S
89
spaces with totally pathwise disconnected fibers. Here, a space is said to be totally pathwise disconnected if it has no pathwise connected subspaces except single points. 16. Covering spaces Throughout the present section, let B be a given connected and locally pathwise connected space. A connected space E is said to be a covering space over B relative to a map p : E + B if the following conditions are satisfied: (CSl) p maps E onto B. (CS2) For each b E B, there is a connected open neighborhood V of b in B such that each component of p  l ( V ) is open in E and is mapped homeomorphically onto V by p. As immediate consequences of (CS2), the following two conditions are also satisfied: (CS3) E is locally pathwise connected. (CS4) p : E + B is an open map. Examples. According to (11; 2.2), the real line R is a covering space over S1relative to the exponential map p : R + S1.Next, let us consider the unit nsphere S n in the euclidean ( n + 1)space Rn+l. If we identify the antipodal points of Sn, we obtain the real projective nspace Pn with natural projection p : S n + Pn. It is easy to verify that S n is a covering space over Pn relative to p and the fibers are the pairs of antipodal points in S”. Covering spaces are the earliest examples of fiber spaces. In fact, we have the following theorem which is an easy consequence of (CS2) and the connectedness of B. Theorem 16.1. Every covering space E over B relative to p : E + B is a bundle space over B relative to p with discrete fibers. For generalizations and converses of (16.1), see Ex. P a t the end of the chapter. Now, let us consider the lifting problem for a map f : X + B of a connected and locally pathwise connected space X into the base space B of a covering space E relative to a projection p : E + B. Pick x, E X and denote b, = f (x,). Let e, be a point of E with p(e,) = b,. The following theorem is a generalization of (11; 5.3). Theorem 16.2. (The lifting theorem). There exists a unique map g : X + E such that g(xo) = e, and p g = f 28 the image of f* : n , ( X , x,) + n l ( B ,b,) is contained i n that of p , : n l ( E ,e,) +7c1 ( B , b,). Proof. The necessity of the condition is obvious. In fact, if a map g : ( X , x,) + ( E ,e,)
exists such that p g = f, then we have p*g, = f* and hence the image off* is contained in that of p , . I t remains to establish the sufficiency of the condition. The remainder of the proof is motivated by that of the special case (11; 5.3).
111. F I B E R S P A C E S
90
To construct the map g , let x be an arbitrary point in X. Then there exists a path n : I , X with n(0) = x, andn(1) = x. The composed map u = f n : I + B i s a p a t h i n Bwithu(0) = f(x,)andu(l) =f(x).Accordingto(15.1), there exists a unique path t : I + E such that t(0)= e, and P t = u. We assert that the point ~ ( 1 of ) E does not depend on the choice of the path n : I f X. To prove this, let n’: I + X be another path in X joining x, to x and let t’: I + E denote the unique path such that ~ ‘ ( 0= ) e, and +t’ = fn’. We are going to prove that t(1) = ~ ’ ( 1 )The . loop A = n ~ . ’ : ~ I + X represents an element a € n l ( X ,x,) and so the loop f I : I + B represents the element f*(a) of nl(B, b,). According to our condition, this element is contained in the subgroup p , g 1 ( E , e,). Hence there exists a loop p : I + E at e, such that p p = f I . Since I = n d  1 , it follows from the uniqueness of the path in (15.1) that t ( t ) = p(#) and t ‘ ( t ) = p(1  i t ) for each t E 1. In particular, t(1) = p(4) = t’(1). This proves our assertion. Because of the preceding assertion, we may define a function g : X + E by setting g(x) = t(1) for each x E X. By the construction of the point t(l), it is obvious that g(xo) = e, and p g = f . By the same method as used in the proof of (11;5.3), one can establish the continuity and the uniqueness of the function g : X + E. I If we omit the condition g(xJ = e, in (16.2), we obtain the result that there exists a map g : X + E with p g = f iff f* maps n,(X, x,) into a group of the characteristic class x (E, b”). Now, let E be a covering space over B relative to p : E f B, E’ a covering space over B’ relative to p’ : E‘ + B’, and f : B + B’ a given map. Let e, E E, b, E B, e’, E E‘, and b,,’ E B‘ be given points such that
p(eo) = b,, The maps diagram:
p , p’
p’(e’,)
=
b’,,
f(b,) = b’,.
k* I*
and f induce homomorphisms indicated in the following n,(E,4
g*
n,(E’, c‘o,
f*
n,(B,4)
nl(B’, b’o)
Theorem 16.3. (The fiber map theorem). There exists a unique map
+ E‘ such that g(eo)= e‘, and p’g that of PI*.
g :E
=
f p i# f* carries the image of p , into
Since X can be considered as the trivial covering space over itself, (16.2) is a special case of (16.3). However, (16.2) also implies (16.3) by considering the map f p : E + B’. This fiber map theorem has quite a few important consequences to which we devote the remainder of this section. First, let us assume that B = B‘ and that f is the identity map on B. Then we obtain the following
16. C O V E R I N G
SPACES
91
Theorem 16.4. (The covering theorem). Let E and E' be two covering spaces over the same space B relative to p : E + B and p' : E' + B respectively, and let b, E B. I/ e, E E and e', E E' are such that
#(eo) = bo = fl'(e'o)* p*[n,(Ele0)l = #J'*rZl(E'* e'J1, then there exists a unique map g : E + E' such that = e'o, P'g = P. Furthermore, E is a covering space over E' relative to g .
Proof. The first assertion is a special case of (16.3). Hence, it remains to prove that E is a covering space over E' relative to g . For this purpose, let us first verify (CSI). Let e E E , b = p ( e ) and e' = g(e). Choose a connected open neighborhood V of b in B such that (CS2) holds for both covering spaces E and E' over B . Let W denote the component of pl(V) containing e and W' that of p'l( V )containing e'. Then the restrictions q=pJw, q'=p')w'
are homeomorphisms onto V . Since p'g = p , we have g I W = q'lq. Hence g maps W homeomorphically onto W'. This implies that g(E)is both open and closed in the connected space E'; so g maps E onto E'. Next, let us verify (CS2). Let e' E E' and b = $'(el). Choose a connected open neighborhoodv of b in B such that (CS2) holds for both covering spaces E and E' over B . Let W' denote the component of p'l(V) which contains e'. Then W' is a connected open neighborhood of e' in E' and gl(W) is the union of a set of components of pl(V). It follows that every component of gl(W') is open in E and is mapped homeomorphically onto W' by g. I In particular, if E is simply connected, then (16.4) implies that E is a universal covering space over B relative to fi : E + B. Here, a covering space E over B relative to p : E + B is said to be ulziversal if, for every covering space E' over B relative to p' : E' + B, there exists a map g : E + E' such that p'g = p and that E is a covering space over E' relative to g. Next, let us define the notion of equivalent covering spaces as follows. Two covering spaces E and E' over a same base space B relative to projections p : E + B and p' : E' + B are said to be equivalent if there exists a homeomorphism g :E + E' of E onto E' such that p'g = p . By (16.2) and (15.4), the characteristic classes X(E, b,) and X(E', b,) are defined for every b, E B. Theorem 16.5. (The equivalence theorem). For any given point 6 , E B , two covering spaces E and E' over B relative to projections p : E + B and p' : E' + B are equivalent ifl X(E, b,) = X(E', b,). Proof. Necessity. Let g : E + E' be a homeomorphism of E onto E' such that p'g = p. Let e, E pl(b,) and e', = g(eo).Then g induces an isomorphism g, of n,(E, e,) onto n,(E', e l , ) . On the other hand, p'g = p gives p', g, = p,. This implies p,[n,(E, e,)] = p',[n,(E', e',)] and hence X(E, b,) = X(E', b,).
92
111. F I B E R S P A C E S
Suficiency. Assume that X(E, b,) and e’, E E‘ such that
=
X(E‘,b,). Then there are points e,
E
E
#(e,) = bo = p‘(e’o)> $ * b , ( E #e0)I = p’*rnl@‘>e’0)l. According to (16.3), there exists a map g : E + E‘ such that g(e,) = e’, and p’g = p . Similarly, there exists a map h : E‘ +. E such that h(e’,) = e, and ph = p’. Now, consider the composed map hg : E +. E . Since hg(e,) = e, and phg = p’g = p , it follows from the uniqueness part of (16.4) that hg is the identity map on E. Similarly, gh is the identity map on E‘. Hence g is a homeomorphism and the covering spaces E , E‘ are equivalent. I In particular, any two simply connected covering spaces over the same base space are equivalent. Next, let us consider the regular covering spaces. A covering space E over a base space B relative to p : E +. B is said to be regular if the fibering p : E +. B is regular in the sense of 5 15. Hence, E is a regular covering space over B iff for every b, E B, X(E, 6,) consists of a single invariant subgroup of n,(B, b,). Now, let E be a given regular covering space over B relative to p : E +. B and let b, be a given point in B. Since X(E, b,) is an invariant subgroup of nl(B,b,), the quotient group
W = ni(B, bo) / X(E, b,) is welldefined. One can easily verify that, as an abstract group, W does not depend on the choice of the basic point b, E B. Then, we have the following Theorem 16.6. T h e group W operates on the right of the regular covering space E. More precisely, to each element w of the group W and each point e of the space E , there corresponds a unique point ew of E such that
(ewJw2 = 4 . F 2 ) and, for each w E W , the correspondence e f ew defines a homeomorphism w* o f E onto itself. Furthermore, the operation has the following two properties : (i) p(ew) = p ( e ) for every e E E and w E W . (ii) For a given e E E , ew = e implies w = 1. In words, the condition (ii) is equivalent to saying that W operates freely on E. Proof. Pick a point e, E $1(b,). According to (15.4), there is a natural onetoone correspondence v : W + pl(b,) of W onto pl(b,) with v ( 1) = e,. For an arbitrary element w E W , let el = v ( w ) . Since E is a regular covering space over B, we have
P*[Zl(E>e0)l = $*[n,(E, e l ) ] . Hence, just as in the proof of (16.5),there is a unique homeomorphism w* of E onto itself such that w*(e,) = el and pw* = p . By the construction of v and w * , one can easily verify that (w1w2)* = w2*w1*,
(w,, W 2 E W ) .
17.
CONSTRUCTION O F C O V E R I N G SPACES
93
If the homeomorphism w* of E admits a fixed point e E E , then it follows from the uniqueness part of (16.3) that w* must be the identity map on E. This implies that el = w*(e0) = e, and hence we have w = vl(e0) = 1. Then, the theorem follows immediately if we set ew = w*(e) for every e E E and W E W . I In particular, if E is a simply connected covering space over a space B, then the fundamental group n,(B) operates freely on E. In the classical theory, the homeomorphisms w* in (16.6) are referred to as the covering transformations (Deckbewegungen) of the regular covering space E. 17. Construction of covering spaces A space B is said to be locally simply connected if, for every point b E B and every open neighborhood V of b in B, there exists an open neighborhood U c V of b in B such that, for any two points u, and ul in U ,every pair of paths in U joining uo to u1 are homotopic in V with end points held fixed. Obviously, every locally contractible space is locally simply connected and hence so is every simplicia1complex. For our purpose in the present section, a slightly weaker condition is enough: we can replace the open neighborhood V by the space B itself. In this case, the space B is said to be semilocally simply connected. Observe that every locally simply connected space is semilocally simply connected and so is every simply connected space. Throughout the present section, we assume that B is a given space which is connected, locally pathwise connected, and semilocally simply connected. Let b, be a given point in B. Theorem 17.1. (The existence theorem). For every subgroup G of the fundamental group nl(B,b,), there exist a covering space E over B relative to a projection p : E + B and a point e, E pl(b,) such that G i s exactly the image of the induced homomo7Phism p., : n l ( E ,e,) +nl(B,b,). Proof. Let us consider the space of paths 52 = [ B ;b,, B ] . As in $ 10, we denote by p , : 52 + B the terminal projection defined by p , (a) = u(1) for each u E 52. We shall define a new equivalence relation in 52 as follows. Two paths u, t E 52 are said to be equivalent modulo G, (in symbols: CJ t mod G), if $,(a) = p,(t) and the element of n,(B, b,) represented by the loop o * T  ~is in G. Let E denote the quotient space of 9 defined by this equivalence relation. Then the points of E are equivalence classes (mod G) of the paths 9.We shall denote by [u] the class which contains the path u E 52. Define a map p:E+B by taking P[u] = $,(a) for every [a]E E. The continuity of p follows from (I; 12.1). Since B is pathwise connected, it follows that p maps E onto B. We shall construct a convenient basis for the open sets of E as follows. For a given e E E and a given open neighborhood U of p(e) in B, choose a path N
94
111. F I B E R S P A C E S
a E e and denote by N(e, U)the subset of E which consists of the classes each containing a path of the form t : I + B such that t ( t ) = a(2t) for every t < 4 and t ( t ) E U for every t 2 4. It is obvious that N(e, U)does not depend on the choice of the representative path a from the class e G E. Since B is locally pathwise connected and semilocally simply connected, it is straightforward to prove that N(e, U)is open in E. Then it follows easily that the collection { N ( e , U ) }for all e E E and all open neighborhoods U of p ( e ) in B constitutes a basis for the open sets of E. An immediate consequence of this result is that p : E + B is open. To prove that E is a covering space over B relative to p , let us establish (CSl) and (CS2). Since p maps E onto B, (CS1) is satisfied. To prove (CS2), let b be an arbitrarily given point of B. According to our assumption on B, there exists a pathwise connected open neighborhood U of b in B such that, for any two points uoand u1in U,every pair of paths in U joining uoto u1 are homotopic with end points held fixed. Then it suffices to prove that pl( U )is the disjoint union of the open sets { N(e, U)I e E $'(a) } in E each of which is mapped homeomorphically onto U by p. For this purpose, we shall first prove that p maps N(e, U)homeomorphically onto U for each e in Pl(b). By the definition of N(e, U), it is obvious that p maps N(e, U )into U.Since U is pathwise connected, there is a path 8 : I + U joining b to u. Choose a path aEl2 with [a] = e and consider the path z = a.8. Clearly [t]E N(e, V) and p [ z ] = u. This proves that p maps N(e, U)onto U.Let us assume that el and eBare any two points in N(e, U) such that #(el) = p(ea). There are two paths t(EB, (i = 1,2), such that [tt] = et, t t ( t ) = a(2t)whenever 1 Q 4, and t t ( t ) E U whenever t 2 4. Since +(e,) = P(ea),we have tl(1) = t g1). ( Call this common terminal point v E U. Denote by 6s :I + U the path defined for each i = 1,2 by ( t E I).
Then t , and [g are two paths in U joining b to v. According to the choice of U, El and t Bare homotopic in B with end points held fixed. This implies that tl and t gare homotopic with end points held fixed and hence e, = eB. This proves that p maps N(e, U)in a onetoone fashion. Since p is both open and continuous, we conclude that p maps N(e, U)homeomorphically onto U. Next, we are going to prove that the open sets N(e, U),e E pl(b), are disjoint. Assume that N(e,, U)and N(e,, U)have a common point x. Choose paths ut E B,(i = 1 , 2 ) , with [at] = et. Then there are paths ti E 8,such that [tt]= x , t t ( t ) = a424 if t Q IJ, and q ( t )E U if 1 2 4. Let 51 : I + U , ( i = 1,2), denote the path defined by (i).Then El and Ea are two paths in U joining b to p ( x ) . By the choice of U,t1and tB are homotopic in B with end points held fixed. This implies that
t,t,lal.t,t~lo;;lul.a$l.
17.C O N S T R U C T I O N
O F COVERING SPACES
95
Since [t,]= x = [ta],t l  t s l represents an element of G and so does a,.a,l. Hence el = e,. This proves that the open sets N(e, U),e E $'(a), are disjoint. Now, we shall prove that pl(U) is the union of the collection { N(e, U)I e E #  l ( b ) }. Since # maps N(e, U ) into U,N(e, U ) is contained in pl(U). Let x E pl(U) be arbitrarily given. We have to prove that there is some e E pl(b) such that N(e, U) contains x. For this purpose, chQose a path z € 8with [t]= x and let 2c = p ( x ) = t(1). Since U is pathwise connected, there exists a path 8 : I + U joining b to u. Let a = t . 8  1 and t = a.8. Set e = [a]. Since a(1) = O(0) = b, we have e E p  l ( b ) . By an easy homotopy, one can prove that t and t are homotopic with end points held fixed. It follows that x = [t]= [t]E N(e, U).Hence, pl(U) is the union of the collection { N ( e , U )I e E +,(/I) }. This completes the proof of (CS2). As a continuous image of a pathwise connected space 8, E is pathwise connected. This completes the proof that E is a covering space over B relative to p . Let 6 E 8 denote the degenerate path at b,, that is to say, 6(I)= b,. Denote e,, = [6]E E. Then p(e,) = b, and p induces a monomorphism 9, :n,(E, e,) +nl(B, bob It remains to prove that the image of p , is esactly the given subgroup G of n,(B, b,). For this purpose, let us first prove an assertion as follows. Let a : I + B be any path with a(0) = b,. According to (15.1), a has a unique covering path a* : I + E with a*(O) = e,. We assert that, for each t E I , a*(t) is the class which contains the path at : I + B defined by at(s) = cr(st) for each s E I . This follows immediately from the fact that the assignment t ,at defines a path t:I ,52 according to (9.10)and that E is a quotient space of 52. Now, let us prove that the image of p , is exactly G. Let a be any element in n,(E, e,). Choose a representative loop u* : I + E for a, then the loop u = pa* represents the element #,(a) of n,(B, b,). According to the foregoing [a] = a*(1) = e, = [d]. assertion, we have This implies that #,(a) E G. Conversely, let p be any element of G. Choose a representative loop a : I + B for p. By (15.11, a has a unique covering path a* : I + E with a*(O) = e,,. According to the foregoing assertion, we have a*(1) = [a]. Since a represents p E G, this implies that a*(1) = e,. Hence a* represents an element a of n,(E, e,). This implies that p = p,(a). Thus, we have proved that G is exactly the image of 9., I In particular, if G is the trivial subgroup of n,(B, b,) consisting of the neutral element 1 of nl(B,b,), then E is a simply connected covering space over B. According to (16.4) this implies that E is a universal covering space over B. By (15.3)it follows that every universal covering space of B is simply connected. Finally, by (16.5) we conclude that every pair of universal covering spaces over B are equivalent. Hence, E is essentially the only
96
111. F I B E R S P A C E S
universal covering space over B. Hereafter, E will be referred to as the universal covering space of B. Combining (16.5) and (17.1), we obtain the following classification of covering spaces. Theorem 17.2. (The classification theorem). For any connected, locally pathwise connected, and semilocally simply connected space B, the equivalence classes of the covering spaces over B are in a onetoone correspondence with the conjugate classes of subgroups of the fundamental group n,(B).
The following simple examples are given t o illustrate the preceding results. (i) Covering spaces of S1. The real line R is the universal covering space over S1 relative to the exponential map p : R f 9. Since nl(S1)is abelian, it follows that every covering space over S1is regular. Since nl(S1)is free cyclic, the nontrivial subgroups of nl(S1) are the free cyclic subgroups Gn,(n = 1 , 2 , * * where Gn is of index n in n,(S1).The covering space which corresponds to Gn is S1itself relative to the projection p , : S1f S1defined by p,(z) = 2% for each z E S1.These are essentially all the covering spaces over S1. This example reveals the fact that homeomorphic covering spaces over the same base space are not necessarily equivalent. (ii)Covering spaces of S n and P" with n > 1. Since S n is simply connected, every covering space over Sn is equivalent to the trivial covering space by which we mean the covering space S n over itself relative to the identity map. On the other hand, S n is the universal covering space over Pn relative to the natural projection p : Sn f Pn. Since the fibers of this covering space are the pairs of antipodal points in 9, it follows from (15.4) thatn,(Pn) is the cyclic group of order 2. Hence, by (17.2), the universal covering space is essentially the only nontrivial covering space over Pn. (iii) Covering spaces of closed surfaces. Let M denote a closed surface other than S2and Pz.Then, by (11; Ex. A), n l ( M ) is an infinite group. It follows from (15.4) that the universal covering space E of M is not compact. Hence, E is a simply connected infinite 2manifold. In fact, it is a classical theorem that E is homeomorphic to the euclidean 2space, [V; p. 1531. As an application of the preceding results, we have the following e ) ,
Theorem 17.3. Assume that X i s a connected triangulable space and E i s the universal covering space over B relative to p : E B. Then, for any x, E X, b, E B and e, E $l(b,), the assignment g + f = p g sets up a onetoone correspondence between the homotojby classes of the maps g : ( X ,x,) + ( E , e,) and those of the maps f : ( X ,x,) .+ ( B ,b,) with f* sendingn,(X, x,) into the neutral element of nl(B,b,,). f
Proof. Let us use the usual notation f* = 0 to denote the fact that f* sendsnl(X, x,) into the neutral element of n l ( B ,b,). If f = p g for some g : ( X , x,) + ( E , e,), obviously we have f * = p*g, = 0. Hence, the assignment g + f = p g defines a function of the liomotopy
97
EXERCISES
classes of the maps g : (X, x,) +. ( E , e,) into those of the map f : ( X , x,) +. ( B , bo) with f* = 0. Let f : ( X , x,) +. (B,b,) be a map with f* = 0. According to (16.2) there exists a unique map g : ( X , x,) + ( E , e,) such that fig = f . Hence is onto. Let g, g’: ( X , x,) +. ( E ,e,) be any two maps such that p g N Pg’ re1 x,. Then there exists a homotopy f t : (X, x,) 3 ( B , b,), 0 < t Q 1, such that fo = p g and f , = pg’. According to (iv) of (3.1), there exists a homotopy gt : (X, x,) f ( E , e,), 0 < t < 1, such that go = g and pgt = f t for every ~ E I Since . gl(xo) = e, = g’(xo) and p g , = f l = pg‘, it follows from the uniqueness part of (16.2) that g, = g’. Hence g I? g’ rel x,. This proves that is onetoone. I In particular, if X is simply connected, then the assignment g +f = p g sets up a onetoone correspondence between the homotopy classes of the maps g : (X, x,) +. ( E , e,) and those of the maps f : (X, x,) .+ ( B , b,). For example, let B = P2,E = S2, and let p : Sz + P2 denote the natural projection. If X is either S2 or Ss, then i t follows that the homotopy classes of the maps f : ( X , x,) .+ ( B ,b,) are in a onetoone correspondence with the set 2 of all integers. Note. In (17.3), the condition that X is triangulable can be replaced by the weaker condition that X is locally connected. In fact, in the proof of (17.3), one may use Ex. P instead of (3.1).
+
+
EXERCISES A. Sliced fiber spaces Let p : E +. B be a given map. By a slicing structure for p , we mean a collection S = { o,+u } of the following entities: (1) a system o = { U } of open sets of B which covers B, called the slicing
neighborhoods. (2) a system of maps { +U I U E w } indexed by the slicing neighborhoods, called the slicing functions, where each +u, is defined on the subspace U x p  l ( U ) of the product space B x E with images in E in such a way that the following two conditions are satisfied: (SFl) P+u(b, x ) = b, ( b E U,x ~ p  l ( U ); ) (SF2) +u(P(x), 4 = x , ( x E p  w ) . If a slicing structure S = { o,+U } for p exists, we say that @ : E B has the slicing structure property (abbreviated SSP). We shall use the abbreviation Para CHP to stand for the covering homotopy property for all paracompact Hausdorff spaces. Prove the following implications, [Hu 8 ; Huebsch 11 : B P SSP s.Para CHP Hence, p : E + B is a fibering if it has the SSP. In this case, E is called a sliced fiber space over B relative to p , and in particular, every bundle space is a sliced fiber space. f
98
111. F I B E R S P A C E S
Let E be a sliced fiber space over B relative to p : E + B. Prove the following assertions: 1. The projection p : E + B is open, that is to say, it maps open sets onto open sets. 2. If two points a and b of B can be connected by a path in B, then the fibers pl(a) and p  l ( b ) are homotopically equivalent. 3. E is a bundle space over B relative to p iff the following two conditions are satisfied, [Griffin 11: (GCI) There exists a space D which is homeomorphic with every fiber +l(b), b E B. (GC2) For each point b E B, there exist a slicing neighborhood U which contains b and a slicing function +u : U x pl(U) + E such that, for every pair of points u and v in U,we always have
4 1 = x, ( x E pl(U)). 4. If E and B are metrizable then p : E + B has the ACHP and hence the + U b , +U(%
PLP, [Curtis 11. Now assume that E and B are spaces such that B x E is a paracompact Hausdorff space and B is an ANR. Let p : E + B be a given map. Prove that E is a sliced fiber space over B relative to p iff, for any map g : X + E of a paracompact Hausdorff space X into E and any homotopy f t : X + B, (0 < t Q l), of the map f = p g , there exists a homotopy g t : X +E, (0 Q t Q I), of g which covers f t and is stationary with f t , [Fox 21. For the definition of the stationary property, see [S ; p. 501. A sliced fiber space E over d relative to p is said to have a unified slicing fumtion if there exists a slicing structure S = { o,C$U 1 for p : E + B such that, for any two slicing neighborhoods U and V in o,we always have +U = +v on the intersection of U x pl(U) and V x pl(V). For such a slicing structure S, we may define a unified slicing function C$ on the union W of the subspaces U x pl(U) for all U E o by taking = +U on each U x Pl(U). The fiber spaces defined by Hurewicz and Steenrod [l] and Fox [2] are those with unified slicing functions. Prove that every metrizable sliced fiber space over an ANR has a unified slicing function, [H ; p. 1701.
+
B. Local path lifting property
A map f : E + B is said to have the local path lifting property (abbreviated LPLP) if, for each b E B , there exists an open neighborhood U of b in B such that the map q : w + u, = pyu), q = p W
w
has the PLP. Prove 1. SSP * LPLP => Para CHP. 2. If B is a paracompact Hausdorff space, then LPLP implies PLP. [Hurewicz 21.
99
EXERCISES
C. Relations between various notions of fiber space
Let us consider a given map p : E f B having one of the following properties : PCHP = CHP for polyhedra, ( 3 2), Para CHP = CHP for paracompact Hausdorff spaces, (Ex. A), ACHP = CHP for arbitrary spaces, ( 3 2), PLP = path lifting property, ( 3 12), LPLP = local path lifting property, (Ex. B), SSP = slicing structure property, (Ex. A), B P = bundle property, ( 3 4), where the abbreviation CHP stands for the covering homotopy property, ( 3 2). The first four properties are of global nature while the last three are apparently local properties. A number of implications among these properties are known and can be summarized by the following diagram B P { para }
U
BP
======

= SSP { para } = LPLP { para } = PLP
.v
SSP
v
LPLP

.ACHP
U
> ParaCHP
U
PCHP where the attached symbol { para } means that the base space B is assumed to be a paracompact Hausdorff space. Check if all of these implications have been established in the text and in the preceding two exercises.
D. Homogeneous spaces Let E be a topological group and let F be a closed subgroup of E . Define an equivalence relation in E as follows: two elements a , b of E are said to equivalent iff there is an element f E F such that af = b. Thus the elements of E are divided into disjoint equivalence classes called the left cosets of F in E . The left coset containing a E E is obviously the closed set a F of E . According to ( I ; fj 12), we obtain a quotient space B = E/F whose elements are left cosets of F in E and a natural projection p : E +. B which maps a E E onto the left coset a F E B. B = E/F is called the quotient space of E by F ; it will be called simply a homogeneous space. Prove the following assertions : 1. B = E/F is a Hausdorff space. 2. The natural projection p is an open map. 3. E operates transitively on B under the homeomorphisms e :B + B , eE E , defined by e(b) = p [ e . p  l ( b ) ] , ( b E B ) . 4. E is a bundle space over B relative to p iff there is a local crosssection of B in E ; by this we mean a crosssection x : V +. E defined on an open neighborhood V of the point b, = F in B. 5. If E is a Lie group, then a local crosssection of B in E exists and hence E is a bundle space over B relative to p , [Che; p. 1101.
111. F I B E R S P A C E S
100
E. Spheres as homogeneous spaces Let Q denote one of the three fields of real numbers, complex numbers, or quaternions. Consider the right vector space Qn whose elements are ordered sets x = (xl, * * * , x n ) of n elements of Q. The inner product x y of x and y in Qn is defined by x y = XIy, * * * + znyn,
+
where Xl denotes the conjugate of xi. The topological group G n of all linear transformations in Qn which preserve the inner product is called the orthogonal, unitary or symplectic grozlp according as the scalars are real, complex or quaternionic. It is a compact Lie group. Let S denote the unit sphere in Q n ; then S is the sphere of dimension n  1,2n  1 or 4n  1 according as the scalars are real, complex or quaternionic. Prove the following assertions : 1. G n operates transitively on S. 2. G n is a bundle space over S relative to the projection p : G n + S defined by P ( f ) = f ( x o ) for every f E G n , where xo = (1,O; * , 0) E S. 3. Let Gnd1 denote the subgroup of G, leaving xo fixed. Then the fibers p  l ( x ) are the left cosets of Gn1 in G n and hence S can be considered as the homogeneous space Gn/Gn1. F. Fiberings of spheres over projective spaces
Consider the nonzero elements of the vector space Q" in the preceding exercise.Define an equivalence relation i n X = Qn\O as follows: for any two elements x and y in X, x y iff there is a q in Q such that xq = y . According to (I, 3 12), we obtain a quotient space M called the projective space associated with Qn and a natural projection n : X + M . Let S denote the unit sphere in Qn and p = 7c I S. Prove the following assertions: 1. S is a bundle space over M relative to p. 2. The fibers p  l ( b ) , b E M , are great spheres of dimension 0, 1 or 3 according as the scalars Q are real, complex or quaternionic. 3. The group G n operates transitively on M in some natural way; and hence M can be identified with the quotient space of G n by its subgroup of the elements leaving a given point of M fixed.

G. Stiefel manifolds
A kframe, v k , in R n is an ordered set of k linearly independent vectors. Let denote the set of all kframes in R n . Let L n denote the full linear group, and let L , k be the subgroup of L n leaving fixed each vector of a given frame vok. Then we may identify vn, = L~I L , V'n,k
and hence V ' n , k becomes a homogeneous space called the Stzefel manifold of kframes in nspace. Let v n , k denote the subspace of V ' n , k consisting of the orthogonal kframes. Prove the following assertions : 1. The orthogonal group O n Operates transitively on V n , k . If O n  k is represented as the subgroup of O n leaving fixed a given orthogonal kframe v,k, then one may identify T / ~ =, 0, ~ I
EXERCISES
I01
2. If k < n, the rotation group R n operates transitively on V n , k and hence one may identify k < n. v n , k = R n I Rnk, 3. V n , k may be interpreted as the space of all orthogonal (k  1)frames tangent to 9  l . In particular, V n , = S n  l and V n , is the space of all unit tangent vectors on S n  1 . 4.V n , k can be identified with the space of all orthogonal mapsSkl intoSnl. 5 . Let v0n be a fixed orthogonal nframe in R n and let vok denote the first k vectors of Let O n  k be the subgroup of O n leaving vok fixed. Then we obtain a chain of Stiegel manifolds and projections
van.
On = Vn,n+Vm,nI + * . * + V n , 2 + V n , i
=Snl.
Each projection or any composition of them is a bundle map, that is to say, the projection of a bundle space over its base space. H. Grassmann manifolds
Let M n , k denote the set of all kdimensional linear subspaces (kplanes through the origin) of R n . The orthogonal group O n operates transitively on M n , k . If R k is a fixed kplane through the origin and R n  k its orthogonal complement, the subgroup of On mapping R k onto itself splits up into the direct product o k x 0 %  k of two orthogonal subgroups the first of which leaves R n  k pointwise fixed and the second leaves R k pointwise fixed. Hence we may identify Mn,k = o n ( o k x onk). Thus M n , k becomes a homogeneous space called the Grassmann manifold of kplanes in nspace. Prove the following assertions : 1. The natural projection O n + M n , k maps the rotation group R n onto M n , k . Let R k and R n  k denote the rotation subgroups of o k , O n  k . Define

Mn,k = Rn
I (Rk
X Rnk).
Then @ n , k is called the manifold of oraented kplanes in nspace. M n , k is a covering space over M n , k relative to the natural projection with 0spheres as fibers. 2 . V n , k is a bundle space over M n , k relative to the natural projection induced by the inclusion o n  k c o k x Onk.The fibers are homeomorphic to o k . 3 . If k < n, v n , k is a bundle space over M n , k with fibers homeomorphic to &. 4. The correspondence between any kplane and its orthogonal (n  k) plane sets up a homeomorphism M n , k t+ M n , n  k . 5 . Mn, is essentially the (n  1) dimensional real projective space P  l and A?,,, , the ( n  1)sphere9  l . 1. Elementary properties of mapping spaces
Let 52 denote the mapping space Yx with the compactopen topology. Prove the following assertions : 1. If Y is a To,T I  ,T 2  , or regular space, then so is 52. Conversely, if 52 is a To,Tl, Tz,or regular space, then so is Y .
I02
111. F I B E R S P A C E S
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 8.On the other hand, if 8 is separable, then so is Y. 3. Assume that X is a compact metrizable space and Y is a metrizable space. Then 8 is an ANR iff Y is such; similarly, 8 is an AR iff Y is such. J. Admissible topologies
A great variety of topologies may be introduced into the set 8 = Y x making it a space. We shall denote by Qr the space obtained by topologizing 8 with a topology t.A topology T of 8 is said to be admissible if the evaluation w : Q rx X + Y is a map. Thus, by (9.1), the compactopen topology of 8 is admissible provided that X is a locally compact regular space. Prove the following assertions, [Arens 11 : 1. Any admissible topology of 8 contains every open set in the compactopen topology of 8. 2 . If X is a completely regular space and Y is a TIspace containing a nondegenerate path, then a necessary and sufficient condition for the compactopen topology to be admissible is the local compactness of X . K. The topology of uniform convergence
Let Y be a metrizable space and let d be a given bounded metric consistent with the topology of Y.There is a natural metric d* defined on 8 = Y x by
r
a*(f , g) = SUPZ,X d f (4 g ( 4 1 for each pair of map f , g E 8.The metric d* determines a topology of 8 called the d*topology, or the topology of uniform convergence with respect to d . Prove the following assertions : 1. The d*topology of 8 is admissible. 2 . If X is compact, then the compactopen topology of 8 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 compactopen topology of 8 to coincide with the d*topology induced by a given bounded metric d on Y is the compactness of X [Jackson 11. 4. The d*topology of 8 depends not only on the topologies of X and Y but also on the metric d chosen for Y.Construct a few examples. 9
L. Maps on topological products Consider three given spaces T. X , Y and the mapping spaces: 8 = yx,@J = yxx T,y = QT. In 5 9, we have defined the association 8 : @ + Y. Prove the following assertions:
EXERCISES
I03
1. If both X and T satisfy the first countability axiom, then 8 sends @ onto Y, [Fox 31. 2. If X and T are Hausdorff spaces satisfying the first countability axiom, then 8 is a homeomorphism of @ Onto Y and hence the exponential law holds in this case. yXxT =
(yX)T
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 = Y x and the subspace B of the mapping space YAconsisting 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 ( f ) = f I A for every f E E. By Exercises C and I, show that this fibering has a unified slicing function. [H; p. 1731. 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 , and B , be two subspaces of X . A deformation of B , into B , in X is a homotopy ht : B, + X , (0 Q t < l), such that h, is the inclusion map and h,(B,) c B,. Such a deformation ht induces a map h # : [ X ; A , B , ] + [ X ; A , B,] described as follows: for each f~ [ X ;A , B , ] , g = h#(f) is
B , is said to be a deformation homeomorph of B , i n X i f there exists a homotopy ht : B , + X , (0 < t < l ), such that h, is the inclusion map and h, is a homeomorphism of B , onto B,. Prove that the spaces [ X ;A , B , ] and [ X ;A , B,] are homotopically equivalent if B, is either a strong deformation retract of B , or a deformation homeomorph of B,. Consequently, if X is pathwise connected, then the homotopy type of the space [ X ; a, b] is independent of the choice of the points a and b. In particular, the spaces [ X ; a, b ] , A, and Ab are homotopically equivalent. 0. The space of curves
In our definition of the space of paths, the domain I is the same for all paths. In his studies on Pontrjagin products, J. C. Moore, has found that it is more convenient to allow different domains for different paths. To avoid possible ambiguity, these will be called curves. Precisely, let Y be a given space. Then, a curve in Y consists of a real number a 2 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
I04
111. F I B E R S P A C E S
tively. A curve f : [0, a] + Y is said to be closed if f (0) = f (a); in this case, f (0) is called the basic point of the closed curve f . Now, let r d e n o t e the set of all curves in Y .To topologize I', let us consider the subspace J of the real line R consisting of the real numbers a > 0 and the space Q = YI of all paths in Y. Define a function : + J x 9 by taking C#J( f ) = (a, uf) for every curve f : [0, a ] + Y, where of : I + Y is the path given by q ( t ) = f (at) for each t E I . This function carries onto a subspace of J x Q in a onetoone fashion. We topologize in such a way that becomes a homeomorphism. Prove that ( J x Q) \ is the subspace 0 x [Q \ i( Y ) ] where i : Y +Q denotes the natural injection j : Y + Q o f $9. Since I = [0, 11, every path in Y is also a curve in Y. The topology of F defined above permits us to consider Q as a subspace of Construct a homotopy ht : + (0 < t < l), such that the following conditions are satisfied : (i) h, is the identity map on (ii) h, is a retraction of r o n t o 52. (iii) ht( f ) = f for every f E 52 and t E I . ( i v ) For every f E I', the initial point and the terminal point of the curve ht( f ) are the same as those of f for each t E I . If f : [0, a ] + Y and g : [0, b] + Y are two curves in Y such that f (a) = g ( O ) , then we may define a product f  g : [0, a + b] + Y by taking
+r + r
+(r)
r +(r)
+
r.
r r,
r.
[ f e d(4 = { f
( 4 1
g(t  4,
(if 0 < t < a), (ifa Q t < a + b).
This multiplication ( f ,g) + f * g of curves is continuous and associative whenever it is defined. Now, let y be a given point in Y and let Ou denote the subspace of consisting of the closed curves with y as the basic point. Then the homotopy ht shows that the space of loops A , is a deformation retract of OY.On the other hand, the multiplication ( f , g) + f  g is defined for every pair f , g E Ou with the trivial curve e : [0, 01 + y as a twosided unit. Hence, if Y is a Hausdorff space, then 0,is a mob with e as a twosided unit in the sense of Wallace [ 13.
r
P. Generalized covering spaces
A space E is called a generalized coveriizg space over a space B relative to a map p : E + B if the following conditions are satisfied: (GCSl) p maps E onto B. (GCS2) For each b E B, there exists an open neighborhood U of b in B such that p  l ( U ) 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 f t : Y + B, (0 < t < l), a homotopy with foq = p g , then there exists a unique hdmotopy gt : X + E , (0 < t < l ) , of g such that ftq = pgt for every t E I and that gt is stationary with f t . In particular, fi : E + B has the ACHP. This assertion is also true for sliced fiber spaces with totally pathwise disconnected fibers, [Griffin 11. Q. Local homeomorphisms
A map p : E + B of a space E onto a space B is called a local 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  l ( 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 S1 x S'. According to (11; Ex. A), the fundamental group n l ( 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, ( m , n = O , 1 , 2 , * * * ) , where G m , , is the subgroup generated by am and bn. Hence, Go,o = 0 and G1.1 =nl(B)* Prove the following assertions : 1. Every covering space over B is regular. 2. Corresponding to G o , o ,we have the universal covering space E = Rz = R x R over B = S1 x S1 relative to the projection Po,, : E + B defined by p0,,(x, y) = ( p x , p y ) , where p : R +S1 denotes the exponential map of (11; Q 2). 3. Corresponding to Go,,, n > 0 , we have the covering space E = H x S' over B = S1 x S1 relative to the projection f i o , n : E + B defined by Po, n ( x , z ) = (fix, zn) for each x E R and z E S1. Similarly, we may get the covering space corresponding to G,,o, m > 0. 4. Corresponding to G m , n , m > 0, n > 0, we have the covering space E = S1 x S1 over B = S1 x S' relative to the projection p m , n : E B defined by P m , n ( u , v) = (urn,v n ) for each (u, v) E E. f
106
111. F I B E R S P A C E S
S. Maps of the torus into the projective plane
Consider the torus T and the projective plane P. Pick arbitrary basic points toE T and Po E P,and study the maps f : ( T ,to) + (P, Po). The fundamental group n,(T, to) is a free abelian group with two free generators a and b, while n,(P, Po) is a group of order 2 generated by c. Hence, there are four possible homomorphisms h , n
:n,(T,to) +ni(P, Po),
with m, n running over 0 and 1, defined by h . n ( a ) = m, h . n ( b ) = nc. By considering T as the unit square with the opposite sides identified, construct for each (m, n ) a map f m , n : ( T ,to)
+
(PIPo)
such that (fin,,,)+ = h,,,,,,. Hence, the homotopy classes of the maps f : ( T ,to) f (P,Po) are divided into four disjoint collections C,, such that / E C m , n i f f f *= h m , n . Next, prove that, for each (m, n ) , the homotopy classes in the collection Cm,n are in a onetoone correspondence with the homotopy classes of the maps (S2,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 f : X + Y as follows. Pick arbitrary basic points xo E X and y o E Y . Two homomorphisms
h, k :n,(X,xo) jn,(Y,Y O ) are said to be equivalent (notation : h N k ) if there exists an element bE~,(Y y ) ,such that k(a) = b  l . l t ( ~ ) . b holds for every a E n , ( X , x,,). Thus the homomorphisms h :n,(X, xo) + n,(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 f : X + Y are in a onetoone correspondence with the equivalence classes C by establishing the following assertions : 1. Every map f : X + Y is homotopic to a map g : X + Y such that d x o ) = Yo. 2. For any two maps f , g : X + Y satisfying f(xo) = yo = g(xo), f N g implies f, N g,. 3. For any ~ E Cthere , exists a map f:X+Y such that f (xo)=yo and f, =h. 4. If f : X + Y is a map with f (xo) = y o and h E C is equivalent t o f a , then there exists a map g : X + Y such’ that g(xo) = yo, g N f and g, = h. 5. I f f , g : X + Y are two maps such that f (xo) = yo = g(xo) and f , = g,, then f N g.
CHAPTER I V HOMOTOPY GROUPS
1. Introduction The basic problem which led to the discovery of “homotopy groups” was to classify homotopically the maps of an nsphere S n 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 (11; 5 4). The same trick was found to work in higher dimensions; in fact, if we pinch the equator of S n to a point, we obtain two nspheres with one point in common. If n > 1, there is a rotation of S n 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 Hn(X, A ) led quickly to the relative homotopy groups nn(X, A , xo), giving a system highly analogous to homology theory. But it differs in several ways: no(X,x,) and n,(X, A , x,) are not ordinarily groups; n,(X, x,) andn,(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 1930thatns(S2)is infinite. This showed that the groups express some very deep topological properties of spaces. A second important fact is that the definition of n n 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 no(X,xo) denote the set of all pathcomponents of X. The pathcomponent of X which contains x, will be called the nezctral element of n,(X, xo) and will be denoted by 0. As in (I1; 5 4), we shall denote bynl(X, xo) the fundamental group of X at x,. For each integer n > 1, the definition of the nth (absolute) homotopy
108
IV. H O M O T O P Y G R O U P S
group n n ( X ,xo) is strictly analogous to that of the fundamental group. We replace the unit interval I by the ncube In, i.e. the topological product of n copies of I . Every point t E I n is represented by n real numbers t = ( t l ,* * , tn), ti E I , (i = 1 , 2 , * * , n), called the coordinates of t. The number tr is called the ith coordinate of t. An (n  1)faceof I n is obtained by setting some coordinate ti to be 0 or 1. The union of the (n 1)faces forms the boundary dIn of I n ; it is topologically equivalent to the unit ( n  1)sphere 9  l . Consider the set Fn = F n ( X ,xo) of all maps
.
f : (I",arn) +
(x, xo).
These maps are divided into homotopy classes (relative to ill.). We shall denote by n n ( X ,xo) the totality of these homotopy classes. We shall also denote by [ f ] the class which contains the map f and by 0 the class which contains the unique constant map do(In)= xo. Topologize Fn by means of the compactopen topology as in (I11 ; 3 9) ; then n n ( X , xo) becomes the set of all pathcomponents of the space Fn. We may define an addition (usually noncommutative) in Fn as follows. For any two maps f , g in Fn, their sum f + g is the map defined by if 0 < t , Q 4, * *tTb), if 4 < t l < 1, for every point t = (t,, *  ,tn) in In. Obviously, f g is in Fn. The homotopy class [f + g] clearly depends only on the classes [ f ] and [g].Hence we may define an addition innn(X, xo) by taking
(f
( 2 t i , t 2 , . tn), + g) (4 = ( fg(2h  1, t2; * **

[fl
+
+ [gl = [ f + gI.
Just as in (11; tj 4) for the fundamental group, one can easily verify that this addition makes n n ( X , xo) a group which will be called the nth homotopy group of X at xo. The class 0 is the grouptheoretic neutral element of n n ( X , xo), and the inverse element of [ f 3 is the class [fe],where 8 : I n + I n denotes the map defined by O ( t ) = (1  t,, t2; * *, tn) for every t = ( t l , t 2 ; .,tn) in In. If the boundary aIn of I n is identified to a point, we get a quotient space which is topologically equivalent to an nsphere Sn with a given basic point so E Sn. It follows that one might equally well define an element of n,(X, xo) as a homotopy class (relative to so) of the maps f : (9, so) + ( X , xo). Since the 4 and t, 2 4 respectively, two halves of I n , defined by the conditions t, correspond to two hemispheres of S n , one can clearly see how to define t h e group operation o f n n ( X ,xo) from this point of view. For details, see [Hu 41. 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 n n ( X , xo)which however will be proved in a different way.

