Lectures on Homotopy Theory (North-Holland Mathematics Studies)

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Author: Renzo A. Piccinini

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LECTURES ON HOMOTOPY THEORY

NORTH-HOLLAND MATHEMATICS STUDIES 171 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK ' TOKYO

LECTURES ON H0MOTOPY TH E0RY

Renzo A. PlCClNlNl University of Milan Milan, Italy

1992

NORTH-HOLLAND - AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas NewYork, N.Y. 10010, U S A .

L i b r a r y o f Congress C a t a l o g t n g - i n - P u b l i c a t i o n

Data

Plcclnini, R e n z o A.. 1933L e c t u r e s on hornotopy t h e o r y / R e n z o A . P l c c l n l n l . p. cm. -- (North-Holland m a t h e m a t i c s s t u d i e s ; 171) An e x p a n d e d v e r s i o n of lectures g i v e n at t h e S c u o l a M a t e m a t l c a Interuniversltaria. in Perug1.a. d u r i n g t h e s u m m e r o f 1989. I n c l u d e s blbllographlcal r e f e r e n c e s and index. I S B N 0-444-89238-9 1 . H o m o t o p y theory. I. T i t l e . 11. S e r i e s PA612.7.P53 1992 91-40793 514'.24--dc20

CIP

ISBN: 0 444 89238 9

0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U S A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher.

No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

To Helena, Marina, Andreas and John


Preface Men can do nothing without the make-believe of a beginning. Even Science, the strict measurer, is obliged to start with a make-believe unit, and must fix on a point in the stars’ unceasing journey when his sidereal clock shall pretend that time is at Nought. His less accurate grandmother Poetry has always been understood to start in the middle; but on reflection it appears that her proceeding is not very different from his: since Science, too, reckons backwards as well as forwards, divides his unit into billions and with his clock-finger at Nought really sets of in medias res. George Eliot, “Daniel Deronda”, Book One. This book is an expanded version of the lectures on homotopy theory I gave at the Scuola Matematica Interuniversitaria, in Perugia, during the Summer of 1989. Its objective is to present a course in the homotopy theory of topological spaces to beginning graduate students who have a good familiarity with some basic topics in point set topology, but have not yet been exposed to algebraic topology. I purposedly avoided any mention of singular homology; this theory and its relationship to homotopy theory can, in my opinion, be taught more profitably later on, when the student has a good grasp of homotopy theory. Section 1.1 of this book is essentially a review of the compact-open topology on function spaces; it also includes a study of the exponential law of maps and of the evaluation map. Although most of the contents of this section may have been studied in a reasonably complete undergraduate course in topology, the student should not entirely avoid reading Section 1.1 as it introduces a good deal of the notation used further down the road. The next section, namely Section 1.2, intro-

...

Vl ll

duces the basic notion of homotopy (free and based), the dual concepts of H-space and CoH-space and shows that the set [ X , Y ] ,of based homotopy classes of based maps from a based space X to a based space Y has a group structure whenever X is a suspension or Y is a loop space; the development of this section is strongly influenced by the ideas of B. Eckmann and P. Hilton (see [ll]).The last section of Chapter 1 deals with the definition of homotopy groups and the proof of the fact that the fundamental group of the circle 5’’ is isomorphic to the group 2 of integers. The bulk of Chapter 2 is devoted to the study of fibrations (in the sense of W. Hurewicx) and of the dual concept of cofibration; in writing Section 2.2 I made generous use of my joint paper [23] and I thank Chris Morgan for letting me quote freely from that paper. In Sections 2.3 and 2.4 I used some ideas and material studied during the preparation of the book “Cellular structures in topology” (see 1151); in particular, the proof of the “gluing theorem” was written for that book, but then discarded for questions of space. I am indebted to Rudolf Fritsch for letting me use this material. Section 2.4 is not necessary for the development of the subsequent sections and so, it could be missed on a first reading. Chapter 3 describes the long exact seqence of homotopy groups associated with a fibration and studies the dual situation for a cofibration. Chapter 4 is devoted to abstract simplicial complexes and polyhedra; in particular, it gives a proof of the simplicial approximation theorem (thereby, showing that the lower homotopy groups of a sphere are trivial) and introduces the notion of Serre fibration. The first section of Chapter 5 is devoted to the homotopy groups of a map, and in particular, to relative homotopy groups; as it was done in [15], the method is taken from the Eckmann-Hilton paper [12]. The next section defines quasifibrations in the sense of A.Dold and R.Thom and connects the long exact sequence of homotopy groups of a pair ( E , F ) - the total space and fibre of a Serre fibration, respectively - t o the long exact sequence of the Serre fibration itself. The last section of Chapter 5 deals with the material which is strictly necessary to prove that the nth homotopy group of the n-dimensional sphere is isomorphic to Z; there, it is also shown that the higher homotopy groups of a sphere are not necessarily trivial.

ix Chapter 6 is entirely devoted to CW-complexes, the spaces first studied by J.H.C.Whitehead in [36]. The first section and part of the second are entirely motivated from the material discussed in the first two chapters of [15]. The definition of CW-complex given here is “constructive”; it coincides with the definition given originally by Whitehead (see [15, Section 2.61). Section 6.2 also contains a proof of the Seifert-Van Kampen Theorem. The chapter is concluded with a section on some special CW-complexes, the Eilenberg-Mac Lane complexes, first discussed in [13] and [14]. Section 7.1 deals with the sets of classes of based homotopies versus the set of classes of free homotopies; it is mainly developed according to ideas given in [25], [2] and [3]. In that section there is a discussion of 7rl(Y,yo)%action on the homotopy groups of a map f : (Y,yo) --$ ( X ,zo) (and hence, the action of the fundamental group of a space Y on its higher homotopy groups). The remaining two sections of the chapter grow naturally from Section 7.1. One could say that Sections 7.2 and 7.3 are devoted to the study of “fibrations from the point of view of their fibres”; through these two sections one can perceive the strong influence of J.Peter May’s classical monograph [22]. The material presented in these two sections appeared, for example, in [5] and [4]. There are two appendices. Appendix A is a brief discussion of colimits in a category and is written to clarify some of the algebra needed mainly in Chapter 6. The second appendix gives a description of the category of compactly generated spaces, a category which features good properties for homotopy theory; it is mostly used in Sections 6.1, 7.2 and 7.3. As a suggested basic text in Topology I selected [24]; another possible (and very good) choice would be [6], which is also more in line with the general thinking of this book. A bare minimum of category theory is used here; of course, the now classical reference to category theory is [21]. Finally, those who would like to study more homotopy theory should consult the books of H.J.Baues [l]and G.W.Whitehead [35]. Throughout the text, “iff” is used in place of “if and only if”; moreover, the symbol 0 stands for “end of the proof” (or that a proof will not be given explicitly). Some of the exercises are preceded by a star *; this means that the exercise in question is difficult and the student should probably look for help in the literature.

X

Many thanks are due to many persons. In particular, I wish to thank my friends and colleagues P.Booth, R.Fritsch, P.Heath and C.Morgan with whom I collaborated over the years and were always a great source of mathematical stimulation; the “class of 1989” at the Scuola Matematica Interuniversitaria, whose zeal and great interest in my lectures convinced me to extend the original lecture notes; C.Pacati, for his excellent suggestions on improvements and exercises; M.Barr, for his “catmacl” TEX macros, responsible for all the diagrams of this book; special thanks are due to my friend Edgar Goodaire for his great patience with my dumb questions on computer text editing, &T+, and English grammar; M.Bonecchi, for rescuing my index file and helping with other computer puzzles. Last, but not least, I wish to thank my wife Nair who, during the many years of our life in common, has always supported and helped me; without her, this work would not have been possible.

R. Piccinini Milano, August 1991.

Contents 1 Homotopy Groups 1.1 Function spaces . . . . . . . . . . . 1.2 H-spaces and CoH-spaces . . . . . 1.3 Homotopy groups . . . . . . . . . .

............ ............. ............

1 1 10 27

2 Fibrations and Cofibrations 2.1 Pullbacks and pushouts . . . . . . . . . . . . . . . . . . . 2.2 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Applications of the mapping cylinder . . . . . . . . . . .

35 35 41 50 63

3 Exact Homotopy Sequences 3.1 Exact sequence of a map: covariant case

73 73

3.2 4

......... Exact sequence of a map: contravariant case . . . . . . .

Simplicial Complexes 4.1 Simplicia1 complexes . . . . . . . . . . . . . . . . . . . . 4.2 Simplical approximation theorem . . . . . . . . . . . 4.3 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Fibrations and polyhedra

5.3

85 85 . 95 100

..................

5 Relative Homotopy Groups 5.1 Homotopy groups of maps

5.2

.

79

106

117 . 117 141

................ Quasifibrations . . . . . . . . . . . . . . . . . . . . . . . Some homotopy groups of spheres . . . . . . . . . . . . . 147

CONTENTS

xii 6 Homotopy Theory of CW-Complexes 6.1 CW-complexes . . . . . . . . . . . . 6.2 Homotopy theory of CW-complexes . 6.3 Eilenberg-Mac Lane spaces . . . . . .

........ ........ ........

153 . . . 153 . . . 174 . . . 197

7 Fibrations Revisited 7.1 Sections of fibrations . . . . . . . . . . . . . . . . . . . . 7.2 3-Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Universal F-fibrations . . . . . . . . . . . . . . . . .

.

215 215 236 .255

A Colimits

267

B Compactly generated spaces

277

Bibliography

285

Index

289

Chapter 1 Homotopy Groups 1.1

Function spaces

In this section we lay the topological foundations necessary for the development of this book. In what follows Top represents the category of topological TIspaces and continuous functions (we note explicitly that if X E Top and z is an arbitrary point of X then {z} is closed in X ) ; the objects and morphisms of that category will be called simply by “spaces” and “maps”, respectively. Similarly, we introduce the category Top, of “based spaces” and “based maps”: a based space is a space X together with a base point z, E X ; a based space is normally denoted as a pair, say In either category, the symbol denotes “homeomorphism”. Given that X , Y E Top, let M ( X , Y ) be the function space of all maps f : X ----t Y with the compact-open topology; recall that for X , Y E Top, the sets WK,U= {f E M ( X , Y ) 1 f ( K ) c U), where K C X is compact and U c Y is open, form a sub-basis for the compact-open topology of M ( X ,Y ) .

(x,~,).

Theorem 1.1.1 Let X E Top be a locally compact HuusdorfS space. Then, i f M ( X , Y ) is given the compact-open topology, the function cXxM(X,Y)-Y

defined by ~ ( z , f )= f(z), for every continuous.

2

E X and f E M ( X , Y ) , is

CHAPTER 1. HOMOTOPY GROUPS

2

Proof - Take arbitrarily ( z , f ) E X x M ( X , Y ) and an open set V c Y which contains f(z). Because X is locally compact HausdorE and f is continuous, there exists an open set U c X containing z and is compact and f ( U ) c V . Now notice that such that the closure (z,f)belongs to the open set U x Wu,,.and that E(Ux W D , ~ c ~V ). 0 The map

E

defined before is an evaluation m a p

.

Theorem 1.1.2 Let X be a locally compact Hausdorff space and let M ( X , Y ) have the compact-open topology. For a n arbitrary 2 E Top, a function f : x zd Y

x

i s continuous iff the function

f

:2

-

M(X,Y)

defined by f(z)(z) = f(z,z ) , f o r every (z, z ) E X x 2, is continuous.

Proof -

+ Take arbitrarily

WK,Uof the sub-basis for M ( X , Y ) which contains f ( z o ) , that is to say, such that f(z,z,) E U ,for every z E K . Hence, K x

(2,)

z, E 2 and an element

c f-'(u) c x

x 2

with f-' (U)open, implying that f - l (U) n( K x 2 ) is open and contains K x {z,}. Because K is compact, there exists an open set W c 2 which contains z, and such that K x W c f-l(U); this implies that f ( W ) c W K ,and ~ thus, f is continuous. Notice that we did not use the fact that X is locally compact Hausdorff this assumption is used in the proof of the sufficiency. + The function f is just the following composition of functions:

x x z

f

-

1,YX.f

X x M ( X , Y ) L Y

Thus, f is a map if and E are continuous; but the latter function is continuous if X is locally compact and Hausdorff, according to the preceding theorem. 0 Remark - Theorem 1.1.2 just proved will be referred to as the exponential law; we shall say that the maps f and J are adjoint under

1.1. FUNCTION SPACES

3

the exponential law. Finally, we note that this theorem shows that there is a bijection between the sets M ( X x 2, Y ) and M ( 2 , M ( X ,Y ) ) as long as X is locally compact Hausdorff; actually, we leave it to the exercises t o show that the hypothesis on X implies that these two spaces are homeomorphic.

Corollary 1.1.3 Let q : Y + Z be a n identification map. If X is locally compact Hausdorfi then qx

1.y

:Y x x + z x x

is a n identification map.

Proof - Recall that q : Y + Z is an identification map iff q is onto and any function g : 2 + W , with W E Top, is continuous whenever g q is continuous (see Exercise 1.1.8). Hence, given any function

with W E Top, we must prove that if h = g(q x 1s) is continuous, so is 9. Theorem 1.1.2 shows that the adjoint

i: Y +M ( X , W ) is continuous, Notice that, for every z E 2, y E q-l(z) and

2

EX

and therefore, we can decompose the map k as k = (kq-l)q. Now, because q is an identification map and k is continuous, the function kq-' is continuous showing thereby that g is continuous. 0

Corollary 1.1.4 Let p : X -+ Y and q : Z + W be identification m a p s with Y and Z locally compact Hausdorfl. Then p x q : xx Z+Yx is a n identification map.

W


4

x x z

Pxq

*YXW

Y X Z FIGURE 1.1.1

Proof - Decompose p x q as in Figure 1.1.1 and apply the previous corollary t o p x 12 and 11. x q. 0 Let us now work in the category Top,. If ( X ,q,), (Y,yo) E Top,, we take M , ( X , Y ) to be the space of all based maps f : (X,a,) 3 (Y,yo) with the compact-open topology.

Lemma 1.1.5 Let ( X ,zo),( Y , y o ) E Top, be given with X Huusdorfl. I f S is a sub-basis for the open sets of Y , the sets

W K , W = {f E M * ( X , Y )I f ( K ) c W } where K C X is compact and W E S , form a sub-busis for the open sets of M , ( X , Y ) . Proof - The sets W K ,=~{f E M , ( Y , X ) I f ( K ) c U},where K c X is compact and U c Y is open, form a sub-basis for the compactopen topology of M , ( X , Y ) . It is enough to prove that if f E WK,.~, with K C X compact and U C Y open, then there exist finitely many compact subsets of S,say K1, - - ,K , and elements Wl, ,W, E 3 such that 9

--


5

the continuity off now implies that there exists an open neighbourhood U, of z in K such that

But K is regular, as a compact Hausdorff space and thus, there is an open set V, c K such that 2

E

v, c v, c u,;

the set (V, I t E K } is an open covering of K and since K is compact, finitely many of them, say Vxl,---,V,,,, suffice to cover K . Let us simplify the notation by replacing z; with i, i = ,n; notice that the spaces K,= are compact and moreover, 1 , a . S

i = 1, * .