1, nn(X, xo) i s an abelian group.
Proof. As in (111; 11), one can prove that F"' is an Hspace with the constant map d,,EFnl as a twosided homotopy unit. Hence, by (111; 9.9), we have we have XP = ~ ~ nx I = l (xI~~)I.
Then one can easily see that nn(X, xo) and n1(Fnl, do) are essentially the same group. Hencenn(X, xo) is abelian. I In the precedingproof, we have incidentally obtained an interesting result : nn(X, xO)
= nl(Fnl,
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 (111; 9.9), we have XI^ = XIPXIQ = (XI~)IQ. By means of this relation, it is easy to deduce the following result:
+
nn(X, ~ 0 = ) ng(FP, do), p 4 = n, where do denotes the constant map do(lg) = xo. In particular, when p = 1, F p becomes the space W of loops in X with basic point xo and do the degenerate loop at xo. Thus we have proved the following proposition which will be used in the sequel. Proposition 2.2. nn(X, x0) = nnl(W, do).
Finally, the following proposition is an immediate consequence of the fact that I n is pathwise connected. Proposition 2.3. If X, denotes the pathcomponent of X containing xo, then
nn(X0, xo) =nn(X, xo), 12 > 0. Examples. If X is a contractible space, thennn(X,xo) Next, by (11; 7.1) we have
= 0 for every
no(S', 1) = 0, nl(S', 1) M 2, nn(S1, 1) = 0, if n > 1. On the other hand, by means of (I1; 3 8), one can prove that n,(Sn, so) = 0, (if m nn(Sn, SO) M 2. Finally, by (111; 6.4), we deduce the result n3(S2,so) M 2.
< n),
n
> 0.
I10
IV. HOMOTOPY G R O U P S
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 n ( X , A , x,). By a triplet ( X .A, x,), we mealn a space X,a nonvacuous subspace A of X I and a point x, in A. If x, is the only point of A , then the triplet ( X IA , x,) will be simply denoted by ( X , x , ) and may be considered as a pair consisting of a space X and a point x, in X . Let n > 0 and define the nth relative homotopy set n n ( X , A, x,) as fbllows. Consider again the ncube I n . The initial (n 1)face of 1%defined by tn = 0 will be identified with I n  1 hereafter. The union of all remaining (n  1)facesof I n is denoted by Jn1. Then we have a I n = I n  1 U Jn1
aIn1 = In1
n Jn1.
By a map f : (I%,I n  1 , Jn1) + ( X ,A , x0), we mean a continuous function from I n to X which carries 1 n  l into A and Jnl into x,. In particular, it sends a I n into A and aZn1 into x,. We denote by F n = F n ( X , A, x,) the set of all such maps. These maps are divided into homotopy classes (relative to the system { In1, A ; Jn1, xo } ). We shall denote by n n ( X , A , x,) the totality of these homotopy classes. We shall also denote by [ f ] the class which contains the map f and by 0 the class which contains the constant map d,(In) = x,. Topologize Fn by means of the compactopen topology; then nn(X,A, x,) becomes the set of all pathcomponents of the space Fn. If n > 1, we may define an addition (usually noncommutative) in Fn. For any two maps f , g in F n , their sum f + g E F n is defined by the formula given in $ 2 for the absolute case. The homotopy class [ f g] depends only on the classes [ f ] and k ] and hence we may define an addition innn(A,X,xo) by taking [fl kl = [f + gl. As in Q 2, one can verify that this addition makesnn(X,A, x,) a group which will be called the nth relative homotopy group of X modulo A at x,. The class 0 is the grouptheoretic neutral element of nn(X,A, x,), and the inverse element of [ f ] is the class [/el, where 8 : I n + I n denotes the map defined in $ 2. If x, is the only point of A , then we have
+
+
F n ( X ,A , x,) = Fn(X,x,).
Hence, in this case, n n ( X ,A, x,) reduces to the absolute homotopy group nn(X,x,) defined in $ 2.
If Jnl is pinched to a point so, ( I n , In1, Jn1) becomes a configuration topologically equivalent to the triplet (En,9  1 , so) consisting of the unit ncell En, its boundary ( n  1)sphereSn1, and a reference point so E 9  l . It follows that one might equally well define an element of n,(X, A , x,) as a homotopy class (relative to the system { Snl, A ; so, xo } ) of the maps of (En, Snl, so) into ( X ,A, xo). Since, when n > 2, there exists a rotation of
3. R E L A T I V E H O M O T O P Y G R O U P S
I11
En which leaves so fixed and interchanges the two halves of En, we see that nn(X,A , x,) is abelian for every n > 2. This commutativity property is also an immediate consequence of the next proposition. n,(X, A, xo) is in general nonabelian. 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, x,)
=
be a given triplet. Consider the space of paths
X' and the initial projection p : X' A' and denote by triplet
E A'
=
[X ; x,X J as defined in (111; 5 10). Let
+X
= Pl(A) =
[X; A , x 0 ] c X'
the degenerate loop x',,(I)
T' = ( X ' , A',
= x,,.
Thus we obtain a
XI,),
called the derived triplet of T , and a map
p
: ( X I ,A',
XI,)
+
( X ,A, x,),
called the derived projection over ( X ,A, x,). Proposition 3.1. For every
> 0 , we have
~t
n,(X, A, xo)
= ~ n l (A',~
' 0 ) .
Proof. If n = 1, this is obvious since each side can be considered as the set of all pathcomponents of A'. Assume n > 1. Since XIn = bY (111; 9.9)' it follows that P ( X ,A, x,) = Fn'(A', XI,).
Hencen,(X, A, x,) andn,,(A', x',) coincide settheoretically. 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 !PUP* Since aIn is pathwise connected for n > I , the following proposition is obvious. Proposition 3.2. If X , denotes the pathcomponent of X containing x, altd A , that of A, then nn(X,A , x,) = nn(X,, A,, x,), n > 1. Finally, the following proposition will be used in the sequel. Proposition 3.3. If a €nn(X, A , x,) is represented by a map f slcch that f (In)c A, then a = 0 .
E Fn(X, A,x,)
IV. H O M O T O P Y G R O U P S
I12
Proof. Since f E F ~ ( XA ,, x,) and f(P)c A , we may define a homotopy f t E F n ( X , A , x , ) , O < t Q 1, bytaking
+
ft(ti..* tn1, tn) = / ( t i , .  8 tn1, t tn  &). Then we have f o = f and f l ( P = ) x,. Hence a = 0. I As an application of (3.3), let us prove the following Proposition 3.4. If
then n,,,(X, A , x,)
=
( X ,A , xo) is a triplet such that ( X ,A ) is a relative ncell, 0 for every m satisfying 0 < m < n.
Proof. By the definition of relative ncells in ( I ; 3 7), X is the adjunction space obtained by adjoining En to A by means of a map g : 9  1 + A defined on the boundary ( n  1)sphere 3  l of En. Let e, denote an interior point of En. Then, by ( I ; Ex. S), A is a strong deformation retract of X \ e,. Let a € n m ( X A , , x,) with 0 < m < n and choose a representative map f : (Im, P  l , Jm1) + ( X ,A , x,)
for a. Applying (11; Ex. C), we can easily prove that we may free e, from the image of f by means of a suitable homotopy of f relative to { Iml, A ; Jm1, x, }. Since A is a strong deformation retract of X \ e,, this implies that there exists a homotopy f t (Irn,Iml, J"') + ( X ,A , xo), (0 < t < I), c A . By (3.3), we conclude that a such that f o = f and fl(P) n,,,(X,A , xo) = 0 for every m satisfying 0 < m < n. I
= 0.
Hence
4. The boundary operator Let ( X ,A , x,) be a given triplet. For every n > 0,we shall define a transformation a : n n ( X , A , x , )+ n 4 A , x o ) . Let a be any element of n n ( X ,A , x,). Then, by definition, a is a homotopy class represented by a map f : (In,In1, Jn1) If n
= 1,
t
( X ,A , x,).,
f(Inl) is a point of A which determines a pathcomponent
p E Z ,  ~ ( Ax,) , of A . If n > 1, then the restriction f 1 Inl is a map of (In1, Wl) into ( A , xo) and hence represents an element p E nnl ( A , xo). Obviously, the element Enni(A, x,) does not depend on the choice of the map f which represents the given element a Enn(X,A , x,). Hence we may define the transformation a by setting a(a) = b. Hereafter, a will be called the boundary operator. The following two properties of a are obvious from the definition. Proposition 4.1. The boundary operator
n n ( X , A ,~
into that o f n f i  i ( A ,~
0 )
Proposition 4.2. I f n
a
sends the neutral element of
0 ) .
> 1, then the boundary operator a is a homomorphism.
5. I N D U C E D
1x3
TRANSFORMATIONS
5. Induced transformations Let (X,A, xo) and ( Y ,B , yo) 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
( Y ,B, Yo). Since f is continuous, it sends the pathcomponents of X into those of Y . Hence, f determines an induced transformation
f : (XI A, xo)
+
f* :no(X,xo) +no(Y,Yo) which obviously sends the neutral element of no(X,xo) into that of no(Y ,yo). Now let n > 0. For any map q5 E F ~ ( XA , xo), the composition fq5 is in F n ( Y, B, yo) and the assignment q5 + fq5 defines a map f#:Fn(X,A,xo) + F n ( y # B , ~ o ) * The continuity of f# implies that f# carries the pathcomponents of Fn(X, A , xo) into those of F n ( Y , B, yo). Hence it determines an induced transformation f* :nn(X,A, xo) +nn(Y, B, Y O ) which obviously sends the neutral element of nn(X, A, xo) into that of nn(Y9 B , Yo). If n = 1, A = xo, B = yo, or if n > 1, thennn(X, A, xol andn,(Y, B , yo) are groups. For any two maps q5, y in P ( X , A , x o ) , one can easily see that
f(+
+ Y ) = 19 + f Y
in Fn(Y, B, yo). Hence it follows that f* is a homomorphism. Thus we have established the following two properties of f*. Proposition 5.1. If n = 0 , A = xo, B = yo, or if n > 0 , then the induced transformation f* sends the neutral element of nn(X, A , x o ) into that of nn(y , B, Yo). Proposition 5.2. If n = 1, A = xo, B = yo, or if n
transformation f* i s a homomorphism.
> 1, then the induced
In the case of (5.2),f* will be called the induced homomorphism. Now, consider the derived triplets (X’, A ’ , (Y’, B‘, yl0) of the given triplets together with the derived projections %IO),
f~: (X’, A’, ~ ‘ 0 )+ ( X ,A , xo), r : (Y’, B’, y’o)
+
(YpB, Y O ) .
The given map f : ( X ,A , xo) + (Y, B, yo) considered as a map from X into Y induces a map f I : X I + Y’ according to (111; 3 14). Since f (A) c B and f (xo) = yo, it follows that ’f carries X’ into Y’, A’ into B’, and into yl0. Hence ’f defines a map xl0
f ’ : (X’, A‘, do)+ (Y’, B‘, yl0)
IV. H O M O T O P Y G R O U P S
114
which satisfies the relation rf ’ = f p and which will be called the derived m a p of f . Let 7 = 1’ I (A’, do).Then j induces
j*
:nn1(A’, x‘o) +nnAB’, Y’o).
Under the identifications in (3.1)it is obvious that f* = j*. Finally, the boundary operator a in 5 4 is essentially a special case of induced transformations. In fact, for any triplet (X, A , xo), consider the pair (A’, do)of the derived triplet and the restriction q = p I (A’,do)of the derived projection p. Then q induces q* :nnl(A’, x’o)
+%I
( A , xo).
Under the identification in (3.1),it is obvious that a
= q*.
6. The algebraic properties The homotopy groups n n ( X ,A , x0), the boundary operator a, and the 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. Property 1. If f : ( X ,A , xo) + ( X ,A , xo) i s the identity map, then f* i s the identity transformation on n n ( X ,A , xo) for every n. Property II. I f f : (X, A , xo) + (Y, B, yo) and g : ( Y ,B, yo) + (2, C, zo) are maps, then for every n 2 0 we have ( g f ) , = g, f*. Hence, for any given n, the assignment ( X ,A , xo) + n n ( X , A , xo) and f + 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. Property 111. I f f : (X, A , xo) + ( Y ,B, yo) is a map and if g : ( A ,xo) + ( B ,yo) i s the restriction of f , then the commutativity relation = g,a holds in the following rectangle for every n > 0 :
nn(X,A , xo)
If*
af,
a
t a nn(Y. B, Yo) +
nn,(A, xo) Ig*
t nnl(B, Yo)
Note that Property I11 is also an easy consequence of Property 11. To see this, let us consider the derived map
f’ : (X’, A ’ , do)+ (Y’, B’, yf0) of f as well as the derived projections
p : (X‘, A‘, x’o) + ( X ,A , xo),
r : (Y’, B’, Y’o)
+
(Y, B, Yo).
7. T H E E X A C T N E S S P R O P E R T Y
115
Let f = f’ 1 ( A ’ , q = p I (A’, do)and s = r I (B’, y f 0 ) .After the various identifications described in (3.1) and 5 5 , the preceding rectangle reduces  form: to the following 4+ nn1(A’, x’o) nnl(A, xo) %IO),
Ii;
J.
s+
nn,(B, Yo) Since f ’ satisfies the relation rf‘ = f p , we have s f = gq. Hence, Property I1 implies that the commutativity relation s,f, = g,q, holds in this rectangle. nnl(B’J’0)
7. The exactness property Let ( X ,A , xo) be any given triplet. The inclusion maps
i : ( A ,xo) = ( X ,xol, j : ( X ,xo) = ( X ,A , xo) induce transformations i, and j, for each n > 0. Together with the boundary
operators a, they form a beginningless sequence:
*A+n n + l ( X A, , xo) a
a , * nn(A,xo) I , n,(X. xo) A+ n n ( X ,A , xo) ...+ ir a , n l ( X ,A , xo) n,(A, xo) I+no(X,xo) which will be called the homotopy sequence of the triplet ( X ,A , xo) and will be denoted by n ( X ,A , xo). Every set in n ( X ,A , xo) has a specified element called its neutral element and every transformation in n ( X , A , xo) carries the neutral element into the neutral element. We define the kernel of a transformation in n ( X ,A , x,,) 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. T h e homotopy sequence of a n y triplet ( X ,A , xo) i s exact. The proof breaks up into the proofs of the following six statements: (1) j,i, = 0,(2) aj, = 0,(3) i,a = 0. (4) If u €n,(X, xo) and j,a = 0, then there exists an element /? € n n ( A xo) , such that i,p = u. (5) If a €nn(X, A , xo) andaa = 0, then there exists an elementp €nn(X,xo) such that j,p = a. ( 6 ) If aEnnl(A, xo) and i,a = 0, then there exists an element p € n n ( X ,A , xo) such that ap = a. 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 € n n ( A xo) , and choose a map f~ Fn ( A , xo) which represents a. Then the element j*i,(a) in n n ( X ,A , xo) is represented by the composition j i f E F n ( X ,A , x o ) . Since obviously j i f ( I n )c A . i t follows from (3.3) that j,i,u = 0. Since a is arbitrary, this implies j,i, = 0. I * *
j.
I 16
IV. H O M O T O P Y G R O U P S
Proof of (2). For each n > 0, let cx E n n ( X ,xo) and choose a map f E F n ( X ,xo) which represents u. Then the element aj,a is determined by the restriction j f I In1 = f I In1. Since f (Inv1)= xo, we have aj,u = 0. Hence aj* = 0. I Proof of (3). For each n > 0, let u € n n ( X ,A , xo) and choose a map f E Fn(X, A , xo) which represents u. Then the element i,du is determined by the restriction g = f I Inl. Define a homotopy gt : Inl + X , 0 < t < 1, by setting
gt(t,,..,tn,)
= j(t1,*.*,tn1,
4.
Then go = g, g 1 ( P  l ) = x,, and gt E Fnl(X, xo) if n > 1. This implies i*au = 0. Hence ;,a = 0. I Proof of (4).Choose a map f E P ( X , xo) which represents u. The condition f*u = 0 implies that there exists a homotopy ft : In + X , 0 Q t Q 1, such that f o = f , f l ( I n ) = xo, and f t E P ( X ,A , xo) for each t E I . Define a homotopy gt : In + X , 0 < t < 1, by setting
Then go = f , g,(In) c A , and gt(aIn) = xo for every t E I . Now, g , represents an element p € n n ( Axo) , and the homotopy gt proves that i*p = u. I Proof of (5). First, assume that n > 1. Choose a map f E F n ( X ,A , xo) which represents a.Then the condition au = 0 implies that there exists a homotopy gt : In1 + A , 0 Q t Q 1, such that go = f I In1, g,(Inl) = xo, and gt(i31nl.n1) = xo for every t E I . Define a partial homotopy ht : dIn + A , 0 Q t Q 1, by setting ( s E In1, t E I ) , gtb) ht(4 = ( s E Jnl, t E I ) . XO,
{
1
Since h, = f I aIn, i t follows from the homotopy extension property that the homotopy ht has an extension f t : I n + X , 0 Q t Q 1, such that f o = f . Since f , ( a P ) = hl(aI*) = x,,, f l represents an element p in n n ( X ,xo). Since f t E P ( X ,A , xo) for t E I , i t follows that j*p = u. For the remaining case n = 1, u is represented by a path f : I X such that f (0) E A and f (1) = xo. The condition au = 0 means that f (0) is contained in the same pathcomponent of A as xo. Hence there exists a homotopy f t : I + X, 0 Q t Q 1, such that f o = f , f t ( 0 )E A , ft(1) = xo, and f l ( 0 ) = xo. Then, f l represents an element /3 E ~ , ( Xxo) , and the homotopy f t implies that j*p = u. I +
Proof of (6). First, assume that n > 1 . Choose a map f E Fnl ( A ,xo) which represents u. Then the condition i,u implies that there exists a homotopy f t : In1 + X , 0 < 1 Q 1, such that f o = f , f 1 ( I n  l ) = xo and ft(aInl) = xo for every t E I . Define a map g : In + X by taking
g(t1,. *
* I
tn1, in) =

ft,,(ll,' ,in1).
8. T H E
HOMOTOPY PROPERTY
117
This map g is in Fn(X,A , xo) and represents an element a of n n ( X ,A , xo). Since g 1 In1 = f , we have aj3 = a. For the remaining case n = 1, a is a pathcomponent of A . The condition i,a = 0 means that a is contained in the pathcomponent of X which contains xo. Pick a point x from a. Then there exists a path f :I + X such that f (0) = x and f (1) = x,. This path f represents an element j3 of n , ( X , A , x o ) . Since f (0) E a, we have aj3 = a. I
8. The homotopy property Consider any two given triplets ( X ,A , xo) and ( Y ,B , yo), and any two given maps f,g:(X,A,x,)+(Y,B,y,). We recall that f and g are said to be homotopic (relative to { A , B ; xo, yo } ) if there exists a homotopy
ht:(X,A,xo)+(Y,B,yo)0 , 0. Hence, by Property IV, this implies that nn(X,A , xo) = 0 for every n > 0.
118
IV. H O M O T O P Y G R O U P S
9. The fibering property Consider two given triplets (X,A , xo) and (Y, B, yo) and a given map
f
:
w, A , xo)
+
( Y ,B , Yo).
We are concerned with the notion of fibering as defined in (111; 8 3). Property VI. If f : X
+
Y i s a fibering and A
= f 1(B), then
the trans
formation sends n n ( X ,A , xo) onto n,(Y, B , yo) ilz a onetoone fashion for every n
is onto, let u be an arbitrary element of %(Y, B, yo).
Proof. To prove that f*
By
> 0.
8 3, a is represented by a map
+ : (In,I n  l , In')
+
( Y ,B, yo).
Since Jn1 is a strong deformation retract of I", it follows from (v) of (111; 3.1) that there exists a map y : In + X such that f y = I# and y(Jn') = xo. Since A = f 1(B), f y = implies that y(I*l) c A and hence we obtain a map p : (In,Inl, Js') + ( X ,A , xo).
+
This map y represents an element /? of n,(X, A , xo). Since f y = I#, we have f,/? = a.This proves that f* is onto. To prove that f * is onetoone, let a and /? be elements of n n ( X ,A , xo) such that f*u = fa/?. Choose representative maps
+, y : (In,P1, 1nl) + ( X ,A , xo) for a and /? respectively. Since f*u = f*/?,the composed maps f + and f y represent the same element of nn(Y,B, yo). Hence, there exists a map
F : (In x I,1 n  l x I, such that F(z, 0) closed subspace
= f+(z) and
T
=
F(z, 1)
X
I) + ( Y .B, yo)
= fy(z) for
each z E I". Consider the
( I n x 0) U (I"' x I ) U (In x 1)
of I n x I and define a map G : T +. X by setting
G(z,t )
+(z), =
xo, Y(Z)>
( z E In, t = O), (2 E p  1 , t E I ) ,
( z € I n ,t
=
1).
Then we have f G = F I T. Since T is clearly a strong deformation retract of I n x I,it follows from (v) of (111;3.1) that G has an extension G* : In x I + X such that fG* = F. Since F maps Inl x I into B and A = f  l ( B ) , the condition fG* = F implies that G*(Inl x I)c A . Hence we obtain a map
G* : (In x I,1nl x I,Jn1 x I ) + ( X ,A , x0)
11. H O M O T O P Y S Y S T E M S
119
+
with G*(z, 0) = +(z) and G*(z, 1) = y(z) for each z E In. This proves that and p represent the same element of n n ( X ,A , x,). Hence a =?!, and f* is onetoone. 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 p : (X’, A‘, x’,) f ( X ,A , x,) By Properties VI and IV, p , and d in the following diagram
.
n,(X, A , x,) +PI nn(X’, A’, x’,) a4 nn1(A’, x’,) are both onetoone and onto. Furthermore, if n n ( X , A , x,) and Z ~  ~ ( A x’,)’ , are identified as in (3.1) one can easily see that 9, = d. On the other hand, the identification in (3.1) may be considered as being effected by the onetoone correspondence :n % ( XA, , ~
X =
+nni(A’,~
0 )
‘ 0 )
which will be called the natural correspondence.
10. The triviality property If X is a space which consists of a single point x,, then, for each n, the constant map f ( I n ) = x, is the only map of I n into X . Hence we have the following Property V I 1. If X i s a space consistingof asingle point x,, thenn,,(X, x,) =0
for every n 2 0. 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 n,(X, x,), we propose to call it the triviality property.
11. Homotopy systems In the preceding sections, we constructed geometrically the homotopy groups n n ( X ,A , x,) 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 no(X,x,) for all pairs ( X , x,) determines all n n ( X ,A , x,) for all triplets ( X ,A , x,) up to onetoone correspondence. A homotopy system H ={n,a,*} consists of three functions n,d and *. The first function n assigns to each triplet ( X ,A , x,) and each integer n > 0 an abstract set n n ( X ,A , x,). The A , x,) and each integer n > 0 second function d assigns to each triplet (X, a transformation :nn(x, A , x,) + n n  l ( ~x,), ,
a
IV. H O M O T O P Y G R O U P S
I20
where, in the case of n = 1, no(A,xo) denotes the set of all pathcomponents of A as in 3 2. The third function assigns to each map f : ( X ,A , xo) + ( Y ,B, yo) and each integer n > 0 a transformation
,
f ,: n n ( X ,A , xo) +nn(Y,B,
YO).
Furthermore, the system H must satisfy the following seven axioms : Axiom I. If f : ( X ,A , xo) + ( X ,A , xo) is the identity map, then f , i s the identity transformation on n n ( X ,A , xo) for every n > 0. Axiom 11. I f f : ( X ,A , xo) + ( Y ,B, yo) and g : ( Y ,B, yo) += (2,C, zo) are maps, then for every n > 0 we have (gf), = g,f,. Axiom 111. If f : ( X ,A , xo) += ( Y ,B, yo) i s a map and g : ( A , xo) += ( B ,yo) is the restriction of f , then the commutativity relation d f , = g*d holds in the following rectangle for every n > 0 :
a
nn(x,A , xo) +
1.
nn(Y,B , yo) +a
nnI(A, xo)
L*
4
nni(B>Y O )
where, in the case of n = 1, g, : no(A,xo) +no(B, yo) denotes the induced transformation in 3 5. Let ( X ,A , xo) be any given triplet and consider the inclusion maps
i : ( A , x O ) c ( X ,xo), j : ( X , xo) C ( X ,A , ~ 0 ) The transformations i,, j* and d form a beginningless sequence as in
57 which will be called the homotopy sequence of the triplet (X, A , x0) in the system H . Axiom IV. The homotopy sequence of any triplet ( X ,A , xo) is weakly exact. This means that, if n n ( X , xo) = 0 for all n > 0, then d sends n n ( X , A , xo) onto Z ,  ~ ( Ax,,) , in a onetoone fashion for every n > 0. Axiom V. If the maps f g : ( X , xo) + ( Y ,yo) are homotopic, then f , for every n > 0.
=
g,
Axiom VI. If p : ( X I ,A ' , d o )+ ( X ,A , xo) is the derived projection over ( X ,A , xo), then p , sends n n ( X ' , A', do) onto n n ( X ,A , xo) in a onetoone fashion for every n > 0. Axiom VII. If X is a space consisting of a single point xo, thennn(X,xo) = 0 for every n > 0. Since the derivedprojectionp : X ' + Xisafiberingby(II1; 13.4),itfollows that the axioms IVII are weaker than the properties IVII respectively. Hence, if we neglect the group operation in n,(X, A , xo),the three functions n,a, as defined in 35 35 constitute a homotopy system. This proves the
,
12. T H E U N I Q U E N E S S T H E O R E M
I21
existence of homotopy systems. One can also easily construct a homotopy system by induction and without using any geometrical representation, (see Ex. A a t the end of the chapter).
12. The uniqueness theorem Two homotopy systems H = { n , a ,* } and H’ = {n’,d’, # } are said to be equivalent if there exists, for each triplet (X, A , xo) and each n > 0, a transformation h, : n n ( X ,A , xo) +n’,(X, A , xo) satisfying the conditions : ( E l ) h, sends n,(X,A , xo) onto n‘,(X, A , xo) in a onetoone fashion. (E2) For each triplet ( X ,A , xo) and each n > 0 the commutativity relation hnl a = d’hn holds in the following rectangle:
a
n n ( X ,A , xo) +
I”.
n’n(X, A , xo) +a’
nnI(A, xo)
(6.1
4
n’nAA, xo),
where, in the case of n = 1, h, denotes the identity map. (E3) For each map f : (X, A , xo) +. ( Y ,B, yo), the commutativity relation hnf, = f#h, holds in the following rectangle:
n,(X, A , xo) f*+ n,(Y, B, Yo) lhn
lh*
J. J. n’,(X, A , xo) f#+ n’,(Y, B, yo). A collection of transformations h = { h, } satisfying the conditions ( E l ) through ( E 3 ) is called an equivalence between the homotopy systems H and H’ a i d is denoted by h : H M H‘. Theorem 12.1. Any two homotopy systems are equivalent. [Milnor 11. Proof. Let H = { n,a, * } and H’ = { n‘,a‘, # } be any two homotopy systems. We are going to construct an equivalence h : H w H’ as follows. Let n 1 and assume that we have already constructed the transformations hm:nn(X,A,x,) +n’m(X,A, %I)
A , xo) such that the conditions ( E l ) for each m < n and each triplet (X, through ( E 3 ) are satisfied. Let us construct h, as follows. Let (X, A , xo) be any triplet. Consider the derived projectionp : ( X ’ , A f , x f o ) +. (X, A , x o ) . By Axiom VI, p, sends 7tn(X’,A ’ , do)onto n,(X, A , xo) in a onetoone fashion and analogously for p#. According to (111; lO.l), X’is contractible to the point xf0. Hence, by Axioms I, 11, V and VII, we obtain
IV. H O M O T O P Y G R O U P S
I22
nm(X’,do)= 0 and nClm(X’,do)= 0 for every m > 0. By Axiom IV, a sends n,(X‘, A,‘ do)onto Z ,  ~ ( A ’do) , in a onetoone fashion, and analogously for a‘. By our assumption of induction, h,, sends Z ,  ~ ( A ’ do) , onto Z’,~(A’d , o )in a onetoone fashion. Hence we may define a transformation h9a :n,(X, A , xo)
by taking
+
n’n(X, A xo) 9
h, = p#a’lh,,ap;’,
where, in the case of n = 1, h, denotes the identity map. Since ( E l ) is obviously satisfied, i t remains to verify ( E 2 )and (E3). Then we have To check ( E 2 ) ,let q = I ( A ’ ,do). a’h n  a‘+#a’lh9a,ap;l =
q#k,,dp;l
=
= q#a’a‘u,,ap;l
h,lq*ap;l
= hn$
by Axiom 111.This proves that 12, satisfies (E2). To check ( E 3 ) , let ( Y ,B , yo) be a second triplet and f : ( X ,A , xo) + ( Y ,B , yo) be any given map. Let r : (Y’,B’,yto)+ ( Y ,B, yo), f ’ : ( X ’ , A ’ ,do)+ (Y’, B’, yl0) denote respectively the derived projection over (Y, B, yo) and the derived map o f f .The relation f p = rf ’is satisfied. Set 7 = f ’ I (A’, d o )Then . we have
f#h,
=
f#p#a’u,,ap;l
= r#f’#a’u,,ap;1
= r#a’lf#Iznlap;l = r#a’u,,af’*p;l
=
r#a’’h,_,f*ap;’
= r#a’lh,lar;lf*
= h,f*
by Axioms I1 and 111.This proves that h, satisfies ( E 3 ) . Thus we have completed the inductive construction of an equivalence h = { h, } between H and H‘ which will be called the natural equivalence h : H w H’. I 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 (12.1) shows that the homotopy system constructed geometrically in $$ 35 is essentially the only homotopy system. As a consequence of this, i t follows that the set of seven properties in §$ 610 is equivalent to the set of seven axioms in $ 1 1 which are apparently weaker. In fact, one can deduce the seven properties right from the axioms without using any geometrical representation of the sets n n ( X ,A , xo). The details of the proof will be left to the reader as an exercise. See Ex.C a t the end of the chapter.
13. T H E G R O U P
STRUCTURES
123
13. The group structures In the last two sections, we have shown that, apart from the group structures in the homotopy sets n n ( X ,A , xo), the homotopy system constructed in 55 35 is completely characterized by the seven axioms in 5 11 and the sets n o ( X ,xo). To complete the axiomatic approach, it remains to determine all the possible group structures that can be introduced into the essentially unique homotopy system
H={n,8,*}.
For this purpose, let us first consider the setsn,(X, xo) in H. According to the uniqueness theorem (12.1), we may assume n l ( X ,xo) to be the underlying set of the fundamental group of X a t xo. The product in n , ( X , xo) as defined in (11; 5 5) will be called the customary product which is denoted by juxtaposition. The reverse of this product, denoted by a dot, is defined by a * @= pa,
(a,@ i n , ( X , xo)).
For any map f : ( X , xo) + (Y, yo), the transformation f* : n L ( X ,xo) + n l ( Y ,yo) is a homomorphism under the customary product as well as its reverse. These are the only group structures in all n , ( X , xo) such that f* is a homomorphism for every map f . In fact, we have the following Lemma 13.1. There are exactly two ways of introducing a group structure into the sets n,( X ,xo) in such a way that f., i s a homomorphism for every m a p f : ( X , xo) f ( Y ,yo). These two group structures are defined by the customary +roduct and its reverse respectively. Proof. Assume that there is a new product, denoted by a 0 @, in each n l ( X ,xo) such that, for each pair ( X ,x o ) , n l ( X ,xo) is a group under this product and that f* is a homomorphism with respect to this product for every map f : ( X ,xo) + ( Y , yo). I t suffices to prove that either a o p = ap for any a, p en,(X,xo) of every ( X ,xo) or a o p = pa similarly. Let 2 be the space which consists of two circles, intersecting a t a single point z,. According to (11;Ex. A5),n,(Z, zo) is a free group on two generators a and b. For any two elements a, p of n , ( X , x o ) , obviously there is a map
f : (2,zo) such that f*(a) = a and f,(b) new product, we have
=
p.
( X , xo) Since f* is a homomorphism under the
f*(a o b)
+
=
a o p.
In terms of the customary group structure of n,(Z, zo), a o b is equal to some word w ( a , b) of the free group. Since f , is also a homomorphism under the customary product, we have f * ( a 0 b) = f * [ w ( a ,41 = w i f * ( a ) ,f * ( b ) l = 4%8,.
This implies that a o p = w ( a , @ ) .Thus it remains to prove that either w ( a , b) = ab or w ( a , b) = ba.
IV. H O M O T O P Y G R O U P S
124
For this purpose, let us first prove that the word w(a, b ) has the following two propeties : (1) w ( a , 1) = a, w(1, b ) = b, w [a, w h c)
(2)
I = w [w(a,b ) , c l ,
where (2) is an identity in the free group on three generators a , b, c. To prove (l), note that the identity element 1 of n l ( Z ,zo) can be defined as the image of the homomorphism
i, :n,(zo, 20) +n,(Z, zo) induced by the inclusion map i : (zo,zo) c (2,zo). It follows that the new product must have this same identity element. Hence W(U,
1)
=uo
1
= a,
~ ( 1b),
=
1o b
=
b.
To prove (2), choose X to be the space which consists of three circles tangent to each other at the same point xo. Then, as in (11; Ex. A5),n l ( X , x o ) 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 ( l) , w ( a , b) # a m . More generally, i t is impossible to have w ( a , b) = amlbni. * .UmkbnkUmk+i

with nonzero (positive or negative) integers m,,nl, mk,n k , and mk,,. To see this, let us assume that w(a, b) were of this form with k > 1. Then, by ( l ) , one can easily see that the reduced words of (w(a,b ) ) m i and ( w ( b , c))n1 would be of the same form, say, a ,
( w ( a ,b))mi = apibq1.s .a%b%a%+l,
( w ( b ,c ) ) n l = b'lC81. * .brjc8jbrj+l,
with nonzero integral exponents. By ( l ) , i > 1 and have w[a, w ( b , c ) ] = amibric81  , +
w [w(a, b ) , c ] = api b*i a%
* * *
as their reduced words. This contradicts (2). Similarly, w ( a , b) # bm and it is impossible 'to have W ( U , b) = bm@i* * .bmkd'kbmk+1 with nonzero integral exponents. Next, assume that w(a, b) =
. .amkbnk
i > 1. Then we would
14.T H E R O L E O F T H E B A S I C P O I N T
125
with nonzero integral exponents. If m, < 0, then the reduced word of begins with b% while that of w [ a , w(b,c ) ] begins with am,. This contradicts (2) and hence we have m , > 0. If n l < 0, then the reduced word of w [ a , w(b, c ) ] begins with amicnk while that of w [ w ( a ,b ) , c] begins with amibni. This contradicts (2) and so we have both m , > 0 and n1 > 0. Then we have w [ a , w(b, c ) ] = amibmicni. w [ w ( a ,b ) , c ]
 .,
W[W(U,
b), C ]
= amibnl. *
*amkbnk.* *
as their reduced words. By (2), it follows that K = 1 and hence w(a, b) = amibni. Then, (1) implies that m, = 1 and n , = 1. Consequently, we have w ( a , b) = ab. Similarly, we can prove that, if w ( a , b) = bmiani * 'bmkank
with nonzero integral exponents, then w ( a , b) possible cases. I
=
ba. This exhausts all
Theorem 13.2. There are exactly tzbo ways of introducing a group structure into the sets n n ( X ,A , xo), 12 > 2, and n,(X, xo), i n such a way that the transformations a and f* are homomorphisms. These two group structures are defined respectively by the customary group operation given in 3 3 and its reverse. [Milnor I]. Proof. Let us denote the customary group operation by juxtaposition and assume that there is a new group operation, denoted by a o /?,in the sets n n ( X ,A , xo), n 2 2, and n,(X, xo) such that the transformations a and f* are homomorphisms. We have to show that a o / ? is equal to a/? or /?a. By (13.1), this is true if a and/? are inn,(X, xo). To prove the theorem by induction on n, consider the natural onetoone correspondence
X =
of
a#;'
:n n ( X ,A , x0) +xnl(A', do)
6 9. Since X is a homomorphism for both group operations, we have %(a/?)= %(a)X(/?) X(a 0 B) = X(a) 0 W),
for any two elements a, /? in n n (X, A , xo). By the inductive hypothesis, X(a) o X(/?) is equal to X(a) %(/?) or %(/?) %(a).Hence X(a o /?) is equal to %(a/?)or X(/?a).Since X is onetoone this implies a o /? = a/?or /?a.I The significance of (13.2) is that the group structure in the essentially unique homotopy system H = { z, a, * } is also essentially unique. This completes the axiomatic approach.
14. The role of the basic point In the notion of the homotopy groupsnn(X, xo) andnn(X, A , xo) 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. H O M O T O P Y G R O U P S
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 a :I+ a(0) = xo, a(1) = x1.
x,
By the definition in
5 2, we have no(X,xo)
=
P ( X ) = no(X,x1)
as the set of all pathcomponents of X . Moreover, since xo and x1 are contained in the same pathcomponent of X , the neutral element of n o ( X ,xo) is the same as that of n o ( X ,xl). Let us denote by a0 : Zo(X, XI) +no ( X , xo) the identity map on n o ( X ,xl) = n o ( X ,xo).
Theorem 14.1. For each n way an isomorfihism
an :n n ( X ,x1)
M
> 0 , every
path a : I
n n ( X ,xo), xo
= a(O),
+
X gives i n a natural
x , = a( l ) ,
which defiends only on the homotopy class of the path a (relative to end points). I f a is the degenerate path u ( I ) = xo, then an is the identity automorphism. I f 4, t are paths with t(0)= a ( l ) , then (at), = a n t n . Finally, for each path a : I + X and each map f : X + Y , we have a commutative rectangle
n n ( X , x1)
If*
A+n n ( X , xo)
I f*
4 4 nn(Y,Yl) A+n n w , Y o ) , where t = fa, yo = f ( x o ) and y 1 = f ( x l ) . As an immediate consequence of (14.l), we deduce the following Corollary 14.2. The fundamental group
as a group of automorphisms.
n , ( X , xo) acts on n n ( X ,x o ) , n
> 1,
To prove (14.1), let us construct un as follows. Let a be any element of n,(X, xl) and choose a representative map f : ( I n , a r n ) + ( X ,xl)
for a. The geometrical idea of the construction is to pull the image of i3In along the path u back to the point xo with the image of I n being dragged in an arbitrary way. The map obtained after this homotopy represents an element /3 of n n ( X ,xo) which depends only on a and the homotopy class of a. Then, we define an(a)= ,!I. The details are as follows. First, let us prove that there exists a homotopy f t : I n + X , (0 < 1 Q l ) , of f such that ft(dZn) = a( 1  1 ) for every t E I . For this purpose, define a partial homotopy+t : 81. + X , (0 Q t < I ) , of f by taking+t(aIn) = o ( l t)
14.T H E R O L E O F T H E B A S I C P O I N T
127
for each t E I . By ( I ; 9.2),aIn has the AHEP in In. Hence the homotopy+t has an extension f t : In + X , (0 Q t Q l), such that f , = f . We call f t a homot@y of f along a. Since f l maps aIn into a(0) = x,, it represents an element p of n n ( X , x,). That ,!?depends only on u and the homotopy class of a is an obvious consequence of the following Lemma 14.3. If f , g : ( I n , ill.) + ( X ,x,) are maps homotopic relative to aIn, a, t : I , X are paths homotopic relative to end points, and f t , gt : In + X , (0 < t Q l ) , are homotopies of .t along a and of g along t respectively, then f l , g, are homotopic relative to aIn. Proof. Define two maps F , G : In x I + X by means of the homotopies f t , gt as usual. Consider the subspace A = (In x 0 ) U (din x I ) of In x I . By the hypothesis f N g and a II t, it follows that F I A abd G I A are
homotopic relative to aIn x 0 and aIn x 1. Since A has the AHEP in In x I by ( I ; 9.2), there is a homotopy Ft : In x I + X , (0 Q t Q l), such that F , = F , F , I A = G I A , and Ft(aIn x 1') = x, for every t E I . The map F , gives a homotopy ht : In + X , (0 Q t Q l), of g along t.Since Ft(aIn x 1) = x, for each t E I , it follows that f l and h, are homotopic relative to aIn. It remains to prove that g, and h, are homotopic relative to a r m . Define a map M : In x I + X by taking g,zg(P),
M(P,q) =
(pEIn,O XOI, w,= [ X i x1, x11 as well as the degenerate loops W,E W , and W , E W,. Then a given path : I + X which connects x,, to x1 induces a map l : W , + W , defined as follows: for each w E W , , [(w) E W , is the loop defined by
{
a(3t),
[ l ( w ) l ( t )= w ( 3 t  I), o(3  34,
(if 0 Q t Q &), (if 4 Q t < 3, (if 3 Q t Q 1).
Intuitively speaking, t ( w ) is the loop traced by running first from x, to x 1 along the path u, next around the loop w once, and then back to x, along the reverse of 0. On the other hand, u also induces a path r j : I + W , defined as follows: for each s E I, rj(s)E W , is the loop defined by 434, [r(s)I(t)= ( 4 S ) J 4 3 s 3 4 ,
(if 0 Q t Q &), (if & < t < $), (if $ < t Q 1).
Then, obviously we have rj(0) = mo and ~ ( 1 = ) t(wl).
15.L O C A L
SYSTEM OF G R O U P S
129
Proposition 14.4. I f 5 : (Wl, wl)+ (W,, f ( w , ) ) and q : I + W,are induced by the path u, the commutativity holds in the following diagram ‘Jn
nn(X, x1)
+
n n ( X ,xo)
where n i s any positive integer and X denotes the natural correspondence given by (2.2). Proof. Let a E n n ( X , xl) and choose a representative map f : (In,din) + (X, xl) for a. Then it is not difficult to see that both an(&)and Xlqnlt*X(a) are represented by the map g : (In,i3P) + (X, xo) defined as follows: 43W) (if e ( t ) G g), g(t) = f (3t1 1; * ,3tn  l ), (if e ( t ) > 4~ where t = ( t l ;  * , tn) is an arbitrar,y point of I n and O(t)denotes the smallest of the 2 n real numbers t 1 ; *  ,tn, 1 tt,;.., 1 tn. I
1
13
The property (14.4) of the operations 0%is characteristic. To formulate this fact precisely, let us define the notion of a system of operations as follows. By a system of operations in a homotopy system H = { n,a, * }, we mean for each path u : I + X in any space X and each integer n > 0 a transformation an : n n ( X ,X I ) +nn(X, xo), xo = U ( O ) , x1 = u(1) such that a,is theidentityonn,(X, xl) = n,(X, x,j and that (14.4)issatisfied. Since (14.4) implies an = Xlqnl[*X, the inductive proof of the following theorem is obvious.
Theorem 14.5. I n a n y given homotopy system H = { n,a, * }, there exists one and only one system of operations. Furthermore, for any two given homotopy systems H and H’, the natural equivalence h : H w H’ commutes with the operations in H and H’. Analogously, for each path u : I + A , one can define the operations on on the relative homotopy group and deduce similar results. See Ex.D a t the end of the chapter.
15. Local system of groups The homotopy groups together with the operations un constructed in the preceding section motivated the notion of a local system of groups in a space X , [Steenrod 11. We shall say that we have a local system of groups { G,} in a space X , if the following conditions are satisfied: (LSG1) For each point x E X, there is given a group G,. (LSGS) For each path u : I + X joining x, to xl, there is given a homomorphism a# : GZ1 + G,.
130
IV. H O M O T O P Y G R O U P S
(LSG3) If u is the degenerate path u(I) = xo, then a# is the identity automorphism on Gm. (LSG4) If two paths u, t : I t X are equivalent, i.e., if u, t have the same endpoints and are homotopic with endpoints held fixed, then a# = t#. (LSG5) If two paths IT,t : I t X are consecutive, i.e., if u(1) = t(O), then (at)# = u#t#. According to (14.1), the collection of the homotopy groups { nn(X, xo) I xo 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 { nn(X,A , xo) I xo 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 a# is an isomorphism. Hence, if X is pathwise connected, then all the groups Gz,x E X, are isomorphic. Since the elements of the fundamental group nl(X, xo) are homotopy classes of the loops [X ; xo, xO] with endpoints held fixed, we deduce as a direct consequence of (LSG3)(LSG5) that, for each xo E X , n,(X, xo) acts as a group of operators (or automorphisms) on , G 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
+
+
h(g1 gz) = k l hgtgzt h,(h,g) = (h,h,)g* lg = g where g, g, g, E G, h, h,, h, E H are arbitrary elements and 1 E H denotes the neutral element. Applying this to the local system of groups {nn(X,x,,) I EX}, where 12 > 0, we obtain (14.2) restated as follows. Proposition 15.1. For each xo E X, tlte fundamental group n l ( X , xo) acts on the nth homotopy group nn(X, xo) as a group of operators.
In the special case n
=
1, one can easily see that, for any two elements
g and h inn,(X, xo),h acts on g as follows:
(15.2)
h(g) = hgh1.
Similarly, for each xo E A , n,(A, xo) acts on the nth relative homotopy group nn(X, A , xo), 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 < l ), be a homotopy such that ho = f and hl = g. Choose a point xo E X and denote yo = f and y1 = g(xo). Define a path a : I +. Y by taking 44 = ht(xo), ( t E 4 ;
(xo)
~~
16. n  S I M P L E then a(0) = yo and a( 1) homomorphisms
= y,.
SPACES
According to
3 5, the
131
maps f and g induce
g* : n n ( X , x o )+nn(Y,Y J ) (Y, YO), f* : n n ( X ,~ 0 +nn 1. On the other hand, the path a determines an isomorphism for each n an :nn(Y,~ Proposition 15.3. f* = ang,.
+nn(Y,Y O ) .
1 )
+
Proof. Let a E T C ~ (xo) X , and choose a map : (In, 81.) + ( X , xo) which represents a. Define a homotopy y t : I n + Y , (0 < t Q l), by taking yt = ht+ for every t E I. Then yo represents f*(a) and y1 represents g*(a). Since yt(8In) = a(t) for each t E I, it follows that f,(a) = crag*(.). I
Corollary 15.4. If f , g : X + Y are homotopic maps such that f (xo) = yo = g(xo),then there exists aiz element w ETC,(Y, yo)such that f* = wg,.
As another consequence of (15.3), we have the following Proposition 15.5. If f : X + Y is a homotopy equivalence and
then
if f (xo) = yo,
f* :n n ( X , xo) +nn(Y,Yo)
is an isomorphism for every n > 0. Proof. Since f is a homotopy equivalence, there exists a map g : Y + X such that gf and fg are homotopic to the identity maps. Let x , = g(yo). Then g induces g, :nn(Y,Yo) +nn(X,4 
Let ht : X + X , (0 < t l), be a homotopy such that h, = gf and h, is the identity map on X . Define a path a : I + X by a(t) = ht(xo)for every t E I. Then, by (15.3), we have g*f* = an. Since an is an isomorphism, this implies that f* 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 f * = g;lan. I Similarly, iff : (X, A ) + ( Y ,B) is a homotopy equivalence and iff (xo) = yo, then f* : n n ( X ,A , xo) +nn(Y, B,Yo) is an isomorphism for every n
> 1.
16. nSimple spaces A local system of groups { Gz } in a space X is said to be simple, if the homomorphism a# depends only on the initial point a(0) and the terminal point a(1) of the path a : 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 E G .
132
IV. H O M O T O P Y G R O U P S
The proofs of the following two propositions are straightforward and hence are left to the reader. Proposition 16.1. A local system of groups { Gx } in a space X is simple i#, for every xo E X , n , ( X , xo) acts simply on Gzo. Proposition 16.2. A local system of groups { Gx } in a pathwise connected space X is simple i# there exists a point xo E X such that n , ( X , xo) acts simply on Gxo. Corollary 16.3. A local system of groups { Gx } in a simply connected space X is always simple.
Let n > 0 be any given integer. A space X is said to be nsimple if the local system { n n ( X ,xo) I xo E X } of the nth homotopy groups in X is simple. The following assertions are immediate consequences of the definition and (16.1)(16.3). Proposition 16.4. A space X is nsimple if, for every xo E X , n , ( X , xo) acts simply on n n ( X , xo). Proposition 16.5. A pathwise connected space X is nsimple i# there exists a point xo E X such that q ( X , xo) acts simply on n,(X, xo). Corollary 16.6. A simply connected space is nsimple for every n
> 0.
Corollary 16.7. A pathwise connected space X is nsimple if n n ( X ) = 0. Corollary 16.8. A pathwise connected space X is 1simple
commutative.
iff n l ( X ) is
Thus, the msphere Sm is nsimple for every m > 0 and n > 0. Now let us consider the unit nsphere Sn and a given point so E Sn. The geometrical meaning of nsimplicity is given by the following Theorem 16.9. A space X is nsimple i#, for every point, x,, E X and any f X with f (so) = xo = g(so), f N g implies f N g re1 so.
two maps f , g : S n
Proof. Assume that X is nsimple. Then, by (16.4), n , ( X , xo) acts simply on n n ( X , xo). Since f N g, there exists a homotopy ht : Sn + X , (0 < t < l ) , such that ho = f and h , = g. According to the remark given in the paragraph which precedes (2.1), the maps f and g represent elements a and p of n,(X, xo) respectively. Define a path u : I +X by taking a(t) = &(so) for each t E I . Since o(0) = xo = a(l),u represents an element w of q ( X , x0). By 5 14 and 4 15, it is easy to see that a = wj3. Since n , ( X , xo) acts simply on n n ( X , x o ) , we have wj3 = j3. Hence a = j3. This proves that f N g re1 so. Next, let us assume that the condition is satisfied. Let w E n , ( X , xo) and choose a loop u which represents w. Let a be any element of n n ( X , xo) represented by a map f : Sn + X with f (so) = xo. Then the element w a of
16. %  S I M P L E
SPACES
I33
n n ( X ,x,) is represented by a map g : Sn + X with g(s,) = x, and satisfying f r g. By our condition, this implies that f N g re1 so. Hence wu = 01. By ( 16.4), X is nsimple. I As a consequence of (16.9), let us prove the following Proposition 16.10. Every fiathwise connected topological group is nsimple
for every n > 0.
Proof. Let X be a pathwise connected topological group and x, its neutral element. Let f , g : Sn + X be any two homotopic maps such that f (so) = x, = g(so). Then these exists a homotopy ht : Sn + X , ( 0 Q t Q l), with h, = f and h1 = g. Define a homotopy kt : Sn + X , (0 Q t < l ) , by taking
kt(4
=
[ht(so)ll. [ht(s)l, (s E S", t E I ) .
Then we have k, = f , k , = g and kt(s,) = x, for each t E I . Hence f N g rel so. By the sufficiency proof of (16.9), n , ( X , x,) operates simply on n n ( X ,x,). By (16.2) this implies that X is nsimple. I This proposition (16.10) can be generalized to the Hspaces as defined in (111; 3 11). See Ex. G. The usefulness of nsimplicity is that, for a pathwise connected nsimple space X , the abstract homotopy groupn,(X) as defined in 5 14 has a natural geometrical meaning as follows. Let us define n n ( X ) to be the set of all homotopy classes of the maps of Sn into X . In other words, n n ( X ) is the set of all pathcomponents of the mapping space @ = XS". Choose an arbitrary basic point x, E X and consider the subspace y / of @ which consists of the maps of (Sn, so) into ( X ,x,). Then the pathcomponents of Y can be considered as the elements of n n ( X ,x,). Hence the inclusion map Y c @ induces a transformation
X :n n ( X , x O ) +nm(X). Lemma 16.11. I f X i s fiathwise connected and nsimple, then X sends n n ( X ,x,) ontonn(X)in a onetoone fashion. Proof. Let a € n n ( X )and choose a map f : S n + X which represents a. Since X is pathwise connected, there is a path a : I + X such that a(0) = x, and a( 1) = x, = /(so). By the method used in the construction of un in 3 14, one can show that there is a homotopy f t : Sn + X , (0 Q t < l ) , such that f o = f and ft(so) = a( 1  t ) for each t E I . Then f l E !P and represents an element /? of n n ( X ,x,). The homotopy f t proves that X(p) = a. Hence X is onto. Next, let M , /? E n n ( X ,x,) be such that X(a) = X(/?). Choose maps f , g : (Sn, so) ( X ,x,) representing a, /? respectively. Since X(a) = X(@), we have f N g. Since X is nsimple, it follows from (16.9) that f N g rel so. This implies that a = /?.Hence X is onetoone. I f
I34
IV. H O M O T O P Y G R O U P S
By means of X , we may define a group structure innn(X)so that X becomes an isomorphism, which will be called the canonical isomorfihism of nn(X, x,,) onto n,(X). To justify our geometrical construction of nn(X)given above, it remains to show that the group structure defined in nn(X) by means of X is independent of the choice of x,, E X. The geometrical meaning of the group operation defined innn(X)by means of X is clearly as follows. Let S: and S! denote the hemispheres defined by 2% < 0 and tn > 0 respectively, where (to,  ,tn) denotes an arbitrary point of Sn. Then the basic point s,, = (1 , 0, *  ,0) is in the equator

sn1 = sn_n s:. Let a and p be arbitrarily given elements of nn(X). Then there exist maps f E a and g E such that f(Sn,) = x,, = g(Sn_). Define a map h : S n f X by taking
Then h represents the element a + ?!t of n,,(X) which does not depend on the choice of f from a and g from @.We call the attention of the reader to the fact that, in case n = l,n,(X) is commutative by (16.8) and hence the additive notation is preferred. Now let x , E X. Since X is pathwise connected, there is a path a : I + X such that a(0) = x,, and a(1) = x , . By a standard method, one can show that there exists a homotopy f t : S n f X, (0 < t < I ) , such that f o = f and ft(.S!) = a(t) for each t E I. Similarly, there is a homotopy gt : S n + X, (0 Q t Q l ) , such that g, = g and gt(S!) = a(t) for each t E I. Define a homotopy ht : Sn +. X, (0 Q t Q l), by setting
ht(4 for each t E I. Then h,, since
+
=
=h
(if s ES?), (if s E S?).
[ ftbh gt(s),
and h,
N
h. Hence h, represents a
+ 8. Also,
this proves that a does not depend on the choice of the basic point x,, E X. Thus, for a pathwise connected nsimple space X,we have freed the basic point from the definition of the nth homotopy group. We have just seen that the homotopy classes of the maps of S n into a pathwise connected nsimple space X form a groupnn(X)which is isomorphic with nn(X, x,,) for every x,, E X. If X is pathwise connected but not nsimple this is not true; in fact, the homotopy classes of the maps S n + X are in a onetoone correspondence with the equivalence classes in nn(X, x,,) under the operations of n,(X, x,,). The proof is left to the reader.
EXERCISES
I35
EXERCISES A. Inductive construction of a homotopy system
a,
Construct a homotopy system H = { n, * } by induction as follows. According to the definition of a homotopy system in 5 11, the homotopy set n o ( X ,xo) and the induced transformation f* : n o ( X ,xo) + no(Y,yo) are welldefined for every pair (X, xo) and every map f : ( X ,xo) + ( Y ,yo). Let n 2 1 be a given integer and assume that we have already constructed the homotopy sets n m ( X , A , xo) for each 1 Q m < n and each triplet (X, A , xo), together with the boundary operators and the induced transformations f* on these homotopy sets, such that the seven axions in $ 11 are satisfied. To construct the homotopy set n n ( X ,A , xo) of a given triplet ( X , A , xo), consider the derived triplet (X‘, A’, do)of ( X ,A , xo) and the derived projection p : (X’, A ’ , d o ).+ ( X ,A , x o ) . We define
a
nn
( X ,A , xo)
Next, let q = p I ( A ’ , nn,(A, xo) is defined by
%IO).
= nni(A’,
~’0).
Then, the boundary operator
a :n n ( X ,A , xo)
f
a = q* :nni(A‘,do)+nni(A, x0).
Finally, let f : ( X ,A , xo) + ( Y ,B , yo) be a given map. Then, f induces a Then A’, do)+ (Y‘,B‘, yl0). Let 7 = f ’ I (A‘, derived map f ’ : (X’, define f * = :nni(A‘, x’o) . + ~ n  ~ ( B~ ’’ ~0 ) . Verify the seven axions of 3 11 for the system H = { n, * } constructed above. %IO).
L
a,
B. The equivalence theorem
Consider two given homotopy systems
H
={n,a,*},
H’
={n‘,a’,#}
together with their natural equivalence h = { hn } : H M H’ constructed in 5 12. By an admissible transformation k = { kn } : H + H’, we mean for each triplet ( X ,A , xo) and each integer n > 0 a transformation
kn : n n ( X ,A , xo) +n’n(X, A , xo) satisfying the conditions : (ATl) For each triplet ( X ,A , xo) and each integer n > 0, we have the commutativity relation kn$ = a’k,, where, in case of n = 1, KO denotes the identity map. (AT2) For each map f : ( X ,A , xo) + ( Y ,B , yo) and each integer n > 0, we have the commutativity relation knf* = knf+ By (El), (E3) and (E4) of 5 12, every equivalence between H and H‘ is an admissible transformation. Conversely, prove the following
IV. H O M O T O P Y G R O U P S
136
Equivalence Theorem. Every admissible transformation k i s a n equivalence between H and H'. I n fact, it coincides with the natural equivalence h, that i s 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 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, 11, 111, 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, xo) +. (Y, B, yo) is a homotopy equivalence, then, for each n > 0 , f* sends n,(X, A , xo) onto n,( Y , B, yo) in a onetoone fashion. 2. If X is contractible to the point xo, thenn,(X, xo) = 0 for every n > 0. 3. For any given triplet ( X ,A , xo), consider the derived projection p : (X', A', do)+. (X, A , xo). Then in the diagram n n ( X ,A, ~ 0 &nn(X', ) A', ~ ' 0 2+ ) nni(A', d o ) , both p , and a are onetoone and onto and hence we obtain a natural cowespondence X = d#;' :nn(X, A, ~ 0 +.7~,1(A', ) do) which sends n n ( X ,A, xo) onto Z ~  ~ ( A do) ' , in a onetoone fashion. 4. For every triplet (X, A, xo) and every n > O,n,(X, A, xo) is nonempty. Furthermore, one can uniquely define a neutral element of nn(X, A , xo) in such a way that X,a and the induced transformations send the neutral element into a neutral element. 5. In the homotopy sequence * . * L~nn+l(X,A,xo) ~ n n ( A , x o ) ~ + . n n ( X , x o ) ~ + . n n ( X , A , xa o ) * *. ,
$5 610