-,n. This shows that, for every i = 1,- - ,n,

and thus, our lemma follows. 0 A very useful example of a based function space is obtained by taking (X,t,) as the one-dimensional unit sphere S1 of R2 (centered at (O,O)), with base point e, = ( 1 , O ) ; the space M,(S', Y )is normally denoted by s2Y and is called the loop space associated to ( Y , y o ) . We can also view s2Y as the space of all maps a : I + Y such that a(0)= a(1) = yo, where I is the unit interval [0,1]. There is an important space associated to two based spaces ( X ,2,) and (Y,yo), namely the smash product X A Y defined as follows: first take the space X V Y = X x (yo} U {zo} x Y - called wedge product of X and Y - regarded as a closed subset of the product X x Y and then set X A Y as the quotient space ( X x Y ) / ( X V Y ) (in other words, the quotient map q : X x Y X A Y is an identification map). We denote the points of X A Y by z A y, with z E X and y E Y ;in particular, X A Y is a based space with base point 2, A yo. A particular case of importance is given by ( X , z , ) = ( S ' , e , ) and (Y,yo) arbitrary; in this --+

CHAPTER 1. H O M O T O P Y GROUPS

6

case, the space 5'' A Y = CY is called suspension of (Y,yo). As we have done with the loop space of the based space (Y,yo), we can use the unit interval I rather than the sphere S1 to define C Y ; in fact, regarding S' as the set of all complex numbers e2"", parametrized by t E I , it is easy to establish a homeomorphism

CY

( I x Y ) / ( Ix {yo} u 8I x Y )

where aI = (0, I}. Whenever we regard CY in this fashion, we denote its elements by [ t , y ] with the understanding that [ t , y ] represents the class of ( t ,y) modulo I x {yo} UaI x Y ;the base point of C Y is denoted by *.

Theorem 1.1.6 For every ( X ,zo),(Y,yo) E Top, there exists tion

a

bijec-

9 : M*(EX,Y)+ M * ( X , R Y ) ; af X is Hausdorf, then @ is continuous.

Proof - Define @ by the condition: for every f E M,(EX, Y), zE X and t E I , (@(f)(z))(t) = f ( [ t 4); , of course, we must prove that @(f) : X -+ RY is continuous in order to guarantee that @ is actually into M , ( X , n Y ) . To this end, take a sub-basis element W K ,with ~ K c S1 compact and U c Y open; we wish to prove that (@(f ) ) - ' ( W K p ) c X is open. For every fixed E (@(f))-'(ww) f ( K A (4)c u and hence, K A {z} is contained in the open set f-l(U) of EX. If q : S' x X -+ EX denotes the identification map,

K x

(32)

c q-l(f-l(u)) c s1x x

and so, as in the proof of Theorem 1.1.2, there exists an open set V c X such that 2 E V and @(f)(V) c Wlc,(,. While it is very easy to show that @ is injective, the proof of its surjectivity requires a little work. Let g E M , ( X , RY) be given. Consider the evaluation map E : S' x O Y -----f Y ; the composite map ~(1s'x g) induces a map f E M , ( C X , Y ) . Then observe that @(f) = g .


7

Now suppose that X is Hausdorff. According t o Lemma 1.1.5, it is enough to study @ - ' ( W K , ~ with ) , K c X compact and U c CIY is of the type W L , i v , where L c S1 is compact and V c Y is open. But @-l(WK,U) =

{f E M * ( C X , Y ) I @(f)(K)c WL,I.}

= (fE M*(Z:X,Y ) I f ( L A K ) C V } = WL~\K,I. ;

hence, the statement will be proved correct if we can show that L A K is compact. This follows from the fact that K being a compact subspace of a Hausdorff space X is closed there and thus, L V K is a closed subset of S1 x X implying that L A K is compact as the continuous image of an identification map from the compact space L x K . 0

EXERCISES 1.1.1 Let X , Y E Top with X Hausdorff, let S be a sub-basis for the open sets of Y and let C = {CA 1 A E A} be a set of compact subsets of X such that, for each A c X compact and each open set U c X which contains A, there is a finite number of elements of C satisfying the inclusions n

A

c U Cx, C U

.

i= 1

Prove that (Wc,i- I C E C,V E sets of M ( X ,Y ) .

S} is

a sub-basis for the open

1.1.2 Prove that M ( X , Y )is Hausdorff (regular) iff Y is Hausdorff (regular). 1.1.3 Prove that if X , Y E Top and A is a subspace of X , then M ( Y , A ) is a subspace of M ( Y , X ) . 1.1.4 Let Y be a locally compact Hausdorff space and let A be a closed subset of a space X. Prove that

{(f,Y)E M ( Y , X ) x y I f ( Y ) is a closed subset of M ( Y ,X) x Y .

E

4


8

1.1.5 If X E Y and Z E W , show that M ( X , Z ) E M(Y,W). 1.1.6

Prove that if X is locally compact Hausdorff and Y,ZE Top are arbitrary spaces, then M ( X x 2,Y )E M ( 2,M ( X , Y)).

1.1.7 Let Q and R be the sets of rational and real numbers endowed with the induced and metric topologies, respectively. Let the number 0 be the base point of both Q and R. Show that the evaluation map E : M,(Q,R) x Q + R is not continuous. (Hint: No compact subset of Q contains all rational numbers of an interval of R.) 1.1.8 Let q : Y --t Z be an onto map. Prove that the following are equivalent: (a) q is an identification map;

(b) F c Z is closed iff q - l ( F ) is closed; (c) U c Z is open iff q-'( U )is open; (d) Z has the final topology with respect to q. 1.1.9

Let X,Y E Top be arbitrary spaces and let B be a closed, compact subspace of Y. If q : Y + Y/Bis the identification map, prove that q x 1,y : Y x + (Y/B) x

x

is

also

x

an identification map.

1.1.10 The following is a counter-example to Corollary 1.1.3 (see also [6, Example 4, Page 1051 ).

Let q : Q + Q / Z be the function which identifies every integer of the rational line to a point. Prove that q is a closed identification map but q x 1~ is not an identification map. (Hint: Let I' be the graph of the function

f:R+R given by:

dF7,

2

$ [-1,11

2

E [-1,1]


9

and let F = ( q x 1Q)(rn Q'). Prove that F is not closed in Q / Z x Q but ( q x l ~ ) - l ( Fis)closed in Q x Q,)

1.1.11 The following problem generalizes Theorem 1.1.6. Suppose that we are given three elements of Top,, say ( X , z , ) , (Y,yo) and (2,2,). Define a function

= f ( z A y), for every f E M,(X A by setting (a(f)(z))(y) z E X and y E Y . Then prove the following statements:

Y,z),

(a) a is one-to-one;

(b) if X is Hausdorff, a is continuous; (c) if

Y is locally compact Hausdorff, a is onto;

(d) if X and Y are both compact Hausdorff, a is a homeomorphism . 1.1.12 Given that ( X , z , ) , (Y,yo) and (Z,z,) are based spaces, prove that ( X V Y ) A Z z ( X A Z) v (Y A 2 ) . 1.1.13 Given that (X,z,), (Y,yo) and ( 2 , ~are ~ based ) spaces with X and Y Hausdorff, prove that

M*(X v Y,2 ) 2 M * ( X ,2 ) x M*(Y,2 ) 1.1.14 Given that ( X , z , ) , (Y,yo) and (Z,z,) Hausdorff, prove that

M * ( X ,Y x 2 )

are based spaces with X

M , ( X , Y ) x M * ( X ,2 ) .

C H A P T E R 1. HOMOTOPY GROUPS

10

1.2

H-spaces and CoH-spaces

We begin this section by discussing the fundamental concept of homotopy in the categories Top and Top,. Two maps fo,fi E M ( X , Y ) are said to be free homotopic (notation: fo w fi) if there is a map

H:XxI+Y (again, I = [0,1]) such that, for every e E X , H(z,O) = fo(e) and H ( e , 1) = fi(z). The function H is called a free homotopy connecting fo and fi; we shall use the notation H : fo f l to signify that H is a homotopy such that H ( - , 0) = fo and H ( - , 1) = f1.