*
4, n , ( X , A ,
x0)
E+ no(Arx0) I+ no(X,x0)
of (X, A , xo) 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. T h e homotopy sequence of any triplet i s exact, that is to say, the kernel of every transformation in the sequence coincides with the image of the preceding transformation. The fibering theorem. I f f : X +. Y i s a j i b e r i n g , A = f  I ( B )a n d x o E f  ' ( y 0 ) , then the induced transformation f* carries n n ( X ,A , xo) onto n,(Y, B, yo) in a onetoone fashion for every n > 0. These two theorems cover the properties I V and VI respectively.
I37
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, x1 connected by a path a:I+A,
For each n
a(0) = xo,
u(1)
= x1.
> 0, define a transformation
an:nn(X,A,xI) +nn(X,A,~o) as follows. Let a €n,(X, A , xl). Choose a representative map f : (In,In1, I"') + ( X ,A , xl) for a. Pull the image of Jn1 retreating along the path a back to xo with the image of I n being dragged in such a way that the image of InI is always in A . The map obtained after this homotopy represents an element p of n n ( X , A , x o ) which depends only on a and 0.Then, we define an(a)= p. Give the details of this geometrical construction as in 3 14, and prove the following assertions: 1. For every t z > 2, an is a homomorphism. 2. For every n > 0, an depends only on the class of the path u. 3. If a is the degenerate path u(I)= xo in A , then un is the identity transformation on n,(X, A , xo) for each n > 0. 4. If a, t are consecutive paths in A , i.e. a(1) = t(O),then (at),= antn for each n > 0. 5. For every n > 0, an carries n n ( X ,A , xl) onto n n ( X ,A , xo) in a onetoone fashion. Hence an is an isomorphism for every n > 2. 6. Each rectangle of the following ladder is commutative * *
Lnn(A,xo) 4 , nn(X,x g ) A+ n n ( X , A , xo) +a
*_a,
kn
n,(A, xl) 2%
bn
n,(X,
XI)
kn
A+ n,(X, A ,
nn1(A, KO)&
* * *
Ion
x1) + a
J. n,,(A,
XI)
4,
* *

7. For any triplet (X, A , x O ) , n 2 ( X , Axo) , is a crossed [ n l ( A ,x o ) , i3lmodule. By this, we mean that the following two conditions are satisfied for every w in n l ( A ,xo) and a, p in n 2 ( X ,A , xo) : (i) a(wa) = w(da)w', (ii) (aa)p = .pa'. Hence j*n,(X, xo) is contained in the center of n , ( X , A , xo) and i,n,(A, xo) acts as a group of operators on j*n,(X, x o ) . See [Hi, pp. 3941]. The significance of the assertion (5) is as follows. If A is pathwise connected and n >, 2, then all the groups n n ( X ,A , xo) for various basic points xo are isomorphic. Hence, as an abstract group, n n ( X ,A , xo) does not depend on the basic point xo and may be denoted simply by n n ( X ,A ) . This abstract group n n ( X ,A ) will be called the nth (abstract)relative lzomoto$y group of X modulo A . For example, we have nm(En,9  1 ) = 0, ( m < n ) , n n ( E n , 9  1 ) = 2, nm(E2,Sl) = 0, (m > 2), n4(E3,9)= 2.
138
IV. H O M O T O P Y G R O U P S
Next, consider the spaces of paths
w,= [X;A,x,l,
W, = [X;A,x,l as well as the degenerate paths wo E W , and w1 E W,. Then a given path u : I  + A which connects xo t o x , induces a map $ : W, + W, defined as follows: for each w E W,, [(w)E W , is the path defined by i f 0 < t 0. an = X%ln,E*X, E. Relative nsimplicity
Let n > 2 be a given integer. A space X is said to be nsimple relative to a subspace A if the local system of groups { n n ( X ,A , x,) I x, E A } in A is simple. In this case, we also say that the pair ( X ,A ) is nsimple. Establish analogues to (16.4)( 16.7), as well as these assertions: 1. If ( X ,A ) is 2simple, then n , ( X , A , x,) is abelian for every xo E A . 2. If, for every x, E A , n , ( X , A , xo) is abelian and i, sends nl(A,x,) into the neutral element of n , ( X , xo) then ( X ,A ) is 2simple. 3. A pair ( X , A ) is nsimple iff, for every point xo E A and any two maps
I , g : ( E n , sn1, so) ( X ,A , xo), 1 implies that f g re1 { Snl, A ; so, x, +
g rel { Snl, A
}. 4. If ( X ,A ) is nsimple and if A is pathwise connected, the elements of the abstract relative homotopy group n n ( X , A ) can be considered as the homotopy classes of the maps of (En,9  1 ) into ( X ,A ) . f
N
N
F. The Whitehead product
Consider a given space X and a given basic point xo E X . Let m
n
> 1 be given integers. For any two given elements a E n m ( X , xo),
B E n n ( X , KO),
> 1 and
I39
EXERCISES
we are going to construct an element [a,/?I of nm+nl(X, xo) which will be called the Whitehead firoduct of a and 8. For this purpose, let us choose representative maps
f : (Im,a r m )
( X ,xo), g : ( I n ,a r n ) + ( X ,xo) for a , 8 respectively. Since Zm+n = Im x In, we have aZm+n = ( I m x aIn) U (aim x In). Hence we define a map h : a P + n + X by taking for each point (s, t ) in dZm+n if t E aZn,
4%1)
+
=
('1, { fg(t)
if s E aZm.
Since the point ro = (0;* ., 0) of aZm+n is in aIm x din, we have h(ro) = xo. Since aZm+n is homeomorphic to Sm+nl, h represents an element y of nm+ni(X,4. Prove that y depends only on the elements a and 8. So, we may define [at81 = Y . Establish the following properties of the Whitehead products : 1. If a e n , ( X , xo) and B e n , ( X , xo), then [ a , b ] is the commutator a/?al/?l of n , ( X , xo). 2. If a € n m ( X ,xo) and ,!I € n , ( X , xo) with m > 1, then [a,81 is the element @aa of n m ( X ,x0). 3. If m > 1, then the assignment a + [a,83 for a given 8 € n n ( X ,xo) defines a homomorphism p* : n m ( X ,xu) +nm+ni(X>xo). 4. If m + n > 2 , then, for every a Enm(X, xo) and 8 € n n ( X ,xo) we have
l)mn[a,81. 5. If u : I + X is a path joining xo to xl, then, for every a € n m ( X ,x , ) and /?E n n ( X ,x , ) , we have
[S, a1
=
(
am+nl[a,81 = [am(a),~ n ( @ ) ] . 6. If : ( X ,xo) + ( Y ,yo) is a map, then, for every a € n m ( X ,xo) and / ? ~ n n ( XX O* ) ~we have +*[a,/?]= [+*(a),+,(~)].
+
7. For any a €n,(X, xo), 8 ~ n n ( Xxo), , r € n q ( X xo), , the following Jacobi identity holds:
+ (
+
l)nm"B, yl, a1 ( l)@""yja ] ,B1 = 0. Whitehead products may be also defined between relative homotopy groups and between n m ( X ,A , xo) and nn ( A , xo). See [Hu 111 and [Blakers and Massey 21. ( l)m@"a,83, yl
G. Homotopy groups of Hspaces
Let X be a given Hspace as defined in (I11 ; fj 11) and let xo be a homotopy unit of X . Then the group operation in n n ( X , xo) is closely related to the multiplication in X as follows.
IV. H O M O T O P Y G R O U P S
140
Let a,/? be arbitrarily given elements of n n ( X , xo) with n any representative maps f, g : ( ~ n , ( x ,xo)
aw
> 0.
Choose
+
for a and /? respectively. By means of the multiplication in X , we may define a map h : (In, 81.) + ( X ,xo) by taking
h(t) = f ( t )* g ( t ) , 1 E In. Prove that h represents the element a + /? of n,(X, xo). Furthermore, if X is a topological group and xo is the neutral element of X , then we may define a map k : ( I n ,a P ) + ( X , xo) by taking
f ( t )* [g(t)]1, t E In. Prove that k represents the element a /? of n,(X, xo). Next, let a €n,(X, xo) and 8 € n n ( X ,xo). Choose representative maps
K(t)
f : (P,
=
a r m ) +
(x, xo),
for a and 8 and define a map h : I m + n h(s, 1 )
=
g ( I %sin)+ , +
(x,xo)
X by taking
f ( s ) * g ( t ) , s E I m , t E In.
Prove that h I d P + n represents [a,81. This implies that [a,81 = 0.
Hence, it follows from the assertion (2) of Ex. F that every pathwise connected Hspace is nsimple for every n > 0 ; in particular, we deduce again that n , ( X , xo) is abelian. H. Semisimplicia1 complexes
A semisimplicial complex K is a collection of elements { a } calledcells together with two functions. The first function assigns to each cell u a n integer m > 0 called the dimension of u, m = dim (a); we then say that u is an mcell. The second function assigns to each mcell u, (m > 0 ) ,of K and each integer i, (0 < i < m ) , an (m  1)cell &u called the ith face of u, also denoted by di),subject to the condition (SSC)
a,a,u = a,,aeU
for m > 1 and 0 < i < j < m. We call a, the ith face operator. It may happen that &a = &a for some i # j. Lower dimensional faces of u may be defined by iterating the face operators. For any two cells u and t of K , we shall write t < u if either t = u or t = .&nu for some set { i1; * * , i, } of integers with 0 < i, < * * * < i n < m. Thus we obtain a proper reflexive partial ordering relation < in K . [Eilenberg and Zilber 13. 1. A simplicia1 complex K is a set whose elements are finite subsets of a given set V , subject to the condition that if U E K and t is a nonempty subset of u, then t E K . Prove that, if V is partially ordered in such a way that every u E K is lineally ordered, then K is a semisimplicial complex.
a,,. 
EXERCISES
141
2. Given a simplicial complex K , construct a semisimplicial complex O(K) as follows. The mcells, m 2 0, are the (m + 1)tuples (v0; * * , urn), repetitions allowed, of the vertices of some simplex of K ; and &(v0, * * , vm) = (v0;
., &,  .

a ,
v,J, where the circumflex over va means that va is omitted.
3. Prove that the singular complex S ( X ) of a space X , [ES; p. 1851, is a semisimplicial complex. 4.Construct the homology and the cohomology groups of a given semisimplicial complex K modulo a subcomplex L over a given abelian coefficient group. I. Degeneracy operators
By a system of degeneracy operators in a given semisimplicial complex K , we mean a function which assigns to each mcell u, (m 2 0), of K and each integer i , (0 < i < m),an (m 1)cell &u of K , satisfying the conditions:
+
eae,(u) = e,+,e,(u), (i < j ) ; a,e,(u) = 0 = a,+,e,(a); (m> 0 , i < j ) ; =
aae,(u) ejlar(u), alej(0)= e,aa,(u), (m> 0, i > j + 1).
We call Ba the ith degeneracy operator of the system. An mcell u, (m> 0), of K is said to be degenerate if u = &(t) for some (m 1)cell z of K and someiwitho < i Gm1. 1. Prove that the semisimplicial complex O ( K )of a simplicial complex K admits a system of degeneracy operators defined b y

 ., va,
,vm). 2. Prove that the singular complex S ( X )of a space X admits a system of ec(vo,
*,
(vo,
Urn) =
degeneracy operators defined by
Of,
+
vt+,,.
a

eao(to, ,L+,) = upo, ,it,, tt ta+l, ,trn+l). 3. Prove that the semisimplicial complex obtained from a simplicial complex by a partial ordering of the vertices admits no system of degeneracy. 4.Prove that every semisimplicial complex of finite dimension admits no system of degeneracy. J. Complete semisimplicia1 complexes
A semisimplicial complex K is said to satisfy the extension condition if , urn+lof K such that given mcells uo,* * , uk,, ak+,, &a, = d ~  p g , (i # k , j # k , i < j ) , for the case m > 0, then there exists an ( m + 1)cell u such that dau = ug for each i # k . A semisimplicial complex K is said to be complete if it satisfies the extension condition and admits a system of degeneracy operators. A complete semisimplicial complex K with a given system of degeneracy operators is called a K a n complex, [Kan 11.

142
IV. H O M O T O P Y G R O U P S
Prove that the singular complex S(X) of a space X is complete and that the semisimplicial complex O(K) of a simplicial complex K may fail to be complete. Hence, the class of Kan complexes is rather limited. K. Homotopy groups of Kan complexes
Let K be a given Kan complex. The Ocells and the 1cells of K will be called vertices and edges respectively. Two vertices vo, v1 of K are said to be equivalent, vo N v l , if there exists an edge e E K with doe = v l and a l e = vo. By means of the extension property of K, prove that this relation is symmetric, reflexive and transitive, and hence the vertices of K are divided into disjoint equivalence classes called the components of K. Let no(K)denote the set of all components of K. Next, let v be a given vertex of K. We are going to define a derived complex.

K'
= D(K, U)
+
as follows. The ncells of K' are defined to be the (n 1)cellsu of K such that v is the only vertex < u and that a0u = 8,*(v), where O0* is the nfold iteration of the degeneracy operator 8,. The face operators a', and the degeneracy operators 8'r in K' are defined by
alfu = a,+,u,
e,+lu. Verify that K' is a Kan complex and v' = 8,v is a vertex of K'. Then, for every n > 0, we define nn(K,v ) inductively by 81,
0
=
nn(K, V ) = nnI(K', v ' ) , no(K,V ) = no(K). Next, let us define the group structure inn,(K, v ) for each n > 0. Because of the inductive definition given above, it suffices to define a group structure in no(K'). Let a,j3 be any two components of K and pick vertices x E a and y E j3. By the definition of K', x and y are edges in K such that aG = v = a1x and a0y = v = a,y. By the extension property of K, there exists a 2cell u in K such that a,u = x and a0u = y. Then z = a, u is a vertex in K . Prove that the component y of K' which contains z depends only on a andj3. Define a/?= y. Prove that nl(K') becomes a group under this multiplication. Prove the following assertions: 1. nn(K, v ) is abelian for each n > 2. 2. If K is the singular complex S(X) of a space X and v the vertex determined by a given point xo E X, thennn(K,v ) andnn(X,xo) are isomorphic for each n.
CHAPTER V THE C A L C U L A T I O N O F H O M O T O P Y G R O U P S
1. Introduction Neither the geometrical construction nor the axiomatic approach to the homotopy groups in the previous chapter leads to effective computation of these groups. In the present chapter, we shall study a few methods which yield successful calculations of the homotopy groups in various special cases. In the first part of the chapter, we give the celebrated Hurewicz isomorphism theorem. For every integer n > 0, there is a natural homomorphism ha of n,(X, xo) into the integral singular homology group H,(X). If 12 2 2 and if X is (n 1)connected, then Hurewicz's theorem states that h, is an isomorphism. Hence the first nonzero homotopy group of a triangulable space is effectively computable. In the second part of the chapter, we give the exact homotopy sequence of a fibering p : E + B together with a few direct sum theorems. By employing the numerous known fiberings in conjunction with this exact sequence, many homotopy groups may be computed. In the last part of the chapter, we introduce Freudenthal's suspension together with the notion of triad homotopy groups. These suspensions are crucial in the calculation of the homotopy groups of spheres, some of which will be given in the final chapter of the book.
2. Homotopy groups of the product of two spaces Let X , Y be two given spaces and xo E X , yo E Y be given points. Consider the product 2 = X x Y and the point zo = (xo, yo) in 2. Let
p
(XIxo), 4 : (ZtZO) (Y, Yo) denote the natural projections defined by p ( z ) = x and q(z) = y for each z = ( x , y) in 2. On the other hand, let : (2,zo)
+
+
i : (X, xo) + (2,z0), j : (Y, yo)
+
(2,zo)
denote the maps (called the injections) defined by i ( x ) = ( x , yo) and i ( y ) = (zo,y) for each x E X and each y E Y. Hence pi, qj are identity maps and p j , qi are constant maps. For each n > 0, the maps p , q, i, j induce the homomorphisms
p* :n,(Z,20) +nn(X,xo), q* :n&, 20) +nn(Y, Yo). i, :n,(X. xo) +n,V, z0), j* : Y,yo) n d Z , zo), +
143
I44
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
satisfying the relations
p*i* = 1, q*j* = 1, $*j* = 0, q*i* = 0. Hence i,, j* are monomorphisms and p*, q* are epimorphisms. Let us consider the direct product of the groups n,(X, xo) and nn(Y , yo). If n > 1, these groups are abelian and their direct product is also called the direct sum. We shall use the additive notation
+
n n ( X ,xo) nn(Y,yo), ( n > O ) , for the direct product even in the case n = 1 where the groups are not necessarily abelian. Theorem 2.1. For every n
> 0 , we have
nn(z,2 0 )
M
n n ( X ,xo)
Proof. Define a homomorphism
+ nn(Y,yo).
+
h :n n ( 2 , zo) +.n.n(X, xo) nn(Y,Yo) by setting h(a) = (&(a), q*(a))for every a ~ n , ( Z zo). , It remains to prove that h is an isomorphism. For arbitrarily given a E n,(X, xo) and p E n,(Y, yo),let y = i,a j*p E n,(2,zO).Then we have
+
Nr) = (P*i*a + P*i*b q*i*a + Q*j*B) = (a,/%. Hence 12 is an epimorphism. On the other hand, let 6 € n n ( Z ,zo) be any element such that h(6) = 0. Then, by definition, we have p,S = 0 and q*6 = 0. Let f : ( I n ,81.) +. (2,z,) be any map which represents 6. Since P,S = 0 and q,S = 0, there exist two homotopies gt : (In,azn) ( X ,xo), ht : ( I n , 81.) ( Y ,Y o ) , (0 t Q l ) , such that go = $f,g,(Zn) = xo, h, = qf and h,(Zn) = yo. Define a homotopy ft : Zn 2, (0 t l ) , by taking +
1 is used only once, namely, to assure that the image of 1 is a normal subgroup of nn(U , uo).In general, this is not true for the case
146
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
n = 1. However, we have anyway the weaker assertion that n l ( U , uo) is an extension of n,(Z, U , zo) by the direct product of n l ( X , xo) and nl(Y,yo). If X and Y are polyhedra, we recall that n l ( U , uo) is the free product of n l ( X , xo) and n,(Y, yo) by (11; Ex. A4). This is also true if X and Y are regular and locally simply connected. As an application of (3.1), we have the following Proposition 3.2 For every
p > 0, q > 0
and n
1.
Let a be an arbitrary element of n , ( X , A , xo) and choose a map
+ : (En,
91,
so) + ( X , A , xo)
which represents a, where En denotes the unit ncell in the euclidean nspace Rn, Snl the unit (n  1)sphere in Rn, and so = (1,0, 0). The natural coordinate system in Rn determines an orientation of Rn and hence a generator &, of the free cyclic homology group Hn(En, Snl). As a map of (En, Sn') into ( X ,A ) ,4 induces a homomorphism

+* : Hn(En,Sn')
a ,
H n ( X ,A ) where H,(X, A ) denotes the singular homology group with integral coefficients. According to the homotopy axiom of homology theory, I$+ depends only on the given element a E n,(X, A , a ) .Hence the assignment a * I$+([,) defines a transformation xn nn(X,A , ~
+
0+ )
Hn(X, A ) .
Proposition 4.1. If either n > 1 or A = xo, then x,, is a homomorphism which will be called the natural homomwfihism of n,(X, A , xo) into H n ( X , A ) . Proof. Let a and /Ibe any two elements of n,(X, A , xo). Then we may choose representative maps y : (En,Snl, so) + ( X ,A , xo) such that
+,
4. T H E +(tl; * ,tn)
=
NATURAL HOMOMORPHISMS
> 0 and y(t,; * ,t n ) ( X , A , xo) by taking
xo if t ,
X : (En, Snl, so)
+
=
xo if
< 0.
t,
I47 Define a map
for an arbitrary point ( t l ; . * , tn) in E n . Then X clearly represents a By a theorem in homology theory, [ES; p. 361, we have
+ @.
+
&(En) = + *( E n ) Y*(ln). This implies xn(a @) = xn(a) + xn(p) and hence xn is a homomorphism. I The following proposition is obvious from the above construction of Xn.
+
Proposition 4.2. For any map f : ( X ,A , xo) rectangle is commutative :
nn(X, A , ~
0
Hn(X,A)
For the case A
=
nn( Y
+ . f )
*'
+
+
( Y ,B , y o ) , the following
, B. yo)
H n ( Y , B)
xo, we have an isomorphism
j * : H n ( X ) % H n ( X ,xo) and hence we obtain a homomorphism hn

.
~ + ' x n: n n ( X ,xo)
+
Hn(X)
which will be called the natural homomorphism of n n ( X , xo) into H n ( X ) . Clearly, h , coincides with the homomorphism h, of (11; 5 6 ) . Sinced : H,,+,(Enfl, Sn) M Hn(Sn),it follows that q n = a[,+, is agenerator of the free cyclic group Hn(Sn). If we identify the boundary S n  l of En to a single point so, we obtain an nsphere S n and a point so. Hence there is a relative homeomorphism
p such that
p*([n) =
fi,
: (En,9  l ) + (Sn, so)
j * ( V n ) , where
: H n ( E n , Snl) M H n ( S n , SO), j * : Hn(Sn)
are the isoniorphisnis induced by the map (Sn,
3.
p
H n ( S n , SO)
and the inclusion map j : S n c
+
Let a E n n ( X , xo) and pick a map : (En, 9  l ) + ( X , xo) which represents a. Then there exists a unique map y : (9, so) ( X , xo) such that = y p . We may consider a as represented by y . As a map of S n into X , y induces a homomorphism y + : H n ( W + H n ( X ) . f
+
Then clearly we have hn(a) = y * ( q n ) . This may be used as the definition of the natural homomorphism h,.
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
148
Then, the following proposition is obvious. Proposition 4.3.
commutative:

For any triplet ( X ,A , xo), the following rectangle is
nn+i(XI A , ~
bn
Hn+i(X,A )
a
0 )
a
nn(A, xo)
+ H n ( A ) .
Now consider the following homotopyhomology ladder: *
*
Lnn+l(X,A , ~ 0 L ) nn(A, x O ) 5nn(X, ~ 0 L ) nn(X, A , xo) +a
1.
I n
Hn+l(X, A ) 2+ Hn(A)
I n
I+ H n ( X ) L
P
Hn(X, A )
* * *
5 * * .
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 (11; § 9), we defined the notion of the nconnected spaces. In terms of homotopy groups, one can easily prove that, for a given integer n 2 0 , a space X is nconnected iff it is pathwise connected and n,,(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 homomorphism hn is an isomorphism. Proof. If X is the nsphere Sn, then the theorem is given already by the Hopf theorem in (11; 3 8). By (3.2), it follows easily that the theorem holds for the case X = SnVSn. Then, by means of (3.1) and finite induction, one can easily prove the theorem for the case where X is the onepoint union of a finite number of nspheres. Next, assume that d i m X Q r t . If we identify the ( n  1)dimensional skeleton Xnl of X to a single point yo, we obtain a quotient space Y with a natural projection p : ( X , X"') ( Y ,Yo). +
If Y is different from yo, then it is obviously homeomorphic to the onepoint union of a finite number of nspheres with yo as the common point. Pick a vertex X ~ Xn1. E Since X is (n  1)connected, Xnl must be contractible to the point xo in X . By an application of the homotopy extension property, it follows that there exists a homotopy f t : X X , (0 t l ) , such that f,, is the identity map, fl(Xnl) = xo, and f t ( x o ) = xo for each t E I . By (I; 6.1), the map /, determines a map q : ( Y ,yo) ( X , xo) such that the
.
+
0. Furthermore, we define for every n > 1
n n ( X ;A , B , xo) =nnl(P, Q, 0 0 ) . If n > 3, then n n ( X ;A , B , xo) is a group which is abelian in case n > 3. This group is called the nth homotopy group of the triad ( X ;A , B ) a t the basic point x,. For completeness, we shall also call n,(X; A , B , xo), n > 2, the nth homotopy set of ( X ;A , B ) at x,. Thus, the homotopy sequence of the triplet ( P ,Q, a,) gives rise to the following exact sequence * * Ln,+l(X;A,B,xo) Lnn(A,C,xo)Lnn(X,B,x,) L z ~ ( X ; A , B , X O )a  +
...Lnz(X;A,B,xo)Ln,(A, C, xo) Ln , ( X , B , xo) which will be called the first homotopy sequence of the triad ( X ;A , B ) at the basic point x,. The homomorphisms
i, :nn(A, C, xo) + n n ( X , B , x,), n
> 2,
are called the excision homomorphisms, [ES; p. 2071. Thus the triad homotopy sets n n ( X ;A , B , x,), n 2 2, measure the extent by which theexcision axiom fails for the relative homotopy groups. The above definition of nn(X; A , B, x,), n 2, gives a geometrical representation as follows. An element of n,(X; A , B , xo) is represented by a
10. T H E H O M O T O P Y G R O U P S O F A T R I A D
161

map f : I n + X such that, for any point t = (t1; * , tn) in the boundary aIn of I n , f (t)E A if t,l = 0, f (t)E B if tn = 0, and f (t) = xo otherwise. Precisely, the elements of n n ( X ;A , B , xo) are the pathcomponents of the space of these maps. See [Blakers and Massey 11. Consider the homeomorphism h : I n + I n defined by
h(t1,. * tn2, in1, in) = (ti,. * * , tn2, tn, tn1). Then the assignment f + f k induces an isomorphism e t
h* : n n ( X ;A , B , ~ 0 M) n n ( X ;B , A , ~ 0 ) for every n > 2 and a onetoone correspondence for n = 2. Hence the first homotopy sequence of ( X ;B , A ) a t xo gives rise to the following exact sequence * *
* L n n + 1 ( X; A,B,xo)a'+,n(B,C,xo)i~~n(X,A,xo)~+nn(X ;A,B,xo)a'+ * * * * *
G+ n 2 ( X ;A , B , x0) X+ n,(B, C, x0) .G.+ n l ( X ,A , x0)
which will be called the second homotopy sequence of the triad ( X ;A , B ) at the basic point xo. Another geometrical representation of n n ( X ;A , B, xo), n > 2, can be described as follows. Consider the unit ncell En in the euclidean nspace Rn and its boundary (n 1)sphere Snl. Let E?l and E71I denote the hemispheres of 9  l defined by tn > O and tn GO respectively. Let so = (1,O; * * , 0). Then the elements of n n ( X ;A , B, xo) are the pathcomponents of the space of maps f : ( E n ;E:, EE,so) + ( X ;A , B, xo).
As an illustrative example of the triad homotopy groups, let us take ( X ;A , B ) to be the triad given by
X =S2, A
=
E:,
B
=
Ea.
Then we have C = S1.Take xo = (1,0,0). Since A and B are both contractible in themselves, it follows from the exactness of the homotopy sequences of the triplets ( A ,C, xo) and ( X , B , xo) that
n n ( A ,C,xo) m nnI(C, xo), n n ( X ,B , ~ for every n > 0. On the other hand,
0 M )
n n ( X ,xo),
nn(A,C,~ 0 M) nnl(S1) = 0 for every n 2 3. This implies that j* in the first homotopy sequence of ( X ;A , B ) is an isomorphism for every n > 3. Hence
n n ( X ;A , B, x0) m nn(S2), n > 3. Finally, it is easy to see that in the following part of the first homotopy sequence of ( X ;A , B)
O+ n,(X,B,%o)tn s ( X ; A,B,xo)t n 2 ( A , C , x o ) Lnc,(X,B,xo)+~ ~ ( X ; A , B J O )  + O ,
1b2
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
the groupsn,(A, C, xo) andn,(X, B, xo) are free cyclic andi, sends a generator of n,(A, C, xo) onto a generator of n,(X, B, xo). Hence i, is an isomorphism. This implies that n a ( X ;A, B, xg) = 0,
n 3 ( X ;A, B, x0) w n,(X, B, x0) w n3(S2)w 2.
11. Freudenthal's suspension Let (X; A, B) be a given triad such that both A and B are contractible in themselves. Set C = A n B and pick a basic point xo E C. Consider the spaces of paths U = [ X ;A , B ] , V = [C; C, C ] and denote by a, the degenerate path uo(I) = xo. Since a , ~ V cU,we obtain the following exact homotopy sequence of the triplet (U, V , uo): '
 &nn+Ju,v,a),
Ln,(V,uo) %nn( U,aJ & nn(u,V ,uo)% ... Ln,(u,v,uo) %no(V,a), 3, no(u,a,).
 *
According to (111; 8 9), there is a homeomorphism 5 : C + V of C onto a subspace 5(C) of V . 5 is called the natural injection of C into V and is defined by taking t ( x ) to be the degenerate path at x for every x E C. By (111; 9. lo), t ( C ) is a strong deformation retract of V and hence we have
5* :n n ( C , xo)
nn(V,no),
2 0. On the other hand, let W = [X; xo, x O ] . Since both A and B are contractible, it follows from (111; Ex. N) that the inclusion map 7 : W c U has a homotopy inverse. This implies that
q* :nn(W, uo)
M

nn(
12
u, a o ) ,
z 0.
Tz
According to (IV; 5 9), we have the natural correspondences
x :nn+1(X,xo) W nn(W, uo),
2 0. Next, consider the space of paths 52 = [ U ; V , uo]. By (111; 9.9), 52 can be considered as the space of maps f : I2 f X such that 12
f ( o , t ) ~ A , f ( l > t ) ~ Bf (, t > o ) ~ Cf (,t S 1 ) 7 x 0 for every t E I . Let Oo denote the constant map Oo(12)= xo. Then, we have the natural correspondences
x :nn+,(u,V , uo)
nn(s2,
eo),
n
> 0.
On the other hand, let P = [X;B, x O ] and Q = [ A ; C, x O ] and consider = [ P ;Q, ao].ThenA can be considered as the space of maps g: I 2 + X with g ( O , t ) E A , g(t,O)EB, g ( l , t ) = x o = g ( t , 1)
A
for every t E I . Obviously Oo E A . According t o the definition in 5 10, we have Z ~ + ~ (AX, B, , xo)
Q, uo) & nn,(n,eo),
= nnp,
12
> 1.
11. F R E U D E N T H A L . S S U S P E N S I O N
Ib3
We are going to prove that Q and A are of the same homotopy type. Let a12 denote the boundary of I 2 and define a map 0 : a I z + a12 by taking for
+(O, t ) = (0,t)l +(t, 1) = (t, I), +(I, t) = (1, 1).
+
On can easily see that is homotopic to the identity map on aI2 and hence has an extension @ : I a + P.For each f E Q, the composed map f @ is in A. The assignment f + f @ defines a map 5 : (Q, 0,) + (A,O0). By constructing another but somewhat similar map Y :I a + I a , one can prove that 5 has a homotopy inverse defined by g g Y for each g E A . Hence we have f
C*
: Z,~(Q, 0,)

n,,(A, O0),
n 1. Now let us consider the following composed transformations
r
=
5;1aX15;1X
:n n + J X ;A , B , xo) +nn(C, xo),
s = %1q;Ii*t* :n,(C, xo) +nn+1(X,xo),
nn+*(X,xo) +nn+a(X;A , B , xo) for each TZ > 0. If n > 0, then all of these are homomorphisms. The homotopy sequence of the triplet (U,V , u0) gives rise to the following exact sequence t
= Xll,Xj,q+X:
...A, n + 2 ( X ; A , B , x o ) ~ n n ( C , x o ) ~ 7 d n( +Xi J ~I+nn+,(X;A,B,xol )
. . . J+ne ( X ;A , B , xo) ".no(C,
* *
( X ,xo) which will be called the suspension sequence of the triad ( X ;A , B ) . In particular, the transformations s are called the suspensions. Thus, the triad homotopy sets nn(X;A , B , xo) measure the extent by which the suspensions s, n > 0, fail to be isomorphisms. Proposition 11.1. For any integer m
equivalent :
xo)
> 2, the following three statements are
(i) n,(X; A , B , xo) = 0 for every n < m. (ii) The suspension s :nn(C,xo) +nn+l(X, xo) is an isomorphism for each n = 1;  ,m  2 and is an epimorphism for n = m  1. (iii) The excision i , : n,(A, C, xo) + z n ( X , B , xo) is an isomorphism for each n = 2; ,m  1 and is an epimorphism for n = m. Proof. The equivalence of (i) and (ii) is a consequence of the exactness of the suspension sequence of ( X ;A , B ) , and the equivalence of (i) and (iii) is a consequence of properties of first homotopy sequence of ( X ;A , B). I If X is the rsphere with r > 2 and A = E:, B = El_,then it will be proved in the sequel that the statement (ii) is true for m = 2 r  2 . See (XI; 2.1).
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
164
Bythe definition s = Xly;li.,,E.+, it is not difficult to see that a geometrical represention of the suspension s :nn(C, xo) +nn+l(X, xo) can be described as follows.For any u € n n ( C ,x o ) , let us choose a representative map f : (9, so)+ ( C , xo). Since both A and B are contractible, f has an extension F : (Sn+l,so) + ( X ,x,) such that F(E,”+l)c A and F(E?+l)c B. Then s(a) is the element of Z ~ + ~ (xo) X represented , by this map F . This beautiful geometrical representation of s is its original definition given by Freudenthal to the special triad (9; E:, EL). As a generalization of the triad (S‘; E:, ET), let C be a given nonempty space. Consider the space X obtained by joining C to two distinct vertices a and b. Precisely, X is obtained from C x I by identifying the subsets C x 0 and C x 1 into single points a and b respectively. If fl : C x I + X denotes the natural projection, then C can be considered as a subspace of X by the imbedding i : C + X defined by i(c) = fl(c, 4) for each c E C. Let
A
={
f l ( ~t ,) I C E C ,0
0, the elements of x, ( X, xo) may be considered as the hornotopy classes of the maps of (An,8dn) into ( X ,x o ) ; for details of this definition see [Hu 121. Let n > 0 and denote by Kn1 the (n 1)dimensional skeleton of the
165
EXERCISES
+
boundary nsphere S n of the unit (n 1)simplex An+, in the euclidean (n 2)space. Consider any given map
+
f : (9, P 
1 ) +.
( X ,xo).
Since f sends the leading vertex of Sn onto x,, f represents an element [ f ] of nn(X, xo). On the other hand, let (i=O;..,n+
e;:A,,+.An+l,
I),
denote the simplicia1map defined in [ES; p. 1851. Then, the composed map fe; : ( A n , adn) +. ( X , xo)
represents an element [fe;] of n n ( X , xo) for every i Prove the following Homotopy addition theorem. For any map f : ( S n , Kfi1)+ ( X ,x o ) , we alway have
72
= 0,
1;

*,
n
+ 1.
>, 2,
n+i
[fl
.x
1)' [fe;I.
= a m 0 (
For the exceptional case, n , ( X , xo) is not necessarily abelian. However, for any map f : (9,KO) +. f X ,xo), the following relation is obvious : In the remainder of this exercise, we shall give an analog for the maps of cubic boundaries. Let rz > 0 and denote by Knl the (12  1)dimensional skeleton of the boundary nsphere a I n + l of the (n + 1)cube P + l , i.e., Knl consists of all points (t , ; . . , I n + ] ) of I n + l such that tf(l it) = 0 for at least two indices i. Consider any given map
f : (ain+l, m  1 )
+
(x,xo).
Since f sends the point (0; * * , 0) of a P + l into xo, it represents an element [f ] of n,(X, xo). On the other hand, for each i = 1 ;* * , n + 1, let yi and ( r denote the homeomorphism of I n into a I n + l defined by qt(ti,.
* I
In) =
(ti,.
*
*,
ti],
0, ti,'
* *,
tn),
t t ( L ** * , tn) = (ti,. * * , ti1, 1, t i , . * * , in). of ( I n , a I n ) into ( X ,xo) represent Then, the composed maps fqc and elements [ f q i ] and [/(*I of nn(X,xo) respectively. Then, prove that nfi
[ f l = 1.x (=1
([/&I  [fytl),
n >2.
For the exceptional case, n , ( X , xo) is nonabelian in general. However, for an arbitrary map f : ( d 1 2 , KO) + ( X , xo), the following relation holds:
166
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
B. The relative homotopy addition theorem
For any triplet (X, A , xo) and any integer n 2 2, the elements of the group n n ( X ,A , xo) can be considered as the homotopy classes of the maps of (An,adn,uo) into ( X ,A , xo), wheie uo denotes the leading vertex of An. Let n 2 2 and denote by Knl the ( n 1)dimensional skeleton of the boundary nsphere Sn of An+l. Consider any map
f : (9, Knl, uo) + (X, A , xo). If we compose f with etn :An +Awl, we obtain a map f e p of (An,adn)into (X, A ) for each i = 0, 1,* ,n + 1. If i # 0, fern maps the leading vertex uo of A , into xo and hence i t represents an element [fern] of n,(X, A , xo). Set x , = f ( u l ) . Then, /eon sends uo into x1 and so it represents an element [/eon] ofn,(X, A , xl). Let u : I + A denote the path joining xo to x1 defined by

u(t) == f ( 1  t , t , O ; . . , O ) ,
tEI.
By (IV; Ex. D), u induces an isomorphism on :nn(X,A , xi) w n n ( X ,A , xo).
Prove the following Relative homotopy addition theorem. For any map we always have
f : (Sn, Knl, uo) +. ( X ,A , xo), n > 2,
L[fl
U+l
=dfeonI
+ 3 (*1
[fan].
where [ f ] i s the element ofn,(X, xo) represented by f as a m a p of (9, vo) into (X, xo) and j* i s the homomorphism induced by the inclusion map j : ( X ,xo) c (X, A xo). State and prove the analogous theorem for the maps of (dIff+l,Kfll, uo) into (X, A , xo). 3
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 mconnected if n,(X, A , xo) = 0 for every n satisfying 1 < n < m. By means of the relative homotopy addition theorem in the previous exercise, prove the following Hurewicz theorem. If n 2 and ( X , A ) i s (n 1)connected, then the natural homomorphism
in (4.1) i s a n epimorphism and its kernel i s the subgroup generated by the elements of the form u  wa, where u ~ n n ( XA,, xo) and w ~ n , f Axo). , For the special case n = 2, prove that the kernel of x 2 contains the commutator subgroup of n,(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 nsimple for a given n 2 2, then 2. If rt
> 2 and X
~n :nn(X,A , xo)
%
Hn(X,A )
is (rc  1)connected,then It, :n,(X, X J w H,(X).

The last proposition generalizes (4.4). As an illustrative example, let us consider the space E = (9 x 2) U (so x R) of 8 7. Then, E is ( n  1)connected and hence n.(E) w H n ( E ) . This implies that nn(E)is a free abelian group with a countably infinite set of free generators. Finally, prove
3. If n > 2 and X is [G. W. Whitehead I].
(n
 1)connected, then
Itn+l is an epimorphism.
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 " . ~ n n ( A , x o ) 4 , n , ( X , x ~ ) ~ n n ( xa, A , x o )  n n  ~ (.Ln,(X,A,xo)"O A,~o)~.. I n
,
lhn
.
I n
kn1
1.
~ H n ( A ) ~ + . H n ( X , ~ + . H ~" (+X. ,~An)  ~ ( A )&+.fZ,(X,A), ~... prove the following assertions : 1. Let m > 0 be a given integer. If i, is an isomorphism for every n Q m, then so is i#. If i* is an isomorphism for every n < m and is an epimorphism for n = m,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 i# is an isomorphism also for n = m, then the kernel of i, inz,,(A, X J is contained in that of hn. Next, let X and Y be pathwise connected spaces. Consider a given map f : ( X ,xo) +. ( Y ,yo) and the induced homomorphisms f* :n n w , xo) +.nn(Y,Yo), f# : Hn(W +. Hn(Y). Using the mapping cylinder M f , and ( I ; 12.1 and 12.2), prove the following assertions : 3. Let m > 0 be a given integer. If f* is an isomorphism for every n < m, then so is f#. Iff* is an isomorphism for every n < m and is an epimorphism for n = m, then so is f#. 4. Assume that both X and Y are simply connected and m > 0 is a given integer. If f# is an isomorphism for every n < m and is an epimorphism for
168
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
n = m,then so is f,. Furthermore, i ff# is an isomorphism also for n = m, then the kernel of f, in Xn(X, xo) is contained in that of hn : n n ( X ,x0) +. Hn ( X ) Note that the condition that there exists a map f : ( X , xo) +. ( Y ,yo) such that f, is an isomorphism for each n > 0 is much stronger than the condition that n n ( X , xo) m z n ( Y ,yo) for every n > 0. In fact, there are pathwise connected spaces which have isomorphic homotopy groups but some of whose homology groups are different [Wang 11.

E. Homotopy groups of adjunction spaces
Let X be a given space and x 0 e X a given point. Consider an indexed family f, : (Sn,so) +. ( X ,xo), p E M . Give M the discrete topology and define a map f:Sn x M + X
by taking f (s, p) = fp(s) for every s E Sn and p E M . Let Y denote the adjunction space obtained by adjoining En+l x M to X by means of the map f, (I; 8 7). The inclusion map i induces the homomorphisms
i, : n m ( X ,xo) + X m ( Y , xo), m > 0. 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 n,(X, xo) is the subgroup generated by the elements wa,, where w E X * ( X xo) , and a, is the element of n n ( X , xo) represented by the given map f , for all p E M . [J. H. C. Whitehead 1, p. 2811. 3. If f,, N 0 for every p E M , the i, is a monomorphism for m = n 1 and its image in nn+l(Y,xo) is a direct summand. The complementary summand is isomorphic to the relative homotopy group nn+l(Y , X , xo) which is a free abelian group with free generators wb,, where w € n , ( X , xo) and b, ~ n n + ~X( ,Yx0) , is the element represented by the map
+
gp
x,xo)
: (E%+l,sn, xo) + ( Y ,
defined by g,(t) = p ( t , p) where fi : En+l x M projection. [J. H. C. Whitehead 1, p. 2851.
+Y
denotes the natural
F. Spaces of homotopy type (a, n)
Let az be a given group and n > 0 a given integer. If n > 1, we assume that x is abelian. A pathwise connected space X is said to be of the homutopy type (n, n) provided Xn(X) w X , n m ( X ) = 0 if m # n. Thus, by (8.4), if f i : E
+. B
is an nconnective fibering over an (n  1)
169
EXERCISES
connected space B , then the fiber F = pllb,) over any b, E B is of the homotopy type (nn(B),n  1). Construct a space X of the homotopy type (n,n) as follows. If n = 1, then n can be represented as the quotient group of a free group F over a normal subgroup R of F . If n > 1, then n is abelian and can be represented as the quotient group of a free abelian group F over a subgroup R of F. Hence IC = FIR.
Let M denote the set of free generators of F and let b, be a space consisting of a single point b,. Give M the discrete topology and adjoin En x M to b, by means of the map g :Snl x M +b,. Let A denote the adjunction space obtained in this way. Then prove that
nn(A)= F , n,(A)
=
0 if m
< 12.
For each r E R c nn(A),choose a map fr : (9, so) + ( A , b,) which represents Let B denote the space constructed by means of the family { fr I Y E R } as in Ex. E. Then we have
Y.
nm(B) = 0 if m < n. Then, by (8.2), there exists a pathwise connected space X which contains B as a closed subspace and nn(B) M FIR
= n,
n,(X) w n, n m ( X ) = 0 if m # 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 x, E X is a space of the homotopy type (n,n  1). G. The realizability theorems
By means of the spaces of homotopy type (n,n), prove the following Realizability Theorem. Let Zz,'
' '
,nn,'
* *
be a sequence of groups. All groups except possibly the first one are abelian andn, operates Onnn 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 51 and [Hu 101: (1) There exists, for each n > 0, an isomorphism hn :n n ( X , x O )
M nn.
(2) For any w € n l ( X ,x,) and a Enn(X,x,), n 2 2, we have hn(Wa) = h, ( W )hn(a). (3) For every a € n m ( X ,x,) and head product [a,!] = 0.
Enn(X,x,), m 2 2, n
> 2, the White
170
V. T H E C A L C U L A T I O N O F HOMOLOGY G R O U P S
. By constructing a cone over X , deduce an analogous realizability theorem for relative homotopy groups.
H. Topological realization of semisirnplicial complexes
For a given semisimplicial complex K as defined in (IV : Ex. H), we can construct a space I K 1 associated with K as follows. To every integer m 0 and each mcell a of K , let us associate an open msimplex 1 a 1, called the open mcell corresponding to a, which is defined to be the topological product I a I = a x Int ( A m ) , Id (Ao) = A,, of u as a single point and the interior of the unit msimplex Am. Then we define the closed mcell Cl I a I to be the set Cllal = U t < ~ l t l . There is a natural function Xu of A m onto C1 I a I defined linearly on each open simplex of the complex A m , [Hu, 101. Give CZ I a I the identification topology determined by Xu, ( I ; 3 6). Next, let us denote by I K I the union of all open cells I u 1, a E K , and define a topology of I K I as follows. A set W in I K 1 is called open iff W n CZ I a 1 is an open set of CZ 1 a 1 for every cr E K . This topology of I K I will be called the Whitehead topology of I K I ; it is the largest topology of I K I such that the topology of every closed cell CZ I a I coincides with the relative topology in I K I. Following a usual custom in combinatory topology, we identify K with I K I and a with 1 a 1. Thus, we may consider any semisimplicial complex K as a topological space which is a union of a collection { a } of disjoint open cells. For example, the singular complex S ( X ) of a space X will be considered as a topological space in this way. For each cell a of K , it follows immediately that the natural function Xo : A m * C1 a c K , where m = dim (a), is a continuous map of A m onto C1 a and the restriction ;C, 1 Id (Am) is a homeomorphism of Int (Am) onto the open mcell a. We shall call Xu the characteristic map for the cell a E K . Prove the following assertions: 1. If X is a compact subspace of a semisimplicial complex K , then X meets at most a finite number of open cells of K . 2. A function f : X + Y of a closed or open subspace X of a semisimplicia1 complex K into a space Y is continuous iff the restriction f I X n C b is continuous for every closed cell Cla of K . 3. A family ft : X + Y , (0 < t < l ) , of functions of a closed or open subspace X of a semisimplicial complex K into a space Y is a homotopy iff the family fl I X n Ckr, (0 < t < l), is a homotopy for every closed cell Ckr of K . 4. Every subcomplex L of a semisimplicial complex K is closed and has the AHEP in K.
EXERCISES
17'
5. A necessary and sufficient condition for a semisimplicial complex K to be a simplicial complex with locally ordered vertices is that, for each mcell U E K , the characteristic map Xo is a homeomorphism of A , onto Cl u and, for any two cells u and t of K , Cl u f l C1 t is either empty or a closed cell of K . 6. For each n > 2, the nth barycentric subdivision, defined in the obvious way, of a semisimplicial complex K is a simplicial complex with locally ordered vertices. Hence, every semisimplicialcomplex is triangulable. 1. The projection w : S(X) + X
For any given space X, there is a natural projection o of the singular complex S(X) onto X as follows. For an arbitrary point p of the space S(X), let 0 denote the unique open cell of S(X) which contains p . Then, u is a singular simplex :A , = dim (a) ~
x,
and the characteristic map X, sends Int A , homeomorphically onto the open cell a of S(X). The natural projection w : S ( X ) + X i s defined b y taking U(P)
= uX,l(p),
p € u c S(X).
Prove the following assertions: 1. The projection w : S ( X ) + X is a map of S(X) onto X. For every subspace A of X, w carries the subcomplex SIA) onto A . 2. The space X is pathwise connected iff S(X) is connected. Next, let (X, A , xo) be any given triplet and let Po denote the vertex of S(X) such that w(po) = xo. Thus we obtain a triplet (S(X),S ( A ) ,Po), and a map w : ( S ( X )S , ( A ) ,Po) + (X,A , xo) defined by the natural projection w . Prove the following important proposition, [Giever 1 and J. H. C. Whitehead 71. 3. The induced transformations W + of w are onetoone correspondences, namely, w* : nm(S(X),S ( 4 , Po) M n,(X, A , xo), w*
:nm(S(X),Po) w n m w , x0)r
a*
:nm(S(A) P O ) I
= nm(A xo). 9
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 , + K , of a semisimplicial complex K , into another such complex K , is a function which assigns to each mcell u of K , an mcell t = T(a)of K , in such a fashion that t ( 0 = T ( u ( ~ ) )(i , = 0 , 1;**, m).
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
172
K , induces a unique map T(a)E K,. A map f : K , + K , is said to be cellular if it carries K,m into K,m for every m = 0, 1; .Thus, the map ffp : K , + K , induced by T is cellular. Consider a given map : K + X of a semisimplicial complex K into a Prove that a cellular transformation T : K ,
+
fT : K , + K , which carries a E K , barycentrically on t =
+
+
space X. induces a cellular transformation
T+ : K +S(X) as follows: For each mcell a E K , msimplex t
t=
T+(a)is defined to be the singular
=+X,:A,+X
whereX,denotes the characteristic map of a. Verify that t ( C ) = T+(a(")for everyi=O,  *  , m . This cellular transformation T+ induces a cellular map +# : K +S(X) which will be called the induced cellular mu# of 4. Prove that w+# = As an application of this, let X = K and take to be the identity map. Then +# 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 T+ defines a chain equivalence. The second assertion implies that the homology and cohomology groups of a semisimplicial complex K are topological invariants. Next, let X , Y be spaces and consider a given map f : X + Y . f induces a cellular transformation T f :S ( X ) + S(Y )
+
defined by
t =
+.
T f ( a )= fa :dm+ Y
for every singular msimplex u E S ( X ) . This cellular transformation T f induces a cellular map f # : S ( X ) + S (Y ) which will be calIed the induced celluEar map of f . Obviously, f# is also the induced map of = f w : S ( X ) + Y . Verify that, in the following diagram, we always have f w = w f # :
+
S ( X ) ___ f # + S ( Y )
I
x
.I
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 xo of
X has been selected as basic point.
I73
EXERCISES
Two singular msimplexes u and t in X are said to be compatible if their faces coincide, that is to say, a(') = t(')for each i = 0; * , m. Hence o and t are compatible iff u I s m  1 = 7 I sm1, P  1 = ad,.

Further, two compatible singular msimplex u and t are said to be equivalent if there exists a homotopy ht : A , + X , (0 < t Q l ) , such that h, = (I, h, = T, and ht(d) = u ( d ) for every d E S m  l and every t E I. For m = 0, any two singular 0simplexes are compatible; and, since X is assumed to be pathwise connected, they are also equivalent. A singular simplex u : A , f X is said to be collapsed if a(&) = x,. The set of all collapsed singular simplexes constitute the subcomplex S(x,) of S ( X ) associated with x, as a subspace of X . A subcomplex L of S ( X ) is called an admissible subcomplex of S ( X ) if the following conditions are satisfied: (ASl) S(x,) c L. (AS2) If u is a singular simplex such that a(') E L for each i = 0, * ,dim (a), then L contains at least one singular simplex t which is compatible and equivalent with u. If, in the condition (AS2), we require that L contains one and only one singular simplex t which is compatible and equivalent with u, then L is called a minimal subcomplex of S ( X ) . Obviously, S ( X ) is an admissible subcomplex of itself and every minimal subcomplex of S ( X ) is admissible. Let the admissible subcomplexes of S ( X ) be partially ordered by means of inclusion. Then a minimal subcomplex M of S ( X ) is clearly minimal in the sense that, if any admissible subcomplex L is contained in M , then L = M . Prove the following assertions : 1. Every admissible subcomplex L of S ( X )contains a minimal subcomplex it4 of S ( X ) . 2. If L is an admissible subcomplex of S ( X ) ,then there is a function D , called a deformation of S ( X ) into L , which assigns to each singular msimplex u a singular ( m+ 1)prismD(u) = P,,[ES; p. 1931, subject to the following conditions :

(i) D(a") = (P,)S, (i = 0; * * , m). (ii) (P,)z= o. (iii) (P,),, is in L. (iv) If u E L, then P,(d, t ) = u(d) for every d E A , and every t E I . Now let A be a pathwise connected subspace of X which contains the basic point x,. An admissible subcomplex L of S ( X ) is said to be relatively admissible with respect to S ( A ) if L n S ( A ) is an admissible subcomplex of S ( A ) . In this case, the deformation D can be so chosen that, for each U E S ( A ) , P, is a singular prism in A .
I74
V. T H E CALCULATION O F HOMOLOGY G R O U P S
3. If L is an admissible subcomplex of S ( X ) which is relatively admissible with respect to S ( A ) , then the inclusion cellular map
1 : (L,L n s ( A ) )t (s(x), SM)) is a homotopy equivalence and defines a chain equivalence. L. The Eilenberg subcomplexes of S(X)
Let X be a pathwise connected space, A a pathwise connected subspace of X , and x, a given basic point selected from A . For each integer n > 0, define a subcomplex S,(X, A ) of the singular complex S ( X ) as follows. A singular simplex u : d , + X is in S , ( X , A ) iff the following conditions are satisfied: (ESl) u sends the vertices of A , into xo. (ES2) u sends all faces of d, of dimension less than n into A . In particular, if A = x,, then we simply write S,(X) = S,(X, xo). S,(X) is called the nth Eilenberg subcomplex of S ( X ) ; and, in the general case, S , ( X , A ) will be called the nth relative Eilenberg szlbcomplex of S ( X ) with respect to A . For completeness, we also define S,(X, A ) = S ( X ) . We thus obtain a descending sequence of subcomplexes of S ( X ) , namely,
S ( X ) = S,(X, A ) 2 S,(X, A ) 2
* * *
2
Sn(X,A ) =I
* *

=I
S,(X, A )
= S,(A).
Prove that, if ( X ,A ) is ( n  1)connected for a positive integer n, then S , ( X , A ) is an admissible subcomplex of S ( X ) and is also relatively admissible with respect to SIA). In particular, S,(X) is a n admissible subcomplex of S ( X ) provided that X is ( n  1)connected.
CHAPTER V I O B S T R UCTlO N TH E O RY
1. Introduction Having studied the homotopy groups in the previous chapters, we are now in a better position to investigate our main problems described in the first chapter. For the sake of simplicity, we shall restrict ourselves to the study of the maps of a finite cell complex K , [ S ; p. 1001, into a pathwise connected space Y . Therefore, we assume throughout the present chapter, that K is a finite cell complex, L a subcomplex of K , and Y a given pathwise connected space with a given point yo E Y . We shall denote by Kn the ndimensional skeleton of K which consists of the cells of dimension not exceeding n and use the notation Kn = L U Kn. For example, let us study the extension problem. Consider an arbitrarily given map f:L+Y.
As described in (I; tj 2), the extension problem for f over K in the restricted sense is t o determine whether or not f can be extended continuously throughout K . Since L has the AHEP in K according to (I;9.2), this restricted extension problem is equivalent to the broadened problem described a t the end of (I; tj 9). Since K is a cell complex and L is a subcomplex of K , there is a natural approach to attack this problem, known as the obstruction method. We first try to extend step by step the map f over the subcomplexes
En, n
= 0,
1,2;
 ..
We shall carry on this stepwise extension until we meet some obstruction to further extension. Then we propose to measure this obstruction and try to change the already constructed partial extension of f so that this obstruction might vanish and hence further extension would be possible.
2. The extension index In order to study the extension problem, let us define the notion of nextensibility. A map f : L + Y is said to be nextensible over K for a given integer n > 0 if it has an extension over the subcomplex E n of K . 175
176
VI. O B S T R U C T I O N T H E O R Y
Since Y is assumed to be pathwise connected, the following proposition is obvious. Proposition 2.1. Every map f : L + Y is 1extensible over K.
For a given map f : L + Y, the least upper bound of the set of integers n such that f is nextensible over K is called the extension index of f over K. Since L has the AHEP in En, the following proposition is obvious. Proposition 2.2. Homotopic maps have the same extension index.
Now let e : Y + Y' be any map and let g : (K', L') + (K, L ) be any cellular map, [S; p. 1611. Then the following proposition is obvious. Proposition 2.3. If a map f : L + Y is nextensible ovev K , then the composed map f ' = e f g : L' + Y' is nextensible over K'. In particular, let ( K ,L) be a subdivision of (K', L') and g be the identity map. If f : L + Y is nextensible over K, then f' = fg : L' + Y is also nextensible over K'. By the classical method of simplicial approximation, one can easily deduce from (2.2) and (2.3) the following Proposition 2.4. If (K, L) is a simplicial pair, then the extension index of any map f : L + Y is a topological invariant, i.e., it does not depend on the triangdation of the pair (K, L ). Because of this it seems that we should prefer simplicial complexes in our treatment. However, to unify the study of the extension problem with that of the homotopy problem and to avoid awkward repetitions, we have to consider cell complexes which are not simplicial.
3. The obstruction c"+'(g) Throughout $3 312, let n be a given positive integer. For the sake of convenience, we assume that the given pathwise connected space Y is nsimple in the sense of (IV; $ 16). In this important case, every map of any oriented nsphere into Y determines an element of the homotopy group nn(Y); the latter group is abelian and will be used as the coefficient group of the cohomology groups in these few sections. If Y fails to be nsimple and n > 1, then one can deduce similar results by using local coefficients. See Ex. A a t the end of the chapter. In the present section, let us consider a given map g:Kn+Y. This map g determines an (n + 1)cochain cn+l(g)of K with coefficients in the homotopy uoupnn(Y) as follows. Let u be any In + 1)cell of K. Then the settheoretic boundary du of u is an oriented nsphere. Since du c Kn, the partial map gu = g I du determines an element [go] of nn(Y). Then cn+l(g) is defined by taking [cn+'(g)l(ul = [gbl Enn(Y)
3.
T H E O B S T R U C T I O N C"+l
+
for every ( n 1)cell c7 of K. This (n the obstruction of the map g.
(g)
I77
+ 1)cochain cn+l(g) of K
Lemma 3.1. The obstruction cn+l(g) is a relative (n modulo L ; in symbols, cn+l(g)E Z ~ + ~ L( ;Kn,n ( Y ) ) .
is called
+ I)cocycle of
K
Proof. Let us first prove that cn+l(g) is in Cn+'(K,L;n,(Y)). For this purpose, let a be any (n 1)cellof L. Since g is defined on C l o c L , g, has an extension over Cla and hence [g,] is the zero element of nn( Y ).This proves that cn+l(g) is a cochain of K modulo L. Next, let us prove that cn+l(g) is a cocycle. For this purpose, let c7 be any (n 2)cell of K . It suffices to show [dcn+l(g)](a) = 0. Let B denote the subcomplex i3a of K and Bn the ndimensional skeleton of B. Then we have the homomorphisms
+
+
Cn+,(B)LZW(B) = Zn(Bn) = Hn(Bn)+nn(Bn) h &+nn(Y), where h denotes the natural homomorphism and k , is induced by the partial map k = g I Bn. 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 , sincenn(Y )is abelian. Hence, in either case, we obtain a welldefined homomorphism
+
&h' : ZW(B)+nn(Y). Since Cn,(B) is a free abelian group, the kernel &(B) of i3 : CJB) + C,,(B) is a direct summand of Cn(B).Therefore, the homomorphism I# has an extension d : Cn(B)+nn(Y). =
+
For every ( n 1)cell t in B , the element [cn+l(g)](r) is represented by the partial map k I at. Therefore, it follows that [c"+l(g)](t) = k,h'(&) = d(&). This implies that
[c"+l(g)](aa) = d(&) = 0. Hence cn+l(g) €Zn+l(K,L ; n , f Y ) ) . I Because of (3.1), cn+l(g) is usually called the obstruction cocycle of g. The following two lemmas are obvious. [dc"+'(g)](a)
=
iff cn+l(g) = 0. + Y has an extension over Lemma 3.2. The map g : Lemma 3.3. Ifgo,g, : Kn+ Y are homotopic mups, then cn+l(g,) = cn+l(g,). Now let (K', L') be another cellular pair and I# : ( K ' , L') + ( K ,L ) be a proper cellular map, [S; p. 1611. For a given map g : Kn + Y , we obtain a composed map g' = g+ : K ' n + Y . Since I# is proper and cellular, it induces a unique cochain homomorphism, [S: p. 1611, +# : Cn+l(K,L;nn(Y )) + C*+l(K',L'; n,(Y)).
178
VI. O B S T R U C T I O N T H E O R Y
Lemma 3.4. cn+l(g') = @cn+l(g).
To prove this, one must show that the two sides have the same value on an arbitrary (n 1)cell u of K'. For this purpose, one should consider the minimal carrier of u and pick an (n + l)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. 1681.
+
4. The difference cochain In the present section, we are concerned with two given maps
go, g, :9. + Y which are homotopic on En1; we shall see that the difference of the obstruction cocycles of go and g , is a coboundary. For this purpose, consider a homotopy ht : R*1+ Y, (0 < t < l), such that h, = go I En, and h, = g, I En1. Regard the closed unit interval I as a cell complex composed of two 0cells 0 and 1 and the lcell I , where 60 =  I and 61 = I . Then the topological product J = K x I is also a cell complex. We shall denote by J n the ndimensional skeleton of J and use the notation
$. = ( L x I ) U Jn Define a map F :
=
( E n x 0) U (Rnl x I ) u (Rnx 1).
+ Y by taking
go(x), ( X E R . , t = O), h:(x), (%€En',t E I ) , ( x €En,t = 1). gl(4, Then, according to 9 3, this map F determines an obstruction cocycle cn+'(F) of the complex J modulo L x I with coefficients inn,(Y). It follows from the definition of F that cn+'(F) coincides with cn+l(g0) x 0 on K x 0 and with cn+l(g,) x 1 on K x 1. Let M denote the subcomplex ( K x 0) U (L x I ) U ( K x 1) ofJ = K x I . Then it follows that F(x,t)
=
(
c*+'(F)  cn+l(gO) x 0  cn+l(g,) x 1 is a cochain of J modulo M with coefficients in nn( Y).Since u + CJ x I is a 1  1 correspondence between the ncells of K \ L and the (n + 1)cells of J \ M , i t defines an isomorphism
k : Cn(K,L ; n n ( Y ) )w Cn+l(J,M ; n m ( Y ) ) . Hence there is a unique cochain &(go, gl; h:) in Cn(K,L ;n,(Y)) such that (i)
k&(go, g , ; ht)
= [
l)m+1{ cn+l ( F )cn+l(go) x 0 Cn+l(gl) x 1 }.
4.
I79
T H E DIFFERENCE COCHAIN
This cochain @(go, g,; ht) will be called the deformation cochain. In particular, if go I En1 = g, I En1 and ht(x) = go(x) for every x E Rn1 and every t, we abbreviate &(go, g,; ht) by &(go, g,) and call i t the di#erence cochain of go and g1. The following lemma is an immediate consequence of (3.2). Lemma 4.1. The homotopy ht : En, + Y has an extension ht* : zn+ Y, ( 0 < t < l), such that ho* = go and h,* = g, i# dn(go, g,; ht) = 0.
The importance of &(go, g,; ht) is shown by a coboundary formula given in the following Lemma 4.2. ddn(go,g,: ht)
= cn+l(g0) cn+'(gl).
Proof. Since 61 = 0, i t follows that the isomorphism k commutes with 6. Hence, we have k6dn(go, gi ; ht) = 6kdn(go,gi : ht)*
On the other hand, since cn+l(F),cn+l(g,), cn+I(g,) are cocycles and since 60 =  I and 61 = I, we may apply 6 to both sides of (i) and obtain
6kdn(go,g,; ht)
= ~ n +(go) l
x Iccn+l(g,)
x I.
Hence we have Since k is an isomorphism, this implies the lemma. I Next, let us consider any given map go : Rfi + Y and any given homotopy Y , (0 < t < l), such that h, = go I En,. We shall establish the ht : following existence lemma. Lemma 4.3. For every ncochain c in Cn(K,L;nn(Y ) ) ,there exists a map 1 En1= h, and &(go, g,; ht) = c.
g, : E n + Y such that g,
K . Then the baundary = (a x 0) u (a0 x II u (u x 1)
Proof. Let u be any ncell of
s
of (J x I is an oriented nsphere and hence there exists a map fa : S + Y which represents the element c(u)of nn(Y ) .Since the subspace T = (a x 0) U (80 x I ) is contractible, any pair of maps defined on T are homotopic. Hence, by the AIfEP of T in S , we may assume that fa(%
t) =
(
gob)
(if x E u, t = 0), (if x E au, t E ;).
Thus, we may define a map g, : K* + Y by taking
Then it is obvious that @(go, g,; ht) = c. I As an immediate consequence of (4.2) and (4.3) we have the following
I80
VI. O B S T R U C T I O N T H E O R Y

Corollary 4.4. For each cocycle z E Zn+l(K,L ;nn(Y))such that z cn+l(go) mod L, there exists a map g, : Rn +. Y such that g, I En1 = h, and cn+l(g,) = z .
In particular, if the homotopy ht is given by ht = go 1 Eni for every t E I, then the corollary gives the existence of a map g, : En + Y such that go I En1 = g , I En1 and cn+l(g,) = z . Next, let us consider three given maps go,g,, g, : an+Y and two homotopies ht, j t : En1 + Y , (0 < t < l), such that go I En,,
h, = g, I En, = j,, jl = g, I En,. Let Kt : En,+Y , (0 < t < l), denote the homotopy defined by Kt = h Z f if t < 8 and kt = j z t  , if t 2 4. Then we have the following addition lemma the proof of which is left to the reader, [S; p. 1731. h,
=
At) = dn(go, g l ; ht) 4dn(gl, g 2 ; i t ) . Finally, let us go back to the maps go,g, and the homotopy ht described a t the begnning of the section. Let (K',L') be another cellular pair and : ( K ' ,L') +. ( K ,L) be a proper cellular map. Let
Lemma 4.5. dn(go, g;,
+
g'c
Since
=
gt+ :E'n
+.
Y , h't
=
h d : E'n1+. Y .
+ is proper and cellular, it induces a unique cochain homomorphism Cn(K,L ; n n ( Y ) )+Cn(K', L';nn(Y)).
The following invariance lemma is an easy consequence of (3.4). Lemma 4.6. dn(g',, g',; h't) = +#dn(g,, g,; ht).
5. Eilenberg's extension theorem As in
5 3, let us consider a given map g:En+Y.
According to 8 3, g determines an obstruction cocycle cn+l(g) in Zn+l(K,L; nn(Y ) )and hence an obstruction cohomology class
yn+'(g) €Hn+l(K,L ; n n ( Y ) ) represented by cn+l(g). Theorem 5.1. yn+l(g) = 0
iff there exists a map h* :
h*
I xn1
=g
I if.,.
f
Y such that
Proof. Suficiency. Assume the existence of h*  and let h = h* IEn. By (3.2), we have cn+l(h) = 0. Since g 1 E m  1 = h 1 Kn1, the difference cochain dn(g, h) is defined. By (4.2),cn+l(h) = 0 implies that cn+l(g)is the coboundary of dn(g, h). Hence yn+l(g) = 0. Necessity. Assume that y"+'(g) = 0. Then cn+l(g) N O mod L. By (4.4), there exists a map h : En + Y such that g I En1 = h I En1 and @+'(A) = 0. Then, by (3.2), h has an extension h* : En+,+ Y . I
6.
T H E O B S T R U C T I O N SETS FOR E X T E N S I O N
181
Now, let us assume that g is an extension of a given map f : L + Y.If the obstruction cocycle cn+l(g)is nonvanishing, then it follows from (3.2) that g cannot be extended over and hence our stepwise extension process faces an obstruction. The significance of Eilenberg’s extension theorem (5.1) is that, if cn+l(g) 0 mod L, then this obstruction is removable by modifying the values of g on the open ncells in K \ L only.

6. The obstruction sets for extension Let f : L +. Y be a given map. We are going to define the (n + 1)dimensional obstruction set On+l(f ) c Hn+l(K,L ;nn(Y)) as follows. I f fis not nextensible over K , we define O,+l(f) to be the vacuous set. Now, suppose that f is nextensible over K . Then there exists an extension g : 8, + Y of f. The cohomology class yn+l(g)in Hn+l(K,L; n,( Y))is called an (n + 1)dimensional obstruction element of f . Then, On+l(f) is defined as the set of all (n + 1)dimensional obstruction elements of 1. The following proposition is obvious. Proposition 6.1. Homotopic maps have the same (n
obstruction set.
+ 1)dimensional
+
Next, let (K’, L’) be another cellular pair and : (K’, L’) + ( K ,L ) be a proper cellular map. This map induces a homomorphism +* : Hn+l(K, L;n,(Y)) + Hn+l(K‘,L‘;n,(Y)). Then the following proposition is an immediate consequence of (3.4).
+
Proposition 6.2.
If 1‘= f+ : L’ +. Y , then +* sends O.+l(f) into O.+l(f’).
+
In particular, if ( K ,L) is a subdivision of (K’, L‘) and if is the identity map, then +* is an isomorphisni and sends On+l(f)ontoa subset of O,+l(f’) in a 11 fashion. Furthermore, if both ( K ,L) and (K’, L’) are simplicial, then the identity map 4l is homotopic to a simplicial (and hence proper cellular) map. This implies that +* sends O.+l(f) onto On+l(f‘).Hence, the (n 1)dimensional obstruction set is smaller on a subdivision and is smallest if ( K , L) is a simplicial pair. Thus, we have the following
+
Corollary 6.3. If ( K ,L) is a simplicial pair and f : L +. Y is a map, then Oa+l(f) is a topological invariant, i.e., it does not depend on the triangulation
( K ,L ) . Now, let ( K ,L ) be a triangulable pair without any given triangulation and f : L +. Y be a given map. Because of (6.3),it makes sense to talk about On+l(f). Further, if : (K’, L‘) + ( K ,L ) is a map of another triangulable pair (K’, L’) into ( K ,L ) , then one can deduce easily from (6.1) and (6.2) that +* sends On+l(f) into On+l(f’),where f’ = f+ : L’ + Y. For the remainder of the section, let us go back to the cellular pair (K, L) and the map f : L + Y given a t the beginning of the section. The following
of
+
182
VI. OBSTRUCTION THEORY
two fundamental lemmas are easy consequences of the definition of On+l(f) and Eilenberg's extension theorem (5.1). Lemma 6.4. The
empty.
m a p f : L + Y is nextensible over K iff On+l(f ) is non
Lemma 6.5. The map f : L + Y is ( n + 1)extensible over K iff On+l(f) contains the zero element of Hn+l(K,L ; n , ( Y ) ) .
By recurrent application of these two lemmas, one can easily prove the following Proposition 6.6. If Y is rsimple and
H'+'(K, L ; n r ( Y ) )= 0 for every r satisfying n < r < m, then the +extensibility of f : L + Y over K implies its mextensibility over K . In particular, if K \ L is of dimension not exceeding m, then the hypothesis of (6.6) implies that a map f : L + Y has an extension over K iff it is nextensible over K . Hence, we have the following Corollary 6.7. If Y is rsimple and H'+'(K, L;nr(Y ) )= 0 for every r > 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 f I L = g I L. As described in (I; 5 8), the homotopy problem (relative to L) is to determine whether or not f and g are homotopic relative to L, in other words, whether or not there exists a homotopy ht : K + Y , ( 0 < t < I), such that h,, = f, h, = g, ht I L = f I L for each t E I . 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 nhomotopy as follows. The maps f and g are said to be nhomotopic relative to L if f 1 Rn and g I En are homotopic relative to L. If f and g are homotopic relative to L, then they are obviously nhomotopic relative to L. Since Y is assumed to be pathwise connected, the following proposition is obvious. Proposition 7.1. Every pair of maps f, g : K + Y with f I L Ohomotopic relative to L.
=
g I L are
8. T H E O B S T R U C T I O N dn(f,g;ht)
183
The least upper bound of the set of integers n such that f and g are nhomotopic 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 3 2. Proposition 7.2. If f II f' and g N g' relative to L, then the pair (f', g') has the same homotopy index relative to L as the pair ( f , 9). Proposition 7.3. I f ( K ,L ) i s a simplicia1 pair, then the homotopy index of any pair of maps f , g : K + 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, flL=glL which are ( n 1)homotopic relative to L. Let ht
:if"'+
Y , (0 < t Q l),
be a given homotopy such that ho = f I zn1,h, = g I and ht I L = f I L for every t E I . Since f and g are defined on En, the construction in fj 4 gives a deformation cochain dn( 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( f , g ; ht) is a relative ncocycle of K modulo L ; in symbols, a n ( / , g; ht) E Z ~ ( KL ,; n n ( Y ) ) . Proof. Since f and g are defined on En+,, it follows that c"+'(f) = 0 = cn+l(g).Hence, by (4.2) we have Gdn(f, g; ht) = c"+' ( f )  cn+l(g) = 0. I Lemma 8.2. The homotopy ht has an extension ht* : En + Y , (0 < t Q I), such that ho* = f I En and h,* = g I Efl i f dn( f , g; ht) = 0. Proof. Consider the pair ( J ,M ) , where J = K x I and M = ( K x 0) u ( L x I ) U ( K x 1). Let
J . = M U J" Define a map F :
f
=
( K x 0 ) U (En, x I ) U (K x 1).
Y by taking
F(x,t) =
1 f (4,
( X E K , t =O),
(xEE"',tEI), ( x E K , t = 1).
ht(x),
g(x)
Then F determines an obstruction cocycle cn+l(F) of the complex J modulo M. By the formula (i) of fj 4,we obtain ( i)
kdn(f, g; ht)
=
(
l)n+'c"+l(F).
Since the homotopy ht has an extension ht* iff the map F has an extension
184
VI. O B S T R U C T I O N T H E O R Y
F* :P+l+ Y , the lemma is an immediate consequence of (3.2) and the preceding formula (i) since k is an isomorphism. I The cocycle dn( f, g; ht) represents an obstruction cohomology class
W f , g ;ht) € H n ( K , L ; n n ( Y ) ) . Analogous to (5.l), we have the following Eilenberg's homotopy theorem. Theorem 8.3. & ( f , g; ht) ( 0 Q t Q l), such that
=0
ie there exists
ho* = f I En, hl* ht* I En' = ht I
=g
a
homotopy ht* : g n + Y ,
I En, (1 E I).
Proof. Consider the map F defined in the proof of (8.2). Then ht* exists Y such that F* 1 p1 = F I and hence iff there is a map F* : iff cn+l(F)is a coboundary by (5.1). Since the isomorphism k in (i)commutes with the coboundary operator, this implies the theorem. I
p1
9. The group Rn(K,L;/) Let f : K + Y be a given map. We are going to construct a group Rn(K,L ; f ) as follows. Consider the space B of all maps of En1 into Y . Let W denote the subspace of D consisting of the maps g : Enl + Y such that g I L = f I L and let woE W denote the restriction w,,= f 1 znl.Then we define RnfK,L ; f ) = n,(W, w,,). An arbitrary element u of Rn(K,L ; f ) is represented by a homotopy ht : Y , (0 Q t Q l), such that
%.I+
ho = f
I En1 = h1, h t I L = f l L ,
(0 < t < I ) .
Hence we obtain an obstruction cohomology class & ( f , f ; ht) E H@(K,L ; z n ( Y ) )which clearly depends only on a. The following lemma is an immediate consequence of (4.5). Lemma 9.1.
morphism
The assignment u + En(u) = dn(f,f;ht) defines a homo[ n : Rn(K,L
; f ) +Hn(K, L ; n n ( Y ) ) .
Note. If one removes the hypothesis that Y is nsimple, then En becomes a crossed homomorphism.
Let J f n = J p ( K ,L ; n n ( Y ) )denote the image of Rn(K,L ; f ) under &,. Then J f n is a subgroup of Hn(K,L ; nn(Y)).We shall denote the quotient group by Q", = Qfn(K,L ; n n ( Y ) )= Hn(K,L ; n n ( Y ) ) / J f n . Lemma 9.2. The subgroup 3p (and hence the quotient group Q p ) depends only on the (n 1)homotopy class of f relative to L ; that i s to say, if f ,g : K + Y are ( n 1)homotopic relative to L , then J f n = Jgn.
10. T H E O B S T R U C T I O N S E T S F O R HOMOTOPY
185
Proof. Let kt : En, +. Y , (0 Q t Q l), be a homotopy such that k, = f I En,, k , = g I Kn1, and kt 1 L = f I L for each t E I . Because of symmetry, i t suffices to prove that J p c J p . For an arbitrary element ci of R n (K , L;f ) , the element tn(ci)E J p is represented by the cocycle dn( f , f ;ht) described as above. Define a homotopy ht* : En, + Y , (0 Q t < l), by taking
( x E ifn1,o Q t
( hst,(x), A, (4 klSf(X),
< #,
( x E E n 1 , i Q t Q # , (xEKn1,# Q 1). 2 Then ho* = g I = h,* and ht* I L = g I L for every t E I . Hence ht* represents an element @ of Rn(K, L ; g). On the other hand, i t follows from (4.5)that ht*(x) =
1. (iii)K is rsimple and Hr(K; n r ( K ) )= 0 for every r satisfying 1 < r dim K . ( i v ) K i s rsimple and Q f ( K ; n r ( K ) )= 0 for every dim K , where i ; K + K denotes the identity map.
Y
satisfying 1 < Y
<
, 1, every ( n 1)homotopy class relative to L contains a certain collection of nhomotopy classes relative to L . For the classification problem, we have to find a reasonable way to count the nhomotopy classes contained in a given ( n 1)homotopy class by means of some homology or cohomology invariant. Throughout the remainder of the section, let 8 be a given ( n 1)homotopy class relative to L of the maps W . We are going to enumerate the nhoniotopy classes relative to L of the maps W which are contained in 8. According to (9.2), 8 determines a subgroup J B n ( K L , ; nn ( Y ) )of the cohomology group Hn(K,L ; n,(Y)) and hence the quotient group Q O w ,L ; ~ ~ ( Y= )HW, ) L ; ~ ~ ( Y ) ) I J PL(;K~ , ( Y ) ) .
Now let us choose a map f : K Y from the class 8 as our reference map. According to (1 l . l ) , every map g E r3 determines a characteristic element X n ( f , g ) in the group QOn(K, L ; n n ( Y ) ) An . element a of Qen(K,L ; n , ( Y ) ) is said t o be fadmissible if there is a map g E 8 with Xn( f , g ) = a. The f
I 88
VI. OBSTRUCTION T H E O R Y
fadmissible elements of QenfK,L ;nn(Y)) form a set Afn, which will be . following proposition called the fadmissib2e set in Qen(K,L ;nn ( Y ) ) The is obvious. Proposition 12.1. For any two maps f, g : K + Y in the ( n 1)homotopy class 8, A f n i s the image of A p under the translation determined by their characteristic element Xn(f, g), that i s to say,
+
A/" = X n ( f , g) Agn. Now we are in a position t o establish the following general classification theorem. Theorem 12.2. Given a n (n 1)homotopy class 8 relative to L of the maps W , the nhomotopy classes relative to L of W which are contained in 8 are in a 1  1 correspondence with the elements of the fadmissible set A f n in Qe"(K, L ;n n ( Y ) )where , f i s a n arbitrarily given m a p in 8. Proof. According to (11.I), every g E 6 determines an element Xn( f, g) in Afn. We assert that Xn( f, g) depends only on the nhomotopy class relative to L which contains g. In fact, if g, h E 8 are nhomotopic relative to L , then i t follows from (11.l) that
Xn( f , h) Xn( f , g)
=
X"(g, h)
=
0,
and hence Xn( f, g) = Xn( f, h). Therefore, the correspondence g + Xn( f, g) defines a function t of the nhomotopy classes relative to L contained in 8 into the elements of Afn. That t is onto follows from the definition of Afn. It remains to prove that T is onetoone. Suppose that Xn( f, g) = X n ( f , h). Then we have X"(g, h) = Xn( f, h) X"( f, g) = 0. According to (1 1. l), this implies that g and h are nhomotopic relative to L. I Remark. Since A f n is, in general, not effectively computable, the general classification theorem given above does not solve the problem. In each particular case, i t remains to compute Afn by special methods.
13. The primary obstructions Throughout $ 9 1317, we shall assume one more condition, namely, that the given space Y is ( n 1)connected for a given positive integer n. According to (V ; 4 4), this means that nr(Y ) = 0 for every 7 < 12. If n > 1, then Y is simply connected and hence nsimple. If n = 1, we assume that Y is nsimple and hence the fundamental group nn(Y )is abelian. Let us consider an arbitrarily given map f:L+Y. According to (6.6),f is always nextensible over K . Hence the first nontrivial obstruction is the ( n + 1)dimensionalOn"( f ) . We are going to prove
13.T H E
PRIMARY OBSTRUCTIONS
189
that O n + l ( f ) consists of a single element of the cohomology group P + ' ( K ,L ; n,( Y )) * Let 8 : L + Y denote the constant map 8(L) = yo. According to (11.2), f and 8 are ( n  1)homotopic grid hence f is homotopic to a map which sends the ( n  1)dimensional skeleton Lnl of L into yo. Since homotopic maps have the same (n 1)dimensional obstruction set, we may assume that f (Lnll = yo. Since f I Ln1 = 8 I Lnl, the difference cochain &( f, 8 ) of 5 4 is defined. By (4.2), &(f, 8) is an ncocycle of L with coefficients inn,(Y). This cocycle dm( f , 8) can be equivalently defined as follows. Let a be any ncell of L. Since f (au) = yo, the partial map fo = f I a represents an element [ f O ] of n,( Y). Then d n ( f , 8) is given by [dn(f, 8)] (a) = [f,,]. The cocycle dn( f, 8 ) represents a cohomology class x"( f ) of L over nn(Y), which will be called the characteristic element of f .
+
Proposition 13.1. T h e ( n
of a single element
+ 1)dimensional obstruction set On+l(f ) consists
o""(f) = 6*x"(f) EHn+l(K,L ; n n ( Y ) ) ,
which i s called the primary obstruction to extending f over K , where 6* denotes the coboundary homomor@hism
6* : Hn(L;n,(Y))+H,+l(K, L ; n , ( Y ) ) . Proof. First, let us prove that 6*x"(f) is in O.+l(f). Since f(Lnl) = yo, f has an extension f * : i?n + Y such that f *(En \ L ) = yo. Let 8* : if, Y denote the constant map 8*(gm) = yo. Then it follows that the difference cochain dn(f*, 8*) is the trivial extension of dn(f,O ) , i.e., for any ncell u of K , we have f
Hence 6*x,( f ) is represented by the cocycle
m ( f * , e*) = C,+yf*)  cn+ye*)
= ,n+yf*).
This implies that 6*xn( f ) = yn+l(f *) E On+l(f ) . Next, let a be an arbitrary element of O.+l( f ). Then there exists an extension g* :En Y of f such that yn+l(g*) = a. By (11.2), g* and  f * are ( n  1)homotopic relative to L. Hence we may assume that g* [ Ks1 = f * I By (4.2), we have cm+l(g*) = c"+'( f *) + 6d"(g*, f *). f
Since dn(g*, f *) is in P ( K ,L ; n , ( Y ) ) according to 5 4, this implies that yn+l(g*) = yn+l(f*). Hence a = 6 * x n ( f ) . I Next, let us turn to the homotopy problem and consider a pair of maps
f , g :K
+Y
, f IL
=g
I L.
VI. OBSTRUCTION THEORY
190
Then their characteristic elements %a(/) and xn(g) in D ( K ;n,(Y)) are uniquely defined and depend only on their homotopy classes. Proposition 13.2. T h e ndimensional obstruction set O n ( / , g) consists of a
single element
o v , g) E Hn(K,L ; n n ( Y ) )
which i s called the primary obstruction to homotopy relative to L of the pair ( f , g) . Moreover x n ( f ) x n ( g ) = j*wn(f,g),
) Hn(K;nn( Y ) )denotes the h o m o m o r ~ h i s minwhere j * : Hn(K,L ; n n ( Y ) + duced by the inclusion m a p j : K c ( K ,L ) . Proof. That On+l( f, g) consists of a single element follows from (13.1) and the definition of the deformation cochain in fj 4.For the second part of the proposition, we may assume that
f(Knl) = yo = g(K"l). Let 0 denote the constant map. Then we have an(/,
e)  dn(g, e) = a n ( / ,g).
This implies that x n c f ) xn(g)
=
j*wn(f,g). I
Corollary 13.3 For any given map f : K
+ Y , we
have
JPW, L ; ~ , ( Y )=) 0 , e f w ,L ; ~ ~ ( Y= )H)W , L , ~ ~ c , ( Y H . Finally, if ( K ,L ) is a simplicia1 pair, then wn+l(f),x n ( f ) , o n ( / , g) are topological invariants according to (6.3) and (10.3).
14. Primary extension theorems By means of the primary obstructions, we shall be able to strengthen the general results in fj 6 and obtain an effective solution of the extension problem for the case where K \ L is of dimension not exceeding n 1. An element of the cohomology group Hn(L ;G) is said to be extensible over K if i t is contained in the image of the homomorphism
+
i* : Hn(K;G)+ Hn(L;G). induced by the inclusion map i : L c K . Theorem 14.1. For a given m a p f : L equivalent : (1) f i s (n I)extensible over K. (2) o n + l (f ) = 0. (3) x*( f ) i s extensible over K .
+
Y , the following statements are
+
Proof. The equivalence of (1) and (2) follows from (6.5) and (13.1), and that of (2) and (3) follows from the exactness of the cohomology sequence. I If K \ L is of dimension n 1, then we have the following corollary which includes Hopf extension theorem (11; fj 8) as a special case. See 3 17.
+
15. P R I M A R Y H O M O T O P Y T H E O R E M S
191
Corollary 14.2. A m a p f : L +. Y i s extensible over K i f its characteristic element x n ( f ) i s extensible over K . The following generalization of (14.2) is an immediate consequence of (6.6). Theorem 14.3. If Y i s rsimple and H r + l ( K , L ; n,(Y))= 0 for every r satisfying n < r < dim ( K \ L), then a necessary and suficient condition for a given m a p f : L f Y to have a n extension over K i s that the characteristic element x n ( f ) i s extensible over K . In particular, the hypothesis of (14.3) holds if ;tr(Y)= 0 whenever n 1 and assume that ( Y ,B) is nsimple in the sense of (IV; Ex. E).Then the nth relative homotopy groupnn = nn(Y,B) is abelian and may be used as coefficient group. For each nnormal map f : ( K ,L ) + ( Y ,B ) , define an ncochain cn( f ) of K with coefficientsinnn in the obvious way. Prove the following assertions, [Hu 9; p. 1941: 1. c"(f) is a cocycle of K modulo L. 2. c @ ( f ) = 0 iff there exists a homotopy f t : K Y , (0 < t < l), such that fo = f , fl[Efi) and t E I. c B, and f&) = f ( x ) for every x E 3. cn(f) is a coboundary of K modulo L iff there exists a homotopy f t : K f Y , (0 < t Q l), such that f o = f, c B, and f t ( x ) = f ( x ) for every x E En2 and t E I. Next, define the ndimensional obstruction set On(f ) in the ndimensional cohomology group Hn(K,L ; nn)in the obvious way and prove the following assertions : 4. The map f : ( K ,L) + ( Y ,B) is (n  1)deformable into B iff O*(f) is nonempty . 5. The map f : ( K ,L) + ( Y ,B) is ndeformable into B iff On( f ) contains the zero element of Hn(K, L ; n n ) . 6 . Every map f : ( K ,L) + ( Y ,B ) is deformable into B if n,(Y, B, yo) = 0 and if, for each n > 1, (Y, B ) is nsimple and Hn(K, L;nn) = 0, where n n == nn(Y,B ) . f
fl(zn)
19s
VI. O B S T R U C T I O N T H E O R Y
7. Assume that K and L are pathwise connected and xo E L. Then, L is a deformation retract of K iff n,(K, L ; xo) = 0 for every n > 1. 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. 2161. 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 Q l), such that h, is the identity map and h,( Y )c B. (iii)nn(Y,B , yo) = 0 for every n > 1.
F.
(n,n) Let Y be a space of homotopy type (n,n) as defined in (V; Ex. F) with a given point yo E Y, and let K be a finite cellular complex with a given subcomplex L and a given vertex vo E L. If n = 1, we assume that both K and L are connected. The extension, homotopy and classification problems Maps into a space of homotopy type
are completely solved for this special case as follows. 1. The case n = 1. Let f : ( L ,vo) + ( Y ,yo) be a given map. Then f and the inclusion map i : L c K induce the homomorphisms
f* : nl(L, vo) +n,(Y,Yo), i* : nl(Lvo) +Zl(K, 210). Prove that f is extensible over K iff there exists a homomorphism h : nl(K,vo) +nl ( Y ,yo) such that f* = hi.,, and that, for any such homomorphism h, there exists an extension g : K + Y o f f such that g, = h. For the homotopy problem, let f, g : K + Y be two given maps. Without loss of generality, we may assume that f (v,) = yo = g(vo). Then, we have the induced homomorphisms
f*, g* :n1(K vo) +Zl(Y,Yo). Prove that f and g are homotopic iff f* and g, are equivalent, that is to say, there exists an element 5 i n n , ( Y ,yo) such that g*(a)= t'.f*(a) . 5 , a €n1(K,yo). For the classification problem, let us consider the set H of all homomorphisms of n , ( K , vl) inton,( Y ,yl). The equivalence relation defined above divides H into disjoint equivalence classes. Prove that the homotopy classes of the maps K + Y are in a natural onetoone correspondence with the equivalence classes of the homomorphisms H. 2. The case n > 1. Then Y is nsimple and we may write n = n,(Y). The following assertions are special cases of f 14.3), (15.4) and ( 16.4). A map f : L + Y is extensible over K iff its characteristic element x n ( f ) is extensible over K . Two maps f , g : K + Y are homotopic iff x n ( f ) = x n ( g ) .
I99
EXERCISES
The homotopy classes of the maps K + Y are in a onetoone corespondence with the elements of the cohomology group Hn(K ; n) determined by the assignment f f xn( f ) . 3. Generalize the preceding results as in Ex. A and Ex. C. G. Homology groups of (n,n)
Let K and L be two semisimplicial complexes of the homotopy type (n,n) and (T,n) respectively. Then, according to the previous exercise, the classification problem of the maps K + L is solved as follows: the homotopy classes of the maps K + L are in a natural onetoone correspmdence with the equivalence classes of the homomorphismsn + t.If n > 1, then n and t are abelian and therefore these homotopy classes are in a natural onetoone correspondence with the homomorphisms n + t themselves. An important direct consequence of this result is that K and L are homotopically equivalent iff n and t are isomorphic. It follows that, for any space X of the homotopy type (n,n) and any coefficient group G, the singular homology group H m ( X ; G) and the singular cohomology group H m ( X ; G) depend only on the integers m, n and the groups n,G. They will be called the mth homology group and the mth cohomology group of the pair ( n , n ) with coefficients in G, denoted by H m ( n , n G) ; and H m ( n , n ;G) respectively. The integer n will be simply deleted from the notation if n = 1. Similarly, we shall omit the group G if it is the group 2 of all integers. In particular, if n = 1 and G = 2, these are denoted by H m ( n ) and H m h ) and called the mth homology group and the mth cohomology group of the group n. Finally, let n = 1 and let G be an abelian group on which n acts as left operators. Choose a space X of the homotopy (n,1). Since the fundamental group of X is n which acts on G, the singular groups H m ( X ; G), H m ( X ; G ) with local coefficients in G are defined. Prove that these groups do not depend on the choice of the space X and hence may be denoted by H m ( n ; G) and H m ( n ;G). H. The complex
K ( n )of a groupn
Let n be a given abstract group written multiplicatively. Define a semisimplicia1 complex K ( R )as follows. [Eilenberg and MacLane I]. 1. The homogeneous definition. For each n > 0, consider the set @n of all ordered sets (xo,* * , x,) of n + 1 elements of n called the nsets of n. Two nsets ( x 0 ; . * , xn) and (yo;*, y,) of n are said to be equivalent if there exists an element x of n such that yt = xxt for each i = 0, * * , n. The nsets @n of n are thus divided into disjoint equivalence classes, called the ncells of n. If n > 0, the ith face of the ncell 0 = [x0; *,x,] of n,0 < i < n, is defined to be the following (n  1)cell


UQ) =
[x0;
*
a ,
xt1, X f + l , '
 %,I. *,
EXERCISES
201
([q,n). In this way, n acts on K as a group of left operators. If [ # 1, prove that the homeomorphism [ has no fixed point. Hence n acts freely on K . J. The influence of the fundamental group
Let X be a pathwise connected space and x,, E X a given point as the common basic point for all homotopy groups. Study the influence of the fundamental group n = n l ( X ) upon the structure of homology and cohomology groups of the space X as follows. According to (V; 8.2), X can be imbedded in a space Y of the homotopy type (n,1) such that the inclusion map i : X C Y induces an isomorphism i, : n , ( X ) w n l ( Y ) . Let n > 1 be a given integer such that n,(X) = 0 whenever 1 < m < n. Verify the following assertions: 1. By the exactness of the homotopy sequence, n m ( Y , X ) = 0 for m Q n and 8, : n,(Y, X ) w n m  I ( X ) for m > n. 2. By the relative Hurewicz theorem (V; Ex. C), H,(Y, X ) = 0 form Q n and the natural homomorphism g :nn+l(Y ,X ) + Hn+l( Y , X ) is an epimorphism. 3. By the exactness of the homology sequence, the induced homomorphism i# Hm(X) + H,(Y) w Hm(n) is an isomorphism for m < n and is an epimorphism for m 4. In the following commutative ladder * * *
* * *
= n.
+nn,](Y, X ) 2 a + n n ( X )A+nn(Y)+
+
1.
% Hn+,(Y, X ) 2
h i k
H n ( X )2 % &(Y)
j

* *
* *

where g, h, k are natural homomorphisms, the spherical subgroup x n ( X ) = Im(h)of H n ( X )coincides with Im(d#) = Key(+). 5. The induced homomorphism
P ( X ) +z#Hm( Y ) w Hm(n) is an isomorphisni for m < n and is a monomorphism for m = n. 6. The subgroup A n ( X ) = Im(i#) of H n ( X ) consists of all elements which annihilate Z n ( X ) . Summarizing these assertions, we obtain the classical theorem : If n l ( X ) w n and n m ( X ) = 0 whenever 1 < m < n, then H m ( X )w H,(n), H m ( X ) H,(X)/C,(X) w H n ( n ) ,

H m ( n ) , m < n, nyx)W Hfl(n).
K. Computation of H m ( n ;G ) and H m ( n ;G) Let G be a given ngroup; that is to say, G is an abelian group on which az acts as left operators. A ngroup G is said to be nfree with { g,, } as a
VI. OBSTRUCTION THEORY
202
nbasis if the elements .$ga for all .$E n and gaE { g.} form a basis of G . A homomorphism f : A +. B of ngroups is called a nhomomorphism i t i t commutes with the operators. A ngroup G is said to be ninjective if, for every ngroup B and every nsubgroup A of B, every nhomomorphism f : A +. G has a nextension g : B +. G. For any ngroup G, we shall denote by I,(G) the subgroup of G which consists of the elements g E G such that 5g = g for all .$ ~ n and , denote by J,(G) the quotient group of G over its subgroup L,(G) generated by the elements &g for all .$enand g E G. Prove the following assertions : 1. H,(n;G) w J,(G), H o ( n ; G )M I,(G). 2. If G is nfree, then H,(n; G) = 0 for each m > 0; if G is ninjective, then H m ( n ; G) = 0 for each m > 0. 3. If n = 1, then Hm(n;G) = 0 = H m ( n ; G) for each m > O ; if n is a free (nonabelian) group, then H,(n; G) = 0 = Hm(n;G) for each m > 2. 4. If n is a free abelian group with r generators and if n operates simply on G, then H&; G) = 0 = H m ( n ; G ) , form > r,
~,(nG ; )W
~(1) w ~ m ( nG; I ,
form

< r,
where GI denotes the direct sum G + * * + G of j terms of G. Now let n be a cyclic group of finite order r and G a ngroup. Let be a generator of n and define a nhomomorphism d : G +. G by taking d(g) = g + tg + + Plg for each g E G. Denote the image of d by M,(G) and the kernel of d by N,(G). Then M J G ) c I,(G) and L,(G) c N,(G). 5. If n is a cyclic group of finite order r, then for each 9 > 0 we have