-

Lemma 1.2.1 B e e homotopy is a n equivalence relation in M ( X , Y ) .

-

Proof - The relation given by “free homotopy” is clearly reflexive. fi gives rise to a free homotopy K : fi fu simply by defining, for every ( z , t ) E X x I , K ( z , t ) = H ( z , l - t ) ; thus, we have symmetry. The transitivity property is proved as follows. Let fo, fi, f2 E M ( X ,Y ) and let H : fu f1 and K : fi f2 be free homotopies; define G : X x I -+ Y by setting, for every (a,t ) E X x I ,

A free homotopy H : fu

N

-

-

Notice that G : fo fi. 0 Free homotopy partitions the set M ( X ,Y ) into equivalence classes called free homotopy classes; the set of all these classes will be denoted by [ X ,YI, Recall that if ( S , z , ) , ( Y , y , ) E Top,, M * ( X , Y )is the space of all based maps f : ( X , z , ) (Y,y,) with the compact-open topology; define the relation “based homotopy” in the underlying set: f , g E M , ( X , Y ) are based homotopic (use the same notation as for free homotopy) if there exists a map N

-

H:XxI+Y

1.2. H-SPACES AND COH-SPACES

11

such that, for every z E X , t E I , H(z,O) = f"(z),H(z,l) = fi(z) and H(z,,t) = yo. Based homotopy is an equivalence relation which partitions the set M , ( X , Y ) into equivalence classes called bused homotopy classes; the set of such classes is denoted by [ X , Y ] , . There is an important generalization of the concept of based homotopy; we are referring to the notion of homotopy relative to a subset: suppose that for a subspace A c X , the maps f,g E M ( X , Y ) coincide when restricted to A ; then f and g are said to be homotopic relative to A - notation: f N g rel. A - if there is a homotopy

H:XxI--+Y such that H ( - , 0) = f, H ( - , 1) = g and, for every (z,t ) E A x I , H ( z , t ) = f ( 4 = S(4. We shall see in Section 7.1 that the sets [ X , Y ] and [ X , Y ] , are related by the action of a group associated to the based space ( Y , y o ) . From now to the end of this section we shall stick to the category Top, and to based homotopies (which we shall simply call homotopies whenever there is no danger of confusion).

Theorem 1.2.2 For every ( X ,zo),(Y,yo) E Top, the bijection @ : M * ( I = X , Y )--+ M * ( X , O Y )

defined in Theorem 1.1.6 induces a bijection

d : [ E X , Y ] *+ [ X , Q Y ] *. Proof - Let H : fo

-

f1 be

map

H ( q x 11) :

a homotopy H : C X x I + Y ; the

s'

x X x I +Y

takes {e,} x X x I and S1 x {zo} x I into the base point yo and thus, it induces a map

H' : SLA (X x I ) + Y. It is now easy to check that @ ( H ' ): @(fO) @(fl). Suppose now that @(fu), @(fi) E M , ( X , Q Y )are homotopic; since

-

@ is a bijection, we may assume that the homotopy connecting these two functions is given by a map @(G),where

G : S1 A ( X x I ) -+ Y

CHAPTER 1. NOMOTOPY GROUPS

12

is such that G(e, A ( 2 ,t ) )= yo, for every ( 2 ,t ) E X x I and moreover, the restriction of G to the space S1 x {z,} x 1 is the constant map t o yo (recall that 9 ( G ) ( z o , t is ) the constant loop at yo, for every t E I ) . Let cj : S1 x ( X x I ) + S' A (x'x I ) be the identification map; the composite map Gq sends {e,} x X x I and S' x (z,} x I into yo and thus, it induces a function G' : EX x I -+ Y . Since I is compact, the map q x 11

:slx X x I -

cx X

I

is actually an identification map (see Corollary 1.1.3) and so, because G'(q x 11) = Gq is continuous, G' is a map. But G' is a homotopy connecting fo and fl. The bijection 5 is defined by @([f]) = [@(f)],for [f]E M,(EX, Y ) . 0

For a given (Y,yo) E Top,, take

-

with the subspace topology induced from the product topology of Y x Y ; the inclusion function i : YVY Y x Y and the function (T : YVY -+ Y defined by (~(y,y,) = a(yo,y) = y for every y E Y are continuous. The map t~ is called the folding map. A space (Y,yo) E Top, is said to be an H-space if there exists a based map 1.1 : Y x Y + Y - called H-multiplication - such that the maps p i and (T are homotopic; in other words, we require that the diagram of Figure 1.2.1 be homotopy commutative i.e., such that there exists a (based) homotopy H : (Y V Y)x I Y

-

satisfying the conditions: H ( - , 0) = u and H ( - , 1)= pi. An H-space (Y,yo) with an H-multiplication p is associative if p ( p x 11;) 4 1 1 - x p ) or, in other words, if the diagram of Figure 1.2.2 is homotopy commutative:

-

Lemma 1.2.3 For every (Y,yo) E Top,, the loop space OY is an associative H-space.

1.2. H-SPACES AND COH-SPACES

13

YVY

YxY,-Y FIGURE 1.2.1

11.- x

YXYXY

p

*YxY

I

I

YXY

P

FIGURE 1.2.2

Proof - Define py : RY x RY -+ OY as follows: for every (cr,P) E

QY and every t E I , PI4Q,P)(t)=

In order to prove that mot opy

{

@t),

o g 5 ;

P(2t - l),

5 5t 51

-

is an H-multiplication construct the ho-

H : ( R YV S l E ’ ) x I OY as follows: take the constant loop c E RY; then, for every cr,P E SlY and every s , t E I , set

H ( ( % c), S ) ( t ) =

i

a(%),

o g (

2)

H(a,2s

1

+ t - 2))

0

5 2s 5 2 - t 5 2

15 2 - t 5 29 5 2

The proof of the theorem is completed by observing that Gio = g and

p(f)G= H .

-

The fibration p ( f ) : X n Y' + Y is the mapping track fibration associated to f : X Y ; the space T ( f )= X n Y' is called mapping track of f.

EXERCISES 2.2.1 Prove that if ( E , p , B ) , ( B , q , A )E Top' ( E ,!lPlA).

are fibrations, so is

2.2.2 Let ( E , p ,B ) be a fibration with B path-connected. Prove that if one fibre of ( E , p , B ) is path-connected, then E is also pathconnected.

C H A P T E R 2. FIBRATIONS A N D COFIBRATIONS

48

2.2.3

* Prove that if p : E -+ B is a covering map, then ( E , p , B ) is a fibration. Use this result to give an alternative proof of Lemma 1.3.3. (You might want to consult some textbook, e.g. [27].)

2.2.4 Let ( E , p ,B ) be a fibration and Y be a locally compact Hausdorff space. Define q : M ( Y , E ) -----t M ( Y , B ) by q( f ) = pf, for every f E M ( Y ,E ) . Prove that the arrow

is a fibration.

2.2.5 Prove that ( E , p , B ) is a fibration if, for every space 2 and maps g :2 E and H : 2 ---f B' such that p g = €OH, there is a map G : 2 + E' such that the diagram in Figure 2.2.5 is commutative. .--)

FIGURE 2.2.5

2.2.6 Let ( E , p , B )be an arrow. A map H:ExI+E

2.2. FIBRATIONS

49

- -

is a fibre homotopy over B if, for every t E I , ( H t , 1 ~: p) p is an arrow-map. An arrow-map f : ( E , p , B ) ( E ' , p ' , B ) is a fibre homotopy equivalence over B if there exists an arrow-map g : p' + p such that g f and f g are fibre homotopic over B to the appropriate identity maps. Now prove that if ( E , p , B ) is a fibration, the map u : E + T ( p ) - the mapping track of p given by u(z)= ( W ~ ( ~ I , Z )where , wP(..) is the constant path at z, is a fibre homotopy equivalence over B .