Hzp+l(n;G) w In(G)/Mn(G)M H'P+'(n; G ) , Hzp+'(n;G) w N,(G)/L,(G) w H2P+l(n;G). In particular,
H,p+l(n) M n M H2P+2(n), H,p+2(n)= 0
=
HQ+'(n).
6. Universal coefficient theorem. If n operates simply on G, then
Hm(n;G) R+Hm(n)8 G
+ Tor (Hml(n),G ) ,
H m ( n ; G) w Hom (Hm(n),G) + Ext ( H m  l ( n ) G). , 7. Kunneth relations. For the direct product n x t of two groups n and t, we have H m ( n x t) = Hp(n)@Hq(t) 2 TOY( H p ( n ) H , q(d). P+q=m
+
fi+q = m1
The assertions 47 allow for a complete computation of the groups
H,(n; C) and H m ( n ; G) for a finitely generated abelian group n which operates simply on G.
EXERCISES
203
L. The EilenbergMacLane complex K ( n , n) Let n be a given abelian group written additively and n a positive integer.
Define a semisimplicial complex K(n,n ) as follows. An mcell u of K(n, n) is an &cocycle of the unit, msimplex A , with coefficients in n. To define the ith face a(') of u, 0 < i < m, consider the cellular map d, :Aml + A m as defined in [ES; p. 1851. It induces a homomorphism (efm)# : Zn(A, ;n) + Zn(Am ;n). Define u(f)= (dn,)#a.If m > 1 and i < j , verify the condition (SSC).Thus we obtain a semisimplicial complex which is called the EilenbergMacLane complex K ( n ,n), [EilenbergMacLane 11. Prove that K ( n , 1) is essentially K ( n ) and that K ( n ,n) is a space of the homotopy type (n,n). M. The influence of nn(x)
Let n > 2 be a given integer and X an ( n  1)connected space. Study the influence of n = n n ( 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 (n,n) such that the inclusion map i : X C Y induces an isomorphism i, : n n ( X )w n , ( Y ) . Let q > n be a given integer such that n,(X) = 0 whenever n < m < q. Verify the following assertions as in Ex. J : l.n,(Y,X) = O f o r m < q a n d d , : n , ( Y , X ) =nm](X) f o r m > q . 2. H,( Y, X )= 0 for m < q and the natural homomorphism g : Y,X) + Hg+,(Y,X )is an isomorphism. 3. The induced homomorphism
i# : H m ( X )+ Hm(Y) M Hm(n.n) is an isomorphism for m < q and is an epimorphism for m 4. In the following commutative ladder * * *
+ng+l(Y)
I
A,3dq+](Y,X+) . a
1
n * ( X )>+
= q.
%(Y) +
1 " P
''*
  .   t H g + ] ( Y ) 1 # , H * , ] ( Y , x )a#+Hg(X)E#,Hg(Y)+..* where f , g , h, k are natural homomorphisms, the spherical subgroup & ( X ) = Im(h) of H p ( X ) coincides with Im(d#) = Ker(z#). 5. The induced homomorphism z# : H m ( Y ) + H m ( X ) is an isomorphism for m < q and is a monomorphism for m = q with A g f X ) as image. Summarizing the assertions (1)  ( 5 ) , we obtain Hm(X)M H,(n, n),Hm(X) M Hm(n,n), m < q, H 4 ( X ) / Z g ( X M) Hgfn,n ) , & ( X ) M H*(n,n).
204
VI. O B S T R U C T I O N T H E O R Y
6. The composed homomorphism
d*glj# : Hg+1(n,4 + n g ( X ) , considered as an element of Hq++l(n,n ;n p ( X ) )does , not depend on the choice of Y and is called the invariant &+l(X) of the space X . The following conditions are equivalent: (i) &+'(X) = 0. (ii) h :n g ( X )+ H g ( X )is a monomorphism. (iii) i# : H g + , ( X )+ Hg+l(n, n ) is an epimorphism. 7. If q = n + 1, then h :n g ( X )+H,(X) is an epimorphism by (V; Ex.C). This implies that Hn+l(n,n) = 0. Furthermore, K",+2(X)= 0 iff h is an isomorphism.
CHAPTER V I I C O H O M O T O P Y GROUPS
1. Introduction Let ( X ,A ) and ( Y ,B ) be any two pairs each consisting of a space and a subspace. As in ( I ; ij 8),we denote by the symbol
n ( X ,A ; Y , B ) the set of all homotopy classes of the maps of ( X ,A ) into ( Y ,B) relative to { A , B }. As described in (I; 5 8),n ( X ,A ; Y , B) is a functor contravariant in ( X ,A ) and covariant in (Y, B). If X is the nsphere Sn, A consists of a single point so of Sn and B consists of a single point yo of Y , then n ( X ,A ; Y , B ) is the underlying set of the nth homotopy group n n ( Y , y o ) of Chapter IV. As shown in the preceding two chapters, the group operation in Itn( Y ,yo) is very helpful in studying the extension and classification problems especially for the maps of Sn into Y . On the other hand, if Y is the msphere Sm and B consists of a single point so of Sm, then n ( X ,A ; Y , B ) will be simply denoted by nm(X,A ) and called the mth cohomotopy set of ( X , A ) . Under suitable conditions on ( X ,A ) , we shall see in 5 5 that nm(X,A ) forms an abelian group which will be called the mth cohomotopy group of ( X ,A ) .This group operation was first sketched by Borsuk [3] and later studied in detail by Spanier [l]. It is useful in studying the extension and classification problems for the maps of X into Sm. Between the homotopy groups and the cohomotopy groups, there is an informal duality given in 3 12 by means of the usual composition operation. Precisely, for every pair (m,n) such that nn(X,A , xo) and nm(X,A ) are abelian groups, the composition operation yields a homomorphism
nn(X,A , xo) @ n m ( X A , ) +nn(Sm, so). In particular, if m = n , then the right side can be replaced by the additive group 2 of integers.
2. The cohomotopy set n m ( X , A )
+
Let Sm denote the unit msphere in the ( m 1)dimensional euclidean space and let so denote the point (1,O; * * , 0). Then, as described in the introduction, the mth cohomotopy set nm(X,A ) of a pair ( X , A ) is defined to be the set of all homotopy classes of the maps of ( X ,A ) into (9, so) relative to A . I t is a contravariant functor in ( X , A ) . If A is empty, then it will be 205
206
VII. COHOMOTOPY GROUPS
simply denoted by nm(X) and called the mth cohomotopy set of the space X . The set nm(X, A ) has an exceptional element, namely, the homotopy class of the constant map 0 : X + so. This exceptional element will be denoted by the symbol 0 and called the zero element of nm(X, A ) . Since { so } is a closed set of Sm, one can easily see that
n m ( X ,A) = n"(X, A ) , where A denotes the closure of A in X. Thus, we may assume that A is a closed subspace of X ; however, we will not so assume unless explicitly mentioned. Finally, nO(X, A ) can be considered as the set of all open and closed subspaces of X not meeting A . If so considered, then the zero element 0 of nO(X, A ) is the empty subspace of X.
3. The induced transformations Sincenm(X, A ) is a contravariant functor in (X, A ) , every map f : (X, A ) + (Y, B ) induces a transforniation
f * :nm(Y, B) +nm(X, A ) , called the induced transformation of f on the mdimensional cohomotopy sets. For any element a in nm(Y, B ) , choose a representative map : (Y, B ) + (Sm, so), then f *(a)is represented by the composed map+f : ( X ,A ) + (Sm,s0). The zero element of nm(Y, B ) is evidently mapped into that of nm(X, A ) by f * ; in other words, f * preserves the zero element 0. Furthermore, f * depends only on the homotopy class off relative to { A , B }. In the remainder of this section, we shall prove some important properties of the induced transformations. Let us consider a pair ( X ,A ) , where the subspace A of X is nonempty. Identifying A to a point q A , we obtain a space XA. Let
+
f
:
(x, A)
+
(XAS qA)
denote the natural projection. Then f is a map which sends X A \ morphically onto XA \ q A .
homeo
Lemma 3.1. The induced transformation f* carries n m ( X ~g, ~ )onto nm(X, A ) in a onetoonefashion.
C#J
Proof. Let a ~ n m ( XA, ) be an arbitrary element represented by a map : ( x , A ) + (Sm,so). We may define a map y : (XA,q A ) + (Sm, so) by
+
The continuity of y follows from that of by (I; 8 6 ) .The map y represents an element p of n m ! X ~q,A ) . Since yf = #, it follows that f*(p) = a. Hence f* is onto.
3.
T H E INDUCED TRANSFORMATIONS
Next, let 6,q : ( X A ,q A ) + (Sm, so) be two maps such that homotopic relative to A . Then there exists a map
207
tf
and qf are
F : ( X x I,A x I) + (Sm, so) such that F ( x ,0) = &) and F ( x , 1) = q f ( x ) for every x E X . Define G : ( X A .X I , q A x I ) + (Sm, so) by taking
Then G is continuous and gives a homotopy between t and 7 relative to q A . This proves that f* is onetoone. I The significance of (3.1) is that, in the computation of the cohomotopy sets of a given pair (X, A ) , we may assume that A consists of a single point a if A is nonempty. For the case m > 1, since S m is simply connected, one can prove without difficulty that the inclusion map i : X c (X, a) induces a onetoone transformation i* of nm(X, a) onto nm(X)provided that a certain homotopy extension properties are satisfied. In particular, these homotopy extension properties are satisfied if X is a paracompact Hausdorff space and a is any given point of X. Theorem 3.2. (The map excision theorem). If f : (X, A ) + ( Y ,B) i s a relative homeomorphism, that is to say, f maps X \A homeomorphically onto Y \ B, and if X is a compact space, A is nonempty, Y is a regular Hausdorf s$ace and B is closed, then the induced transformation f * sends nm(Y, B ) onto nm(X,A ) in a onetoone fashion. Proof. Since A is nonempty, so is B. It follows that we may identify A and B to single points q A and q E respectively. Let
(XA, q A ) , v : ( y , B ) (YE, q E ) denote the natural projections of identification. One can easily verify that f induces a onetoone map g : ( X A , q A ) * (YE, q E ) of (XA,q A ) onto (Ye, 46) such that qf = g t . Since X is compact, so is X A . Since Y is a regular Hausdorff space and B is closed, it follows that Y Eis a Hausdorff space. Hence, g is a homeomorphism. By a property of contravariant functors, we have f*q* = l*g*. Since g is a homeomorphism, i t is obvious that g* is a onetoone correspondence. On the other hand, [* and q* are onetoone correspondences according to (3.1).Therefore, f * = t+g*q*lsendsnm(Y, B ) ontonm(X, A ) in a onetoone fashion. I The auxiliary conditions in (3.2) are essential. In fact, we have the following counter examples. First, let Y denote the unit mcell of the euclidean mspace and B its boundary (m 1)sphere. Let yo, y1 E B where y o # yl. Let X = Y \ yo
t :(x,A )
+
+
VII. COHOMOTOPY G R O U P S
208
and A = B \yo. Then clearly there is a deformation retraction of ( X , A ) into the pair (yl, yl). Therefore nm(X,A ) = 0 although nm(Y,B) m 2.This shows that the induced transformation i* :nm(Y ,B) +nm(X,A ) of the inclusion map i : ( X ,A ) c ( Y ,B ) , which is a relative homeomorphism, cannot be onetoone. In this example, X fails to be compact. Next, let Y denote the unit ( m + 1)cell of the euclidean ( m + 1)space and X the boundary msphere. Let A consist of a single point of X and let B = ( Y \X) U A . Then nm(X,A ) m 2 while nm(Y,B) = 0. This shows that the induced transformation i* :nm(Y , B ) + f l ( 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 puir where X i s a compact Hausdorfl space and A is a closed subspace of X . If V is any open set oi X contained i n A , then the induced transformation e* of the inclusion map e:(X\V,A\V)c (X,A) carries nm(X, A ) onto nm(X \ V ,A \ V ) in a onetoone fashion. Proof. Since e is a relative homeomorphism, (3.3) follows from (3.2) if A \ V is nonempty. If A \ V is empty, then A = V and hence A is both open and closed in X . Therefore, every map I# : X A \ + Sm has a unique extension y : ( X ,A ) +(Sm, so). This proves that e* is a onetoone correspondence. I
Remark. The condition in (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.
4. The coboundary operator In the present section, we are going to construct a coboundary operator 6 for cohomotopy sets which will be analogous to the coboundary operator for cohomology groups. Unfortunately, this coboundary operator 6 is not defined for a completely arbitrary pair ( X ,A ) . Since, in the construction of 6, one has to use some forni of Tietze's extension theorem, it is necessary to assume a normality condition on the pair ( X ,A ) . By a binormul pair ( X ,A ) , we mean a binormal space X , (I; 8 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 semisimplicial complex and A is a subcomplex of X , or more generally, if X is a paracompact Hausdorff space and rl is a closed subspace of X . Throughout the present section, we assume that (X, A ) is an arbitrarily given binormal pair. Let Sm+l denote the ( m + 1)sphere obtained by joining Sm to two points, the north pole and the south pole. Denote by ET+l and E?+l the north and the south hemispheres respectively. Then we have Sm+l Em+l U Em+l s o ~ S m= Em+l + n Em+l. ~
4. T H E C O B O U N D A R Y O P E R A T O R
209
Consider the following diagram
n ( X ,A : Sm+', so)
n(A:S m )
Iy
i'
n ( X ,A ; E?+l, Sm) y.+n ( X ,A ; Sm+l, E?+l) where the transformations are natural, namely, a is defined by restriction while t9 and y are induced by inclusions. Since E?+l, ET+l are contractible to the point so and (X, A ) is a binormal pair, it is straightfonvard to verify that both u and y are onetoone correspondences. Thus, we obtain a transformation
6 = y1pu1: n y A ) + n m + l ( X , A ) , which is called the coboundary operator. Geometrically, the coboundary operator 6 can be determined as follows. Let e € @ ( A ) . Choose a map : A + Sm which represents e. Since ET+l is contractible,
+ + has an extension
+# : ( X ,A ) + (ET+l,S m ) . Then, +# represents ul(e). Let tt
: (EY+l,Sm) + ( S m + l , Etn+1), 0
xm1, xm).
Let ET and Etn denote the hemispheres of Sm defined by xm > 0 and xm < 0 respectively. Then r maps E$ onto Em and E!? onto E$. Let ht : (Sm, so) + (Sm, so), 0 < t < 1, be a homotopy such that ho is the identity map and hl(E?) = so. Then hl+ represents a and h,r+ represents /?. Let M , = ++(E?) and M , = +l(Etn),then we have
A c M , n M,, hl+(M2) = so = hlr+(Ml). Hence the map hl+ x h,r+ sends X into B and the composed map y = j(h,+ x h,r+) represents the element a + /?.Furthermore, it is obvious that y(x) = h,+(x) if x E M I andy(x) = h,r+(x) if x E M,.Then, define a homotopy yt : ( X ,A ) + (Sm, so), 0 < t < 1, by taking
X
= MI u M,,
5. T H E G R O U P O P E R A T I O N
IN
n m ( X ,A )
+
Then yo also represents u /I. Since h, is the identity map, it follows that yo sends X into ET and hence yois homotopic (relative to A ) with the constant map e ( X ) = so. This proves that u j3 = 0. I For the special cases m = 1 or 3, Sm is a topological group with so as neutral element. Then there is a natural multiplication in the cohomotopy set n m ( X , A ) of any pair ( X , A ) defined as follows. Let u,j3 be any two elements of n m ( X ,A ) represented by 4, y : ( X ,A ) + (Sm,so) respectively. Let X : ( X ,A ) f (Sm,so) denote the map defined by X ( x ) = + ( x ) y ( x ) . Then X represents an element [ X I of nm(X,A ) which obviously depends only on u and j3. This element [XI will be called the product of u and j3, and will be denoted by up. Obviously, n m ( X ,A ) forms a group with this multiplication as group operation.
+
Proposition 5.3. If m = 1 or 3 and if ( X ,A ) i s a ( 2 m  1)coconnected cellular +air, then uj3 = u + j3 for a n y pair (a,j3) of elements of n m ( X ,A ) .
Proof. Let ,u : ( Y ,yo) +. (Sm,so) denote the map defined by p(s, t) = st for any pair (s, t) of elements in the topological group Sm. By (5.1), there exists a homotopy f t : ( X ,A ) +. ( Y ,yo), 0 < t < 1, such that fo = x y and f l ( X )c B, where 4, y are representative maps of u,j3 respectively. Then ,ufo represents cij3 and j f l represents u j3. Since f l ( X )c B, it follows that pf, = ifl. Hence the homotopy ,uft implies that aj3 = u + j3. I In the remainder of the section, we shall establish that the induced transformations and coboundary operators are homomorphisms.
+
+
Proposition 5.4. If ( X ,A ) and ( X ' , A') are two ( 2 m  1)coconnected cellular pairs and if f : ( X ,A ) + ( X I ,A ' ) i s a map, then the induced transformation f * : @(XI, A') +.nm(X,A ) i s a homomor#hism.
Proof. Let U ,/?beany two elements ofnm(X', A ' ) and choose representative maps y : ( X ' ,A ' ) + (Sm, so) for u,j3 respectively. By (5.1) there exists a homotopy gt : ( X ' , A') + ( Y ,yo), 0 < t Q 1 , such that go = x y and g,(X) c B. Then jg, represents u + j3 and hence j g l f represents f * ( u + j3). On the other hand, since+/, yf represent /*(u), f*(B) respectively and since go/ = +f x y f , it follows that j g l f also represents /*(a) + /*(/I). Hence / * ( a j3) = /*(a) /*(PI. I
+,
+
+
+
Proposition 5.5. If ( X ,A ) i s a ( 2 m + 1)coconnected cellular pair such that A is ( 2 m  1)coconnected, then the coboundary operator 6 :nm(A) +nmtl ( X ,A ) is a homomorphism.
Proof. Let a, @ be any two elements of @ ( A ) and choose representative maps 4, y : A +Sm respectively for u,B. By (5.1), there exists a homotopy
214
VII. COHOMOTOPY GROUPS
=+
g t : A + . S m x Sm, 0 < t < 1, such that go x y and g l ( A ) cS m V S m . Then j g , : A +.Sm represents the element a B. Since ET+1 v ET+1 is contractible over itself to (so, so), g , has an extension
+
v ET+l,Sm v Sm).
gl# : ( X ,A ) +. (ET+l
Let j # : ET+l v E T + l + . ET+l denote the map analogous to j : Sm 'd Sm + Smn; then j#gl# is an extension of jgl. Hence, Elj#gl# represents 6(a+B). Let pi : ET+l x ET+l +. ET+l, (i = 1,2), denote the projection defined by P Zm, then the following part of the cohomotopy sequence i s exact: '* '* nyx, A ) L nyx)5 n m f A )+6 n m + y x , A ) L n m + l ( x )5 * .
10. Higher cohomotopy groups Theorem 10.1. If ( X ,A ) i s an mcoconnected cellular pair, t k n n m ( X , A )=O. Proof. Since Sm is (m 1)connected, every two maps q5, y : ( X ,A ) + (Sm, so) are ( m 1)homotopic relative to A . Then, by (VI; 11.3), q5 and y are homotopic 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 0.
X i s an mcoconnected triangulable space, t k n n m ( X ) = 0. I n particular, the corollary includes the obvious special case : If X consists of a single point, then n m ( X ) = 0 for every m > 0 ; on the other hand, n"(X) clearly consists of two elements. Corollary 10.2. If
11. Relations with cohomology groups I n (V; 9 4), we constructed a natural homomorphism hm from the homotopy group n m ( X , A ) into the homology group H m ( X , A ) over integral coefficients. In the present section, we shall define a similar operation
h m : n m ( X , A ) + H m ( X , A ; n m ( S m , ~ o ) ) ,m > 0 , which will be a homomorphism if ( X , A ) is an ( 2 m  1)coconnected cellular pair. Since nm(Sm, so) is free cyclic and the identity map on S m represents a generator, it can be identified with the group Z of integers in a natural way. Hence we may denote the cohomology group H m ( X , A ;%(Sm, so) ) simply by H m ( X ,A ) . I n order to construct the natural homomorphism hm, let us denote by Xm the characteristic element of the cohomology group Hm(Sm, so) as defined in (VI ; 9 17). This characteristic element Xm can also be defined by the natural homomorphism hm :nm(Sm, so) + Hm(Sm, so) as follows. By Hurewicz's theorem, (V; §4),hm is an isomorphism and hence the inverse hG1 is a welldefined homomorphism of Hm(Sm, so) into nm(Sm,so) = 2. Therefore, hi' determines a generator of Hm(Sm, so) which can be easily proved to be Xm.
11. R E L A T I O N S W I T H C O H O M O L O G Y G R O U P S
Now let us define the natural operation hm as follows. Let be an element represented by a map 4 : ( X ,A ) + (Sm, so). homomorphism +* : H m ( s m , so) +. P ( X ,A ) .
223 01
4
~ n m ( XA,) induces a
If we use singular cohomology, than #* depends only on u according to the homotopy axiom. In this case, we define hm(a) = #*(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 i s a homomorphism.
Proof. Let fit : Sm x S m +.Sm, (i = 1,2), denote the projections defined by Pt(y,, y z ) = y t ; let i : S m V S m + S m denote the map defined in 3 5 ; and consider the inclusion map k : S m V Sm c Sm x Sm. These maps induce the homomorphisms pt* : Hm(Sm, so)
+.
x
Hm(Sm
s m , (so, so)),
i* : Hrn(Sm,so)  + H m ( S m V S m , (so, so)), k* : Hm(Sm x Sm, (so, so)) + H m ( S m V S m , (so, so)).
+
It is easy to see that i*(Xm) = k*Pl*(Xm) k*pz*(Xm). Now let a, /?~ n m ( XA, ) be represented by#, y : ( X ,A ) + ( S m , so) respectively. By (5.1), 4 x y is homotopic to a map g : (XIA ) + (Sm V S", (so, so)) relative to A . Then we have # r p,kg and y N fizkg relative to A . Hence hm(a) hm(p)= #*(xm) y*(Xrn) g*k*p,*(Xm) g*k*pz*(xm)
+
+
+
= g*i*(X,).
On the other hand, since ig : ( X ,A ) +. (Sm, so) represents a we have This implies that hm(u
+
b y definition,
+ /?) = hm(a) + hm(p).I
proposition 11.2. For a n y m a p f : ( X ,A ) +. ( Y ,B ) , the following rectangle
i s commutative :
nyx,A ) + L n m ( Y , B )
lhrn
4.
lhrn
4.
H y X , A ) tf' Hm( Y , B ) . Proof. Let u ~ n mY(, B) be represented by f *(a)is represented by # f and hence
4 : ( Y ,B) +. (Sm,SO).
hmf *(a) = (+f)*(Xm) = f *#*(Xm) = f *hm(a).I
Then
VII, COHOMOTOPY GROUPS
224
Proposition 11.3. For any binormal pair ( X ,A ) , the following rectangle i s
k
commutative :
p
nm(A)d nm+1 ( X ,A ) Hm(A)2 4Hrn+l(X,A ) .
Proof. Consider the map
El of 5 4. Then we have
Hm+l(E?+l, Sm) +!.L Hm+l(Sm+l,so). Hm(Sm, so) By the definition of the characteristic elements Xm, i t can be seen that 6*(Xm) = 51*(Xm+,). Now let a be any element of n"( A ) . Choose a representative map : A +Sm. Then, by 5 4, 6(a) is represented by E1y: ( X ,A ) + (Sm+l,so), where y : ( X ,A ) 3 (E?+l, Sm) is any extension of +. Hence we have hm+l&a) = (5,y)*(X,n+,) = w * E I * ( x m + l ) = y*6*(Xm) = &$*(Xm) = Ghm(a). I As a consequence of the last two propositions, we have the following Proposition 11.4. For every binormal triple ( X ,A , B ) , the natural operators
hm, m = 1,2; , define a transformation of the cohomotopy sequence of ( X ,A , B ) into its integral cohomology sequence, that i s to say, each rectangle of the following ladder i s commutative: nl(X,A)
i ,
..*
i"l
Hl(X,A) +
* * .
1"
nrn(X,A)
i ,
+
p
@ ( X , B ) + n"(A,B) /hm
+
k+.
nm+l(X,A) +
+ Hm(X,A) + Hm(X,B) + H m ( A , B ) + Hm+l(X,A) +
*..
*
Theorem 11.5. (Hopf theorem). If m i s a positive integer and ( X ,A ) i s a n (m + 1)coconnected cellular pair, then hm sends nm(X, A ) onto Hm(X,A ) in a onetoone fashion. This is merely a special case of (VI; 16.6).The significance of this theorem is that, for an ( m + 1)coconnected cellular pair ( X ,A ) , we have @ ( X , A ) w Hm(X,A ) , n n ( X ,A ) = 0 for n > m.
12. Relations with homotopy groups If we neglect the group operations, nn(Sm,so) and nm(Sn, so) are identical. If n < 2 m  2, nm(Sn,so) has a group operation as defined in 5 5 ; on the other hand, nn(Sm, so) is always a group. We are going to see that these two group operations are the same. Let a, be any two elements of nm(Sn, so). Let E: and E!! denote the north and the south hemispheres of 9.As in (IV; $2), we can choose representatives y : (9, so) + (Sm, so) of a, p respectively such that
+,
+(Elf)= SO
= y(ET).
12. R E L A T I O N S W I T H H O M O T O P Y G R O U P S
225
Then the map g = C$ x y sends X into Sm V S m and jg represents the sum of u and t9 in &(Sn, so), where j denotes the map defined in 4 5. On the other hand, since if x E ET, ( C$(X)I =
jg also represents the sum of following
if x E E!, 01
and t9 in nn(Sm, so). Hence we obtain the
Proposition 12.1. The cohomotopy group nm(Sn, so) i s exactly the homotopy group nn(Sm, SO), that is nm(Sn, SO) = nn(Sm, SO).
Now let us consider a pair ( X ,A ) and a given point xo E A . Let u ~ n m ( X ,A , xo) and /?~ n n ( XA,) be represented by the maps
q5 : ( I m , Im1, I"') + ( X ,A , xo), y : ( X ,A ) (Sn,so). The composition y$ is a map of (Im,dim) into (Sn,so) and therefore represents an element [y$] of nm(Sn,so) which obviously depends only on u and 8. +
We denote
PO.
=
[y$l
and call /I0a the composition of u and t9. If m = n, then 10u ~ n m ( S mso) , and hence is an integer. The following proposition is an immediate consequence of the definition of addition in n m ( X ,A , so) and in nm(Sn,so).
B ~ n n ( XA,) , then we have B 0(a1 + a2) = (B 0El) + (B 0a2). we assume that n n ( X ,A ) forms a group according to 4 5, then
Proposition 12.2. If u l , u 2 ~ n , ( X ,A , xo) and
Now, if we have the following
Proposition 12.3. If
0:
+
e n m ( X ,A , xo) and
0 Proof. Let+ : (Im, I m  l , I"') (81
B2)
= (PI
pl, /I2~ n n ( XA,) , then
we have
04 + ( B 2 04.
+ ( X ,A , xo) represent u andy,, y 2 : ( X , A ) + (9, so) represent PI, B2. Let g : ( X ,A ) + (Sn V Sn, (so, so)) be a normalization of y1 x y 2 ;then gq5 is a normalization of
(y1 x y!J+ = y14 x yz+ (Irn, aim) (Sn,so). This and (12.1) imply the proposition. I Combining (12.2) and (12.3), we obtain that the composition operation defines a homomorphism +
n m ( X ,A , xo) @ n n ( X A > ) +nm(Sn, so). The following properties of the composition operation can be easily verified.
2 26
EXERCISES
Proposition 12.4. Let f : (X,A,x,) + (Y,B,y,) be a map and aEnm(X,A,xo),
B ~ n nY(, B). Then we have
( f *B) 0a that is, the induced transformations f
=
B 0( f * a ) ,
* and f* are "dual" to each other.
Proposition 12.5. Let ( X ,A , B ) be a binormal triple with x, a €nm(X, A , x,) and fi ~ n n  ' ( AB , ) , then we have
(d*B)
0a
=
E B.
If
x [B 0@*a)1,
w k r e Z :n,,,l(S*l) +nm(S") denotes the suspension of Freudenthal, (V; 9 11). Thus, in case X is an isomorphism, a, and d* can be looked on as dual operations.
EXERCISES A. Generalizations to compact pairs
For the sake of simplicity, most of the operations and results in this chapter are formulated and proved for triangulable pairs. However, by using Cech cohomology theory, these can be generalized to the finite dimensional compact pairs ( X ,A ) , that is to say, X is a finite dimensional compact Hausdorff space and A is a closed subspace of X . The pair ( X ,A ) is said to be ncoconnected if the integral Cech cohomology group Hg(X, A ) = 0 for every q > n. Prove that, if ( X , A ) is a (2m  1)coconnectedfinitedimensional compact pair, then it is possible to define a commutative group operation in nm(X,A ) by the method of fj5. See [Massey 1, p. 2831. Also prove that the induced transformations f * and the coboundary operator 6 are homomorphisms for finitedimensional compact pairs satisfying similar conditions concerning coconnectivity . Now let ( X ,A ) be a compact pair with dim X < 2m  1. Consider any finite open covering a = { U ) of X . Let K,, denote the geometric nerve of a and La the geometric nerve of the open covering a n A = { U n A } of A . Then (K,,,La) is a finite simplicia1 pair. Since dim X < 2m  1, the finite open coverings 3: of X such that dim K,, < 2m  1 form a cofinal subset M of the directed set of all finite open coverings of X . For eacha E M , nm(K,,,La) is an abelian group; and hence we obtain a direct system of abelian groups { nn,(Ka,La) I a E M }. Prove that nm(X,A ) is isomorphic to the limit group of this direct system, [Spanier 1 ; p. 2271. Consequently, the cohomotopy groups satisfy the continuity axiom. This can be used to extend results from finitely triangulable pairs to compact pairs. For example, generalize (9.3)to compact pairs.
EXERCISES
B. Connection with
227
Freudenthal's suspension
Consider the inclusion maps #:(Sm+l,s o ) c (Sm+l,E??+l)and q:(Ey+l,Sm) c (Sm+l,E??+l).By (3.3), q* is onetoone and onto and hence q*l is welldefined. Prove the commutativity of the following diagram:
I(=)
nn(Sm,sol
E+. nn+l(ET+l,Sm)4+1+ nn+l(S"+1,E!?+') e+ n"+'(Sm+l, so) [(=I z
nm+l(Sn+l,so) where 2 denotes the suspension. Since p* is also onetoone and onto, this shows that the coboundary operator 6* is essentially the suspension X. Now consider the following part of the cohomotopy sequence of the triple (E?+', 9,so): nn(E?+l,so) 2 nn(Sm,so) E+nn+l(Ey+l,Sm)i'+ zn+l(E~+l,so). Since E?+' is contractible, we have Im(i*) = 0 and Ker(i*) = nn+l(EY+l,Sm). If m = 2 and n = 1, then (6) of 9 6 is false sincen,(S) = 0 whilen3(S2)M 2. On the other hand, if m = 3 and n = 2, then (5) of tj 6 is false since n,(S3) M 2, while n3(S2)M 2. %n(Sn, so)
+
C. Connection with cochain groups
Let ( K ,L ) be a finite cellular pair. Denote E m = K m U L. Let a be any element of @(Em, Em1) and pick a representative map : ( E m ,f f m  l ) + (9, so) for a. For each simplex u p of K , the partial map +I = I ( u p , sup) represents an element [+t] of nm(Sn,so) which depends only on a and u p . Hence a determines a n mdimensional cochain y ( a ) of K modulo L with coefficients in nm(Sn,so). Prove that @(Em, Em1) forms an abelian group with addition defined as in tj 5 and that the correspondence a +y(a) defines an isomorphism y :n n ( E m , Em1) M C y K , L ;nm(Sn,so)).
+
+
Furthermore, verify that y commutes with the induced homomorphisms. Prove also that the following rectangle is commutative:
C y K , L;nm(Sn, so)) A+Cm+1(K,L;nm+l(Sn+l, so)). D. Connection with the obstruction
Let ( K ,L ) be a pair as in Ex. C. Let a be any element of n n ( E m ) and +Sn be a representative map of a. By (VI; 9 4), determines an obstruction cocycle C"+l(+) E Cm+l(K,L ; nm(Sn,so)),
+
r j :E m
which depends only on a. Prove that
y6(a) = Z (cm+'(+)),
EXERCISES
228
where 6 :n n ( E m ) +nn+l(Em+l,E m ) is the coboundary operator, y denotes the isomorphism in Ex. C, and X is the homomorphism determined by the suspension in the coefficient group. Hence, if X is an isomorphism, @(a) is essentially the obstruction cocycle cm+l($). Therefore, has an iff 6(a)is the zero element of nn+l(Em+l,E m ) . extension over
+
E. The structure of n n ( X , A )
Let ( X ,A ) be a given (2n  1) coconnected cellular pair. Define a sequence of subgroups nn(X, A ) = Dn13 Dn 2 . . 3 p n  2 = 0
.
as follows: an element of n n ( X ,A ) is in Dm iff i t can be represented by a map f : ( X ,A ) f (9, so) such that / ( E m )= so. As in [Hu 61 and [Chen 13, define the #resentable subgrou# and the regular subgroup of H m ( X ,A ; nm(Sn,so) ) and consider their quotient group
J*"(X, A ;nm(Sn,so)) = P*m(X, A ;%(Sn, ~ g ) ) / R * ~A( ;xnm(Sn, , so) ). Then prove that DmllDm w J*"(X, A ;nm(Sn, SO) ) for every m
=
n,

a ,
2n  2 .
F. Relations between induced homomorphisms
Let ( X ,A ) be a triangulable pair such that H n ( X ) = 0 = Hn(A) for every > 2m  1, where m is a given positive integer. Prove that the following four statements are equivalent: 1. The induced homomorphism i* : H n ( X ) + Hn(A) of the inclusion map i : A c X is an isomorphism for every n > m and is an epimorphism for n = m. 2. H n ( X ,A ) = 0 for every n > m. 3 . n n ( X , A ) = 0 for every n > m. 4. The induced homomorphism i* : f l ( X )+nn(A) of the inclusion map i : A c X is an isomorphism for every n > m and is an epimorphism for n = m. Next, let X and Y be any two (2m  1)coconnected triangulable spaces and f :X + Y a map; prove that the following two statements are equivalent. 5. The induced homomorphism f * : H n ( Y ) + H n ( X ) is an isomorphism for every n > m and is an epimorphism for n = m. 6. The induced homomorphism f* : nn( Y ) + n n ( X ) is an isomorphism for every n > m and is an epimorphism for n = m. n
CHAPTER V l l l E X A C T C O U P L E S A N D SPECTRAL S E Q U E N C E S
1. Introduction Spectral sequences were originally devised by Leray to exhibit relations among the Cech groups of the various spaces of a fibering; but they have proved useful, indeed crucial, in numerous other investigations. Serre, for example, established the same results for the singular groups, and, using relations between homotopy groups and singular groups, he was able to obtain important information about the homotopy groups of spheres; and Eilenberg, Massey, and Spanier found spectral sequences to be useful in the study of the homotopy groups of a complex. Leray’s original device involved imbedding the homology or cohomology groups in question in a much larger system of groups and homomorphisms, namely the spectral sequence; this sequence, after the first term, is itself an invariant of the fibering. Being quite complicated, it is not amenable to successful computation except in special cases ; but it always provides much information, and it furnishes a pattern for deeper investigation. Massey improved Leray’s device by imbedding in a still larger but much more versatile system which he called an exact couple. The present chapter will be devoted to this formulation of the machinery of spectral sequences; an outline of Leray’s direct construction appears in Ex. A. There follows in Chapter IX Serre’s version of homology and cohomology theory of a fibering, together with several immediate applications ; results from Chapter IX will then be used in the computation of certain of the groups n,(Sn).
2. Differential groups Let A be an abelian group. An endomorphism
d:A+A is said to be a di#erential operator on A if dd
=
0,
i.e., d[d(a)] = 0 for each a E A . An abelian group A furnished with a given differential operator d is called a differential group, or simply a dgroup. With the dgroups as objects and the homomorphisms which commute with d as mappings, we obtain a category 93’a called the category of differential groups. For the definition of a category, see [ES: p. 1091. 229
230
If f :A
V I I I . E X A C T C O U P L E S AND S P E C T R A L S E Q U E N C E S
B is a mapping in g d such that A c B and f ( a ) = a for each A is called a subgroup of the dgroup B. Let C = B/A : then we can define a differential operator on C so that the projection g : B 3 C +
a E A , then

becomes a mapping in g d . When furnished with this differential operator, C is called the quotient dgroup. We obtain an exact sequence (2.1)
0
A f,B A+C
0.
Conversely, if an exact sequence (2.1) of dgroups and mappings is given, then A can be identified with a subgroup of the dgroup B b y means of the monomorphism f and the quotient dgroup BIA can be identified with C by means of the epimorphism g. Given a dgroup A, we shall denote b y S ( A ) the kernel of d (called the group of cycles of A), and by g ( A ) the image of d (called the group of boundaries of A). The condition dd = 0 implies that g ( A ) c S ( A ) and that they are subgroups of the dgroup A with d = 0 on each of them. Hence we may define the quotient dgroup
&'(A) = d ( A ) / N A ) with d = 0 which is called the derived group of the dgroup A. Let f : A +B be a mapping in g d . Then the commutativity df  fd implies that f m a p sd (A) intoZ(B) a n d a ( A ) intoB(B) and hence f induces a mapping S ( f:&'(A) ) +&'(B). One can easily verify that the operation &' is a covariant functor from g d to itself, [ES; p. 1111. Two mappings f , g :A + B in are said to be homotopic (notation: f a? g) if there is a homomorphism 6 : A + B such that
f g
=
d6
+ 6d.
The homomorphism 6 is called a homotopy (notation: 6 : f N g). One can = &'(g). easily prove that f cxg implies &'(I) Let an exact sequence (2.1) in g d be given. We shall define a mapping
a :&'(C)
+&'(A)
as follows. Let aE&'(C) and choose x ~ a ( Cwhich ) represents a. There is some Y E B with g(y) = x . Since gd(y) = d(gy) = 0, there is a Z E A with f ( z ) = d(y). Since f d ( i ) = d( f i ) = 0 , d(z) = 0 and z represents a @ €#(A) which depends only on a. Then a is defined bya(a) = /I. Since the differential operators on&' (C) and&'(A) are trivial, a is a mapping in g d . Thus we obtain a triangle in Yd: &'(A) + Wf) X ( B )
3. G R A D E D A N D B I G R A D E D
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 , of A such that A can be written as a (weak) direct sum
A
==
ZAn. U
The elements of the subgroup An 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 Am,,, of A such that A can be written as a direct sum
A
= Z Am,%. m,n
The elements of the subgroup Am,, are said to be homogeneous of degree (m, 4. When dealing with graded or bigraded groups, only a certain limited class of homomorphisms are of interest, namely, the homogeneous homomorphisms. If are bigraded groups, then a homomorphism f : A + B is said to be homofor every pair (m, n). With bigraded groups as objects and homogeneous homomorphisms as mappings, we obtain a category %a called the category of bigraded groups. Similarly, one can define the category ggof graded groups. If f : A + B is a mapping in g b of degree (0,O)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 = B / A . Then one can verify that C is isomorphic to the direct sum of the groups Bm,n/Am,n. We agree to identify these naturally isomorphic groups. Then C is bigraded and the projection g : B + C is a mapping in g b of degree (0,O). With this bigraded structure, C is called the qiiotient bigraded group. Note that both the kernel and the image of a mapping in 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 differentialoperator 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 [ES; p. 1871.C ( X )has a natural graded structure
C ( X ) = X Cn(X). n
V I I I . EXACT C O U P L E S A N D S P E C T R A L SEQUENCES
232
where C,(X) is the group of nchains if n > 0 and C,(X) = 0 if n < 0. The boundary homomorphism a : C,(X)+ C,,(X) extends t o a differential operator d = : C ( X ) +C(X)
a
which is homogeneous of degree  1. Hence the kernel Z ( X ) of d and the image B(X)of a' are subgroups of the graded dgroup C ( X ) with graded structures Z ( X ) = Z Z , ( X ) , B(X)= Z B%(X). n
n
Furthermore, the derived group H ( X ) of C(X)has a graded structure
H ( X ) = ZH,(X), n
where H n ( X ) is the ndimensional singular homology group of X if n and H,(X) = 0 for n < 0.
>0
4. Exact couples By an exact couple, we mean a system
Y =(D,E;i,j,k> which consists of two abelian groups D and E , and three homomorphisms
i:D+D, j:D+E, k:E+D such that the following triangle is exact:
D
i
+D
There is an operation which assigns to an exact couple V another exact couple V' = < D',E'; i', j', k' > called the derived couple of morphism
Y constructed as follows. Define an endod:E+E
b y d = jk. Since k j = 0 byexactness, i t follows that ad = j k i k = j ( k j ) k = 0 . This implies that d is a differential operator on E . Let D' = i ( D ) , E' = & ( E ) ; then D' is a subgroup of D and E' is the derived group of the dgroup E. Since D' c D, i(D')c D'. We may define an endomorphism i' : D' + D'
i I D'. Since one can easily verify that k [ J ( E ) ]c D', K[&'(E)j = 0, k induces a homomorphism k' E' D I . by i'
=
~
233
4. E X A C T C O U P L E S
Let x E D' and choose a y E D with i ( y ) = x . Then i ( y ) is in 9 ( E )and the coset of j ( y ) mod B ( E ) does not depend on the choice of y. Denote this coset by it(%). The assignment x + j ' ( x ) defines a homomorphism
j' : D'
+
E'.
This completes the construction of W. The verification that i', j' k' form an exact triangle is straightforward and is left to the reader. This process of derivation can be applied to v' to obtain a second derived coztple V", and so on. In this way, we obtain a sequence of exact couples
Vn = ( D n , E n ; i n , j n ,kn>, n
=
1,2;..,
defined inductively by
V 1= V, V n = ( W  I ) ' ,
n > 1.
The sequence { W n } has two important properties. Firstly, the groups Dn form a decreasing sequence D = Dl 2 0 2 2 . . =I Dn =I Dnil I>
.
. ..
with i n : Dn + Dn defined by the restriction of i to Dn. The intersection of the groups Dn is denoted by Dm. Secondly, i t has been shown above that the endomorphisni dn = jnkn : En +En is a differential operator on En and En+' is the derived group of En with respect to dn. Hence we obtain a sequence of differential groups E = E l , E 2 , . . . , En , ... such that En+l = .%'(En) for each n > 0. This will be called the spectral sequence associated with the exact couple V. In the spectral sequence { En } there exist natural homomorphisms
It, : I ( E n ) + En+' defined by assigning to each element of 9 ( E n ) its coset modulo .%f(En). Thus hn is an epimorphism of a subgroup of En onto En+l. We can define an epimorphism ht of a subgroup of En onto En+P by the formula hi
=
ha+plhn+p2
* * *
hn.
Then hf, = hn. The precise domain 9% of definition of hf: can be defined inductively as follows: 3; =%"(En), 9f:= { u E 9 2  l I h$'(a) E Z(En+Pl) },
p > 1. Let E n denote the intersection of all subgroups9{, p = 1,2; ., then, for each a E E n , ht(a) is defined for all values of p . Define 2%: E n B n + l +
to be the restriction of hn to E n . Then the sequence of groups { } and homomorphisms { i n } constitutes a direct sequence of groups in the usual
234
V I I I . EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
sense, [HU'; p. 1321. The limit group Ew of this direct sequence of groups will be called the limit grozcp of the spectral sequence { En }. In particular, if there is an integer r > 1 such that dn = 0 for each n 2 r, then I ( E n ) = En, hn : En M En+' and therefore E n = En and En = hn for each n > r. This implies that E" w Er. Returning to the general case again, we can express the limit group E* in terms of the original exact couple V as follows. Consider the epimorphisms i(n) = inin1. .il : D + Dn+l c D.
.
Since i n = i 1 Dn for each n > 1, i(n) is actually the nfold iteration of the homomorphism i : D + D . Let 01
Dw = n Dn n=i
03
=
W
n Im [icn)], Do = n U 1 Ker [;(")I, n=i
where Im [ W ]and Key [icfi)]denote the image and the kernel of i(n) respectively. Then it is easy to verify that
E"
M
klD"ljDO.
In the following chapters as well as in the exercises a t 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)
0 =
X,C
x o cx , c * . *
c
x, = x.
Using total singular homology groups over a given coefficient group G,we define D = X H ( X p ) , E = I:H ( X p , Xp1). P
P
Then the homology sequence of the pairs ( X p ,X p  l ) give rise to an exact couple VH(@) = < D,E ; i, j , k > called the homology exact couple of the filtered space X .
5. Bigraded exact couples In the exact couples V = < D , E ; i, j , k > which we shall deal with in the sequel, the groups D , E are usually bigraded and the homomorphisms i, j , k are homogeneous. In this case, in the successive derived couples %n of V, the groups Dn, En are also bigraded, i.e., and the homomorphisms in, i n , kn, and dn
=
jnkn are homogeneous. The
5. B I G R A D E D EXACT C O U P L E S
235
elements of D;,q and E:,q are said to be homogeneous of degree (p, q). We shall call p the primary degree, q the complementary degree, and p q the totd degree. It is easy to verify the following relations about the degree of homogeneity:
+
deg (in) = deg (i), deg (kn) = deg (k), deg (in) = deg (j)  ( n  1) [deg (i)].
Hence the degree of the differential operator deg (an)
=
deg ( j )
dn
is given by the formula:
+ deg (k)  (n 1) [deg (41.
The following two special classes of bigraded exact couples are important. A bigraded exact couple V = < D , E ; i, j , k > will be called a acouple if i is of degree (1,  I), j is of degree (0,O)and k is of degree ( 1,O). In this case, the degrees of in, jn, kn, and dn are listed as follows: ( dl ) i ni s of deg ree(l ,1 ); (d2) jn is of degree (n 1, n 1); (d3) kn is of degree ( 1,O); ($4) dn is of degree ( n, n  1).
+
Similarly, V will be called a 6couple if i is of degree ( 1, I), j is of degree (0,0), and k is of degree (1,O). Then the degrees of in, in, kn, and d n are listed as follows: (61) i n is of degree ( 1, 1); (62) jn is of degree (n 1,  n 1) ; (63) kn is of degree ( 1, 0) ; (& dn is I) of degree (n, n 1).
+
+
A %couple V = < D , E ; i, j , k > is a quite elaborate structure; it can be developed into a “latticelike’’ diagram as follows:
ii
ii
*
*
li k
k
* * *
* *
’
k
EP,*il
DP+I,Q
EPil,Q+
li k
i
DP,4+1
DPi2,qI
li
k
i
+
k
Ep+z,q1+
ii k
i
DP1,*+12, EpI,Q+l
DP2,(1+1+
li
41.4
DP*Q
li
DPil*Ql’+
li
li
i > EP,Q
EP+I,PI
k
*
DP,,l
li
*
i ...
 li k

i
* * *
236
V I I I . EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
The steps from upper left to lower right are exact sequences, for example,
...R
i i R i i D,r,,+1D**, E,.,+ DPl,,+ D*.q1+ " ' is an exact sequence. Similarly, one can develop the derived couples W, n = 2, 3; ,into analogous diagram. It is more or less evident that the notions of &couple and 8couple are dual to one another. Thus, let %?= < D, E , i, j , k > be any given bigraded exact couple. We shall construct another bigraded exact couple %f* = < D*, E * ; i*, j*, k* >, called the dual of W by merely reindexing the groups D,,, and E p , g as follows:
.
D;,, = D,.,, E;,, = E,,q. Then we obtain D* = D and E* = E ; therefore, we may take i* j* = j , and k* = k. Their degrees are given by deg (i*)
=  deg
[i), deg ( j * )
=  deg
=
i,
( j ) , deg (k*) =  deg ( k ) .
Hence the dual of a &couple is a &couple and vice versa. For example, the homology exact couple VH(@)of $ 4 is a &couple Indeed, the groups D and E are bigraded, namely,
DP,, = % * ( X P ) , E,,, = %*(XP> X P  1 ) ; and the homomorphisms i, j , k are obviously homogeneous of degree ( 1,  l), (0, 0), ( 1,O) respectively.
6. Regular couples A %couple 5f = < D , E ; i, j , k > is said to be regular provided: (Rdl) DP,, = 0 if p < 0; (Ra2) E,,, = 0 if q < 0.
(Ral) and the exactness of
i
R
DmlEP,, DP14 imply that Ep,q = 0 if p < 0. Hence, for any positive integer +
1z,
we have
En = 0 = Egg, if p < 0 or q < 0. (Ra3) P.4 If n > p, each element of E;A is a cycle since dn is of degree ( n, n  1). If n > q + 1, no nonzero element of E;,q can be a boundary under dn. Hence En = E n f l =...= Egg if n > max (p, q I),
maq
P4
+
Ps4
Since each ir is of degree (1,  l ) , we have Dn9.4 = in1 in2 . . . a'1 (D p  n + i , q + n  J . W5) This and (Rdl) imply
(Rd6)
D"Pd = O
=
Dw ifn>p P,9
+ 1.
6. R E G U L A R Consider the homomorphismi : D,,, k
237
COUPLES
+
D p + l , q  l (Rd2) . and theexactnessof
i
i
EP+1,4 DP,, DP+l,,l+ EP+199l imply that i is an epimorphism if q < 0 and an isomorphism if q we obtain a sequence +
+
< 0. Thus
where i : Dm, + D,+,,, is an epimorphism. This suggests defining a graded
group
S ( V ) = ZXm(V), x m ( V ) = Dm+l,] m
=
D&+1,1.
Then D:+l,l is a subgroup of Sm(%) and (Rd5) gives a homomorphism

A,,, : DP,,
+.@P+,(%)9
+
(q
> 019
where A,,, = ig+1 iq * il is actually the ( q 1)fold iteration of the homomorphism i : D + D. Denote the image of A,,, by
D;$;+l,l. If p < 0, then S P , , ( V )= 0 since D,,, = 0. If q = 0, then X P , , ( V )= .x?~,($?)since A,,, reduces to the epimorphism i. Hence, for each m 2 0, we obtain a finite decreasing sequence: = AP,Q(DP,,) =
=@P,,(W
(Rd7) X m ( v ) =
~ m , o ( 3~x )m  1 ,
i ( g )2 *
*
* = S o , m ( W = ' S  i , m + i ( g ) = 0.
The first row is exact since it is a part of W . Since i maps Dp+q+r,r isomorphically onto Dp+q+r+l,   I  l for each 7 > 1, we obtain two natural isomorphisms u and@.y denotes the inclusion and the rectangle is commutative. Since n is greater than max ( p , q 2), we get
+

E$+ni,qn+Z = 0, Eg,q = E$,qr Dg1,q according to (Rds), (Rd4), and (Rd6). Hence we have
=
0
  @ P , P ( ~ ) I ~ P  l , , + l ( ~ ) E&.
(Ra8)
In other words,XP,,(V) is an extension of X p  l , p + l ( % by )E&. Similarly, a &couple V = < D , E ; i, j , k > is said to be regular provided:
< 0; + < 0.
(R61) D P , , = 0 if q (R62) E,,, = 0 if
By methods dual to those used above, one can prove the following assertions.
(R63)
E;.,
=
0
=
E&
if
p < 0 or q < 0.
238
VIII. EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
E;,q
(RW
.
E;;' = = E;,p i f n > ma x (p ,q Dpn,q = in1 in2 . 2'1 (D p + n  i , q  n + i ) .
=
..
+ 1).
(RS5) (RW DRq = 0 = D;,,if n > q 1. The homomorphism i : D p , q+ Dp1,9+1is an isomorphism if we obtain a sequence
+
i
i
i
i
i
i
p < 0. Thus i
O + D m , o  + D m  l , 1   + * * *    + D O , m w D  1 , m + l ~w Dr,m+rw *
*
a
*
a
*
This suggests defining a graded group
x ( v )= xxm(v), xm(v) m
=
Do,, w D4i,m+l
denote the subgroup D;,;+, ofSPM(V).Then, for each m we obtain a finite decreasing sequence:
> 0,
(R67) x m ( v ) = x o , m ( v ) I x i , m  i ( q ) I * * . = ' ~ m , o ( v I) x m + i ,  l ( % ) = O which satisfies the relations :
X P ,,(a)IS,+] .QlW)= E;, Q' Let us consider again the &couple W H ( @ )of 3 5. (R8l) is obviously true while (R82) is false in general. However, in case X is a finite simplicia1complex and X, is a subcomplex of X containing the pdimensional skeleton of X, then (R82) is also satisfied. Hence, in this case, 'ip~(@) is a regular 8couple. If, in particular, X , is the pdimensional skeleton of X , then Ep,q = 0 for each q # 0 and E,,o = H p ( X p ,X,J is the group of pchains of X over G. One can verify that the differential operator d : E p , o+ E p  l , o is precisely the boundary operator on pchains. Hence we obtain
(RW
E&o = H,(X), E i , q = 0 if q # 0. If n > 2, dn = 0 since it increases the complementary degree q. Hence we obtain H ( X ) = E2 = E3 = * * * = Em.
7. The graded groups It(%')and S(%) Let Q = < D , E ; i, j , k > be a regular 8couple. Define a graded group R(W) =
x4 Rq(Q),
Rq(W
=
El),,.
Since dn is of degree ( n,n  l), each element of E t , 4 is a cycle. Thus we obtain epimorphisms Rq(W) = Ei,q?k+ E:,,?!, . %+EQ+z = Em O*Q 0.4.

3 6, we have EEQ * xO*Q(v) =*dW*
Let x denote the composition. By (R88) of Denote the composed monomorphism by
(Rag)
1.
Then we have
Rq(5f)X+ E C Q L Xg(%?).
7. T H E
GRADED GROUPS
R(V) A N D S(V)
239
On the other hand, define a graded group
S(V) = ZSS,(V), S,(V) P
,
=
JZ;,,.
If n 2 2, no nonzero element of Ep"* can be a boundary under dn. Thus we 0btai.n the monomorphisms
EZ,,
=
E g 7 t  L EpP,,
* * *
lJ,E;,,%
Let L denote the composition. By (Rd8) of Eg.0
7 w
E;,,
= SP(V).
5 6 , we have
J f P , O ( ~ ) / Z I J  l l(%) . =s P ( ~ P ) / ~ P  l * l ( ~ ) .
Let x denote the natural epimorphism. Since
X*(W= S,(W = E;,,. we have j z : J f P ( V )+SP(V).One can verify commutativity in the triangle q + 1 *  1 9
(RalO)
Thus is is factored into the composition of an epimorphism and a monomorphism. If V is a regular &couple, then we define the graded groups R ( V )and S(V) exactly as above. Since dn is of degree (n, n + l ) , no nonzero element of E:,, can be a boundary under dn and every element of Epn,o with n 2 2 is a cycle. Hence, one may establish the epimorphisms x and the monomorphisms L in an analogous way as above with the roles of R(%)and S(V) interchanged. Furthermore, since Jfg(V)= and &('if) = E,,,, we have j : Z q ( V )+ I?,(%'). Thus we get
(R69)
S p ( V ) If+ EEo LX p ( V )
and a commutative triangle (R610)
Note. One might define I?,(%) to be E ; , , instead of EO,g.Then all the results in this section stand as they are except that the j in (R610) should be replaced by j'. Our choice of E,,, is based on the fact that it gives a natural expression for %(a)if V is defined by a filtered graded differential group. See 3 1 1 below.
240
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
8. The fundamental exact sequence Let Q = < D, E ; i , j , k > be a regular acouple and v > 0 an integer. Under certain conditions on the bigraded group E”,we shall obtain a useful exact sequence which will be referred to as. the fundamental exact sequence. Theorem 8.1. If Ea has only two rows which might be nontrivial, more precisely, if there are two integers a < b such that
E;4 = 0 if p # a and p # b. then we obtain a fundamental exact sequence
. .+~ i A!+ , ~ .sr ~n(Q L ) E&,,.b
Xm
Ehm1.a
&L+ sml(Y)+

Proof. The hypothesis implies that E;,q = 0 for each n > 2 and hence E& = 0 if p # a and p # b. Then (Rd7) and (Rd8) of 9 6 give rise to an exact sequence
Since dn is of degree ( n, n  1) and a < b, every element E:,ma is a cycle if n > 2. Hence we obtain epiniorphisms xr Ei,m.a+ %* Ei,m.a+ . . . E+En+’ a,ma = E*a,ma)
where n = max (a,m  a + 1). Similarly, if n > 2, no nonzero element of Et,mbcan be a boundary under dn and we obtain monomorphisms
In
n
ECm.6 = GLtb Eb,m.) + * ’ 4 E&& where n = max (b, m  b+ 1). Thus we obtain an exact sequence
~ i ,““*rn(Q) ~ +
bn1
S
19
Ei,mb*
To determine the kernel of +m, we have to study that of x r , r = 2, * * ,n. The kernel of xr consists of those elements of which are boundaries under dr. There are only two terms of Er of total degree m + 1 which might be nontrivial, namely, EL,m+laand Ei,m+lb,The elements of the first are cycles; and dr maps the second into Ei,maiff r = b  a. Hence & is a monomorphism if b  a = 1. Now let r = b  a > 2; we have an exact sequence Furthermore, E&,,a = Ei,maand ELZa = E&,. A similar argument shows that EL,,+,, = Elm+,,. Thus we obtain an exact sequence

E:,m+l, Xm+i E:,ma A zrn(Q) where&,,+, = O i f b  a = 1 andXrn+, = d r i f b  a = r >2. By similar methods, one can prove the exactness of the sequence sm(a)
tpm,E:,m.b
%m +
I
Ea,m.la*
The fundamental exact sequence is obtained by putting the various parts together. I
8. T H E
FUNDAMENTAL EXACT SEQUENCE
I t was proved above that Xm+, following
=
0 if b  a
=
Corollary 8.2. If in the hypothesis of (8.1), b  a we have a n exact sequence
o+
241
1. Hence we have the =
1, then, for each m,
h z m ( V )2E ; , ~+ . ~0.
H e n c e Z m ( V ) i s a n extension of E:,maby Ei,m.b. Analogously, one can prove the following Theorem 8.3. If E2 has only two columns which might be nontrivial, more precisely, if there are two integers a < b such that E;,¶ = 0 , if q # a and q # b,
then we obtain a fundamental exact sequence

bm1 + x m  i ( w )+ * '* Ernlb,b I Corollary 8.4. I f , in the hypothesis of (8.3), b  a = 1, then, for each m, we have a n exact sequence *
9
+ E;+,b
h z m ( V ) 3E;,,& +Xm
0 + Ekbsbh z m ( V ) EL.,, + 0. Hence S m ( % ' ) i s a n extension of ELb,bby EL,,,. Although the preceding two theorems cover most of the applications, it is sometimes necessary to deal with fundamental exact sequences obtained under weaker conditions. T h e twoterm condition. Let 2, p, v be integers such that 1 < p and v 2 1. We shall say that V satisfies the twoterm condition { I , p ; v } if the bigraded group E' has the following properties. For each integer m such that il < m < p, E;,c = 0 if p + q = m and ( p , q ) is different from two given pairs (am, bm) and (cnl, dm), where am bm = m = Cm dm, am < Cm.
+
+
Moreover, we require that the following two conditions should also be fulfilled : = 0 if p q = m  1, p Q am  v , and il < m Q p ; (1)
(2)
+ E;,¶ = 0 if p + q = m + 1, p > cm + v, and I
Q m Qp.
With some obvious modifications of the proof of (8.1) one can prove the following theorem which includes (8.1) and (8.3) as special cases. Theorem 8.5. If V satisfies the twoterm condition { 1,p ; v }, then we have a fundamental exact sequence
E&,bp
+
* * *
+
Eim,bm  + m ( % ) Erm9dm Eim.l,bm.1
Corollary 8.6. Let v
+
> 1 and { a,,
+
+
'''
+
&A,dA
} be a sequence of integers such that
< am1
+v
242
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
for each m 2 0 . If Ei4
p # am, then
=0
Jfm(Y)
Proof. Put Cm = a,
El,,b,,
+q = m
( p , q) such that p
for every pair bm
=
and
marn.
+ 1 and dm = bm  1. Then V satisfies the twoterm
condition { 0, p ; v } for every p > 0. Since Erm,d, = 0 for each m > 0, the fundamental exact sequence implies the conclusion of this corollary. I If U is a regular &couple, then the twoterm condition is stated by precisely the same words. The fundamental exact sequence in (8.5) becomes
E:,,,b,, * * ' f Eim.bm+ J f m ( q ) + EIm.dm+ Elrn1,brn.l f ' * *+EZn,di and similarly in (8.1) and (8.3). Regular 3 and &couples are considered further in Ex. B at the end of the chapter. f
9. Mappings of exact couples Let 'ip, = < Dr, Er; if, jr, mapping
Kf. >, r
=
(+, y ) : Vl
we mean a pair of homomorphisms +il = i2+,
If dr = jrkr, then y d , a homomorphism Since
=
1,2,, be two exact couples. By a
yi1
+
v,
+ : D,+ D , and y : E ,
+ E ,
= i2+>
+kl
=
such that
k2y.
d,y and y is a mapping in the sense of
3 2. y induces
y' : E ; +EL.
+(D;) = +il(ol) = i2+(DJ C
i2(D2) =
0;)
+
we may define a homomorphism +': D;+D; by +' = 10;. It can be readily verified that the pair (+',I$) of homomorphisms constitutes a mapping of the derived couples. This mapping y') : U; + U; is called the derived mapping of (+, y ) . Thus, the set of all exact couples and their mappings forms a category, and the operation of derivation is a covanant functor. If we iterate this process, we obtain a sequence of successive derived mappings ( r p , yn) : v,'z + V$, (n = 1,2; ). (+I,
In particular, the homomorphisms yn : Eln + E2n, ( n = 1, 2; ), of the spectral sequences commute with the differential operators dn, and yn+' is the induced homomorphism of yn. If the exact couples Wl and V, are bigraded, then only a limited class of mappings are of interest, namely, the homogeneous mappings. The mapping (4, y ) is said to be homogeneous of degree ( p , q) if both and y are homogeneous of the same degree ( p , q ) . By reindexing one of the given exact couples, we may always assume that (+, y ) is of degree (0,O); in this case,
+
9.
M A P P I N G S O F EXACT C O U P L E S
243
we say that (+,y) preserves degree. Then each of the derived mappings (+n, yn) also preserves degree. Now let Vl, V, be regular dcouples, and (+, y ) : V, + V, a mapping which preserves degree. Then the homomorphism +z defines a homomorphism
(4,Y)* :%(V1)  + S ( @ z ) which carries Xm(V1)into X m ( V 2and ) S p , q ( V l )into 2f?p,q(V2). Furthermore, y and y2 define homomorphisms
RWI)
+
W f Z ) ,
S(531)
+
S(532).
These obviously commute with the epimorphisms x and the monomorphisms I in 3 7. Proposition 9.1. (4, y)* is a n isomorphism if either of the following two eqzcivalent conditions is satisfied:
(i)+2 : D12M Dz2; (ii)y2 : E l 2 w Ez2. Proof. I t is obvious that (i) implies that (+, y)* is an isomorphism. That (i) e (ii) is an immediate consequence of the "five" lemma, [ES; p. 161. Finally, that (ii) e (i) follows from an easy induction on the primary degree by using (Rdl) 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 3 6, that (i)implies that (+, y)* is an isomorphism while the equivalence of (i) and (ii) is proved exactly as above. Obviously, similar results can be obtained for regular dcouples. Now, let us go back to the general case where V, V, are any two exact couples and consider two mappings
(+, y ) , (a, z) : Vl
+
V,.
They are said to be homotopic (notation: (+, y ) 3~ (a,z)) if there is a homomorphism 5 : E l + E , such that
4%) +@) 4 Y ) Y(Y) for every x E D,and y but important
E E,.
=
=
kZEjl(4,
Eddy)
+ dzE(Y)
Then one can easily prove the following obvious
Proposition 9.2. If the mappings (4,y ) and (a, z) are homotopic, then the (+I, y') and (a', z') are equal.
derived mappings
Hence, (p, y") = (a",7%) for every n 2. In particular, if V, W, are regular &couples or regular &couples and if (4, y ) , (a, t)preserve the degree, then (4, y ) N (a,z) implies that (+, y)* = (a, z)*.
244
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
10. Filtered differential groups Let A be a dgroup. By an increasing filtration in A , we mean an increasing sequence of subgroups { A D } of the dgroup A , p ranging over all integers, with their union equal to A ; in symbols, we have UpAP
=A,
AD c AP+',
d(AP)c A p .
If an increasing filtration { Ap } is given in A , A will be called an increasing filtered dgroup. Let A be an increasing filtered dgroup. For each a E A , the greatest lower bound of the integers p such that a E AD 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 dgroup A with integral values (including  03) which has the properties given above, we can define an increasing filtration { Ap } by taking
A ~ = ( u E A I w ( u l ) ,
and the graded groups
D"
=
X D;,
En
P
=
E E;. P
The inclusions Zpn C Zpn+l and B$:A c B $ Z ~ define + ~ a homomorphism in : Dn + Dn which is homogeneous of degree 1. The inclusions Zpn+l c Z;n+l and B $ I ~ c+ (2;:: ~ B ; Y ~ +define ~ ) a homomorphism j n : Dn +En which is homogeneous of degree  (n  1). Finally, since d(Z$)c Zpn, d(Zg:i) = B$:k, and d(B;]) = 0, the differential operator d induces a homoniorphism kn : En + Dn which is homogeneous of degree  1. Prove that < Dn, Elk; in, i" kn, >
+
is an exact couple for each n = 1,2, and in fact is precisely the associated exact couple @ ( A ) defined in 5 10. The differential operator d also induces the homomorphism dn = jnkn 1 En +En a ,
of degree n. Thus one obtains Leray's direct construction of the spectral sequence { En } of the filtered dgroup A . See [Leray 11. Formulate a similar construction for the case that the filtration { Ap } is decreasing.
250
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
B. Regular couples satisfying twoterm conditions
Let V = < D, E ; i,j , k > be either a regular &couple or a regular &couple. Prove the following assertions, [Moore 1 ; p. 3271: 1. L e t p > Oandv >2. IfE;,q = 0 wheneverp # 0 , q # 0 , a n d p p, then V satisfies the twoterm condition { 0, p ; v } with
a,
=
 1, 6,
=
1,
C,
=
+q
I , @ , > 9, > p , > 0, and qo > 0. Then V satisfies the twoterm condition { 0, 9, + qo  2; v } am
=
0, bm
=
if the following two conditions are satisfied:
(i) E;,q = 0 if p < pol p # pl, and p # 15,; (ii)EXq = 0 if q < q,. 3. Let v, fi,, pl, qo and q1 be integers such that v > 1, Po > p1 > 0 and qo > q1 > 0. Then V satisfies the twoterm condition { 0, Po + qo  1 ; v } if the following two conditions are satisfied:
(i)E;,q = 0 i f p < p , and p # p l ; (ii)E;,q = 0 if q < qo and q # q l . C. Multiplicative structures
Let A be a regular &complex. A multiplication in A which makes A a ring is said to be allowable if it satisfies the following three conditions : (Ml) If x E A , and y E Aa, then x y E A p + q . (M2) If x E A , and y E A,, then x y E A,+*. (M3) d is an antiderivation, i.e., for x E A , and y
+
EA ,
we have
d ( x y ) = ( d x ) y ( I)Px(dy). Assume that an allowable multiplication has been given in A . By using (M l)(M3), define multiplicative structures in the bigraded groups D n and E n of the exact couples V n ( A )= < Dn, E n ; in, jn, kn >

for each n = 1,2; * , Prove the following assertions, [Massey, 21 : 1. If x is homogeneous of degree ( p , q) and y is homogeneous of degree (7, s ) , then x y is homogeneous of degree ( p + r, q s). 2. i n is a transducer of Dn, i.e., in(xy) = x [ i n ( y ) ]= [in(x)]y.
+
3. jn is multiplicative, i.e. jn(xy) = P ( x ) jn(y). = jnkn is an antiderivation of En, i.e., for x E E,,, we have dn(xy) = [dn(x)]y ( l)Ptqx [dn(y)]. 4. dn
+
and y
E En,
EXERCISES
5. For x E D;,, and y E En, we have kn[jn(x).y]= ( I)P+qx.kn(y), k n b . j n ( ~ )=] kn(y).x. D. The exact couples of a bundle space over a finite simplicia1 complex
Let X be a bundle space over a base space B with projection o : X + B and fiber F as defined in (111, 4). Assume that B is a finite simplicia1 complex with B p denoting its #dimensional skeleton and that G is an abelian group. Define the bigraded groups D and E by taking the singular homology groups Dp,q = H p + q ( X p ; G), Ep,q = Hp+p(Xp,X p  1 ; GI, where X , = ml(BP), and the homomorphisms i : D + D , j : D + E , and k : E + D as in the example of 5 5. Thus one obtains an exact couple %f
=(D,E;i,j,k>
which is called the homology exact couple of the bundle space X over B. Consider the derived couples ben = < Dn, En; in, i n , kn> of C and prove the following assertions : 1, W n does not depend on the triangulation of B if n > 2. 2. Ei,q is naturally isomorphic to the homology group H,(B; H,(F; G)) of the complex B with local coefficients in H,(F; G). 3. V is a regular acouple with D,,, M H,+,(X; G) if q < 0. Similarly, one can construct the cohomology exact couple and prove the analogous assertions. E. The homotopy exact couple
Let K be a connected cellular complex and v be a vertex of K . Denote by Kn the ndimensional skeleton of K. Then, for each pair (KP,K p  l ) , there is an exact homotopy sequence
...
f
R nm(KPl,u)2+ nm(KP,u) L nm(KP.KP1,u) + n*1(KP1,u) +
Define two bigraded groups D and E as follows:
DP,,
=
E,,,
=
4, [ n,+,(KP, o,
(
n,+,(KP, Kpl, v ) , j[n,+*(KP,v ) ] , 0,
* * *
.
+
i f p ZOandp q >2, for other values of p and q ;
+ +
if p Z 1 a n d p q Z 3, if p Z 1 a n d p q = 2, for other values of p and q.
The homomorphisms i, j , k in the homotopy sequences of the pairs ( K p ,Kp I) may be extended in a unique way to define homomorphisms
i:D+D, j:D+E, k:E+D.
252
V I I I . EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
Prove that the following triangle is exact :
D+D i
Thus we obtain an exact couple Q ( K ,v ) = < D, E ; i, j , k > which is called the homotopy exuct couple of K at v. Prove the following assertions [Massey 11: 1. The homomorphisms i, j , k in Q ( K ,w ) are homogeneous of degree (1,  l), (0,0), ( 1,O) respectively. Hence Q ( K ,v ) is a &couple in the sense of 5 5. 2 . D P , , = 0 if p < 2 or p + q < 2 ; D p , , w n , , ( K , v ) if q < O and p+q>1. 3. E P , , = 0 if p < 2 or q (0 or p q < 2. 4. Q ( K ,v ) is a regular &couple in the sense of 5 6. 5. If vo and v1 are any two vertices of K , and if u : I +. K 1 is a path joining v,, to v l , then [ induces a natural isomorphic mapping
+
Q(8= (+€, yd : Q(KV J

V ( K ,Vo)
in the sense of 3 9. Furthermore, if two paths [, 7 : I +. fl from vo to w1 are homotopic in K with endpoints fixed, then Q([) = Q(7).These facts can be concisely expressed by saying that the set of homotopy exact couples V ( K ,v ) for various vertices v of K constitutes a local system of exact couples in K . The same remarks hold for the successive derived couples @ ( K , v l . 6. Let K , L be connected cellular complexes, v, w be vertices of K , L respectively, and f : ( K ,v ) +. (L, w ) be a cellular map. Then f induces a mapping Q ( f ) = (+j, yr) : Q ( K ,v ) +. WL, w ) . Let X denote the category in which the objects are all pairs ( K ,v ) of a cellular complex K with a vertex v E K and the mappings are all cellular maps. Let Va denote the category of all regular &couples and their mappings. Then the operation ( K , v ) + Q(K,v ) and f f Q ( f ) is a covariant functor from .fto %‘a. 7. Let k be a connected covering complex over K with projection o :I? + K and 3 be a vertex of I? with o(3)= v. Then g(o) :
W(k,Z )
w Q ( K ,v ) .
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 V ( K ,v ) and its derived couples. 8. The derived couple Q2(K,v ) = < 0 2 , E2; i2, j 2 , k2 > is an invariant of the homotopy type of K . 9. Di,q = 0 if p < 3 or p + q < 2 .
253
EXERCISES
+
10. D;,4 w nP+,(K,v ) if q < 0 and p q > 1. 11. E;,q =O if p < 2 or q < 0. 12. If K is simply connected, then Eb,o is isomorphic to the homology group H,(K) of the complex K for each p > 2. 13. If K is 7connected, then Db,q = 0 for p < 7 1 and E;,, = 0 for < 7. 14. The exact couple V ( K ,v ) contains the following exact sequence of J. H. C . Whitehead: 'kf D ; , o S D$+1,1& E i 3 0 S D&I,oi' . . * .
+
...
If K is simply connected, then, by 10 and 12, there are natural isomorphisms Db+,, 1 w n p w ,4, Eb, 0 = H p ( K ) for each p 2 2. Hence we obtain the following exact sequence for a simply connected I 1, then h commutes with d and hence induces a derived homomorphism Prove that the images of g,
X ( g ) and h, in X P ( A are )
=
Ego. Next, consider h, together with the boundary homomorphism d : g*[zP(A)I
=
EgJlt k,[.@P(A/AO)I
=
h a X P ( A )+ XP(A/AO) + IX p 1 ( A O ) , ( p > 2). Denote by J the image of h,, K the kernel of h,, L the image of d, and M
the kernel of 3. Let X E J . Choose Y E X ~ ( A / A O withh,(y) ) = x , and consider d(y). It is an element ofXPl(AO) which, when y varies, describes a coset mod d ( K ) . Hence we obtain a homomorphism
T :J %p1(AO)/d(K) called the transgression. As the image of h,, we have J = EPP,o.By 3 7, E&,l is a quotient group of i@Pl(Ao).Denote the projection by x . Prove that the rectangle +
is commutative and d(K)is the kernel of x . Hence x induces an isomorphism ($ > 2). After this identification, the transgression T reduces to the differential operator d P : EPP,o + E&l. x*
i@pI(A)/a(K)M
P
E0.pI'
For an arbitrary dcouple, we may define the transgression to be this differential operator d P . Analogous to the transgression, define a homomorphism
x :L
3
Xp(A)/h*(M)
called the suspension. If XP(A)= 0 = i @ P  l ( A ) , then d becomes an isomorphism and the suspension reduces to
x = h,W
: XPl(AO)+ X p ( A ) .
In this case, the transgression diagram
dp =
T is an isomorphism and in the
258
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
the image of X coincides with the image of the monomorphism i and the kernel of Z is precisely the kernel of the epimorphism x . Furthermore, prove that the following relation holds:
z = I (dq1 x. Establish analogous results for a regular &complex A . 1. Properties of Steenrod squares
Let (X, A ) denote any pair consisting of a space X and a subspace A of X , and 2, denote the cyclic group of order 2. Reproduce the definition of the Steenrod square operations Sq' : @ ( X , A :2,)+ Hn+'(X, A ; Z,),
i
> 0,
and prove the following properties of these operations, [Steenrod 2 and Cartan 11: 1. Sq' o f* = f* o Sq' for any map f : ( X , A ) + (Y, B). 2. Sq' 0 6 = 6 0 Sq', where 6 denotes the coboundary operator in the cohomology sequence. 3. Sq'(a
u B) = j+k=s I: . S q W u sqk(B).
4. Sqf(a) = 0 if dim (a) < i.
5. Sq{(a)= a u a if dim ( a ) = i. 6 . SqO(a)= a. 7. Sql coincides with the coboundary operator induced by the exact sequence 0
+z,
2,
+z,+o
and hence we have an exact sequence * *
.  + H n ( X , A ; z ~ ) + H n ( X , A ; ZH1n) +~l~( X , A ; Z z )  + ~ n + l ( X , A ; Z* '4 ) ~ .
CHAPTER IX T H E SPECTRAL S E Q U E N C E O F 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 Sene, 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. I t 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 E2 of this exact couple is computed ($9 310; the main result appears in §$ 56). A variety of applications follow. Kelations among the Poincark 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 ( $ 5 1518), 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 auxillary 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 simplicia1singular homology theory, [ES; pp. 1852 1 11, the unit 12simplex An is used as the antiimage in defining the nchains 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 ncube 1" plays the role of An. In the present section, we shall give a sketch of the cubical theory. As to the equivalence of the cubical theory and the simplicia1 theory, a proof is given in [Eilenberg and MacLane 21. By a singular ncube in a space X , we mean a map u : I n + X.If n = 0, then u is interpreted as a single point in X.If fa > 0, we define the ith lower and @per faces Afou and A& of u to be the singular (n  1)cubes given by 259
260
IX. T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
(2r".u) (t1; for every i we have
=;

1, 2;
a ,
tn1)
* *,
n, E
= u(t,;
= 0,
* *,
1, and (tl;
ti1,
E,
ti;
., tn,)
*
', tn1)
*
E
In1. Then, for i
< i,
1; 17 = 45 1;
where E and 7 may be either 0 or 1. Define Qn(X)t o be the free abelian group generated by all singular ncubes in X if n 2 0 and Qn(X) = 0 if n < 0. Then the operation
au =
n
;r; ( i)yjzeU  1pU)
i=I
determines a homomorphism
a
e n ( X ) + en](x)
aa
= 0. This yields a chain for every n. It is straightforward to verify that complex { Q n ( X ) , }, [ES; p. 1241. Unlike the simplicia1 theory, this chain complex { Qn(X), } does not give the "correct" homology groups of X ; for example, if X consists of a single point, a simple computation reveals that the nth integral homology group of { Qn(X), } is infinite cyclic for every n 2 0. Hence, we have t o "normalize" { Q n ( X ) , }. For each singular ( n 1)cube u in X , n > 0, we define a singular ncube D u in X by taking (Dzc) (t1; * * , tn1, tn) = u(t,; *  ,t n  1 ) .
a
a
a
a
A singular ncube v in X is said to be degenerate if v = Du for some u. In other words, v is degenerate iff it does not depend on the last coordinate tn of the point (tl; * t n )in In. The degenerate singular ncubes in X , n > 0, generate a subgroup D n ( X ) of Qn(X).Since a ,
it follows that
i < n, An'D
=
n I
ItODu) = Z (i=I
1,
l)t(Dl"u  DL&)
and hence carries D n ( X )into Dn,(X). So, the degenerate singular ncubes for all n form a subcomplex { D n ( X ) , } of the chain complex { Q n f X ) ,a }. For each integer n, the quotient group
a
C n ( X ) = Qw(X)/Dn(X) is obviously a free abelian group and will be called the group of normalized cubical singular nchains in X . Since carries D n ( X )into D n  l ( X ) ,i t induces a homomorphism
a
a : c,(x)+ cn,(x)
for every n, which will be called the boundary homomorphism. Since aa = 0, we obtain a chain complex { C,(X), a}. For an arbitrarily given abelian group G,the homology group H n ( X ; G) and the cohomology group H % ( XG) ;
2. C U B I C A L S I N G U L A R H O M O L O G Y T H E O R Y
261
of this chain complex over G are called respectively the ndimensional cubical singular homology and cohomology group of the space X over the coeficient. group G . Let C ( X )denote the direct sum of the groups C,(X) for all n. Then, C ( X ) is a graded differential group with a homogeneous differential operator
a : C ( X )+ C ( X ) of degree  1. We shall call C ( X )the group of all normalized cubical singular chains in X . For any subspace X , of X , the group C,(X,) can be considered as a subgroup of the group C,(X). The quotient group
C,(X, X,)
=
C,(X)/C,(X,)
is obviously isomorphic to the free abelian group generated by the nondegenerate singular ncubes in X not contained in X,. Since i3 cames C,(X) into C,,(X) and C,(X,) into C ,  l ( X o ) , it induces a boundary homomorphism a : C,(X, X,) + C,,(X, X,). Since di3 = 0, we obtain a chain complex { C,(X, X,), a } and hence the relative homology groups H n ( X ,X,;G) and the relative cohomology groups P ( X ,X,; G) over a n arbitrarily given abelian group G. Furthermore, if G = { G, I x E X 1 denotes a local system of abelian groups in X , then one can define the groups H,(X,X,; G) and H n ( X , X , ; G) with local coefficients in G in a way similar to that in the traditional singular homology theory. See [Steenrod 11 and [Eilenberg 31. The direct sum C ( X ,X,) of the groups C,(X, X,) is a graded differential group, called the group of all normalized cubical singular chains in X modulo X,. The direct sum H ( X , X,; G) of the groups H n ( X ,X,; G) will be called the total singular homology group o! X modulo X , over G , and the direct sum H * ( X , X,; G) of the groups P ( X , X,; G) will be called the total singular cohomology group of X modulo X , over G . For the important special case that both X and X,, are pathwise connected, pick a point x, of X as basic point and assume x, E X , if X , is nonempty. Just as in the traditional singular theory, it may be proved that we may consider only the singular cubes with all vertices a t x,. Let G = { G , I x E X } denote a local system of abelian groups in X . Since all vertices of the singular cubes are a t x,, the coefficients of the chains and the cochains with local coefficients in G are all in the group , G on which the fundamental group n , ( X , x,) 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 n , ( X , x,) acts as left operators. For each n, consider the group Cn(X,Xo; G) = C n ( X , Xo) 8 G,
262
T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
where Cn(X,X,) denotes the group of normalized cubical singular nchains in X modulo X , (defined by the ncubes with all vertices at x,). Define a boundary homomorphism
a : Cn(X,Xo; G )
+
Cni(X, Xo; G)
as follows. Consider a generator u 8 g of the group C n ( X ,X,) 8 G, where u : I n + X is a nondegenerate singular cube not in X , with all vertices a t x, and g is an element of G. For each i = 1, * , n, we define a loop afu:Z + X

by taking
where ti = t and = tj0 for every j # i. The loop uiu represents an element [atu] of n , ( X , x,,). Then is defined by
a
n
a ( u 8 g)
=
x
l){ { Ir'u 8 [Ufulg IrOu €3g }.
(
i 1
It can be easily verified that ad = 0. Hence we obtain a chain complex { Cn(X,X,; G ) , }. The nth homology group of this chain complex is defined to be the ndimensional homology group H n ( X , X,; G) of X modulo X , with local coeficients in G. Analogously, one may define the ndimensional cohomology group H n ( X , x,; G) of X modulo x, with local coeficients 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 :I n + X is said to be of degeneracy q iff there exists a nondegenerate singular cube v : Znq + X such that u = Dqv, where Dq denotes the qfold iteration of the operation D. Hence the degeneracy of a nondegenerate cube is 0, and degeneracy of u : I n + X is not less than q if
a
~ ( t , ,*. * , for every point (ti, *
*
a ,
tn)
In) = U(t1,. *
*,
.*,
t n q , 0,.
0)
of In.
3. A filtration in the group of singular chains in a fiber space Throughout $9 314, let w : X + B be a given fibering as defined in (111; $ 3). In other words, X is a fiber space over a base space B with w as projection. Pick a point x, E X and let b, = w(xo)E B. The subspace
F
=
w'(b,)
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, or b, 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. T H E
263
A S S O C I A T E D EXACT COUPLE
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 fi iff u(t,, * * *, t n ) remains within a fixed fiber as tPtl; * * , tn vary, but moves from fiber to fiber as t p varies. For each singular cube u in X , the weight w(u)satisfies the condition f
0
(3.1)
< w(u) Q dim (u).
It is also straightforward to verify the following two relations : (3.2) (3.3)
w(&"u)< w(u)if 1
< i < w(u)and E = 0, 1 ;
w(&"u) = w(u)if w(u)< i and its derived couple %?(A) = < D2,E z ; i2, j2, ka >. Since C p is a direct summand of C, it follows that = APIAP1 = t P
8 G.
Next, let us consider the group K p =
Cp(B,B,) 8 C ( F )@ G.
Define a differential operator dF on K
dF(b8 f 8 g)
=
p
(
by 1)pb 8
a/ 8 g
for every generator b 8 f 8 g of Kp. Then
s p + g ( K P )= Cp(B, Bo) 8 Hq(F; G),
270
IX. T H E SPECTRAL SEQUENCE O F A F I B E R SPACE
w k r e H,(F; G) denotes the qdimensional singular homology group of F with coefficients in G. The homomorphism p of 5 4 defines in this case a mapping y : A p + Kp and hence induces the homomorphisms &,q
The lemmas of
: E p , q +Cp(B, Bo) 8 Hq(F;G).
5 4 imply the following
Ep,q onto C,fB, B,) 8 H,(F; G). By the lemmas C and D in 5 5, n l ( B ,b,) acts on H ( F ; G) as a group of left operators. Let d B denote the boundary operator in C(B,B,) 8 H ( F ;G) as chains with local coefficients in H ( F ; G). Then, one can easily verify that Theorem 6.1. Xp,, is an isomorphism of
(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
xp,q
( p , q) of integers an
: Ej,q * Hp(B, Bo; Hq(F; G))
of E;,q onto the $dimensional singular homology group of B modulo B, with local coeficients in H,(F; G).
An important special case of (6.2) is that A = C ( X )8 G. Then B, is empty and x,,, is an isomorphism of E;,q onto the homology group H,(B; Hq(F;G)) with local coefficients in H,(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 n , ( B , b,) 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 1; p. 4451. Now assume that the coefficient group G is either the additive group of integers or a field. Then the isomorphism x P , , in (6.2) is actually an isomorphism of Gmodules. If n,(B, b,) operates simply on H,(F; G), then the group Hp(B,B,; Hq(F,G)) in (6.2) reduces to the singular homology group with coefficients in H,(F; G) in the usual sense. Hence, by the Universal Coefficient Theorem, [ES; p. 1611, we obtain the following Theorem 6.3. If n,(B. b,) operates simply
+
011
H,(F; G), then
Ei,q * Hp(B, B,; G) @cHq(F;G) T o Y G ( H ~  ~B,; ( BG), , H & F ; G)). The torsion product TOYG has the important property that Torc(L,M ) = 0 if L or M is a free Gmodule, [ES; p. 1341. Hence we have the following Corollary 6.4. If H P  ] ( B ,B,; G) or H,(F; C ) is a free Gmodule, then
Ei,q w Hp(B, Bo; G) @GHq(F;G). Note that the hypothesis of (6.4) is always true if G is a field.
7. T H E
S P E C T R A L HOMOLOGY S E Q U E N C E
271
7. The spectral homology sequence Let us consider the regular 8complex
A
=C8
G, C = C ( X , X,) of the preceding section. As in (VIII; 3 4), we denote by
@(A)
= < Dn,En; in,
p,kn >
the successive derived couples of the associated exact couple % ( A ) = W ( A ) of A . Then, En is a bigraded differential group with dn = jnkn as differential operator and is the nth term of the associated spectral sequence {EnIn=l,2,.} of Y = % ( A ) , which will be called the sfiectral homology sequence of the fiber space X modulo X , over the coefficient group G. Since A is a regular &complex, we may apply the results of (VIII ; §§lo1 1). In particular, #(%) = # ( A ) = H ( X , Xo; G).
Then H ( X , X,; G) is filtered with E* as its associated graded group; more precisely,
Hm(X, x,; G) = Z m , o ( % )
= %m1,1(%)
2 ''
*
= X o . m ( V ) = 2PI,rn+1(%)
= 0,
(U) = EZq. Hereafter, we shall use the notation H p , q ( X X,,; , G) = 2Pp,q(%). For further studies in this section, we have to specify whether or not X , is empty. Let us first consider the case that X , is empty. According to (VIII; 3 I l ) , Rq(%)= S q ( A 0 ) . A singular cube G 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 a t x,, this single point must be b, and hence u is in F . Therefore, Ao = C(FI 8 G. Then it follows that /%P1,9+1
#,*(1(%)
I?*(%)
=
Hq(F; G).
Next, S,(%) =#,(A) with A, = ED,,. However, by means of Xp,o, we may identify E p ,o with C,(B) 8 H,(F; G). Since F is pathwise connected, H,(F; G) may be identified with G on which n l ( B ) operates trivially. Hence A = C(B)8 G and S p ( % ) = H p ( B ;G ) . Then, we may apply the results of (VIII; two commutative triangles
Hq(F;G)
\
'\EZq
& ( X , G)
/
3 7)
and obtain the following
H p ( X ;G) W. H P P , G)
\ J/ Ego
where 8*, o* are induced by the inclusion 8 : F c X and the projection o : X + B, the x's are epimorphisms, and the L'S are monomorphisms. It is
272
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
an immediate consequence of these triangles that H,,,(X; G) is the image of 8, and H p  l , l ( X ;G ) is the kernel of we. It remains to study the case that X , is not empty. Then F c X,. One can easily see that A0 = 0 and hence E:.q = 0 = EC,; in particular, Rq(U) = 0. On the other hand, we have S p ( V ) = Hp(B, Bo; G). Thus, we still have a nontrivial commutative triangle
Hp(X, Xo; G) 0.Hp(B, Bo; G)
\
EZO
J/
The kernel of w* is H p  l , l ( X ,X,; G ) and the image of o* is isomorphic to EZo. 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 = 0.Then the singular cube v reduces to the point x, ; and the problem reduces to constructing, for each singular cube u : I p + B , a singular cube z = M(u, v ) : I p +. X such that wz = u. Let V,denote the leading vertex (0; * ., 0) of I p . Since V , is a strong deformation retract of I p , we may apply (111; 3.1) and obtain a map y :I p f X such that wy = u and y(Vo) = x,. Let V , denote the various vertices of Ip. Since u(Va)= b, and wy = u, it follows that y(Va) is a point of F. Since F is pathwise connected, there exists a path a. : I +.F such that a,(O) = y(Va) and aa(l) = x,. Let Q denote the subspace of I p which consists of all vertices of I p . Define a homotopy f t : I p +. B, (0Q t Q I ) , and a homotopy gt : Q +. X , (0 Q t Q l) , by taking ft =
gt(Va) = 4
for every t E I and every vertex V , of go = Y I
e,
Ip.
4
Then we have
wgt
= ft
I
e
for every t E I . According to (111; 3.1), there exists a homotopy gt* :I p + X , (0 Q t Q l ) , such that
e,
Wgt* = f t go* = y , gt = gt* I for every t E I . Let z = g,*. Then z is a singular cube in X with all vertices a t x, and such that wz = u. This proves the lemma for q = 0. Next let q > 0 and assume that we have constructed M ( u , v ) for all u and all v with dim (v) < q satisfying the conditions of the lemma. We are
8. P R O O F O F
LEMMA A
273
going to construct a singular cube z = M(u, v ) for a given pair of singular cubes u : I P + B and v : I q + F as follows. To construct the singular cube z = M ( u , v ) , let us first dispose of the special case where v is degenerate. Then v does not depend on the last variable and Aqo(v) = Aql(v). Define z = M(u, v ) by means of the formula: z(t1,.
tp+g) =
*
M ( u , &Ov) (ti). * *
t
tptq1).
We have to verify the conditions (Alk(A5) of the lemma. The conditions (Al), (A4) and (A5) are obviously satisfied. The condition (A2) is verified as follows: B p ~ ( t , ; * * , t p ) = wz(t,, * * * , tp, 0, * * * , 0 ) = wM(zt, A ~ O V ) ( t , ; . * . t p , O ; * * , O ) = up,, * , t,) ;

Fpz(t,,
*
*,
tq) =
z(0, * '  ,0, t,,
* * *,
tq)
M ( u , A*%) (0; * ,0 , t,; * tpl) = &O"(t,, * * * , = "(t,, * * * , tq). =
* )
To verify the condition (A3), we consider two cases. If i
=
q, then we
have
G+k (tl, If i
* * *
,tp+q1)
=
< q, we have
A;+$
(t,, *
= Z(t1,
* *,
tp+qJ
* * *
I
tpq1, E )
M ( u , &%) (t,;
= Z(t,, '
* *
,tp+i,,
* *,
tp+ql).
E , tp+a,

.
* *>
tp+qJ
M ( u ,AqO") (t,, * * , t p + ;  , , E , t p + z . * = n;+i M ( u , &On) (4,* * * , t p + q  2 ) = M ( u , /?a"*") (tl,* * * ) tp+*J. =
* *
,t p + q  , )
On the other hand, since i < q and v is degenerate, it follows that A~'v is also degenerate and so is M(u,A(ev\. Therefore, we have
M ( u , Atv) (tl, * * * , t p + *  1 )
M ( u , Afv) (4,* * * , t p + q  2 , 0 ) = A;+4l fif(.u, A t 4 ( 4 , . * * , tp+*2) = M ( u , il;,La"v) (11,* * * , t p + q  2 ) . =
Since AaeAqOv= A:lAzev, this completes the verification of the condition (A3). It remains to construct z = M ( u , v ) for a nondegenerate v . Let P = Z p + q = Z p x I q and Q = V , x Zq U Z p x a P , where V , denotes the vertex (0; * . , 0) of I p and d Z 4 the settheoretic boundary of IQ.ThenQ is contractible, because it is clearly deformable into V , x I 4 U V , x = V , x Zq which is contractible. Hence, Q is a strong deformation retract of P and we may apply (111; 3.1) once again.
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
274
Define maps f : P + B and g : Q + X as follows: f(tl;
g(0; g(t,,
*
tp, s1;
*,
 ,0,s]; * * *
= M(u,
, t p , S,'
* *)
sg) =
uft,;
*,
sg) =
v(sl;
*
* * *
Ar8v) (t,, *
*
9
si1, 5, Sf,
., tp, s1;
*
*,
 .)t p ) ;

sg),
a ,
* * *,
on V , x Ig;
+I)
sgJ, on I p
x i3D.
I t is straighforward to verify that g is onevalued and hence, by (I; 5.1), g is continuous. Besides, it is obvious that wg = f 1 Q. Therefore, according to (111; 3.1), there exists a map z :I p + g + X such that wz = f and z 1 Q = g. Since q > 0, Q contains all vertices of I p + g . This implies that z = M(zl,v) is a singular cube in X with all vertices at x,. The conditions (Al) through (A5) are all obviously satisfied. This completes the inductive construction of M(u, v ) and proves the lemma.
9. Proof of Lemma B The proof of this lemma is analogous to that of Lemma A and is based on a construction u + Dpu by induction on the integer q. It will be sketched as follows. First assume that q = 0. Let
P
= I,+, = I p
x I, Q
= (Ip
x 0) U (V, x I) U ( I D x 1) c P .

where V , denotes the leading vertex (0;  ,0) of I p . Then Q is a strong deformation retract of P. Define two maps f : P + B and g : Q + X by taking
g(t,;
f (4,* * * ,
t p , t ) = wu(t,,
g(t1, *
*,
t p , 0) =
g(0,
',
0,t )
=
I
tp),
onIp x I ;
* *>
tp),
o n I p x 0;
* * *
u(4, *
onV, x I ;
x,,
 ., tp, 1) = M(Bpu,Fpu) ( t l ;
+, tp), on I p
x 1.
Applying(III;3.1),weobtainamapv:P  + X s u c h t h a t w v = f andv IQ=g. Since Q contains all vertices of P, v is a singular cube with all vertices x,. Let Dpu = v and the conditions (Bl)(I37) are obvious. Next let q > 0 and assume that we have constructed Dpu for every u with w(u) Q p and dim(u) < p + q. Now let u be a singular cube in X with w(u) Q P and dim(u) = p + q. To construct v = Dpu, let us dispose of the special case where u is degenerate. Then u does not depend on the last variable and Ai+qu = A;+,. Define v = D,u by taking "(t,, *

*
,tp+g+J
=

DpA;+qu (4,  ,t p + g ) .
By means of the hypotheses of induction, one can easily verify the conditions (Bl)(B7).
10. P R O O F O F LEMMAS C A N D D
275
It remains to construct v = D,u for a nondegenerate u with w ( u ) and dim(u) = p + q. For this purpose, let p = I , x I x 14 = IP+(l+l, Q
=
(I, x o x 14)
u (I, x
1 x IQ) u
Qp
(v,x I x IQ) u ( I P x I x 3141,
where V , denotes the leading vertex of Ip and 319 the settheoretic boundary of I@.Define two maps f : P +. B and g : Q + X by taking
f (t,;
t, SIP* * ,
BpN1,' * *, t p ) . g(tl,...,1,,O,Sl,...,Sq) = U(tl,"',tp,Sl,...,Sq), g ( t 1 , . * * , t p , 1, ~ 1 , '* * , Sq) = M(Bpu, Fpu) (ti,. * t p , ~ 1 , ' . Sq), g(0; * * , 0 , t, s]; * *, sq) = ~ ( 0 *;* , 0 , sl; * * , ~ 4 )= F ~ U ( S *, ;* , Sq),  8
tp,
Sq) =
' 8
t, S1" * * , S t  1 , E , .Q>* B&+,,iu ( t i , .* t p , t, ~ 1 * , *
g(t1,.
*,
tp,
* t
*
* )
,e  1 )
=
Sq1).
Since Q is a strong deformation retract of P and wg = f I Q, there exists a map v : P +. X such that o v = f and v 1 Q = g. Since Q contains all vertices of P , v is a singular cube with all vertices at x,. Let v = D,u and the conditions (Bl) through (B7) can be easily verified. This completes the proof.
10. Proof of Lemmas C and D To prove Lemma C, we consider u as a singular 1cube in B . For each singular ncube v in F , we take D,v = M(u, V ) given by Lemma A . Then D,v is a singular (n + 1)cube in X . The assignment v + D,v defines a homogeneous homomorphism D, : C(F)+. C ( X ) of degree 1. The conditions (Al)(A4) imply the conditions (Cl)(C4). This proves Lemma C. To prove Lemma D, let u, t be two loops in B representing the same element of n , ( B , b,). Then there exists a singular 2cube u : I 2 + B such that u(0, t)
=
b,
=
u(1,t ) , u(t, 0)
=
up), u(t, 1)
= t(t)
for each t E I . Let D, and D, be deformations of C ( F ) covering the paths u and z respectively. For each singular ncube v in F , we shall construct a singular (n + 2)cube Qv in X satisfying the conditions: (D1) ~ ( Q v
f ( s , t, t,,.
* *)
=+
43,
E , tt,.
* )
tn1) =
Q&'v(s, t, ti;
* *,
in 1 ) .
Since of I A and A is a strong deformation retract of In+a, f has an extension g : In+2 + X covering 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
+
+.
Ja, J r
:C(F)
+
C(F)
defined by the deformations Do, D, respectively. Let K denote the homogeneous endomorphism of C ( F ) of degree 1 defined by
( K v ) ( t , t i , . * * , t n ) = (Qv)( 1 , t , t , , * ' * , t n ) for each singular cube v : I n
+
aKv
F . Then, one can easily verify that
+ Kav = JIv Jav.
Hence, Ja and J , are chain homotopic. This completes the proof of Lemma D. Remark. If X is a bundle space over B with respect to the projection w : X f B , then the proofs of the lemmas AD can be simplified as follows.
11. T H E
POINCARBPOLYNOMIALS
277
For a given singular cube u : I p 7B, one considers the bundle space U over IP induced by u. By a theorem of Feldbau [S; p. 531, U is equivalent to
the product space I p x F and hence the constructions can be easily carried out in U instead of X .
11. The Poincard polynomials Throughout this section, let G be a field. We shall consider vector spaces Mover G graded by the spaces M p ; the dimension of M p is called the pth Betti number of M and is denoted by R p ( M ) Then . M is of finite dimension iff the Betti numbers { R p ( 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 M p = 0 if $ < 0. For such an M , we define the Poincart! polynomial y ( M ) and the Euler characteristic X(M) by
y ( M ) = 2: Rp(M)tP, X f M ) = P
x (P
l)PRp(M).
A few elementary properties of the PoincarC polynomials are listed as follows. Proofs are left to the reader. (1) If L is a subspace of M and N = M / L , then y ( N ) = y ( M )  y ( L ) . (2) For any two graded vector spaces M and N , we have y ( M @ G N )= y ( M ) y ( N*) (3) If M is a graded vector space with a linear differential operator d : M + M o f d e g r e e  I , thenwe havey(&'(M)) = y ( M ) ( l + t ) y ( d ( M ) ) . For any two polynomials f and g, the symbol f Q g will mean that g  f is a polynomial with nonnegative 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 X ( M ) represented by the cycles N and y ( L )y ( N ) < t y ( 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 X ( M ) represented by the cycles L , then we have y ( L )y ( N ) Y ( d ( W). Let ( Y , Yo)be a pair of a space Y and a subspace Y oof Y . If H ( Y , Y o ;G) is of finite dimension, then its Betti numbers, its Poincark polynomial, and its Euler characteristic will be defined to be those of the pair ( Y , Yo) over G denoted by R p ( Y ,Y o ;G ) , y ( Y ,Y o ;G ) , and X(Y, Y o ;G) respectively. Now let us consider the spectral homology sequence of 5 7. Assume that n,(B, b,) oprates simply on H ( F ; G) and that H(B, B,; G) and H ( F ; G ) are of finite dimension. Then we have the following two theorems: Theorem 11.1. R,(X,
X,; G) Q X Rp(B, B,; G)R,(F; G ) . P+q=m
Theorem 11.2.
X(X, X,; G) = X(B, 23,; G)X(F;G).
In words, (1 1.1) states that the Betti numbers of the fiber space X cannot be greater than those of the product space B x F and (1 1.2) states that the
278
IX. T H E SPECTRAL SEQUENCE O F A FIBER S P A C E
Euler characteristic of the fiber space X is the same as that of the product space B x F. These two theorems are immediate corollaries of the general theorem below on PoincarC polynomials. Since H(B, B,; G ) and H ( F ; G) are assumed t o be of finite dimension, their PoincarC polynomials are of the form :
y(B,B,; G) = bata + ba+lt"+l+
   + bptp,
y ( F ;G) = 1 + c1t + c2t2 +
+
(@ > a
> 0),
( y >O).
~$7,
We shall also consider the following four polynomials:
P
=
b,+2ta+2 + *
u = Clt + C2t2+
* *
*
+ bptp,
+cy,
+ bp2tfl2, v = 1 + C1t + + CVltY'. Q
=
bata + *
* *
* * *
If M is a vector space over G bigraded by the subspaces M p , p ,then M can be graded by means of the total degree p + q and its PoincarC polynomial y ( M ) is defined in terms of this grading. Now let
A
=
Z dn(En). ?I>;!
Then A is a bigraded vector space over G of finite dimension. The general theorem mentioned above can be stated as follows. Theorem 11.3. If H ( B , B,; G ) and H ( F ;G) are of finite dimension, then so
i s H ( X ,X , ; G) and its Poincart polynomial i s given by
+ t)y(d).
y ( X , X o ; G ) = y ( B ,Bo; G)y(F;G) (1
Furthermore, the polynomial ~ ( dsatisfies ) the following inequalities:
y(A) < t  ] W , y ( A ) G Q U . Proof. Consider the spectral homology sequence of
3 7. By (6.4), we have
E2 w H ( B ,B,; G ) @ G H ( FG). ; This implies that Em is of finite dimension and so is H ( X , X,: G ) . According to (2), we have y ( E 2 ) = y ( B ,Bo; G ) y ( F ;G ) . By (3), we obtain y(En+l) = y(En) f 1 + t)y(dn(En)). Hence i t follows that
y ( X , Xo; G)
= y(Em)= y ( E 2 ) (1
+ t)y(d).
This proves the first part of the theorem and it remains to establish the two inequalities. For each n 2 2, consider the subspace
11. THE
PO IN CAR^
POLYNOMIALS
279
of En. Since dn is of degree ( n, n  l ) , every element of M n is a cycle and hence we have the epimorphisms M2+M3+...
By (5), we have
+MQ.
+Mn+Mn+l+...
y(Mn)y(M"+') Q v(dn(En))
for each n Z 2 and hence
Y ( M 3 < Y W r n ) + Y ( 4 y(E2) Qu
Hence we obtain y ( d ) Corollary 11.4.
< QU. I
R P + ~ (XXo, ;G)
=
R,@, Bo;G) R y ( F ;G).
Rp+yl(X,Xo; G) = Rp(B, Bo; G)R A  ~ ( F G);
+ R@i(B,B,;G)Rn(F;G).
As an application of (1 1.3), 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 y ( B ;G) = 1 t P , y ( F ; G) = 1 tQ
+
+
with p > 1 and q > 0. Leaving aside the critical case q = p  1, the PoincarC polynomial of X over G is completely determined by B and F : Proposition 11.5.
y ( X ;G)
=
(1
+ t") (1 + tQ) if q # fi  1.
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
280
Proof. P = tP, Q = 1, U = tg, V = 1. According (1 1.3), we have
y(X;G) = (1
+ tP) (1 + t*)  (1 + t)y(d)
with y(d) < tpl and y(d) < tQ. Since q # p  1, the latter conditions on y(d) imply that y(d) = 0. I For the critical case q = p  1, the conditions on y(d)imply that either p(d) = 0 or y(d) = tpl. Hence we obtain the following Proposition 11.6.
+ t2p1.
or 1
If q
=
p  1, then y ( X ; G) i s either
(1
+ tp) (1 + tpl)
A consequence of (11.5) and (11.6) is that the only possibility for a homology nsphere X to be a fiber space over a simply connected homology psphere B with a homology qsphere F as fiber is the critical case q = p  1 and n = 2 p  1, [S; p. 1451. Examples for p = 2, 4,8 are the Hopf fiberings, (111; 5 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 rsphere for some r > 1 and that q ( B , b,) operates simply on the total homology group H ( F ; G) and the total cohomology group H * ( F ; G). Then we have the following Theorem 12.1. There i s an exact sequence
*H m ( X , X , ; G )
i. Hm+l(B,BO;G)__* Hmr(B,Bo;G)
HmfB,Bo; G) ( . ,* * *
called Gysin's homology sequence, where w* i s induced by the projection w : (X, X,)+ ( B ,Bo). Proof. Since F is a homology rsphere, i t follows that E2 contains only two columns which might be nontrivial, namely
Hp(B,B,:G), Ei,, M Hp(B,B,;G) @G Hr(F;G) w Hp(B,B,;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 M
Theorem 12.2. There i s an exact sequence a
*
.
+ 6.
H m ( B , B o ; G ) " l ~ H m ( X , X , ; GH)m~ r ( B , B , ; G ) L
Hm+l(B,B,;G)
x+ 
called Gysin's cohomology sequence, where w* is induced by the projection : (X,X,) + (B,B,). If B, = u, then Ho(B;G) w G. Let s = +*(1) E H ~ + ~ (G). B ;Then it can be proved that +*(x) = x u s = s u x
12. GYSIN'S E X A C T S E Q U E N C E S
281
for every x E Hmr(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 1; p. 4701. 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 thls purpose, assume that X itself is a homology nsphere for some n > r and consider the cohomology ring H*(B). By the exactness of Gysin's cohomology sequence, we deduce that the homomorphism +* : H"'l(B) + H y B ) is a monomorphism if m = 0 or m = n, is an epimorphism if m = 1 or m = n + 1, and is an isomorphism for other values of m. It follows immediately that
+
Hm(B) m 2, m = O mod (r l ) , 0 Q m < n, Hm(B) = 0 , m =/SOmod ( r + l), 0 < m < n. The structure of Hm(B) with m > n depends on the relation between n and Y. If 12 = p ( y 1) q, 0 Q q < 7,
+ +
then it is easy to verify that the cohomology ring H*(B) is generated by three elements 1 E Ho(B) w 2, s = +*(1) E Hr+l(B),and t E Hn(B) with o*(t) as a generator of H n ( X ) w 2. More precisely, the cohomology group H*(B) is free abelian and has a free basis (12.3)
{ 1, s, s2;

* )
t, st, s2t;
* *
},
where juxtaposition denotes cup product. If n = p(r
+ 1) + r ,
we have to consider the exact sequence
o LH ~ ( B2 ) H ~ ( xL ) HP(r+yB)LHn+l(B)%0, where we have H n ( X )w 2 and HP(r+l)(B)m 2. Since Hn(B) is isomorphic with a subgroup of H n ( X ) ,it follows that either Hn(B) M 2 or Hn(B) = 0. If Hn(B) w 2, it follows that y* = 0 and hence w* and +* are both isomorphisms. In this case, H*(B)is free abelian and has (12.3) as a free basis. If H n ( B ) = 0, then i t follows that Hn+l(B) is a cyclic group of finite order k 1. In this case, the groups Hm(B),m > n, are given by
Hm(B) M z k , m=O mod ( r + l), H y B ) = 0, m+O mod ( Y + 1).
Hence H*(B) has a system of generators (12.4)
{ 1, s, s2;
* *
>,
where si is free if i Q p and s{ is of finite order k if i
> p.
282
IX. T H E SPECTRAL S E Q U E N C E OF A F I B E R S P A C E
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 rsphere over G for some r 2 2. First, let us consider the case B , = o. Theorem 13.1. There is an exact sequence
.
* *%
Hmr+l(F;G ) P++ Hm(F;G ) 8 ' + H m ( X ;G)'+
Hmr(F; G)
*
*
called Wang's homology sequence, where 8* is induced by the inclusion 8 :F c X . Proof. Since B is a homology rsphere, it follows that E2 contains only two rows which might be nontrivial, namely
(p = 0 , ~ ) . the commutative triangles i n 5 7, we obtain
w H p ( B ; G) @GHq(F;G ) w Hq(F; G),
Applying (VII; 8.1) and one of the theorem. I Similarly, one can establish the following
Theorem 13.2. There is an exact sequence T* H ~ XG);e* + H ~ ( FG ;) P', ~ m  r + y G) ~ ;5 H ~ + I ( xG) , B',
. . .+

called Wang's cohomology sequence, where 8* is induced by the inclusion 8 : F c 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 antiderivation if r is even, that is to say, p*(x
u y ) = p * ( x ) u y + (
l)@+l)P x u p*(y)
for each x E HP(F;G ) and y E H* ( F ;G ) .The proof of this result is left to the reader. [Serre 1 ; p. 4711. Next, consider the case B, = b, and hence X , = F . Since B is a homology rsphere, Hr(B, b,; G) m G and Hm(B,b,; G) = 0 for m # r. Then E2 and E*2 contain only one row which might be nontrivial, namely
E:A w H,(F; G ) , E;; w Hg(F; G ) . Hence, by (VIII; 8.1) we obtain the following Theorem 13.3. For every integer m, we have
H m ( X , F ; G ) w Hmr(F; G ) , Hm(X, F ; G) w Hm+(F; G). As an application of these theorems, let us consider the fiber space
X
=
[ B ;B , b,]
13. WANG’S E X A C T S E Q U E N C E S
283
over B with the initial projection w : X .+ B as defined in (111; fj 10). Then we have F = [B;b,, b,]. Since n,(B) = 0, it follows from ,(IV; fj 2) that F is pathwise connected. Therefore, we may apply (13.1) to this case. Since X is contractible, we obtain Hm(F) M Hmr+l(F) for every m > 1. Together with H,(F) theorem of Morse. Theorem 13.4. If
[B; b,, b,l, then
M
2, this implies the following
B i s a simply connected homology rsphere and F
=
H m ( F )  2 ifm=Omod(r1), Hm(F) = 0 if m 0 mod ( r  1).
+
For another application, let us consider the sphere bundles over spheres. Assume that the fiber F is a homology ssphere for some s > 1 and consider the integral homology groups H m ( X ) . Assume that s > 2. Then, by the exactness of Wang’s homology sequence, we deduce that the induced homomorphism
e* : H ~ ( F.+) H ~ ( X ) if m = Y  1 or m = r + s  1, is a monomorphism if
is an epimorphism m = Y or m = 7 + s, and is an isomorphism for other values of m.Therefore, it remains to compute H m ( X )for the four critical values 7 1, r, r + s  1, r s of m. First, let us compute Hrl(X).If Y  1 # s, then it follows that
+
Hri(X) = e*[Hri(F)]= 0 since F is a homology ssphere. If 7  1 = s, we define a numerical invariant k of this fibering as follows. Consider the homomorphism
p* : Ho(F) + Hs(F).
If p* = 0, then k is defined to be zero; otherwise, the quotient group Hs(F)/ p*[H,(F)] is a finite cyclic group and k is defined to be the order of this finite cyclic group. Hence, if Y  1 = s, we have if k = 0, ifk # O .
Next, let us compute H,(X). From the exact sequence e
it follows that
0 PI.+ Hr(F)A+ & ( X ) A+H,(F) A+0,
Hr(X) M
2,
(Z.2.
ifr # s , if Y = s.
284
IX. T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
Then, let us compute Hr+a](X).Since r > 1, we have Hr+aI(F) = 0. Hence, we obtain Hr+aI(X) = O*[Hr+a,(F)I = 0. Finally, let us compute Hr+a(X).In the following exact sequence ~r+a(x) A+H ~ ( FA ) Hr+al(F),
Hr+s(F)
we have Hr+,(F) = 0 = &+81(F)and hence
Hr+a(X)M Ha(F) M 2. For the remaining case s
=
1, the induced homomorphism
0, : H,(F) .H,(x)
is an epimorphism if m = r  1, is a monomorphism if m = r + 1, and is an isomorphism if m < r  1 or m > r + 1.Therefore, it suffices to compute the group H m ( X ) for three critical values of m,namely, m = r  1, r, and r + 1. The computation is similar to the case s > 1 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 Let G denote either the additive group of integers or a field. Assume that X , = and that n,(B, b,) operates simply on H ( F ; G ) . Theorem 14.1.
0
If H,(B, G) = 0 for 0 < m
< m < q, then we have a n exact sequence
Hp+Ql(F; G)*. +
e
*
and a spectral cohomology sequence { E*n } of the regular covering space X over B. In particular, (15.3) E*2 M H*(P;H * ( X ; G)) = H * ( n ; H * ( X ;G)) and Z ( V * ) = H * ( B ;G) is filtered with E*m as its associated graded group.
R,(Q*) = H g ( X ; G), S,(V*) and we have two commutative triangles
HU(B;G) *+ Hg(X; G)
\ /
=
H P ( n ; G) 2HP(B; G)
\\
E;;
HP(n; G)
E;;
/
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 5 15. Theorem 16.1. If a discrete groul, n acts freely on a locally contractible acyclic space X of finite dimension, then n has no element of finite order other than 1. Proof. Assume, on the contrary, that n has an element 6 of order Y # 1. The subgroup of n generated by 6 is a cyclic group of order Y and acts freely on X.Thus we may assume that n itself is a cyclic group of order Y . Let B = X ! n denote the orbit space. Then X is a regular covering space of B with n as quotient group. This implies that B is locally contractible and finite dimensional. Let G = 2 be the additive group of integers and let n operate simply on G. Consider the spectral homology sequence of 3 15. Since X is acyclic, it follows that E& M H,(n) and E;,q = 0 if q # 0. Then, by (VIII; 8.3), we obtain H,(B) H p ( n ) , ( P > 0).