2.2.7 An arrow ( E , p , B ) E Top' is said to be a weak fibrution if given any arrow ( A ,g , E ) and any homotopy H : A x I + B such that H ( -, 0) = p g , there is a homotopy G : A x I E such that G(-, 0) g and p G = H . Prove that if (Y, f , X ) E Top' is such that the map u : Y + T(f) defined in Exercise 2.2.6 is a fibre homotopy equivalence over B , then (Y,f , X ) is a weak fibration. -----)

N

2.2.8 Prove that ( E , p , B ) is a weak fibration iff it has the covering homotopy property with respect to all homotopies H : A x I + B which are stationary on [0, i],that is to say, such that H ( z , t ) = H ( z , O ) for every (x,t) E A x [0,f]. 2.2.9 Let ( E , p ,B ) E Top' be defined by the spaces E = {0} x I U I x (0) (with the topology induced by R2)and B = I x {0}, with p : E + B the projection on the first factor. Let A be a space, g : A -, E be a map and let H : A x I -, B be a stationary homotopy such that p g = H ( -, 0). Define D : E x I + E , ((z,s),t)H(z,S-tS) for every ( ( z , s ) , t ) E E x I . Construct the map G from A x I to E by the formula

to prove that ( E , p , B ) is a weak fibration. ( E , p , B ) is not a fibration.

Next, prove that

50


2.2.10 Let ( E , p , B ) , ( E ' , p ' , B ) E Top' be given. Then ( E ' , p ' , B ) is dominated by ( E , p , B ) if there exist arrow-maps (u,1 ~: p) + p' and ( u , 1 B ) : p' -, p such that uu is fibre homotopic to 1p over B. Prove that if ( E ' , p ' , B ) is dominated by a weak fibration ( E , p ,B ) , then (E',p',B ) is also a weak fibration.

2.3

Cofibrations

-

Let X E Top be given; a subspace A c X is a strong deformation retract of X if there exists a homotopy H : X x I X such that

x

H ( z , O ) = 2, 2 E H(z,l ) E A , 2 E X H(a,t) = a, (a,t) E A x I

.

The homotopy H is a strong defomation retraction of X onto A. The map r = H ( - , 1) : X -+ A is a retraction and A is a retract of X . Thus, a retract A of X with retraction r : X + A is a strong deformation retraction of X if

is homotopic rel. A to 1s.

Theorem 2.3.1 Let A be a closed subspace of X . The following statements are equivalent: 1) for any two given maps f : X x ( 0 ) -+ Z and G : A x I -+ Z which coincide when restricted to A x ( 0 ) there is a map F : X x I -+ Z such that F restricted to X x (0) is f and F restricted to A x I is G; 2) the space X = X x (0) U A x I is a retract of X x I ; 3) X is a strong deformation retract of X x I . Proof - 1) e 2): Let i : A -+X be the inclusion map; because A is closed in X , the space is a pushout of 1 , x~io and i x l{o), where

2.3. COFIBRATIONS

51

io denotes the inclusion of (0) into I . Then X x I is a weak pushout of these two arrows 2 is a retract of X x I . 2) j 3): Let T : X x I +X be a retraction; for every (2,t ) E X X I , write

and notice that r,y(z,O) = z , r ~ ( z , O= ) 0 for every z E X and, for every a E A , ~ . y ( a , t )= u , r l ( a , t ) = t. Now define

R : ( X x I ) x I +x x I by setting: R ( ( z , t ) , s )= ( T S ( z , t S ) , t ( l - s ) + s T I ( z , t ) ) . This is a strong deformation retraction of X x I onto X. 3) + 2): Obvious. 0 An arrow (A,i, X ) with A c X closed and i the inclusion map is said to be a cofibration if it satisfies any one of the equivalent conditions of Theorem 2.3.1. Condition 1) of Theorem 2.3.1 is the so-called homotopy extension property for the cofibration ( A , i ,X ) . Figure 2.3.1 expresses this situation pictorially.

I

ix

l{O)

FIGURE 2.3.1: homotopy extension property

CHAPTER 2. FIBRATIONS A N D COFIBRATIONS

52

Before we state a corollary to this theorem, we describe a simple example of cofibration. In the proof of Lemma 2.2.3 we have seen that there exists a retraction T

:Ix I

-I=I

x

(o}uar X I ;

the second statement of the previous theorem shows that (aI,j, I ) is a cofibration.

Corollary 2.3.2 I f ( A , i , X ) and ( B , j , Y )are cofibrations, then so is

( A x B,i x j , X x Y ). Proof - The hypotheses imply that there are two retractions :X x

TI

I

d

X x (0) U A x I

and :Y x

I + Y x (0) U B x I which, as we did in Theorem 2.3.1, can be decomposed into two maps T I = ( r l x , r l l ) and r2 = ( T ~ I , - , T ~ INow ). define r2

T :

( X x Y )x I

T((2,Y)J)

+( X

x Y )x {O}U(A x B ) x I

= ((T1.~((2,t),TZ]’(Y,t)),

1 i(T”(X,t)

+T2I(Y,t))) -

The map r is a retraction and so, the result follows. 17 The previous Corollary shows, in particular, that if (A,i,X) is a cofibration and Y is an arbitrary space, then

( A x Y,ix 1y,X x Y ) is a cofibration. The following is a very useful characterization of cofibrations (see 1301). Theorem 2.3.3 Let A be a closed subset of X . Then ( A , i , X ) is a cofibration if there exist a map q5 : X -+ I such that A = q5-’(0) and a hornotopy relative to A

H : X x I + S such that H ( z , O ) =

w*

2, for

all z E X and H ( a , t ) E A whenever t

>

53

2.3. COFIBRATIONS

Proof - =+-: By Theorem 2.3.1 there is a retraction X ; define: H : x I + , H(z,t)= T.y(z,t)

x

T

of X x I onto

x

and

$(.)

= SUP I t t€ I

- +,t)

I

*

We begin by proving that q5 is well-defined and continuous. Take the map e x I. I , ( z , t )-1 t - T I ( z , t ) I

:x

and note that, for every z E X , O({z} x I) is compact and thus, bounded; this implies that q5 is well-defined. Now fix a point z, E X ; suppose that 0 < q5(zo)< 1 and take E > 0 so that (q5(zO)-~,q5(z0)+e). From [24, Lemma 3.5.81 (the “tube lemma”) we conclude that there exists an open set U1 C X such that

hence, for every z E Ul,$(z) 5 q5(zo)t E . Now take to E I so that O(z,, t o ) E (q5(zo) - E , q5(zo) E ) ; again, by the tube lemma, there exists an open set U2 C X such that

+

and so, for every z E U2, +(z) 2 q5(zo)- E . It follows that qJ(U1 n Uz) C ( $ ( a o )- ~,q5(z,) E ) and thus, q5 is continuous at 2,. If q5(zo) = 0 or q$(zo)= 1 we make similar considerations. We now prove that A = q5-’(0). If z E A

+

and so, T ~ ( zt ), = t for every t E I , implying that $(z) = 0. Conversely, if q5(z) = 0, for every t > 0, ~ ( z , t E) A x I ; since A x I is closed, r(a,O)= ( a,O ) also belongs to A x I and so, z E A . Finally, suppose that t > &(z) for a certain 2 E X . Then r ~ ( z , t>) 0 and so, ~ ( z , tE) A x I implying that H ( z , t ) E A . Note that since A is closed ~.y(z,q5(z))= H ( z , $ ( z ) )E A .

54

CHAPTER 2. FIBRATIONS AND COFIBRATIONS

H and by the conditions:

+=: Suppose that

I

+X

4 are given; define the retraction

T

:X x

The following result is an application of this theorem:

Theorem 2.3.4 Let ( A , i , X ) and ( B , j , Y ) be cofibrations. Then

where

L

is the inclusion map, is a cofibrution.

-

Proof By the characterization of cofibrations given in Theorem 2.3.3 there exist maps 4 : X -+ I , $J : Y -+ I and homotopies H : X x I -, X , K : Y x I + Y satisfying the conditions explained in that theorem. Now define

and

L : ( X x Y )x I + X

XY

L((z,y),t) = ( H ( z : , m i n ( t , ~ ( y ) ) ) , K ( y , m i n ( t , + ( z ) ) ) ) for every (z,y) E X x Y and t E I . It is easy to verify that r-'(O)= X x B U A x Y ;it is also routine to verifythat L((z,y),O) = (z,y)and that,forevery(z,y) E X x B U A x Y and t E I , L((z,y),t) = (z,y). To conclude the proof, observe that L((z,y),t)) E A x Y U X x B whenever t

> y(z,y).

0

Corollary 2.3.5 If ( A , ( X ) is a cofibration, the arrow

( Ax I U X x d I , ~ , x I ) is a cofibration.

2.3. COFIBRATIONS

55

Proof - Just after the proof of Theorem 2.3.1 we proved that (8I,j, I)is a cofibration; now use Theorem 2.3.4. 0 Lemma 2.3.6 Let ( A , i , X ) be a cofibration and let ( A , f , B ) be a n arbitrary arrow. T h e n ( B ,i, B U j X ) is a cofibration. Furthermore, i f A is a strong deformation retraction of X , then B is a strong deformation retraction of B Uf X .

Proof - In order to simplify the notation, write Y for B U j X . Let T ' : Xx I

-+

x ' x {O}UA x I

be the retraction obtained from the fact that ( A , i , X )is a cofibration (see Theorem 2.3.1). If j :B x

I -+

Y x (0) U B x I

denotes the canonical inclusion, j(f x I[) = (fx l{o) U f x ll)r'(i x 11). Lemma 2.1.1 shows that Y x I is a pushout space for the arrows f x 11 and i x 11 and so, there exists a unique map T

:Y

X

I

-$

Y

X

(0) u B

X

I

such that ~ ( f 11) x = j and r ( f x 11) = (f x l{o) U f x 11)~'; the map r is a retraction and the first result follows from Theorem 2.3.1. All this is illustrated in the commutative diagram of Figure 2.3.2. For the second part, let H : X x I + X be a strong deformation retraction of X onto A. We know, from Lemma 2.1.1, that Y x I is a pushout space for the arrows given by f x 11 and i x 11;now take the maps p r $ x 11) : B x 1 4 Y (here p ~ isl the projection on the first factor) and

JH

x x I +Y

and notice that they induce a unique map K : Y x 1 -+ Y such that K ( i x 11) = p~l(ix 1 1 ) and K ( f x 11) = fH. This is a strong deformation retraction of Y onto B. 0 Observe that by default a space A and the empty set 0 viewed as a closed subset of A define a cofibration (8,i0, A ) ; now from the previous Lemma we conclude that for any two spaces A and B , the arrows defined by inclusions ( A ,2.4, A U B ) and ( B ,is,A U B ) are cofibrations.


56

AxI

i x 11

WXXI

Y x (0) u B x I FIGURE 2.3.2

Corollary 2.3.7 If ( D , i , X ) is a cofibration, A c D and ( A ,f , B ) is a n arrow, then ( B U j D,i,B UfX ) is a cofibration. Proof - The law of vertical compositions gives rise to the commutative diagram of Figure 2.3.3; the first part of Lemma 2.3.6 applied to the bottom square of that diagram proves the statement. 0 We are going to give another example of cofibration which will be very useful later on; however, we first define the cone of a based space: given (Y,yo) E Top,, the cone of (Y,yo) is the space CE’ = ( I x Y ) / ( Ix {yo}

u (0)

x

Y);

-

we use the notation [ t , ~to] indicate the points of CY. The space Y is embedded into CY as a closed subspace by the map i : Y CY taking any y E Y into [ l , y ] E CY. Notice that the cone CY is intimately related to the suspension of Y : in fact, CY = C Y / Y .

2.3. COFIBRATIONS

57

-(B

x

Uf

D)UfX z B u j x

FIGURE 2.3.3

Lemma 2.3.8 For every (Y,yo) E Top,, the arrow (Y,i,CY) is a cofibration.

-

Proof- Let g : C Y x(0) X and H : Y X I X be two maps su c htha tg( [l, y ] , O ) = H ( y , O ) , f o r ev er y y E Y . D e f i n e G : C Y x I + X by:

for every [ t , ~E] CY and every s E I. This proves the desired result. 0

The cofibration (Y, i, CY)gives rise to another important cofibrazo); we are tion, this time related to a based map f : (Y,yo) + (2, referring to the mapping cone cofibration of the based map f: (Z,:, 2 u,jC Y ) . Notice that (2,i, 2 U j C Y ) is a cofibration because of Lemmas 2.3.6 and 2.3.8. In the sequel we shall refer to the space 2 U j C Y as the mapping cone of the map f and will denote it simply by C j . Theorem 2.2.7 has a counterpart for cofibrations:

58


Theorem 2.3.9 Every map f : A + B can be factored as a cofibration followed by a homotopy equivalence. Proof - Because ( d I , i , I ) is a cofibration, it follows that ( A x a I , 1 ,x~i , A XI ) is a cofibration by Corollary 2.3.2; call 1~x i = j . Now apply Corollary 2.3.7 with D = A x 81, X = A x I and with A viewed as a closed subset of A x dI by the identification A = A x (0); we obtain that ( BU j ( A x 81),j,B U j ( A x I ) ) is a cofibration. Apply the law of horizontal compositions to the inclusion 0 c A and the arrows (@,;@,A and ) (A,f, B ) to obtain that

BUj ( A U A ) = B U j ( A x a I ) 2 B U A and therefore, ( B U A,?, B Uf ( A x I ) ) is a cofibration. The situation described is reflected in Figure 2.3.4.

AxI

+

B Uf ( A x I )

FIGURE 2.3.4

Now observe that the inclusion (A,i,i,B U A ) is a cofibration and thus, writing i ( f ) = j i , . ~we , conclude that ( A , i ( f ) B , U j ( A x I))is a

2.3. COFIBRATIONS

59

cofibration (see Exercise 2.3.6). Notice that geometrically i(f) is just the map taking A into A x (1) c B U j ( A x I ) ; in other words, if i l : A +A

xI

,u

H

(a,l)

and f : A x 1 -+ B Uf ( A x I ) is a characteristic map for the adjunction space B U j ( A x I ) , then i(f) = fil. Consider once more the pushout diagram of f and the inclusion of A onto A x (0) c A x I which will be denoted by io from now on; we shall also write z0 for the map j i ~ Take . the identity map 1~ : B --+ B and the map f : A x I -+ B given by j ( a , t ) = f(a); these give rise to a unique map T j : B U j ( A x I ) +B which satisfies the properties: r j q = 1 B and r , i ( f ) = f. To complete the proof we have only to show that ~ r is fhomotopic to the identity self-map of B U j ( A x I ) . Define the homotopy

H :B U j ( A x I ) x I by

--+

B U j (Ax I )

H ( [ a , t ] , a )= [ a , ( l - ~ ) t,]( a $ ) E A x I

and

H ( [ b ] , s )= [b] , b E B

.