288
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
Since R is locally contractible, H p ( B )= 0 if p > dim B. However, H,(n) w x 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 R, (V; Ex. I), we may assume without loss of generality that B is locally contractible. Then the universal covering space X of R is defined and the fundamental group n = n,(B) operates freely on X . Theorem 17.1. If
morphisms
np(B)= 0 for 1 < p < r, then we have the natural iso
p*:H,(B; G) M H,(n; G ) , ,u*:HP(n;G) M HP(B;G) for each p
< Y and every coeficient group G o n which x operates simply.
Proof. Consider the spectral homology sequence of 3 15. Then the assumption implies that & ( X ; G) = 0 if 0 < q < r. By (15.2),we have
E:,q = 0, if p + q < r and q # 0. Then it follows from (VIII; 8.5) and one of the commutative triangles in 9 15 that ,u* is a n isomorphism whenever p < r. By considering the spectral cohomology sequence, one can also show that ,u* is an isomorphism whenever p < r . I Thus we see that the fundamental group of B determines the homology groups and the cohomology groups of B for all dimensions less than r if n p ( B )= 0 for every p satisfying 1 < p < Y . To study the groups of the critical dimension r, let us consider the homomorphisms ( o , : H r ( X ;G) + H r ( B ;G ) , w*:Hr(B;G)  + H r ( X ;G) induced by the projection o : X + B. Denote by & ( B ; G) the image of o* and by k ( B ;G) the kernel of o*. Theorem 17.2. If n p ( B )= 0 for 1
< p < r, then we have the exact sequences
(a)
H r ( X ;G) a Hr(B, G) PI+ H,(n;G) +. 0 ,
(4
W ( X ;G ) +fW ( B ;G ) +"IW ( n ;G) c 0 ,
H r ( B ;G)/&(B;G) w H r ( n ;G ) , Ar(B;G) M H r ( n ; G ) . Proof. Consider the spectral homology sequence of the terms of E 2 with total degree r  1 arid r. Let
am
= 0,
bm
=
m, C,
=
m, dm
=
3 15 and in particular
0 , (m = Y  1, Y ) .
17.I N F L U E N C E O F T H E
FUNDAMENTAL GROUP
289
Then it is easily verified that the twoterm condition { r  1, r ; 2 } of (VIII; 4 8) is satisfied. Since E&l = 0 and E:,o w H r ( B ;G ) , we obtain an exact sequence Hr(B;G) 2tH r ( n , G) + 0. (4
According to 3 15, the kernel of !I* isXrl,l(%’) and the image of w * :Hr ( X ; G) + Hr(B;G) is X o , r ( g ) . Since E m contains only two terms of total degree r which might be different from zero, namely, E& and E20, it follows that Xr.l,l(%) =Zo,,(5f). Hence we obtain the exact sequence (a). By considering the spectral cohomology sequence of 3 15, one can also establish the exactness of the sequence ( b ) . I In the proof of (17.2), we did not make full use of the twoterm condition { Y  1, r ; 2 }. Indeed, we actually have longer exact sequences given in the following Theorem 17.3. I f n p ( B ) = 0 for 1
< p < r, then we have the exact sequences:
G)%+J,(Hr(X; G ) ) L H r ( B G)&+Hr(n; ; G)+0,
( d ) Hr+l(B;G)”+Hr+l(n;
(el H‘+~(B;G) +@:~r+l(n ;G)L I , J G) H +CH~(B ~(x ;G) s;~ r ( nG)+o. ; For the definition of the operators I , and J n , see (VI; Ex. K). Proof. Applying the twoterm condition { r  I , r ; 2 } as in the proof of ( 1 7.2), we obtain instead of (c) a longer exact sequence
(4
E20.7 Hr(B;G) 3t H r ( n ;G) + 0, where Ei,rw H o ( n ;H r ( X ;G ) ) is isomorphic with J,(Hr(X; G)) according to (VI; Ex. K l ) . For further extension of this exact sequence, we have to consider the terms of E 2 with total degree Y + 1. There are only three such terms which might be different from zero, namely +
E;,r+l, E ? p G + , , O w &+An; GI. Let n > 2. Since the differential operator dn in En is of degree ( n, n  l), the elements of E$+, and ET,r are cycles and dn sends EY+l,,, into E& iff n = Y + 1. Since every element of E& is a cycle, we get an exact sequence Er+I EY+1 ctET+2 + O , (ii) r+1,0

+
0,r
Since E,2+1,0= E:$:,,, E;,, = E$‘, and EZr and (ii) so that we obtain an exact sequence
o,r
=
E;;t2, we can combine (i)
*+
Hr(n; G) +o E,2+1,2A+ ~2 on*%+ Hr(B; G) with +* = &+I. Since the kernel of dr+l is obviously EL$?,o = E,*;,,, and since E:+I,Ow Hr+l(n;G ) , one of the commutative triangles in 8 15 gives an exact sequence H ~ ( +B ;G) ~ 3+Hr+l(n;G) A+ (iv) Combining (iii) and (iv), we obtain the exact sequence ( d ) . Similarly, one can deduce (e). I
(iii)
~2,~.
290
IX. T H E S P E C T R A L S E Q U E N C E O F A FIBER S P A C E
Note. If G is the group of integers, then the groups&(B; G) andAr(B; G) are essentially the groups Zr(B) and k ( B ) as defined in (VI ; Ex. J). See also [EilenbergMacLane 13.
18. Finite groups operating freely on s' Let n be a finite group operating freely on the rsphere X = Sr. Let B = Srln denote the orbit space and w : S + B the projection. By the finiteness of n,one can easily show that Sr is the universal covering space of B with o as projection and that n can be considered as the fundamental group of B. To apply the results of 3 17, let us take the additive group 2 of integers as the coefficient group and let n operate simply on 2. By (17.1) and (17.3), we have the natural isomorphisms (i) ,u* : Hp(B) H p ( n ) , /A*: HP(n) M HP(B) for each p (ii)
< r and the exact sequences 0 + Hr+l(n)!L J,(Hr(Sr))>+ Hr(B)Ifi+ Hr(n) + 0,
+c
Hr(B) +fHr(n)+ 0, 0 c Hr+l(n) I,(Hr(Sr)) (iii) since dim (B) = r and therefore Hr+,(B) = 0 = H*+l(B). Proposition 18.1. If there is an element 6 EZ which changes the orientation of Sr, then r must be even.
Proof. Since 6 changes the orientation of Sr, 5 is of order h > 1. The subgroup of n generated by 6 is a cyclic group of order h and operates freely on S'. Therefore, we may assume that n itself is the cyclic group of order h generated by 6. Since 6 changes the orientation of Sr, we have t ( x ) =  x for every element x of the free cyclic group Hr(S*).It follows that I,(Hr(Sr))= 0 and the exact sequence (iii) reduces to
0 c Hr+l(n)+ 0 c Hr(B)
Hr(n)c 0.
This implies that (iv) Hr+l(n) = 0, p* : Hr(n) w Hr(B). By (VI; Ex. K5),the first equality implies that r must be even. I Proposition 18.2. If n contains ait element 6 # 1 which preserves the orientation of Sr, then r must be odd.
Proof. We may assume that n is the cyclic group generated by 5. Since 6 preserves the orientation of Sr, n operates simply on Hr(S*).It follows that ],(Hr(Sr)) = Hr(Sr)and the exact sequence (ii) becomes 0 Hr+l(n)+ Hr(Sr) + Hr(B) + Hr(n) + 0. (4 This implies that Hr+l(n)is isomorphic to a subgroup of the free group Hr(Sr)and hence it is also free. According to (VI; Ex. K5), we conclude that Hr+1(n)= 0 and so r must be odd. I +
18. F I N I T E
GROUPS OPERATING FREELY O N
sr
291
Proposition t8.3. If r is even and n contains more than one element, then n is a cyclic group of order 2 and, for each p with 1 Q p < r, we have:
H,(B)
= 0,
HB(B) M n, if p is even;
H,(B) w n, HP(B) = 0 , i f p is odd. Proof. Let 6 # 1 and 7 # 1 be any two elements inn, then, by (18.2), both t and 7 must change the orientation of S r and hence t7l preserves the orientation of Sr. Using (18.2) once more, we deduce that 6ql = l. This proves that n is cyclic of order 2. For the rest of the proposition, (i)and (iv) give all the homology and cohomology groups except Hr(B).The latter can be computed either from the relations between homology and cohomology groups or as follows. In the exact sequence (ii), since both Hr+l(n)and J,(Hr(Sr)) are of order 2, +* must be an isomorphism. On the other hand, we have Hr(n) = 0. Therefore, it follows easily from the exactness that Hr(B) = 0. I Proposition 18.4. If Y is odd a n d n is abelian, thenn must be cyclic, Hr(B) M 2 m Hr(B),and for each p with 0 < p < r we have:
H,(B)
= 0,
H,(B)
M
p i s even; if p is odd.
HP(B) w n, if
n, HP(B) = 0,
Proof. According to [ 18.1), every element of n preserves the orientation of 9. I t follows that J,(Hr(Sr)) = Hr(Sr),
and hence the eFact sequence (ii)becomes
(4
H,(s~)2tHr(B)&+Hr(n) + 0. Hr+i(n) Since every element of n preserves the orientation of Sr, B is an orientable manifold of dimension r and hence we have Hr(B) w 2 M Hr(B). Assume 0
+
that n is of order h. Then, by the definition of o*, one can easily see that we can choose generators a E Hr(Sr) and B E Hr(B)such that o * ( a ) = hp. Hence the exact sequence ( v i ) implies that Hr(n)is a cyclic group of order h. According to ( i )and (VI; Ex. K), it remains to prove that n is cyclic. Let us assume that n is not cyclic. Then there is a prime number p such that n contains a subgroup of the form Z, + Z,, where Z, denotes the cyclic group of order p . We may assume that n itself is of the form Z, 2,; then n is of order Pz. Therefore, Hr(n) is a cyclic group of order pz. Since Hr(Zp) w Z,, i t follows from (VI; Ex. K) that 2, is a direct summand of Hr(n).This is impossible. I
+
Note. 1. If n is a discrete group operating freely on the rsphere 9,then the compactness of Sr implies that n is finite. 2. One can deduce all the homology and cohomology groups of the real projective spaces from (18.3) and (18.4).
292
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
EXERCISES A. Cohomology with arbitrary coefficients
Let G be an abelian group and C denote the regular &complex C ( X ,X,). Consider the group A* of singular cochains of X modulo X,with coefficients in G. Then A* is a graded group with A: = Hom(C,, G) and a differential operator 6 of degree 1. Since Hom (C, G) is the direct product of the groups Horn(&, G ) , [see ES, p. 1471, A* is a subgroup of Hom(C, G). Define a decreasing filtration { A*P } in A* by taking
A*P={f€A*If(CPl) = o > . Then A*P is a subgroup of the graded group A* with A t 9 = A*P n A:. Define a weight function w on A* by taking w ( f ) , f E A*, to be the least upper bound of the integers p such that f E A*p. Prove that 0 < w(f ) < dim ( f ) for every nonzero homogeneous element f E A*. Therefore, with the filtration { A*P }, A* becomes a regular &complex. Study the associated exact couple
%'(A*)= < D*, E * ; i*, j*, k* > and its derived couples, as follows: prove that
E;,,
Hom(Cp(B,B,), Hq(F;G ) ) , E *2m M HP(B, B,; Hq(F;G)), M
where HP(B, B,; H @ ( FG; )) denotes the $dimensional singular cohomology group of B modulo B, with local coefficients in H @ ( FG) ; ; and that if G is a commutative associative ring with a unity element, then these are ring isomorphisms.
B. The spectral cohomology sequence Consider the regular &complex A* of the preceding exercise. Denote by
$+(A*)
=
< D*n, E*n; i*n, j*n, k*n >
the successive derived couples of the associated exact couple V = %'(A*). Then the associated spectral sequence { E*n I n = 1,2; * } of W will be called the spectral cohomology sequence of X modulo X , over G. Prove that &'(%') = &'(A*) = H * ( X , X , ; G) is filtered with E** as its associated graded group; more explicitly,

Hm(X,Xo;G)=*o,m(%')
3 s i , m  i ( % ' ) 3 * .  3*m,o(%')
&'PA
%'I /&'P
+ I . q 1
(%')
= q,;.
3
zm+i,
i('+4= O ,
Hereafter, we shall use the notation Hp,q(X,X , ; G) = &'p,q(%'). Next, assume X , = and prove that &(W) = H@(F;G ) and Sp(V) = HP(B; G). Then we obtain the following commutative triangles
H q ( X ;G )
e*
293
EXERCISES
HU(F;C)
\ J E;P
HP(B; G) "'+
\\
H p ( X ;G)
J
Ego
where 8*,w* are induced by 8 : F c X and o : X +. B, the x's are epimorphisms, and the i s are monomorphisms. Prove that H 1 ~ g  l ( XG) ; is the kernel of 8* and Hpv0(X; G) is the image of w * . Now, consider the transgression. According to (VIII; Ex. H), there are equivalent definitions : (1) The transgression is the differential operator d*m : E:,; +. E:,:, (2) In the following homomorphisms
H ~  I ( Fc) ; 2tH ~ XF ,; G) + C H ~ ( B G), , (m 2 21, let M denote the image of w* and K the kernel of w *. Then the transgression is the homomorphism T : dl(M) + H m ( B ;G)/K defined as in (VIII; Ex. H). A comparison of these definitions shows that E:; is isomorphic to the image of Hm(B; C) in H m ( X ,F ; G) under the induced homomorphism w*. Let G be the group of integers mod 2. Prove that the transgression T commutes with the square operations and the reduced powers of Steenrod. Investigate analogously the transgression in the spectral homology sequence of Q 7. C. The maximal cycle theorem
In addition to the assumptions of 9 3, assume that G is a field. Then prove the following Theorem. If Hm(B, B,; G) = 0 for each m > fi and Hm(F;G) = 0 for each m > q, then we have:
+
H,(X, X,; G) = 0, (m> fi q ) ; H p ( B , Bo; G) @G Hq(F; H p + q ( x ,xo;G) Deduce the following assertions : 1. If G # 0 and H m ( X ;G) = 0 for each m > 0, then a t least one of the following three statements must be true: (a) H m ( B ;G ) = 0 = H m ( F ;G) for each m > 0. ( b ) H,(B; C) # 0 for infinitely many values of m. (c) H m ( F ;G ) # 0 for infinitely many values of m. 2. If a euclidean space X = Rn has a bundle structure over a base space B with a connected fiber F , then both B and F are acyclic.
D. An isomorphism theorem In addition to the assumptions of Q 3, assume that X , # B, # o. Prove the following
and hence
294
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
Theorem. If Hm(B,B,; G)
0
< m < q, then we have
=0
for each m
0 , then Hm(B; G) M H m ( X ;G) + HmI(X; G ) .
F. Relations between the cohomology algebras of a space and
i t s space
of loops
Let B be a pathwise connected and simply connected space and b, a given point in B . Let X = [B; B, b,]. Then, X is a contractible fiber space over B with fiber F = A(B),the space of all loops in B with b, as basic point. Prove the following assertions: 1. If K is a field and the cohomology algebra H * ( F ;K ) is isomorphic to an exterior algebra over K generated by an element of odd degree n, then the cohomology algebra H * ( B ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n + 1. [Serre 1 ; p. 5011. 2. If K is a field of characteristic zero and the cohomology algebra H * ( F ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n, then the cohomology algebra H*(B;K ) is isomorphic to an exterior algebra generated by an element of odd degree n + 1. [Serre 1 ; p. 5011. 3. If K is a field and the cohomology algebra H * ( B ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n > 2, then the cohomology algebra H * ( F ;K ) is isomorphic to an exterior algebra generated by an element of odd degree n  1. [Serre 1 , p. 4921. 4. If K is a field of characteristic zero and the cohomology algebra
EXERCISES
295
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. 4891. 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 1z  1 and a polynomial algebra generated by an element of degree 2(n  1). [Serre 1; p. 4891. 6. If K is a field of characteristic p # 0 and Hm(B; K ) w Hm(Sn; K ) for every m < 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 { 1, y, y2; ., y p  l , z }, where [Serre 1 ; p. 4941 deg ( y ) = n  1, deg (z) = p(n l), y p = 0. 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,* * .,y p  l , z }, where deg (Y)= q, deg (4= pq, yp = 0, then the subspace of H * ( F ;K ) formed by the elements of degree not exceeding p q  2 admits a homogeneous basis which consists of the elements { 1, u, v }, where [Serre 1 ; p. 4951 deg (H) = q  1, deg ( v ) = #q 2. G. The cohomology algebra of (Z, n)
Let 2 be the free cyclic group. Prove that the cohomology algebra H * ( 2 ) 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 rt 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 (1(S")
Consider the sphere Sn of dimension n > 2 and study the cohomology algebra of the space S2n = A ( S n ) of loops in Sn with so as basic point. Let T = [Sn; S n , so]. Then T is a contractible fiber space over Sn with projection o : T + Sn and fiber Q n = ol(so). From (13.2), deduce an isomorphism p* : H y Q n ) M Hmn+l(Qn)
296
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
for each m ; p* is a derivation or an antiderivation according as n is odd or even. Thus, a homogeneous basis { ec }, i = 0, 1,2; * ,of H*(SZn) is given by e, = 1, et = p*l(eil), i = l , 2 ;  . , with dim (et) = i(n 1). Prove the following Themem. T h e mzlltiplicative structure of the integral cohomology algebra H*(SZn) i s given by the rule epeq = cp,qep+q,where cp,q i s a n integer given as follows: (i) If rt is odd, then (9 + q ) ! cplq =
p!4!
*
(ii) If n i s even, then cp,q = 0 if both 9 and q are odd; otherwise
where [XI denotes the largest integer not exceeding x. Then prove 1. If n is odd, then (el)P = (p!)e,. If n i s even, then (el), = 0, (ez)P = ( p ! ) e 2 p ,and ele2p = e2pel = e2p+1. 2. If n is even, then H*(SZn) is isomorphic to the tensor product of the algebras H*(Snl) and H*(Q2nI). 3. Let K be a field of characteristic zero. If n is odd, then H*(@; K ) is isomorphic to a polynomial algebra generated by an element of degree n  1. If n is even, then H*(Q,; K ) is isomorphic to the tensor product of an exterior algebra generated by an element of degree n  1 and a polynominal algebra generated by an element of degree 2(n  1). 4. Let K be a field of characteristic 9 # 0. If n is odd, then H*(Q,; K ) is isomorphic to a polynomial algebra with infinitely many generators f g , (i = 0 , 1; * *), modulo the ideal generated by (f@, (i = 0 , 1 ; *), f t being of degree 94(n  1). 5. The cohomology algebra H*(Snl x Q2al) is isomorphic to the tensor product H*[Snl) 18H*(P'nl). 1. The connective fiber spaces of S3
For each IZ >, 3, let X , denote an nconnective fiber space over S3.Verify the following results on the cohomology algebra H*(X,; 2,) due to [Serre 31 : 1. For the dimensions < 11, H*(X,; 2,) has a homogeneous basis { 1, a , b, c, d } where d i m ( a ) = 4, d i m (b) = 5 , d i m (c) = 8, d i m ( d ) = 9, and where b = Sqla, c = a,, d = ab. 2. For the dimensions < 8, H*(X,; 2,) has a homogeneous basis { 1, e, f , g, h, i 1 where d i m (e) = 5, d i m ( f ) = 6 = d i m (g), d i m (h) = 7 , d i m ( 2 ) = 8, and where f = Sqle, h = Sqlg = Sq2e, i = Sq2f, Sq2g = 0. 3. For the dimensions < 7, H*(X,; 2,) has a homogeneous basis { 1, i, k } where d i m ( j ) = 6 , d i m ( k ) = 7 , and where Sqlj = 0, Sq2j = 0. 4. H7(X,; 2,) has a basis formed by a single element m with S q l m # 0.
CHAPTER X CLASSES O F A B E L I A N GROUPS
1. Introduction Our principal remaining object is the computation of certain of the groups nm(Sn). To facilitate these computations, a digression on Serre’s class theory
of abelian groups is necessary. It is to this digression that the present chapter is devoted; the study of the groupsnm(Sn)is deferred to the next (and final) chapter . In $$ 26, we introduce the formal definitions. In $4 710, certain immediate topological consequences are obtained ; the most important of these is the generalized Hurewicz theorem, from which one deduces that the homotopy groups of a simply connected finite polyhedron are finitely generated.
2.The Definition of Classes A collection V of abelian groups is called a class if the following four conditions are satisfied: (CG1) V contains a group which consists of a single element. (CG2)If a group A is isomorphic to some group in V, then A is in V. (CG3)If a group A is a subgroup or a quotient group of some group in %, then A is in V. (CG4)If an abelian group A is an extension of a group in V by a group in %, then A is in V. Examples of classes are listed as follows. Verifications are left to the reader. (1) The class d of all abelian groups. (2) The class 0 of all groups consisting of a single element. (3)The class dfof all finitely generated abelian groups. (4)The c l a s s 9 of all finite abelian groups. (5) The class of all finite abelian groups with order not divisible by any prime number of a given family of prime numbers. In particular, if the given family consists of all prime numbers except p , this reduces to the class alp of all finite abelian groups of order p r , r = 0, 1,2; * * . (6) The class 9 of all torsion groups. An abelian group A is called a torsion group if every element of A is of finite order. Other examples of classes will be given in the next section and in Ex. A at the end of the chapter. 297
298
X. C L A S S E S OF A B E L I A N G R O U P S
I t is an easy exercise to prove that a nonempty collection %' of abelian groups is a class iff, for every exact sequence L f M f N of abelian groups, the following condition is satisfied: (CG5) L E%' and N E W imply M E%'. Furthermore, it can also be verified that every class V has the following properties : (CG6) I f A E Y a n d B E W , thenA + B E % ' . (CG7) If A E W and B is finitely generated, then A @ B and Tor(A,B ) are in V. (CG8) If A is finitely generated and A EW,then H m ( A )E%'for every m. See (VI; Ex. G ) . It is, of course, necessary to observe that a class V cannot be a set; thus the usual precautions must be taken to avoid contradictions. See [ES; p. 1201.
3. The Primary Components of Abelian Groups For any given prime number p, the #primary component of an abelian group A is defined to be the subgroup of A consisting of the elements of order fir, Y = 0, 1,2, * . If A is a torsion group, then it is the direct sum of its pprimary components for all p. Hence, in order to compute a certain torsion group A , it suffices to compute all of its primary components. See [Ka]. If, for some prime number p, an abelian group A reduces to its pprimary component, that is, if all elements of A have orders which are powers of p, then A is said to be a 9primary group. A finitely generated abelian group is pprimary iff i t is of order for some r. It is evident that, for a given prime number p, the pprimary groups form a classgp. Let F be a given family of prime numbers. Then the torsion groups with null pprimary components for each prime number p in the family F constitute a class. If F contains all prime numbers, then this class reduces to the class 8;if F consists of all primes except a given prime number p , then it becomes the classPp; if F is empty, then it is the classy.

4. The %'Notions on Abelian Groups For a given homomorphism f : A f B, let I m ( f ) , Key(/), and Coker(f) denote respectively the image, kernel, and cokernel of f ; this latter is defined by Coker(f) = B/Im(f). The following sequence is obviously exact :
o
f
K e y ( / )+ A ! B f coker(f)f 0.
4. T H E VNOTIONS ON ABELIAN
GROUPS
299
Furthermore, any pair of homomorphisms
A ~ . B & C gives rise to a natural exact sequence S(f, g) :
0
f
Key(/) f Kerhgf) + Ker(g) + Coker(f) + Coker(gf) + Coker(g) + 0.
In the applications of class theory, the groups in a class V are usually to be neglected in a certain sense. Thus we are led to the following terminology: Let V be a given class. A group A is said to be Vnull if A E V. Let f :A + B be a homomorphism. Then f is said to be a Vmonomorphism if Ker(f ) E %', a Vefiimmphism if Coker(f)EV, and a Visomorphism if it is both a Vmonomorphism and a Vepimorphism. If a %isomorphism f : A + B exists, then A is said to be Visomorphic to B . If V 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 Vequivalent if there exists an abelian group L with two Visomorphisms f : L + A and g : L + B. Proposition 4.1. Two abelian groufis A and B are Vequivalent if there exists an abelian group M and two Visomorphisms h : A + M and k : B + M .
+
Proof. Suficiency. 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 f : L + A and g : L + B by / ( a , b) = a and g(a, b) = b. Then one can verify that f and g are Visomorphisms. Necessity. Let A and B be Vequivalent. Then there exists an abelian group L together with two %'isomorphismsf : 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 (f(l),g(1)) for all 1 E L . Let p : A B + M denote the natural projection. Define homomorphisms h . A + M and k : B + M by h(a) = $(a, 0) and k(b) = fi(0, b ) . Then one can verify that h and k are Visomorphisms. I The relation of being Vequivalent is obviously reflexive and symmetric. It is also transitive. To verify this, assume that A , B and B , C are both Vequivalent. By definition and (4.1), there are two abelian groups L,M and four %'isomorphisms f : L + A , g : L + B , h : B + M , k:C+M.
+
+
Since hg : L + M is aVisomorphism, it followsthat A andC are Wequivalent. This proves the transitivity and the relation of being Vequivalent is an equivalence relation.
300
X. C L A S S E S O F A B E L I A N G R O U P S
5. 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 E%?implies that Hm(A)E%? for every m > 0. V is said to be complete if A E%? implies that A @B E 9 and Tor(A,B) E%? for every B. %? is said to be weakly complete if A E %? and B E %? imply that A 8 B E %? 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 i t can be verified that every strongly complete class is complete and perfect. See Ex. C a t the end of the chapter. The usefulness of these completeness conditions can be illustrated by the following Proposition 5.1. If %? i s a comfilete class, X a pathwise connected space, X , a subspace of X , and G = { Gx I x E X } a local system of groups in X with each GxE %?, then H m ( X , X,; G) is in %? for each m > 0. Proof. Pick x, E X as in (IX; fj 2) and denote Gxoalso by G. Then Hm(X,Xo; G ) is isomorphic with a quotient group of some subgroup of Cm(X,Xo)@G which is in %?. Hence Hm(X,X,; G) is in V . I As to the examples of classes (1)(6) in fj 2, one can verify that ( l ) , (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 P p of the 9primary 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, i t suffices to assume that the class Vinvolved is weakly complete (and, sometimes, perfect). For the second kind, we have to assume that V 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 V is a given class and Vf is the class consisting of those abelian groups all of whose finitely generated subgroups are in V, then %?f is strongly complete and
w f n d f =undf. 6. Applications of Classes to Fiber Spaces Let us go back to the notation of (IX; 3 3) and assume that n,(B, b,) 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
O F CLASSES TO FIBER SPACES
301
Let V be any given class of abelian groups. Let us consider the spectral homology sequence of (IX; 3 7). Lemma 6.1. I f , for some
n >, 1, Ep”,qE V , then EZq EV.
Proof. Since E;,:’ is the quotient group of a subgroup of Eg,q, it is in V. Then it follows by finite induction that E& is in V. I
+
Lemma 6.2. I f , for a given pair ( 9 , q ) , E T j € V whenever i j =p +q and i Q p , then H p , q ( X X,) , E V. I n particular, if ETj E W whenever i + j = P q, then Hp+q(X,X o ) E V. Proof. The lemma follows from the fact that Hr,j(X, X,) is an extension of H L ~ , ~ + ~X,) ( Xby , ETj and that Ho,m(X, X,) = E&,. I
+
Proposition 6.3. If %? is a weakly complete class and i f , for some integer
> 0 , Hm(B, B,) E V and H m ( F )E V whenever H m ( X ,X,) E V whenever 0 < m < r. 7
0
< m < r, then we have
Proof. According to the above lemmas, it suffices to show that Ei,qE V whenever 0 < p q < r. By (IX; 6.3),
+
+
Ei,q % Hp(B, Bo) 8 Hq(F) Tor(Hpi(B, Bo),Hq(F)). If p > 1, q > 0, and 0 < p + q < r, the weak completeness of V implies E& E V. If q = 0 and 0 < p < r, we have Ei,o m H p ( B ,B,) E V since H,(F) w 2. To verify the proposition for the cases p = 0 and p = 1, let us first assume B, # o . Then H,(B, B,) = 0 and hence Ei,m= 0 , Ef,m1 M H , ( B , Bo) 8 Hml(F) E V whenever 0 < m Q r. Next, assume that B, = a. Then H,(B, B,) M 2 and hence E& M Hm(F)EV, E:,ml whenever 0 < m < r . I
M
H , ( B , Bo) 8 HmI(F) EV
Proposition 6.4. If V is a complete class and if, for some integer p > 0 and q > 0 , Hm(B, B,) E V whenever m 2 p and Hm(F) E V whenever m > q, then H m ( X , X,) E V whenever m 2 p q.
+
Proof. By the completeness of V, it follows that E t j € V whenever, i j >, fi + q. Hence the proposition is an immediate consequence of the above lemmas. I Throughout the remainder of the section, we assume that B, # and hence H,(B, B,) = 0.
+
Theorem 6.5. If %? i s a weakly complete class and
if, for some integers
p > 0 and q > 0 , we have H 1 ( B ,B,) = 0 , Hm(B, B,) E V whenever 1 < m
1. Since
E t j M H t ( B ,B,) 8 H A F )
+ Tor(Ht,(B,B,I, H A F ) ) ,
i t follows that Ei,j = 0, Ef,j = 0, and E&E V whenever i > 1, j and i j Q r. This proves that w* is a Vmonomorphism for m < r. It remains t o prove that w, is a Vepimorphism whenever m Q r By (IX; 5 7), the image of w, is E;,". Furthermore, we have
+
E:,o = E mm,O+ l c
*
*
c
En+' c E" c m,O m,O
* * *
1,
+ 1.
c Ef,o= H,(B, B,),
EZ,o/EZ$lw dn(E:,o) c E:n,nl,(2 Q n Q m). Since EZn,nl is of total degree m  1 Q r, it is in % for 2 Q n Q m.This implies that the cokernel Ei,o/EE,:l of 0,is in V. I Note. If p < q + 1, the condition H,(B, B,) = 0 may be replaced by H,(B, B,) E V. In fact, only this is used in the proof of Et,rl E V.
q 0
Theorem 6.6. If V is a complete class and if, for some integers p > 0 and and Hm(F)E % whenever
> 0 , we have H m ( B ,B,) E %? whenever 0 < m < p < m < q, then the induced homomorphism
: H m ( X , Xo) + Hm(B, Bo)
i s a Visomorphism whenever m Q r and is a Vepimorphism whenever m = r + 1, where 7 = p + q  1. The proof of this theorem is analogous to that of (6.5). In the remainder of the section, we are concerned with the important special case where the subspace B, consists of a single point b,. Then we have
H,(B, Bo)
=
0, Hm(B,B,)
Hm(B), (m > 1).
Proposition 6.7. If V i s a weakly complete class, if H m ( X )E V for each m > 0, and if H , ( B ) = 0 and Hm(B) 6 % whenever 1 < m < for some given integer # > 0, then Hm(F) E V whenever 0 < m < p  1 and the homomorp hisms a Hp(B, bo) w Hp(B) Hp1 ( F ) + H p ( X ,F ) >+
define a Vequivalence of Hpl(F) and H p ( B ) . Proof. We shall prove this proposition by means of induction. The case
1 is trivial. Assume that p > 1 and the proposition is true for #  1. Then, by the hypothesis of induction, we have Hm(F) E V whenever 0 < m < p  2 and H,,(F) is %equivalent to Hp,(B) and hence is in V.
p
=
6.
303
A P P L I C A T I O N S O F C L A S S E S TO F I B E R S P A C E S
Applying (6.5) with q = 9  1 and B, = b,, we deduce that H p ( X ,F ) is Visomorphic to H,(B, b,) under w,. In the exact sequence a
’+H p ( X ) + H p ( X , F ) + H p  l ( F ) + H p  l ( X ) we have H p ( X )E V and H p  l ( X ) E %. This implies that morphism. I * *
*
+ a
a
., is a %iso
Proposition 6.8. If V is a complete class, Hm(X)E Y for each m > 0 , and Hm(B)E % whenever 0 < m < p for some given integer p > 0 , then Hm(F)E V whenever 0 < m < p  1 and the homomorphisms
a
Hm(F) Hm+l(X,F ) %+Hm+l(B,bo) w Hm+l(B) define a %equivalence of Hm(F)and Hm+l(B)whenever 9  1 < m < 2p  2. The proof of this theorem is analogous to that of (6.7). A space X is said to be %acyclic if H,,,(X) E % for each m
> 0.
Theorem 6.9. If Y is a weakly complete class, H l ( B ) = 0 , and two of the spaces X, B , F are Vacyclic, then so is the third. Proof. If the two spaces are B and F , then it follows from (6.3)with r = co and B , = 17that X is %acyclic. If the two spaces are X and B , then it follows from (6.7) with p = co that F is %acyclic. If the two spaces are X and F , then we shall prove H,(B) E % by induction on p . The case p = 1 is trivial. Assume 9 > 1 and Hm(B)E % if 1 < m < p . By (6.7), H,(B) is %equivalent to H p  I ( F )and hence H,(B) E % . I As an application of these results, let us consider the special case where X = [ B ;B , b,] and o : X + B is the initial projection. In this case, the fiber F becomes the space A ( B ) = [ B ;b,,b,l
of all loops in B with b, 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 , b,] is contractible, the following theorem is an immediate consequence of (6.7) and (6.8). Theorem 6.10. If % is a class and Hm(B)E % whenever 0 < m < p , then Hm(A(B))is Vequivalent to Hm+l(B)for the following values of m: ( i )0 < m < p if Y is weakly complete. ( i i ) 0 < m < 2p  2 if % is complete. Therefore, for a weakly complete class %, A ( B ) is %acyclic iff B is
Vacyclic. In particular, if B is a space of the homotopy type (n,n) with n > 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 Hm(n, n  1) w Hm+,(n,n ) , O < m < 2n  2. Furthermore, we have the following obvious
X. C L A S S E S O F A B E L I A N G R O U P S
304
Proposition 6.11. For an abelian group n,an integer n > 1, and a weakly complete class V, the following two statements are equivalent : (i) H m ( n )E 'i$ for each m > 0. (ii) Hm(n, n) E V fm each m > 0.
7. Applications to nconnective f'iber spaces Let B be a pathwise connected space and b, a given point in B. According to (V; 3 8), we may construct inductively a sequence of spaces (B, n), n = 0, 1,2; , and a sequence of maps

fin: (B,n) + ( B , n 
l), n
=
1,2,3;.,
as follows. Let (B, 0) = B. For each n > 0, let (B, n) be an nconnective fiber space over (B, n  1) with projection &. This system { (B, n), pn 1 will be referred to as a connective system of the space B. If B is locally pathwise connected and semilocally simply connected, then of course we may take (B, 1) to be the universal covering space over B with as the projection. Now let { (B, n),Pn } be any connective system of B. Since ( B ,n) is an nconnective fiber space over an (n 1)connected space ( B ,n  1) with /?, as projection, i t follows that the fiber of this fibering is a space F,, of the homotopy type (nn(B),n  1). Next, let
wn
=/lj?2...pn:(B,n) +B,
n
=
1,2;.,
then it is clear that (B, n) is an nconnective fiber space over B with wn as project ion. Applying (6.9) and (6.11) to the fibering pn, we obtain the following Proposition 7.1. If V is a perfect and weakly complete dass, n n,(B) E %?, then the following two statements are equivalent: (i) (B, n) is Wacyclic. (ii) (B, n  1) is %acyclic.
Applying (6.6) with q
=m
> 1, and
to the fibering pn, we obtain the following
Proposition 7.2. If V is a perfect and complete class, n
then
(Bn)* : &(B, 4 is a Visomorphism for each m > 0.
+
> 1, and n,,(B) E V,
Hm(B, 92  1)
Finally, if B is (n  1)connected, then we may take (B, m) = B for every m Q n  1. The following proposition w ill be used in the sequel. Proposition 7.3. Let V be a perfect and weakly complete class. IfB is (n  1)connected, nn(B)E V, n > 1, and p > n is an integer such that Hm(B)E %f whenever n < m < p, then the induced homomorphism
(A)*: H m P , n) Hm(B) is a Visomorphism for m Q p and is Vepimorphism for m +
=
p
+ 1.
8. T H E
GENERALIZED HUREWICZ THEOREM
305
Proof. Since the fiber F = F n of the fibering /?n is of the homotopy type (n,(B),n I), it follows from (6.11) that HnL(F)E % for every m > 0. Let B,c B consist of a single point, then we have H , ( B , B,) = 0 and Hm(B,B,) E V for each m < 9. Thus we may apply (6.5) with q = co. Hence the induced homomorphism
(&)# : Hm(X, F ) + Hm(B, Bo), X
=
( B ,4 ,
is a Visomorphism whenever m < $ and is a Vepimorphism whenever = p + 1. Since Hm(F)E V for each m > 0, it follows from the homology sequence that i : Hm(X) Hm(X, F ) is a Visomorphism for every m > 0. Finally, using the isomorphisms
m
+
k : Hm(B)M Hm(B,E,), we obtain (&)*
=
k'(fin)#j.
m > 0,
I
8. The generalized Hurewicz theorem Theorem 8.1. Let % be a perfect and weakly complete class. If X i s a simply connected space and n > 2 is a n integer such that nm(X)E V whenever 1 < m < n, then the natural homomorphism
hrn :nm(X) i s a %isomorphism whenever 0 m=ntl.
+
Hm(X)
< m < n and is a
Vepimorphism whenever
Proof. We are going to prove this theorem by induction on n. If n = 2, then this is implied by the usual Hurewicz theorem since n o ( X ) = 0 and n l ( X ) = 0. See (V; 4.4) and ( V ; Ex. C). Let p > 2 be an integer and assume that the theorem is true for n < p. Let us prove the theorem for n = p. By the inductive hypothesis, i t follows that hm is a Visomorphism whenever 0 < m < p and is a Vepimorphism for m = p. I t remains to prove that hp is a Vmonomorphism and hp+l is a Vepimorphism. Consider a connective system { ( X ,r ) ,/?,.} of the space X as defined in 5 7. Since X is simply connected, we may assume ( X , 1) = X . Then ( X , p  1) is a ( p  1)connective fiber space over X with 0 =
p2ps.
*
.ppl
: ( X ,p  1) +
x
as projection. Thus we obtain a commutative rectangle
nm(X. p  1) &+ Hm(X, p  1)
1*
n m( X )
,*
1#
+
Hm(X)
306
X. C L A S S E S O F A B E L I A N G R O U P S
where w*,w# are induced by w , and gm, hm are the natural homomorphisms. By the definition of ( p  1)connective fiber space, w* is an isomorphism for m 2 p. Since ( X , p  1) is (p 1)connected, it follows from the usual Hurewicz theorem that g, is an isomorphism and g, is an epimorphism. Hence, it suffices to prove that w# is a Visomorphism form = p and a Vepimorphism for m = p + 1. Since w = p J S . *j3pl, it suffices to prove that, for each r = 2,3; * , p  1, the induced homomorphism,

9
: Hm(X, r)
+
H m ( X , 7  1)
+
is a Wisomorphism for m = p and is a %‘epimorphism for m = p 1. For this purpose, let B = (X, r  1). Then, B is ( r  1)connected and we may take a connective system of B with ( B ,r  1) = B and (23, r) = ( X , Y ) . Since B is simply connected and nm(B)E V whenever 1 < m < p , it follows from the inductive hypothesis that Hm(B) E V whenever 0 < m < p . Therefore, by (7.3), (j3r)# is a %isomorphism for m = p and is a Vepimorphism for m = p + 1. I Corollary 0.2. Let % be a Perfect and weakly complete class. If X is a simply connected space and n > 2 is an integer such that H m ( X ) E V whenever 1 < m < n, thenn,(X) E V whenever 0 < m < n.
This corollary follows from (8.1) by finite induction on m. Therefore, if X is simply connected and %acyclic, then X is Vasphericd, i.e. n m ( X )E V for all m > 1. In particular, we have the following Corollary 0.3. The homotopy groups of any simply connected finitely triangulable space are finitely generated.
9. The relative Hurewicz theorem Theorem 9.1. Let V be a perfect and complete class, X a simply connected space, and X , a simply connected subspace of X . If n , ( X , X o ) = 0 and n > 2 is an integer such that n m ( X ,X,) E % whenever 2 < m < n, then the natural homomorphism hm n m ( X , Xo) H m ( X , Xo) is a %isomorphism whenever 2 < m < n and is a Vepimorphism whenever mn+l. +
Proof. We prove this theorem by induction on n. If n = 2 , the theorem is true since H , ( X , X,) = 0 and h, is an isomorphism by (V; Ex. C). It follows from the hypothesis of induction that k, is a %isomorphism whenever 2 Q m < n. Hence H m ( X , X,) E % whenever 0 < m < n. It remains to prove that hn is a %isomorphism and hn+, is a Vepimorphism. Pick a point x, E X , and consider the space of paths Y = [ X ;X , x,]. Then, Y is a fiber space over X with projection w : Y + X defined by
307
10. T H E W H I T E H E A D T H E O R E M
o(t)= l ( 0 ) for each 5~ Y . Let Y o = [ X ;X,, x,]. Then Yo is pathwise connected and zm(Yo) =nm+l(X, X,), (m > 1). Hence, nl(Yo)= 0 andnm(Yo)E %? for 2 < m < n  1. By (8.I), the natural homomorphism g, :Z m ( Y 0 ) + H m ( Y o )is a %?isomorphismfor m = n  1 and is a Vepimorphism for m = n. Since F = ol(xo) = A ( X ) is pathwise connected and H m ( X ,X,) E Y whenever 0 < m < 12, we may apply (6.6) with p = n and q = 1. Hence the induced homomorphism o# :Hm(Y, Yo)+. H m ( X ,X,) is a %?isomorphism for m = n and is a Qepimorphism for m = n 1. In the diagram
+
w
nm(X, Xo) +*nm(Y,
a
Yo)A Z m  l ( Y o )
K n ( X , X,)6 w# ~ t n ( yyo) , +a# Hmi(y0) we obtain hm = w#a#l gmlll*o*'. Therefore, h,, is a %?isomorphismfor m = n and is a %?epimorphismfor m = n + 1. I Corollary 9.2. If X and X , are simply connected, n,(X, X,) Hm(X, X,) is in a $erfect and complete class V whenever 2 < m n m ( X , X,) i s also an V whenever 2 < m < 12.
=
0 , and
< n, then
Remark. The theorem (9.1) does not hold if we merely assume that Y is perfect and weakly complete. For example, let X = A x B and X , = A x b, where A and B are simply connected, B is %?acyclic,and b E B. By making various choices of A , one can see that (9.1) is true for a given class Q iff %? is perfect and complete. Similarly, (8.1) is true iff %? is perfect and weakly complete. 10. The Whitehead theorem Theorem 10.1. Let %? be a perfect and complete class. If X and Y are simply connected spaces, f : X + Y i s a map such that f* : n 2 ( X )+n,(Y)i s a n epimorphism, and n 2 2 i s a given integer, then the following two statements are equivalent : (1) f* : n m ( X )+nm(Y) i s a %?isomorphism for m < n and is a %?epimorphism for m = n. (2) f# : H m ( X )+H m ( Y ) i s a %isomorphism for m < n and i s a Qepimorphism for m = n. Proof. Consider the mapping cylinder Zfof the map f . By ( I ; 5 12), both X and Y can be naturally imbedded in Zfand Y is then a strong deformation retract of Zf.Thus the map f is decomposed into the composition r i of the inclusion map i : X c Zf and a strong deformation retraction r :2, + Y . Since Y induces isomorphisms on homotopy and homology groups, (1) and (2) are equivalent respectively to the following two statements:
308
X. C L A S S E S O F A B E L I A N G R O U P S
(1’) i, : a ( X )+ n m ( Z f ) is a $9isomorphism for m < n and is a 55‘epimorphism for m = n. (2‘) i# : H m ( X ) + H m ( Z f )is a %isomorphism for m < n and i s a Vepimorphism for m = n. Then i t follows from the homotopy sequence and the homology sequence of ( Z f ,X ) that (1‘) and (2’) are equivalent respectively to the following two statements: (1”) nm(Zf,X ) E V whenever 2 < m < n. (2”) H m ( Z f ,X ) E %whenever 2 Q m f n. By (9.1), we conclude that (1”) and (2”) are equivalent. This entails the equivalence of (1) and (2). I
EXERCISES A. Examples of Classes of Abelian Groups.
In addition to the classes given in 2 and 5 3. we give the following examples : 1. The class of all abelian groups with power not exceeding a given infinite cardinal number N,. In particular, if w, is the power of the set of natural numbers, this reduces to the class 01, of all countable abelian groups. Verify that this class is perfect and weakly complete but not complete. 2. The class of all abelian groups A such that there is an integer N depending on A with Nu = 0 for every a E A . Verify that this class is perfect and complete but not strongly complete. 3. The class of all abelian groups satisfying the descending chain condition. Verify that this class is perfect and weakly complete but not complete. B. Composed Homomorphisms
By using the natural exact sequence S ( f ,g) of two homomorphisms f : A + B and g : B + C, prove the following six assertions for a given class V: 1. I f f and g are Vmonomorphisms, then so is gf. 2. I f f and g are %?epimorphisms, then so is gf. 3. If gf is a Vmonomorphism, then so is f. 4. If gf is a Vepimorphism, then so is g. 5 . If gf is a Vmonomorphism and f is a %‘epimorphism, then g is a Vmonomorphism. 6. If gf is a ‘Xepimorphism and g is a Vmonomorphism, then f is a Vepimorphism . C. O n Perfectness and Completeness
Prove the following assertions: 1. For any class V of abelian groups, the following three statements are equivalent :
309
EXERCISES
(a) Q is complete. (b) A E W implies A @ B E V for every abelian group B. (c) For any A E V, every finite or infinite direct sum of groups isomorphic to A is in V. 2. Every strongly complete class is perfect and complete. It is unknown whether there is a class which is not perfect or not weakly complete. D. The CGeneralization of the “Five” Lemma
Prove that the “five” lemma, [ES; p. 161, remains true modulo a class %‘. Precisely, if we have two exact sequences each with five terms and five homomorphisms of the groups of the first sequence into the corresponding groups of the second sequence with the commutativity relations being satisfied, and if the four extreme homomorphisms are Visomorphisms, then the middle homomorphism is also a Visomorphism. E. The CInverse Homomorphism Theorem
Consider a class %? and an exact sequence A , >+ / A , f l + A,f”+ A,+1.
A,
Assume that there exist two homomorphisms g, : A + A , , and g, : A , t A , such that the endomorphisms f , g , and f4g4 are Visomorphisms. Define a homomorphism 12: A , A , + A 4 by taking h(x, y ) = f,(x) g,(y). Prove that h is a %isomorphism.
+
+
F. The Products of CEquivalent Groups
Let V be a complete class. Prove that, if A and B are %‘equivalent respectively to A‘ and B‘, then A 8 B and Tor(A,B) are %equivalent respectively to A‘ @ B’ and Tor(A’,B‘). G. CExact Sequences
Let V be a class and A , B two subgroups of an abelian group G. We say that A and B are Vequal if the inclusion homomorphisms A n B + A and A n B + B are Visomorphisms. Replacing equality by %?equality, one can define the notion of a Vexact sequence. 1. Establish the elementary properties of 9exact sequences as in [ES ; p. 501. 2. Generalize the results in (VIII; 3 8) to obtain various fundamental Vexact sequences. H. O n Induced Homomorphisms
Let X , Y be simply connected spaces, f : X + Y a map such that f* : n z ( X )+ n2(Y)is an epimorphism. Assume that the homology groups are finitely generated.
310
X. C L A S S E S O F A B E L I A N G R O U P S
1. Let 9 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) f # : H m ( X )+ H m ( Y ) is an Yisomorphism for m < n and is an 9epimorphism for m = n. (b) f# : H m ( X ) + H,(Y) is a 9isomorphism for m < n and is a Yepimorphism for m = n. (c) f# : H,(X; G) + Hm(Y; C) is an isomorphism for m < n and is an epimorphism for m = n. (d) f# : H m ( Y , G) + H m ( X ; G ) is an isomorphism for m < n and is a monomorphism for m = n. 2. Let F Pdenote the class of all finite abelian groups of order not divisible by a given prime number p, Y pthe class of all torsion groups with null +primary component, and G a field of characteristic p. Prove that the following four statements are equivalent : (a) f# : H,(X) + H,(Y) is an gpisomorphism for m < n and is an Spepimorphism for m = n. (b) f# : H m ( X )+ H m ( Y )is a .Tpisomorphism for m < n and is a Ypepimorphism for m = n. (c) f # : H,(X; G) + H m ( Y ;G) is an isomorphism for m < n and is an epimorphism for m = n. (d) f#: H m ( Y ; G) + H m ( X ; G) is an isomorphism for m < n and is a monomorphism for m = n. The usefulness of these two propositions is that, on many occasions, we may replace the calculus mod W by the calculus with coefficients in a field. Secondly, since Y and Y Pare perfect and complete, we may apply (10.1). Finally, the statements (b,), (c), (d) are equivalent even if the homology groups are not finitely generated.
CHAPTER XI H O M O T O P Y GROUPS OF SPHERES
1. Introduction Finally we come to the determination of certain of the homotopy groups of spheres. The calculations are particularly based on the results of the previous two chapters, and, since they are quite technical, we will not attempt to summarize them here. However, in the course of the development, several topics of independent interest appear: Freudenthal's suspension theorem (stated in 3 2 and proved in $9 25), pseudoprojective spaces and Stiefel manifolds ($3 101 l), and the Hopf invariant of a map f : S2nl + Sn ( 3 14). We have already seen that if Y < 0 then nn+r(Sn)is the zero group, and, that if Y = 0 then this group is free cyclic; and now in $8 1517 we shall settle the cases Y = 1,2,3, and 4.A brief report of the cases 5 < Y < 15 is given in the final section.
2. The suspension theorem
+
Let n > 1 and consider the nsphere Sn as the equator of the ( n 1)sphere Sn+l with u and v denoting respectively the north and south poles of Sn+l. Pick a point so in Sn and consider the space
w = n(sn+l) of loops in Sn+l with so as basic point. There is a natural imbedding i : Sn + W described as follows. For each x E Sn, i ( x ) is the loop in Sn+l joining so to u,u to x , x to v , and v back to so, all by shortest geodesic arcs. That a is a homeomorphism of Sn into W is obvious. Furthermore, the loop i ( s o ) is homotopic to the degenerate loop W,E W which maps I into so by means of a natural homotopy; in other words, the points W , and i(so) of W are connected by a natural path u in W . Hereafter, we shall identify x and i ( x ) for every x E Sn. Thus, Sa becomes the subspace i(Sn) of W and i : Sn + W reduces to the inclusion map. For each m > 0, i induces a homomorphism
i, :7cm(Snt so) +nm(W, so), the path u induces an isomorphism u* : nm(W,so) M nm(W,W,). 311
XI. H O M O T O P Y G R O U P S O F S P H E R E S
312
and, according to (IV; 2.2), we have an isomorphism
h* :n m ( W , wo) M
nm+l(Sn+', SO).
Composing i,, a,, and h,, we obtain a homomorphism
C
=
h,o,i,
:nm(Sn,so) +nm+l(S"+l, so)
for each m > 0, called the suspension. One can verify that this definition is equivalent to the more general one given in (V; 4 11) for this special case. Theorem 2.1. (The Suspension Theorem). The suspension I: i s an isomorphism if m < 2n  1 and is an epimorphism if m = 2n  1.
Proof. Since a* and h, are isomorphisms, it suffices to prove that i, is an isomorphism if m < 2n  1 and is an epimorphism if m = 2n  1. Thus, according to Whitehead theorem, ( X ; lO.l), it suffices to prove that the induced homomorphism i# : Hm(Sn) +. H m ( W )
is an isomorphism if m < 2n  1 and is an epimorphism if m Since Hm(W) M 2, if m 0 mod(n),
=
2n  1.
Hm(W)= 0, if m f 0 mod(n), by (IX; 13.4), i t remains to prove the following Lemma 2.2.
i# : Hn(Sn) M Hn(W).
This lemma will be proved in the next three sections; we conclude this section with one immediate consequence of the suspension theorem. Let U and V denote respectively the north and south hemispheres of Sn+l; then, using (V; 1l.l), we have the following Corollary 2.3. The excision homomorphism
e, :nm(U,Sn)+n,(Sn+l, V )
is an isomorphism whenever 2 m
=
2n.
< m < 2n and i s an epimorphism whenever
The identity map on Sn extends to a map f : (En+l,S n ) +. ( U ,9). If we compose this map with the excision e : ( U ,Sn) c (Sn+l, V ) ,we obtain a map g
=
ef : (En+l,Sn) + (Sn+l, V ).
Since V is contractible to the point so, g is homotopic in
(Sn+l,
V ) to a map
h : (En+l,S n ) f (Sn+l, so) which represents a generator o f n n + l ( S n + l , so). For any element a ofnm(Sn,so), choose a map : S m + S n which represents a. The map has an extension
+
+
y : (Em+l, Sm) + ( E n + l , 9).
3. T H E
313
CANONICAL MAP
Then it can be seen that the composed map hy represents the element X(ct) in ~ ~ + ~ ( Sso). n+l,
3. The canonical map Consider the space of paths X = [Sn+l;so, Sn+l]. According to (111; 13), X is a fiber space over Sn+l with a projection o : X +Sn+l defined by w ( x ) = x ( 1) for every path x E X and with fiber W = col(s,). Let U and V denote the north and the south hemispheres of Sn+1 respectively and let
Q
x, = wl(S*),
x u = wl(V),
x, = w'(V).
We are going to define a map x : (U x
w,S"
x W ) ( X u ,X,) +
which will be called the canonical map. For each point b E U , let y(b)E X u denote the path joining so to u and then u to b by geodesic arcs. The assignment b + y ( b ) defines a crosssection y : U + X u . Then x is defined by x ( b , f ) = f . y ( b ) for each b E U and f E W , where f.y(b) denotes the product of the paths f and y(b). As a consequence of the construction, we have
wx(b, f )
=
b, (6 E U , f E W ) .
Lemma 3.1. The canonical map x is a homotopy eqzlivalence. Proof. Let
A : ( X w ,X,)
+
( U x W ,Sn x W ) be the map defined by
A(%) = (w(x),X ' [yw(x)l'),
( X E Xa).
where [yo(x)]ldenotes the reverse of the path yo(%). Then we have xA(x) = x(w(x),X ' [yw(x)]l)= ( x . [yw(x)]1)* y w ( x ) ,
[r(411)'
W h f ) = W Y ( b ) )= ( b , [ i * Y ( b ) l *
Hence, x l and ;Ix are both homotopic to the identity maps. I Next, let us define a map
pusnx
w+w
by taking p ( b , f ) = f.i(b) for each b E S and ~ f notes the imbedding in 3 2.
E
W , where i : Sn + W de
Lemma 3.2. The m a p p is homotopic in X v to the map v : Sn = x I Sn x W .
x W + X ,
defined by v
Proof. Intuitively, a homotopy of p to v is accomplished by "unwinding" the path i ( b ) to half its original length. More precisely, define a homotopy Itt : Sn + X,, ( 0 < t < I), by taking
XI. H O M O T O P Y G R O U P S OF S P H E R E S
314
Then Ito = i and k, = y . Define a homotopy Kt : Sn x W + X,, (0 < t by taking K & f ) = f.ht(b), ( b E S " , f E W , t E I ) . Then KO
=p
and k ,
= Y.
< I),
I
4. Wang's isomorphism p* In the present section, we shall construct for each integer q 2 0 an isomorphism p* Hn(Sn)8 Hg(W) M Hn+g(W)
+
Let m = n q + 1. The construction of p* will be made in six steps as follows. Step 1. Since the south hemisphere V of Sn+l is contractible to the point so, an application of the covering homotopy theorem proves that W is a strong deformation retract of Xu. Hence the inclusion map induces an isomorphism t : H m ( X , W ) W Hm(X, Xu).
Step 2. The inclusion map induces a homomorphism Let
q : Hm(X,, X,) + H m W , Xu)* D , = sn+1\ v , D, = sn+l\ U , D = D ,
n D,; Y , = w  l ( ~ u )Y, , = O  ~ D , ) , Y = yUn Y,.