Note that at level 0 this homotopy is just the appropriate identity map , = [u,O] and H ( [ b ] 1) , = [ b ] . On the and at level 1, we have: H ( [ a , t ] 1) other hand, G q ( [ a , t ] )= zof(a) = f((.,O) = [a, 01 and G.j([bl) = G(b) = [bl and so, the restriction of H to B Uf ( A x I ) x (1) coincides with G T ~ . 0

The cofibration ( A ,i(f),B Uf ( A x I))is the mapping cylinder cofif ; the space B U j ( A x I ) , usually denoted simply by M ( f ) ,is the mapping cylinder of f . As we did for the mapping track fibration, we describe the mapping cylinder cofibration with the aid of a diagram (see Figure 2.3.5). bration associated to

60

CHAPTER 2. FIBRATIONS AND COFIBRATIONS A

AxI

f

f FIGURE 2.3.5

EXERCISES 2.3.1 Prove that the arrow ( A , i , X ) is a cofibration if, given any f : X + 2 and any G : A + 2' such that fi = E,G, there is a map F : X --+ 2' such that the diagram of Figure 2.3.6 commutes.

i

X

f FIGURE 2.3.6

2.3.2 Prove that ( e o , i ,S")and ( S f ' , i , B " + l (defined ) by inclusions) are cofibrations. 2.3.3 Prove that for every (Y,yo) E Top,, the cone CY is contractible (it contracts to its vertex).

2.3. COFIBRATIONS

61

2.3.4 Let I" be the hypercube obtained by multiplying I = [0,1] with itself n times. Let 8I" be the boundary of I" and let J"-l be the subspace defined by: J"-I

= dI"

x I u In-1x (0)

.

Finally, consider the inclusion maps

i : 81n -+ I" and j : J"-l Prove that the arrows ( a I " , i ,I n ) and tions.

+-c

dl"

( P - l ,

.

j , d I n ) are cofibra-

2.3.5 Let ( A , i , X ) and ( B ,j , Y ) be cofibrations. Prove that the arrow

( A X Y U X x B,i x

ly Ulx

x j,X x Y )

is a cofibration.

2.3.6 Prove that if ( A , i , Y ) and (Y,j,Y') are cofibrations, then the arrow ( A ,j i , Y') is also a cofibration. 2.3.7 Let ( A , i , X )be a cofibration. Prove that for any compact space Y , ( M ( Y , A ) , i # ,M ( Y , X ) )is a cofibration.

2.3.8

* Let

A , B and C be given spaces with A c B closed and B C C closed; take the arrows ( A , i ,B ) and ( B ,j , C) with i and j denoting the inclusion maps. Prove that if ( A ,ji,C) and ( B ,j, C) are cofibrations, so is ( A , i , B ) . (See [32].)

2.3.9 Let ( A , i , X ) be a cofibration. Let A A and A X be the diagonals of A x A and X x X , respectively. Prove that if ( A X , i . y , X x X ) is a cofibration, then (AA,i.A,A x A ) is also a cofibration. (Note: Our definition of cofibration forces A X to be closed in X x X and so, X is Hausdorff. In the literature, spaces X such that ( A X ,is,X x X ) are cofibrations are called (Hausdorff) LEG spaces.

2.3.10 Prove that, for every n 2 1, the spheres S" and the balls B" are LEC spaces.


62

2.3.11 Let X = {l/n 1 n E N \ (0)) U (0) and A = (0). Prove that the arrow given by the inclusion A c X is not a cofibration. 2.3.12 Consider the cofibration (Sn,i,Bn+')and let f : S" + B be a given map. Prove that

( B Uf B"+1)\ B

s B"+l\

S" ;

furthermore, prove that B U j Bntl is a normal space if B is normal. 2.3.13 Prove that a based map f : ( X , z , ) + (Y,yo)is homotopic to the constant map cy, if, and only if, f factors through the cone

cx.

2.3.14

* We defined ( A , i ,X ) E

Top' to be a cofibration whenever A is a closed subset of X , i is the inclusion map and X is a retract of X x I . Let us now drop the assumption that A is closed and define ( A , i , X ) to be a (non-closed) cofibration if X is a retract of X x I . Prove that the following generalization of Theorem 2.3.3 holds true: ( A , i , X ) is a (non-closed) cofibration iff there exist a map 4 : X + I such that A c 4-'(0) and a homotopy H : X x I + X such that

H(2,O) = 2, 2 E x H ( a , t ) = a, a E A , t E I , and such that H ( z , t ) E A whenever t > $ ( a ) . Moreover, if A is a strong deformation retract of X , we may assume that 4 is everywhere less than 1. (See [31].) 2.3.15

* Let A and B be disjoint closed subsets of X such that ( A , i . A , X )

and ( B ,ig,X) are cofibrations; prove that if ( A n B , i d n ~X ,) is a cofibration, then so is ( A U B , z , X ) . (See [20].)

2.4. APPLICATIONS OF T H E MAPPING CYLINDER

2.4

63

Applications of the mapping cylinder

In this section we shall discuss some of the applications of the mapping cylinder.

Theorem 2.4.1 Let ( A ,i, X ) be a cofibration. Then i is a homotopy equivalence iff A is a strong deformation retract of X . Proof - j : Let j : X t A be a homotopy inverse of i and let J : la4 ji and K : 1s ij be the homotopies. Because ( A , i , X ) is a cofibration, there exists a homotopy L : X x I -+ A such that L(-,0) = j and L(i x 11) = J . This shows, in particular, that j is homotopic to a retraction of X onto A; thus, assume from the beginning that j is a retraction. N

N

Define the homotopy

M : ( A x I U X x 81)x I

--+

X

by the following conditions:

M(z,O,t) = 2 M(a, 1,t) = K ( j ( z ) ,1 - t ) M ( a , s , t ) = K(a,(I - t ) s ) M(z,s,O) = K(a,s) for all z E X , a E A and s,t E I . Now we use the fact that

( A XI U X x ~ I , L , xX I ) is a cofibration (see Corollary 2.3.5) to extend M to a homotopy

N : (Xx I ) x I

---f

x

whose restrictions to ( A x I U X x 01)x I and ( X x I ) x (0) are, respectively, M and K . The homotopy

H :X x I

---t

X

, H(z,s) = N(z,s,~)

is a strong deformation retraction of X onto A . +: Conversely, suppose that H : X x I -+ X is a strong deformation retraction of X onto A . The map j = H ( -, 1) : X +A is a homotopy inverse of i.


64

Corollary 2.4.2 A map f : A + B is a homotopy equivalence i f l A is a strong deformation retract of M ( f). Proof - We know from the construction of the mapping cylinder that the map f factors out as f = ~ f i ( fwhere ) ~j is a homotopy equivalence and ( A , i ( f ) M , ( f ) )is a cofibration. Using these statements and the previous theorem, we see that f is a homotopy equivalence iff i(f)is a homotopy equivalence iff A is a strong deformation retract of

M(fh

0

Theorem 2.4.3 Let ( A , i , X ) be a cofibration and let f : A homotopy equivalence. Then any characteristic map

f :X

-+

--+

B be a

Y =BUf X

is also a homotopy equivalence.

Proof - In view of the previous corollary, it is enough to prove that X is a strong deformation retract of M ( f ) . , ( f ) ) and ( A , i , X )are cofibrations, Because ( A , i ( f ) M

M(f) U(f)x

2

x ui M ( f )

and since A is a strong deformation retraction of M ( f ) , from Lemma 2.3.6 we conclude that X is a strong deformation retract of M ( f )U;(f)X. We now apply the law of vertical compositions to the arrows ( A ,f,B ) , ( A , i o , A x I ) and ( A x I , L , Ax I U X x (1)) to obtain:

M ( f ) U f ( Ax I U X x (1))

BUJ ( A x I U X x {I}).