Since Y, and Y , are open sets whose union is X , the excision theorem holds and hence the inclusion map induces an isomorphism
Hm(Yu, Y) Hm(X, Y v ) . Since U , V ,Sn are strong deformation retracts of D,, D,,D respectively, an application of the covering homotopy theorem proves that Xu, Xu, X , are strong deformation retracts of Y,, Yo, Y respectively. Hence 7 is an isomorphism. Step 3. Since the canonical map x is a homotopy equivalence, it induces an isomorphism x* : Hm(U x W ,s n x W ) M Hm(Xu, XO).
Step 4. By the Kiinneth theorem, we get an isomorphism
C : Hn+l ( U ,Sn) 8 Hg(W) w Hm(U x W ,Sn x W ) . Step 5. Since X is contractible, we have an isomorphism
a : H m ( X , W ) w Hm1(W).
Step 6. Since U is contractible, we have an isomorphism Hn+i(U,Sn) M Hn(Sn). Taking tensor products, we obtain an isomorphism 0 : Hn+l(U,Sn) 8 Hg(W) M Hn(Sn) 8 Hg(W).
5. RELA'I'lON
BETWEEN
p*
AND
i#
315
Composing these steps, we get an isomorphism p*
: Hn(Sn)€3 HQ(W)M Hn+g(W).
= dt'qx,@'
This isomorphism p* is the same as the homomorphism p* in Wang's exact sequence (IX; 13.1) for the fibering o : X +Sn+l. See also [Wang 21.
5. Relation between p,, and i# The space W of loops has a continuous multiplication
M:WxW+W defined in (111; 5 11). The total homology group
H(W)=
m
x Hm(W)
m=O
becomes a ring under the Pontrjagin multi~licationdefined as follows. Let U E H ~ ( W and ) BEH*(W).By the Kunneth theorem, u and determine a unique element u x B of Hp+Q(Wx w).The map M induces a homomorphism M , : Hm(W x W ) Hm(W) for every m. Then the Pontrjagin product of u and /Iis defined to be the element a.B = M,(a x B) E H ~ + Q ( W ) . Proposition 5.1. For every u E H n ( S n ) and @ E HQ(W), we always have
P*(Q €3B)
= B.i#(.),
where i# : Hn(Sn) + Hn(W) denotes the homomorphism induced by the imbedding i : S n + W of 5 2 . Proof. Consider the diagram
H m (u x
w,S n x
W )5 Hm(X,, XO) A H m ( X , X")
I.
I
HmV, W)
+
I.
where u and z are induced by inclusion maps and the homomorphisms d are boundary operators. The rectangules are all commutative and hence (1)
apqx, = ultv,a.
By (3.21, we have
(2)
tv,
= ap,.
By the Kunneth theorem, we have an isomorphism
XI. H O M O T O P Y G R O U P S O F S P H E R E S
316
and a commutative rectangle
Hn(Sn)@ Hg(W)5Hn+q(Sn x W ) .
Hence we obtain (3)
x
= age1.
Using ( l ) , (2) and (3), we deduce p*
= at%x*cei
=
altv,agel
=
cl*agel
=
cl*x.
Then it follows from the definition of p that
P * b @ B)
= p*%
@
B) = Pi#(.).
I
In particular, if q = 0, then H,(W) is a free cyclic group generated by the element e represented by wo as a 0cycle of W . For each a E Hn(S"),we have $+(a) = e*i#(a) = p*(a@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].
nm(T) =nm+i(W,Sn)
for every m. Hence the homotopy sequence of the pair (W,9)gives rise to an exact sequence
*..+%(T) +%(Sn)
T'
+nm+l(Sn+')
+nffl1(T)+*...
This is essentially the suspension sequence of the triad ( S n + l ; U ,V ) ,ndm(T) being essentially the triad homotopy group ~ + ~ ( S n + U l ;, V ) . See (V; $8 1011). Because of this exact sequence, it is desirable to determine the triad homotopy groups h ( T ) .The following lemma is an immediate consequence of the suspension theorem (2.1). Lemma 6.1. n m ( T ) = 0 for every m
< 2n 2.
To determine the higher homotopy groups of T , let us study the space of paths Q = [W;Sn, W ] which is of the same homotopy type as Sn. Consider the projection w : Q +W defined by w(a) = a(1) for every a E Q; then Q becomes a fiber space over W with fiber w'(s0) = T.
7. 0
317
F I N I T E N E S S O F H I G H E R HOMOTOPY G R O U P S
Since Hm(W)= 0 whenever 0 < m < n and Hm(T) = 0 whenever ( I X ; 14.1) an exact sequence
< m < 2n  1, we have by
H3nn(T) + *
*
.+Hm+l(Q)+Hm+!(W) +Hm(T)+Hm(Q) + *
*
*+H1(W)+O.
Since Q is of the same homotopy type as Sn, we have Hm+l(Q)= 0 = Hm(Q) for every m > n. This implies that H m ( T ) M Hm+,(W)whenever n < m < 312  2. Hence, we deduce the following Lemma 6.2. H2n1(T)M Z and Hm(T)= 0 whenever 2n  1 < m < 3n 2 .
Choose a map f : S2n1+ T which represents a generator of the free cyclic group n2nl(T)M H2nI(T). Then f induces an isomorphism
f# : H2n1(S2nl) M HanI(T). Then, by (6.1) and (6.2), it follows that f# : Hm(S2%l)M H m ( T ) for every m < 3n  2. An application of Whitehead's theorem proves that the induced homomorphism f* :n m ( ~ 2 n  ~ ) + n m ( ~ ) is an isomorphism if m < 3n  3 and is an epimorphism if m Hence we have proved the following
to nm(S2n') for every m is isomorphic to a quotient group of r~,~,(S~nl).
Proposition 6.3. nm(T)i s isomorphic
andn,,,(T)
=
3n  3.
< 3n  3
7. Finiteness of higher homotopy groups of odddimensional spheres In the present section, we are concerned with an odddimensional sphere
Sn. Since the homotopy groups of the 1sphere S1 are completely computed in (IV; 5 2), we may assume that n > 3. Consider an nconnective fiber space X over Sn with a projection w :X 4%. By definition,
n m ( X ) = 0 , (m G n ) , o,,,: n m ( X )mnm(Sn),
(m> n).
Then it follows that the fiber F is a space of the homotopy type (2, n  1). By ( X ; 8.3), nm(Sn) is a finitely generated abelian group for every m. An application of the generalized Hurewicz theorem proves that H m ( X ) is finitely generated for every m. Let K be a field of characteristic zero. Since n  1 is even, it follows from (IX; Ex. G) that the cohomology algebra H * ( F ; K ) is isomorphic to a polynomial algebra over K generated by an element of degree n  1. Let a E Hn'(F; K ) be a generator of H * ( F ;K ) . Consider Wang's cohomology sequence (IX; 13.2):
.  .  t P ( X ; K ) + H " ( F ; K ) P*+
Hmn+yF;K)+Hrn+'(X;K) +'

*,
318
XI. H O M O T O P Y G R O U P S O F S P H E R E S
where p* is a derivation since n is odd. Since H n ( X ; K ) = 0 = H n  l ( X ; K ) , p* sends Hnl(F; K ) isomorphically onto H o ( F ;K ) . Therefore, p*(a) is a nonzero element of H o ( F ;K ) w K . Now let us prove that p* :HP(nl)(F;K ) H(p9(nl)(F;K ) for every positive integer p . I n fact, the vector space Hp(rl)(F;K ) over K admits a p as a basis. Since p* is a derivation, we have
p*(kap) = pkp*(a)apl, (k E K ) . Hence p* is an isomorphism for every m > 0. Then an exactness argument proves that H m ( X ; K ) = 0 for every m > 0. Since &(X) is finitely generated and K is of characteristic zero, this implies that H m ( X )is finite for every m > 0. An application of the generalized Hurewicz theorem (X ; 8.1) proves that n m ( X )is finite for every m. Thus, we have proved the following Theorem 7.1. I f Sn is an odddimensional sphere and m
i s finite.
> n, then n,,(Sn)
8. The iterated suspension The natural imbedding Sn+l c L I ( S ~ + of~3) 2 induces an imbedding j : A(Sn+l) +.A2(Sn+z). Composing with the natural imbedding i:Sfi+A(Sn+l), we obtain an imbedding k = ji :Sn +.A2(Sn+z). For each m, k induces a homomorphism k , :nm(Sn, SO) +.nm(A2(Sn+z),SO). As in 3 2, we have a natural isomorphism I , :%(AZ(Sn+z), so) M n m + p + z , so). Proposition 8.1. I,k, is equal to the iterated suspension Proof. By
P.
3 2, there is an isomorphism
a = h,a, :n,(A(Sn+l), so) Similarly, there are isomorphisms
M
~ ~ + ~ ( S so). n+l,
p :nm(AZ(Sfi+3,so)
w nm+l(A(Sn+z),so), y :nm+l(l.i(Sn+2), so) w n m + p + z , so). Then 1, = yp and k , = j*i,. The proposition is a consequence of the commutativity of the diagram:
nm(SB,so) L+ nm(A(Sn+l),so) Jbnm(AZ(Sn+Z),so)
X P
7cm+l(S"+1,so)
1
nm+l(A(s*+z),so)
The following proposition is an immediate consequence of (8.1) and (2.1).
9.
T H E PPRIMARY C O M P O N E N T S O F Z ~ ( S ~ )
319
Proposition 8.2. The homomorphism k , is an isomorphism if m < 2n  1 and is an epimorphism if m = 2n  1. If we study the pprimary components instead of the whole homotopy groups, then we can deduce more detailed information from the iterated suspension X2. Theorem 8.3. Let n > 3 be an odd intcger, p a prime number, and V the class of all finite abelian groups of order prime to p. Then the iterated suspension
X2
nm(Sn) +nm+s(Sn+')
is a %fisomorphismif m < p(n + 1)  3 and is a Vepimorphism if m p(n 1)  3.
+
=
Proof. According to (8.1), it suffices to prove the theorem for the homo
k , :nm(Sn) +nm(A2(Sn+2)) morphisms induced by the natural imbedding k : Sn + L I ~ ( S ~ + ~ ) . Let K be a field of characteristic 9. Then k induces the homomorphisms k# : H ~ ( L I ~ ( S ~K) ++ ~H ) ;m ( S n ; K).
By Whitehead theorem ( X ; 10.1) and (X; Ex.H2), it suffices to show that k# is an isomorphism for every m Q p(n 1)  3. By (8.2) k , is an isomorphism if m < 2n 1 and is an epimorphism if m = 2n 1. An application of ( X ; 10.1) and (X;Ex.H 2 ) proves that k# is an isomorphism for m < 2n  1. By ( I X ; Ex.F6 and F7), we have Hm(A*(Sn+2); K ) = 0, ( n < m Q p(n 1)  3). Since n >3, it follows that k# is an isomorphism for every m < p(n + 1) 3. I
+
+
Corollary 8.4. If n > 3 i s an odd integer, p a prime number, and m < n + 4p6, then the pprimary components of nm(S") and nmn+3(S3) are isomorphic.
Proof. We shall prove the corollary by induction on n. When n = 3, there is nothing to prove. Assume that q > 5 is an odd integer and the corollary is true for every odd integer n with 3 Q n < q. By (8.3), the pprimary components of nm(Sq) and nm,(Sq2) are isomorphic if m  2 < p ( q  1)  3. Since q > 5, we have ( p  1) (q  5) >O and hence 4 + 4 p   8 Q P ( q  1) 3.
This implies m  2 < p ( q  1)  3 whenever m < q + 4p  6. By the hypothesis of induction, the #primary components of nma(S@2) and 7 ~ ~ * + ~ ( Sare 3 )isomorphic if m < q + 4p  6 . Hence, the corollary is also true for n = q. I
9. The pprimary components of nm(S3) The corollary (8.4) reveals the importance of finding the +primary components of the homotopy groups of the 3sphere S3.
3 20
XI. H O M O T O P Y G R O U P S O F S P H E R E S
Consider a 3connective fiber space X over S3 with a projection w : X + S3 and fiber F which is a space of homotopy type ( Z , 2 ) . Lemma 9.1. T h e integral homology groups of X are as foZlows: H m ( X ) = 0 if m i s odd; H,,(X) i s cyclic of order n for every n > 0. T h u s , the first few homology groups are: z,o, o,o, z,, o , z 3 , o,z,, O,Z,,‘

Proof. Consider Wang’s cohomology sequence (IX; 13.2) : *
*
+ Hm(X) + Hm(F) P’+ HmZ(F) + Hm+l(X)+*
* *
where p* defines a derivation of H*(F).By (IX; Ex. G), H*(F)is isomorphic to a polynomial algebra over the ring of integers generated by an element of degree 2. Since n m ( X ) = 0 for m < 3, we have Hm(X) = 0 for m < 3. Hence we obtain p* : H 2 ( F )M Ho(F)= 2. Let 0: denote element of H 2 ( F )with p*(u) = 1. Then M generates the algebra H * ( F ) and p*(un) = nc1nl. Let m = 2 n with n > 2. The sequence becomes 0 + HZ%(X)+ H y F ) p’+ HZfiZ(F) + HZ”+l(X) + 0. Since H2fi(F)is free cyclic with a” as generator and p*(un) = i t follows that p* is a monomorphism and its cokernel is cyclic of order n. Then the exactness implies that H2n(X)= 0, H2n+l(X)M 2,. The lemma follows from this and the duality between homology and cohomology. I Theorem 9.2. If p i s a prime number, then the $primary component of 4 S 3 )i s 0 if m < 2 p and i s Z, if m = 2p.
Proof. Let %? denote the class of all finite abelian groups of order prime to p. By (9.1), we have H m ( X )E V whenever 0 < m < 2p. An application of the generalized Hurewicz theorem proves that n m ( X )E V whenever 0 < m < 2 p and n z p ( X )is %?isomorphicto 2,. Since X is a 3connective fiber space over S3, we have nm(S3)= n m ( X ) for each m > 3. This implies the theorem. I Corollary 9.3. I f n 2 3 i s a n odd integer and p a p r i m number, then the pprimary component of nm(Sn) i s 0 if m < n + 2p  3 and i s 2, if m = n + 2p3.
+
+
+
Proof. Since > 2, we have n 2 p  3 < n 4$  6. Then it follows (8.4) that the pprimary components of nm(Sn) and nmn+3(S3) are isomorphic. I
10. P S E U D O  P R O J E C T I V E S P A C E S
321
10. Pseudoprojective spaces If we adjoin to the nsphere Sn an (n + 1)cellEn+l by means of a map
$ : i?En+l = Sn + Sn of degree h as in ( I ; 5 7), we obtain a space P
= e+1
which is called a 9seudoprojective space, [AH; p. 2661. We shall assume that h > 0; in this case, the homology groups of P are
H,(p) Wz, H n ( p ) W z h ; Hm(P) = 0, m # 0, m # n. Lemma 10.1.
For every m < 2n  1, we have a n exact sequence
0 j7Zm(Sn) 8 zh +.nm(P) + TOr(nm,(Sn),zh)+ 0. Proof. The map $ extends to a map y : En+l+ P in an obvious way. We obtain a commutative diagram: a a * * *+.nm+](P,Sn) + nm(Sn)+7zm(P)+n,(P,Sn) + nm](sn) +' * *
iv*
n,+l(En+l,Sn)
 F* a
nm(S@)
F.
nm(En+l,Sn)
a j
Id*
nm,(S*)
where the top row is the homotopy sequence of the pair (P,Sn) and
$* nm(Sn)  j n m ( S n ) , y* :nm(E"+', Sn) + n m ( P , Sn) are induced by the maps $ and y. Since q5 is of degree h, we have $*(a) = ha for each u €nm(Sn)whenever m < 2n  1 ; on the other hand, y* is an isomorphism for every m < 212  1. See Ex. A and Ex. B at the end of the chapter. Hence we may replace the homotopy sequence of (P,9) by the exact sequence 4 n*(Sn) L+. nm(S@)nm(P)+nmI(S")
nml(Sn)
for every m < 2n  1. Since the kernel and the cokernel of $* :nm(Sn)+ nm(Sn) are isomorphic to Tor(nm(Sn),zh) and nm(Sn)8 zh respectively, the exactness of this sequence implies the lemma. I Let X denote a 3connective fiber space over S3with projection o : X + S3. Since the pprimary component of n,,(X) is cyclic of order p, there exists a map f :S 2 p +. X which represents a generator [f] of this pprimary component of n,,(X). Consider the pseudoprojective space P = P;p+'. Then S 2 p c P. Since f~[ f ] = 0, f can be extended to a map g : P + X . Composing with o : X +. S3, we get a map X = wg : P + S 3 which induces the homomorphisms X* :nm(P)+nm(S3).
XI. H O M O T O P Y G R O U P S O F S P H E R E S
322
Lemma 10.2. X , is a monomorphism for m < 4p  1 and sends nm(P) onto the #primary component of nm(S8)for m Q 4p  1. Proof.
It suffices to prove the lemma for the induced homomorphisms g* :nm(P)+ n m ( X ) .
It follows from the generalized Hurewicz theorem (X; 8.1) that nm(P) is a +primary group for every m. Hence g, sends nm(P) into the $primary component of n m ( X ) . Let W denote the class of all finite abelian groups of order prime to $. It remains to prove that g, is a %isomorphism for m < Irp  1 and is a Wepimorphism for m = 4p  1. From the construction of g, one can see that g#: Hzo(P)M H a p ( X ) . Then, by (9.1), g# : Hm(P)+ H m ( X ) is a Wisomorphism for m < 4p. An application of the Whitehead theorem (X; 10.1) completes the proof. I This lemma reveals the importance of finding the homotopy groups of p = PZP+'. P
If p is a $rime number and m Q 4p 2, then (P) if m is digerent from 2p, 4p  3, and 4p 2, while: Theorem 10.3.
=0
n,p(P) M 2,; n,3(P) w zp; nu,(P)w z, if p > 2. Proof. According to Hurewicz's theorem, we have n m ( P ) = 0 for each
m < 2p and nzp(P)w 2,. Applying (10.1) with n 4p  3, we obtain an exact sequence
=
2$, h
=
p, and m
3 is odd, the Pprimary component of nm(S") is 0 whenever n + 2p  3 < m < n 49  6. By (2.1), nm(S2P) M nm+l(S'P+') for every m < 9, 2. Hence, the $primary component of nm+1(S'P+l) is 0 whenever 4p  3 < m < 6p  6. If p > 2, then 4p  2 < 69  6 and hence the @primarycomponent of Z ~  ~ ( S is ~ P0.) By (10.1) with n = 2p, h = p, and m = 4p 2, we get n,,(P) 2,. I
+
323
11. S T I E F E L MANIFOLDS
Corollary 10.4. If p is a #rime number, then the #primary component of nm(S8)is 0 if 2 p < m < 4 p  3 and is 2, is m = 4p  3 . If p > 2 , then the pprimary component of n4p,(Sa)is 2,. Corollary 10.5. If n Z 3 is an odd integer and p a prime number, then the #primary component of %(Sn) is 0 if n 2p  3 < m < n 4 p  6 and that of nn+apn(S") is 0 or Z p .
+
+
11. Stiefel manifolds Let n > 4 be an even integer and consider the Stiefel manifold V = Vn+l,a of all unit tangent vectors on 9,(111; Ex. G). Then, V is simply connected and its homology groups are as follows:
H,(V)

2, Hn1(V)
= z,,
Ha?&#)
=z
and all other homology groups are zero, [Stiefel 1 , 2 ] and [S; p. 1321. Since V is the tangent bundle of Sn, it is a fiber space over Sn with a projection w : V +.Sn and fibers homeomorphic to 9  l . According to (V; § 6), this fibering gives an exact sequence * *
d.
*  + n m ( ~5 ) nm(Sn) +n m  I ( ~ n  1 )
nmi(V) +.*.
*.
This exact sequence is usually used to deduce properties of the homotopy groups of V . Here, on the contrary, it will be used to study the groups nm(Sn). Let um denote the generator of n m ( S m ) represented by the identity map. Consider the following part of the sequence: * *
.+.nnfV)+
d
nn(Sn) L+nnl(Snl)
>+ n?&l(v) +. 0 .
By Hurewicz's theorem, Z %  ~ ( Vfi:) 2,. Since z, is an epimorphism, the exactness of the sequence implies that the image of d , is a subgroup of nnl(Snl) of index 2 . Hence we deduce that d*(un) = f 2un1. I n fact, it is , we shall not need this refinement. known that d,(un) = 2 2 ~ , ,  ~but The structure of the homomorphism d , with m > n is described by the following Lemma 11.1. If X is a fiber space over Sn with a pathwise connected fiber F , then the homomorphism d , i n the exact hmotopy sequence *
*+.nm(X)+ X m ( S n )
d
A+ nml(F)  + n m  l ( X ) +
*
sends the suspension X(a) of any element a ~ n m  l ( S n  l )into the composition d*(un) 0 a.
If a map k : 9
K, :nml(Snl)
+. F represents d*(un), then k induces a homomorphism nm,(F) which gives k,(a) = d,(un) 0a for each a in
 1 +.
XI. HOMOTOPY G R O U P S O F S P H E R E S
324
nmPl( 9  1 ) . Hence the relation stated in the lemma means that the following triangle is commutative : nmi(Snl) A %(Sn)
nm1 ( F ) Proof. The projection w : X + Sn induces an isomorphism w, : n m ( X ,F ) M nm(Sn). By definition, d , is wsl followed by the boundary homomorphism : n m ( X ,F ) tnml(F). A representative map k : Snl+ F of d,(u,) can be constructed as follows. Let h : (En,9  l ) + (Sn, so) be a map which represents un. It follows from the covering homotopy theorem that there exists a map H : (En, 9  1 ) f ( X ,F ) such that wH = h. Then the restriction k = H 1Snl represents d , @n) * Let a ~ n ~  ~ ( S nIt  l remains ). to prove &(a) = d,X (a). Let : Sml+ Snl represent a ; then k,(cr) is represented by A+. Extend to a map y : (Em, 9  1 ) (En,9  l ) . By 3 2, hy represents X ( a ) . Since wHy = hy, H y represents X (a) and hence k+ = H y I Sml represents d,X (a).I Next, let us study the homotopy groups of the Stiefel manifold V = Vn+,,z. Using the generalized Hurewicz theorem, we can deduce that: (1) n m ( V ) is finitely generated. (2) nm(V) = 0 if m < n  1. (3) nndV) 2,. (4) nm(V)is a finite 2primary group whenever n  1 < m < 2% 1. (5)n2n1(V)is isomorphic to the direct sum of 2 and a finite 2primary group.
a
+
f

Lemma 11.2. If %? denotes the class of all finite 2primary groups, theiz there exists a map q : S2nl + V such that the induced homomorphism
:nm(S2nl) +nm(V)
i s a %?isomorphism for every m. Proof. By (5), there is a map q : S2nl +V which represents a free element a of nsnl(V)such that the free cyclic subgroup generated by a is of index some power of 2. Since the natural homomorphism of Z ~ ~  ~into ( VHznl(V) ) is a %isomorphism, it follows that the induced homomorphism
4% : Hm(S2nl) + Hm(V)
is a %'isomorphism for m = 2n  1 and hence for every m.An application of Whitehead's theorem (X; 10.1) proves the lemma. I We list the following additional results on the homotopy groups of V ; these are immediate consequences of (10.2). (6) nm(V)is finite if m > 2% 1.
13.
THE PPRIMARY C O M P O N E N T S O F HOMOTOPY G R O U P S
325
(7) If p is an odd prime number, then the +primary component of h ( V )is isomorphic to that of nm(S2nl).
12. Finiteness of higher homotopy groups of evendimensional spheres Theorem 12.1. If S n i s a n evendimensional sphere and m i s a n integer such that m > n and m # 2 n  1, then n,(Sn) i s finite and nz,,(Sn)i s isomorphic to the direct sum of Z and a finite group. Proof. Since nm(S2)mnm(S3) for every m > 2 , the theorem is true for n = 2. Hence we may assume n 2 4 and apply the results of 3 11. If m > n and m # 2 n  1, then both n,(V) and nml(Snl) are finite. Therefore, the exactness of the sequence n,(V) +n,(Sn) +nml(Snl) implies that n,(Sn) is finite. To study the critical case m = 2 n  1, let V denote the class of all finite abelian groups. It follows from the exact sequence that
: n z n  i ( V +zdanI(S") is a %isomorphism. This implies that n,nl(Sn) is isomorphic to the direct sum of Z and a finite group. I
13. The pprimary components of homotopy groups of evendimensional spheres In the present section, we are concerned Cith the pprimary components of the homotopy groups n,(Sn) of an evendimensional sphere Sn. Since nm(S2)m nm(S3) for every m > 3, we may restrict ourselves to the case n > 4 and apply the results of 8 11. Consider the following part of the exact sequence appearing in 3 11 :
n,+l(Sn)
d
d n,(Sn1) A+ n,(V) 5tn,(Sn) >+ n,l(Snl).
I+
Using the homomorphism o* and the suspension X, we define a homomorphism :n m ( ~ + ) z m  l ( ~ n  l )+ n , ( ~ n )
r
by setting r ( a ,p)
= o,(ct)
+ Z(p) for each ct E n,V) and p ~ n ,  , ( S ~  l ) .
Lemma 13.1. If V denotes the class of all finite 2primary groups, then r i s a
%?isomorphism. Proof. According to ( 1 1 . l ) , the suspension Z followed by d , is the induced endomorphism k , onnmI(Snl) of a map k : S n  l + Sn' of degree d = & 2. By Ex. A6 at the end of the chapter, k , is a Vautomorphism. Hence the lemma follows as a consequence of ( X ; Ex. E). I Theorem 13.2. If V denotes the class of all finite 2primary groups, thex the homotopy group n,(Sn) of a n evendimensional sphere Sn i s %?isomorphic to the direct s u m of nn(S2nl) and nrn,(Sn').
X I . HOMOTOPY G R O U P S O F S P H E R E S
326
Proof. Since nm(S2) w %(Ss) for every m > 3, the theorem holds for n = 2. If n > 4, then ( 13.2) is a direct consequence of (13.1) and ( 1 1.2). I The importance of (12.1 ) and (13.2)is that the calculation of the homotopy groups of an evendimensional sphere, except for their 2primary components, reduces to that of the homotopy groups of odddimensional spheres. Precisely, we have the following Corollary 13.3. If n is even and p an odd prime, then the #primary component of nm(Sn) is isomorphic to the direct sum of those of nm(S2nl) and n, 1 (91). 14. The Hopf invariant
In order to strengthen f13.2), we propose to present Serre's version of the notion of Hopf invariant. Consider the nsphere Sn and a given point so E Sn. Let Qn = A(Sn)denote the space of loops in Sn with so as basic point. Consider the natural isomorphism j :nsns(S2n) w and the natural homomorphism h :nsn&2") + H,,(@)
w 2.
Let f :9 n  l +Sn be a given map representing an element [f] En,nl(Sn). The Hop/ invariant of f is the integer H ( f )uniquely determined by where ug denotes the generator ~ * ~ ( ofl ) Hzna(@) 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 (111; Q 5). See [Hopf 21 and [S; p. 1131. Theorem 14.1. Let n be an even integer and f : 9 n  l +Sn a map with HOP/ invariant H ( f) = k # 0. Let V denote the class of all filzite abelian groups of order dividing some power of k. If
Xf :nml(Sn1)
+ nm(S2nl) +nm(Sn)
is the homomorphism defined by
B) = V a ) + f&),
xf(a,
a Enrni(S"'),
then Xf is a Visomorphism for every m
B E~c,(S"'),
> 1.
Proof. The theorem is obvious if n = 2. Hence we assume n > 4. The map f defines a map g : Qzn1 +S2n in the obvious way and g induces a homomorphism g* :H*(SZn) + H*(BZnlj
14. T H E
327
HOPF INVARIANT
of the cohomology algebras with integral coefficients. According to (IX;
Ex.H), H*(@) admits a homogeneous basis { ai } with dim (ai) = i(n 1) for each i = 0, 1, * such that a0 =
1, ( a J z = 0, (as)'
=
(p!)aap,alaap
=
aspal
= aap+1.
Similarly, H*(Bafil) admits a homogeneous basis { bi } with dim (bi) = i ( 2 n  2 ) for each i = 0, 1, * * * such that
bo
=
1, (b1)P = (P!)bp.
Since H ( f ) = k , it follows that g*(a,)
(P!)g*(a,)
= g*(a,p)
=
=
kb,. Then we get
kp(b1)P = k p ( p ! ) b p ;
and therefore g*(am) = Mbp. Since obviously g*(aZp+J= 0, it follows that g* is completely determined. Now let i : Snl +Qn denote the natural imbedding in 4 2 and consider the induced homomorphism
i* : H*(@)
+ H*(Sn1).
By (2.2), e = i*(a,)is a generator of Hn1(Sn1). By means of the multiplication in Qn, we may define a map : Snl x Sd2nl+ Bn by setting +(x, w ) = i ( x ).g(w) for each x E Snl and w ~ S 2 ~ n  l . By (IX; Ex. H), H*(Snl x J2wl) is naturally isomorphic to the tensor product H*(Snl) @ H*(B2nl). This enables us to determine the induced homomorphism +# : H*(Qn) f H*(Sn1 x P  1 ) as follows: +#(asp) = k p *1 @ b p ,
+
+#(asp+J = +#(a1)+#(aap)
=
@ bp.
Hence, +# : Hm(On) + Hm(Snl x L P  l ) is a monomorphism and its cokernel is finite of order equal to a power of k for every dimension m. By duality, this implies that +# : Hm(Snl x Qznl) + H m ( P ) is a Wisomorphism for every m. An application of the Whitehead theorem shows that the induced homomorphism
+* :nm(Snl
x
Qznl) +nm(.Rn)
is a Visomorphism for every m. The group nm(Snl x Q2nl) is isomorphic to the direct sum 7cm(Sml) Z , ( Q ~ ~ ;~on ) the other hand, n m ( 8 n ) M nm+l(Sn).Then, the construction of shows that +* reduces to the homomorphism Xf. I
+
+
Corollary 14.2. If H ( f ) = f 1, then X, is an isomorphism and hence the suspension Z :nml(Snl) +nm(Sn) is a monomorphism for every m > 1.
Because of the existence of a map f with H ( f ) = 2, (14.1) implies (13.2).
328
XI. H O M O T O P Y G R O U P S O F S P H E R E S
15. The groups mn+,(Sn) and J Z , , + ~ ( S ~ ) Since nm(Sl) = 0 for each m we may assume n 2 3.
> 1 and nm(S2)M nm(S3)for every m > 2,
> 3.
Theorem 15.1. i ~ ~ + ~ (i S s cyclic n ) of order 2 for every n
Proof. Let
we have
X denote a 3connective fiber space over S3. Then, by (gal), n4(S3) w n4(X) w &(X) w 2,.
By the suspension theorem (2.1), we deduce n4(S3)M n5(S4)M M nn+l(SN)M

*

*.
Hence nn+l(Sn) w 2, for every n > 3. I One constructs the generator of Z ~ + ~ ( as S ~follows. ) Let us consider the Hopf map p : S3 +S2 defined in (111; 3 5 ) . According to (V; 3 6), p represents the generator of n3(S2).Since the suspension X : n3(S2)+n4(S3) is an epimorphism, the suspended map Zp :S4+ S3 represents the generator of n4(S3). Then the generator of nn+l(Sn) is represented by the (n 2)times iterated suspension Zn2p of the Hopf map p. Theorem 15.2. nn+,(S@)i s cyclic of order 2 for every n
Proof. Applying (10.1) with n = 4, h = 2, m
sequence
=
> 3.
5 , we obtain an exact
0 + 2, 8 2, +n5(Pi)+ Tor(Z,Z,) + 0.
Since 2,g 2, M 2, and Tor(2,2,) = 0, this implies n5(Pz)M 2,. Then, by (10.2), the 2primary component of n5(S3)is isomorphic to Z,.Since the $primary component of x5(S3)is 0 if p > 2 by (9.2), it follows that n5(S3)M 2,. Since the Hopf map S7 + S4 is of Hopf invariant 1, we may apply (14.2) with n = 4 and m = 6. Hence, (2.1) and (14.2) imply that X :n5(S3)M 7z6(S4).Thus, n,(S4) M 2,. Finally, by (2.1). we deduce
n,(S4)
M
n,(S?
M

* *
M
nn+,(Sn) M
* * *.
Hence nn+,(Sn) w 2, for every n > 3. I To obtain the generator for nn+a(Sn),let i :S4 + Pz denote the imbedding given by the definition of Pz. Then, by the proof of (lO.l), the generator of n5(Pi)is represented by the composition of i and X2p :Ss+ S4. Composing with the map X : Pi + S 3 in 4 10, we obtain a representative map Xi C2p for the generator of n5(S3).This implies that Xi represents the generator of n4(S3)and hence is homotopic to Zp : S4+S3. Therefore, the generator of n 5 ( S 3 )is represented by = ~p 0~ 2 :p ss + ~ 3 , where p : S3 + S2 denotes the Hopf map. Then it follows that the generator of nn+,(Sn) is represented by the ( n  3)times iterated suspension Cn39 of q for every n Z 3.
16. T H E
G R O U P S ntn+3(S")
329
Corollary 15.3. n3(S2)M 2,n,(S2)M Z,, n,(S2) M 2,.
By (V; tj 6 ) , the generators of these cyclic groups are represented by respectively the maps
p
: s3 +s2, p
0cp : s4 +s,, p 0cp 0c2p :ss +SZ. 16. The groups n,,+3(Sn)
Theorem 16.1. n,(S3) M Z,,. Proof. Applying (10.1) with n = 4, h
sequence
=
2, m
= 6,
we obtain an exact
0 + 2, 8 2, +n,(P5,)+ Tor(Z,, 2,) + 0.
Since 2, 8 2, M 2, and Tor(Z,,2,)M Z,, n,(P;) is isomorphic to an extension of 2, by 2, and hence has 4 elements. Hence, by (10.2), the 2primary component of n6(S3)has 4 elements. By (9.2), the 3primary component of n,(S3) is isomorphic to 2, and the $primary component of n,(S3) is 0 for every prime p > 3. I t follows that n6(S3) has 12 elements and hence is isomorphic to either Z,, or 2, + 2,. Suppose that n,(S3) w 2, 2,. Let X denote a 5connective fiber space
+
and it follows from the universal coefficient theorem [ES; p. 1611 that
W ( X ;2,) M Hom ( H , ( X ) ;2,) M 2, + 2,. This contradicts to ( I X ; Ex. I ) ; hence, we conclude that n,(S3) M Zlz. I Examination of the first paragraph of the proof reveals that the composition of the maps S6 ZSfi + Sb P P S4 s3 f
=f
represents an element of n,(S3) of order 2. A generator of n,(S3) is represented by the characteristic map 5 : S6 + S3 of the fiber bundle Sp(2) over S6 with Sp( 1) as fiber, [Borel and Serre 1 ; p. 4421. For the definition of the characteristic map, see [S; p. 971. Corollary 16.2. n,(S2)w Z,,. A generator of n,(.S2) is represented by the composed map Theorem 16.3. n7(S4)w 2
pt : S6+ S2.
+ Z,,.
Proof. Let us denote by q : S 7 + S 4 the Hopf map in (111; 5 5). Since H ( q ) = 1, we may apply (14.2) with n = 4, m = 7, and f = q. Thus, we obtain an isomorphism
& :n,(S3)+ n7(S7)
n7(S4).
Since n7(S7) M 2 and n,(S3) Z,,, this proves the theorem. I From the preceding proof, it follows that q represents the generator of
330
XI. HOMOTOPY G R O U P S O F S P H E R E S
the free component Z of n7(S4) and the suspended map Xt : S7 + S4 represents an element of order 12 which generates the torsion component Z,, of n7(S4). Theorem 16.4. nfl+3(Sfl) F=Z2, if n 2 5.
Proof. By the suspension theorem (2.1), Z maps n7(S4)onto n,(Sb). According to Ex. D at the end of the chapter, the kernel of X is the free cyclic subgroup of n7(S4)generated by the Whitehead product [e, e l , where e denotes the generator of n4(S4) represented by the identity map on S4. On the other hand, it follows from a theorem on characteristic maps, [S; p. 1211, that
where E = f 1 depends on the conventions of orientation. Hence, in n8(S6),we have P [t]= E2 E [q]. This implies that n8(S6)is isomorphic to 2, with X [q] as a generator. Finally, by the suspension theorem (2.1), we deduce n,(S6)
M
n p )M
* * *
M
nfl+3(sfl) m
 . *
Hence ~ ~ + ~ ( w S Z,, f l ) for every IZ 2 5. I Obviously, a generator of Z ~ + ~ ( Sn~2) ,5, is represented by the (n4)times iterated suspension D  4 q : Sn+3 +Sn of the Hopf map q : S7 + S4.
17. The groups a,,+&”) Theorem 17.1. n7(S3) w 2,.
Proof. By (9.2), the pprimary component of n7(S3)is 0 for every prime
fi > 3. By (10.4), the 3primary component of n,(SS)is also 0. Hence n7(S3)
is a 2primary group. Next, consider a 6connective fiber space X over S3. Then n7(S3)M n7(X)rn H 7 ( X )and hence we have
Horn (n7(S3), 2,)w H , ( X ; 2,). By ( I X ; Ex. I4), H 7 ( X ;2,) M 2,. This implies that n7(S3)is isomorphic to a cyclic group 2, with q = 2h, h > 1. If h > I , every homomorphism of n7(S3)into 2, can be factored into
n,(S*)+ 2, + 2,. Then it follows from the exact sequence in (VIII; Ex. 17) that Sq’a for every element a E H 7 ( X ;2.J.This contradicts ( I X ; Ex. 14). Hence h andn7(S3)2,. I According to [Hilton 2; p. 5491, the two maps
6 0c4p : S’
9,
z p 0q : s7 +s3
=0 =
1
17.THE
GROUPS ~ c ~ + ~ ( s f l )
331
are both essential. Hence they are homotopic and represent the nonzero element of n7(S3). Corollary 17.2. n7(S2)w 2,.
The nonzero element of n7(S2) is represented by the homotopic maps : S7+S2, f~0X P
P 05 0 Theorem 17.3. n,(S4)w 2,
+ 2,.
0q : S7 +S2.
Proof. As in the proof of (16.3),we obtain an isomorphism
%* :n7(S3)
+
ns(S7) W
n,(S4).
Since n7(S3)w 2, and n8(S7) m Z,, the theorem is proved. I The group n,(S4)is generated by two elements a and @ of order 2. a is represented by the homotopic maps
I:
(loI:”) :
S8
I:(I: 0q) : S8 +S4,
+S4,
and @ is represented by q 0ESP :S8 + S4. Theorem 17.4. ng(S5)w 2,.
Proof. Consider the following part of the suspension sequence in
n,(T )
*
+
+
X
X
n,(S4) + n,(S5) JLn,(T ) + n7(S4)
+
5 6:
n,(SS)
+ 0.
As mentioned in the proof of (16.4), the kernel of X :n7(S4) +n6(S5) is a free cyclic group. By (6.3),n7(T ) w 3 4 . 9 ) w 2. I t follows from the exactness of the sequence that :n,(T) +n7(S4)is a monomorphism and hence I: :n,(S4)+n,(Ss) is an epimorphism. Since n,(T) M n,(S7) w Z,, the kernel of I: :n6(S4) + n,(S6) contains at most two elements. On the other hand, consider the element a of n6(S4) represented by X (6 0X4$) : S8+S4. Inn,(Sy, we have
+
I:b) =
0W P 1 = (&2I::[qI)0 (CYPI) = (&I:hI)0 (21:6[Pl)= 0.
Hence the kernel of Z :n8(S4)+ n,(Sb) consists of exactly two elements, namely, 0 and a. This implies that n,(SS) w 2,. I The nonzero element of n,(S6)is I:(,!?) represented by the map I:(qOI:Sfi): Sg +Ss. On the other hand, the Whitehead product [e, e] of the generator e of n,(S5) is also nonzero and hence [e, el = X(@), [Serre 3; p. 2301. Theorem 17.5. ~ ~ + ~ ( = S 0f ifl )12 2 6. Proof. By the suspension theorem (2.1), X maps n,(Ss) onto n,,(S6). According to the delicate suspension theorem in Ex. D at the end of the chapter, X [e, el = 0. Hence we obtain n,,(Ss) = 0. Finally, by (2.1), we deduce nl0(S6) n,,(S7)w * * M n,+,(Sn) M * * .

Hence ~ ~ + ~ ( = S 0f lfor ) every n Z 6. I
X I . HOMOTOPY GROUPS O F SPHERES
332
18. The groups nn+r(Sn), 5 Q r
< 15
In this final section of the book, we will list the groupsn,,+r(Sn)for the cases r = 5, 6, 7, 8.For more detailed information, see [Serre 5, 61. H. Toda has computed the groups nn+r(S") for 9 < I Q 15. We will not list his results here; the interested reader should refer to poda 1,2] with recent corrections given in poda 31. I = 5.
r = 6.
I=
Y
7.
= 8.
EXERCISES
333
EXERCISES A. The distributive laws
Let u €nm(Sn,so) and /?E ~ , ( X xo). , If u and p are represented by the maps
f : (Ern,sm1)
f
(Sn, so), g : (9, so) + ( X ,xo)
respectively, that the composed map gf represents an element y of n,(X,xo). Prove: 1. The element y depends only on the elements u,p and will be called the composition /I0u of u and /?. 2 . The right distributive law. For a given Enn(X, xo), the assignment u +/? 0u defines a homomorphism. In fact, this is the induced homomorphism g,. 3. The left distributive law. If X is an Hspace with xo as homotopy unit or if u is the suspension X(6) of some element 6 ~ n ~  ~ ( Sso), n then ~ , the assignment B +/I 0u defines a homomorphism. [S; p. 1221. I n particular, let ( X , xo) = (9, so). Consider a map g : (Sn, so) + (Sn, so) of degree d and study its induced homomorphism
g, :nm(Sn, so) 'nm(Sn, so). Prove : 4. g,(u) = du for every u E n m ( S n , so) if n = 1,3, 7 or if m < 2n  1. 5 . If m = 3 and n = 2 , then g,(u) = d2ufor every u €n,(S2,so). [Hopf 1 1 . 6 . If V denotes the class of all finite abelian groups of order dividing a power of d, then g, is a %?automorphism for every m > 0 and n > 3. 6. On relative (n
+ 1)cells +
Let (X, A ) be a relative (rt 1)cell obtained by adjoining En+l to A by means of a map g : Sn + A . Then g has an extension f : (@+I, Sn) + ( X ,A ) defined by f ( x ) = x for every x E En+l\ Sn = X \ A . This map f is called the characteristic map of ( X ,A ) . If we identify A to a single point, we obtain an ( n + 1)sphere Sn+l as quotient space with projection h : X +Sn+l. Choose so E Sn and let xo = f (so) E A . Use so and xo as basic points of the homotopy groups. Verify that the rectangle
nm(En+l,Sn)+ '
+
.
T CI( ~ Sn)
z
+
n m ( X ,A )
nm(Sn+')
is commutative, i.e. h,f, = Ed. Prove 1. If 2 is a monomorphism, so is 1,. If 2 is an epimorphism, so is h,.
334
X I . HOMOTOPY G R O U P S O F S P H E R E S
2 . If m < 2n, then f , is a monomorphism, h, is an epimorphism, and
n,(X, A ) decomposes into the direct sum of I m ( f ) *and Ker(h,).
3. If A is rconnected for some Y < n, then f , is an epimorphism whenever < n + Y . See [J. H. C. Whitehead 6 ; p. 141 and [Hilton 1 ; p. 4641. As an application of these results, consider the relative ( p q)cell ( X ,A ) with X = SP x SQ and A = SP SQ. By 2 and (V; 3.1), we obtain natural isomorphisms
m
+
nm(SP x SQ,SP Z,~(SP v SQ)M Z,~(SP)
for every m < 2 p min ( p , q) 2 .
+
SQ)M nm+,(Sp+Ql)Ker(h,),
+ nml(SQ) + nml(Sp+Ql)+ Ker(h,)
+ 2q 2 . Finally, by 3, Ker(h,) = 0 if
m
0, define a homomorphism H* :nm(S%)+nm+l(s2fi)
by taking H* to be composition of the sequence h
n,(S%) &+ 7c~ ( s ~ V S L " )+ n m + l ( ~ n X S ~ , S ~ V S  L~+ )z ~ + I ( s ~ ~ ) , where g, h, are induced homomorphisms and p denotes the projection of Prove the nm(Sn Sn) onto its direct summand nm+l(S"x Sn, Sn 3). following relations : (i)H* = XH whenever m