From the law of horizontal compositions applied to the arrows ( A , i , X ~ ( l ) ()A, , i l , A x I ) , ( A x I , f , M ( f ) ) and the equality fil = i(f) we obtain the homeomorphism

M ( f ) U f ( A x I u x x (1)) 2 M ( f )Ui(f) x. Figures 2.4.1 and 2.4.2 illustrate the law of vertical compositions applied to the relevant arrows; furthermore, Figure 2.4.1 shows - because A x I U X x (1) is a strong deformation retract of X x I - that M ( f )U;(J) X is a strong deformation retract of B UJ (Xx I ) while Figure 2.4.2 shows that M ( f ) 2 B UJ (X x I ) . Altogether, the previous results prove that X is a strong deformation retract of M ( f ) . 0

2.4. APPLICATIONS OF THE MAPPING

A

+

X X I

f

cmmm

65

*B

+ -- ( M ( f )U; X ) U j ( X x I )

B Uj ( X x I )

FIGURE 2.4.1

A

f

--B

i0

FIGURE 2.4.2

Theorem 2.4.4 (The gluing theorem) Suppose that we are given the cofibrations ( A , i , X ) and (A',i',X'), the maps f : A --+ B and f' : A' 3 B' and the homotopy equivalences hA : A + A', hB : B 4 B' and h x : X + XI such that

Then B U j X and B' U j t XI have the same homotopy type.


66

Proof - Let us denote B Uf X and B' Up X' simply by Y and Y', respectively; now take the map h : Y + Y' determined by the universal

property of the pushout diagram determining Y and which satisfies the properties hi = z'hB and h f = f h - y .

The commutative diagram of Figure 2.4.3 probably expresses more clearly the situation.

i

i'

X FIGURE 2.4.3 Our aim is to prove that h is a homotopy equivalence. For this, we shall consider four different cases.


67

Case I- Suppose that A and A‘ are, respectively, strong deformation retracts of X and X’; then, according t o Lemma 2.3.6, B and B’ are strong deformation retracts of Y and Y’, respectively; the same Lemma also shows that ( B , i , Y ) and (B’,i’,Y’) are cofibrations. Now, use Theorem 2.4.1 to conclude that ;t and ;i/ are homotopy equivalences. The equality h%= i‘hs and the fact that hB is a homotopy equivalence now imply that h is also a homotopy equivalence. Case 11- Suppose that the attaching maps f and f’ are homotopy equivalences (notice that because h,A and h~ are homotopy equivalences, f is a homotopy equivalence iff f’ is a homotopy equivalence). Use Theorem 2.4.3 and the equality h f = 7 h - y to show that h is a homotopy equivalence. Case 111-Suppose that (A’,f’,B’) is a cofibration. We construct the commutative diagram of Figure 2.4.4 in a stepwise fashion as follows: Step 1 - Begin by constructing the trapezoid labelled 1 as a pushout of h,A and i; then, according to Theorem 2.4.3,h.A is a homotopy equivalence. Because h-yi = i’h.4, there exists a unique map g : X“ + X’ such that g6.4 = hay and i’ = gi”; the first of these equalities implies that g is a homotopy equivalence, while the second shows that the trapezoid below trapezoid 1 is commutative. Step 2 - We now construct the rectangle 2 as a pushout o f f ’ and i“. In view of Lemma 2.3.6 the arrow ( X ” ,f’, Y”)is a cofibration; moreover, from the equality = ?f’we obtain the equality T‘f‘ = J’gi“ and thus, because 2 is a pushout, there exists a unique map : Y’’+ Y‘ such that ij%“ = Z’ and ijf = f’g. The law of vertical compositions shows that the rectangle labelled 3 is a pushout. Furthermore, from the fact that (X”,f,Y”) is a cofibration and the hypothesis that g is a homotopy equivalence we conclude that i j is a homotopy equivalence. Step 3 - The commutativity of the figures labeled 1 and 2, equality hBf = f’hA and the universal property of the pushout for i and f give rise to a unique map

pi‘

LB : Y

Y”

such that h ~ = f f h , ~and h ~ =i ;t’’h~.This takes care of the commutative trapezoid labelled 4. Because of the law of horizontal compositions, the two figures labelled 1 and 2 together give rise to a pushout diagram which, in view of

68


f

A

\

i

1

A‘

i‘I

t

f ’ /

*B

B’

2

4

ill

I

7

FIGURE 2.4.4

the equality k ~ = f fh,4, can be split into the two commutative squares of Figure 2.4.5. The left hand square of Figure 2.4.5 is a pushout; then the law of horizontal compositions shows that the right hand side square is also a pushout; in other words, trapezoid 4 of Figure 2.4.4 is a pushout! This and the fact that h s is a homotopy equivalence, prove that k~ is a homotopy equivalence. Finally, because g % H f = f’hs and i j h B i = Z’hB, the universal property of pushouts applied to the pushout of i and f shows that h = i j h ~ . Since both i j and h~ are homotopy equivalences, so is h.

2.4. APPLICATIONS OF THE MAPPING CYLINDER

.

I

T

-

i

i”

a

X

-Y

f

69

I;.B

- Y”

FIGURE 2.4.5

Case IV : General Case - Observe that A and A’ are strong deformation retracts of A x I and A’ x I , respectively; then, applying Case I to the cofibrations ( A ,iu,A x I), (A’,ih, A’ x I ) , the maps f, f’, and the homotopy equivalences h A , hg, h.4 x 11,we obtain a homotopy equivalence hA1

: M(f)

-----)

M(f‘)

such that hnIf0

where

= a:hg

, h,-,If^= f’(h.4

f^ : A x I

+ M(f)

f’ : A‘ x I

+M ( f ‘ )

(2.4.1)

x 11)

and are characteristic maps. From the equalities 2.4.1 given before, and the definitions of the maps T $ , ~ yi (, f ) and i ( f ’ )we obtain the equalities hBTf

Tjihm

, h M i ( f ) = i(f’)h,i .

Now we apply Case 111 to the cofibrations ( A , i , X ) , ( A ’ , i ’ , X ’ ) , ( A , i ( f )M , ( f ) ) , (A‘,i(f’),M ( f’)) and the homotopy equivalences h.4, hhr and h-y to obtain a homotopy equivalence

i;. : M(f) U(f)x

-

Wf’) U*(f’)x

*

To simplify the notation, let us write

M ( f ) U,(f) X = 2 and M ( f ’ ) U,(p) X‘ = 2’

.

CHAPTER 2. FIBRATIONS A N D COFIBRATIONS

70

Apply Case I1 to the cofibrations ( M ( f ) , i , Z ) ,(M(f’),z’,Z’),the ho-motopy equivalences T J , T J , and the homotopy equivalences hnc, hg, h to obtain a homotopy equivalence

k : B U, Z

+ B‘ U,

2‘

,

Finally, the law of horizontal compositions and the equalities TJi(f)

, T J , i ( f ’ ) = f’

=f

prove that

B U, 2 2 B U j X and B’ U,

2’ E B‘ Uf‘X’

.

For an alternative proof see [ l ] .

EXERCISES 2.4.1 Let ( A , i , X ) E Top‘

be a cofibration and let f 0 , f I : A + B be homotopic maps. Prove that there is a homotopy equivalence

x

B ujo + B Uf,

x

rel. B.

2.4.2 Let ( A , i , X ) be a cofibration. Prove that X / A and the mapping cone space Ci have the same homotopy type. 2.4.3 Let ( Y , y o ) E Top, be such that ( { y o } , i , Y )is a cofibration. Let C Y

= (I

x Y ) / ( { O }x Y )

and

C Y = ( I x Y ) / ( Ix {yo} u {O} x Y ) be, respectively, the unreduced cone of Y and the cone of Y . Prove that CY C Y . N

2.4.4 Let (Y,yo) E Top, be such that ({yo}, i, Y ) is a cofibration. Let

UY = ( I x Y ) / ( d I x Y ) = ( c Y ) / ( { l } x Y ) be the unreduced suspension of Y . Prove that UY

N

EY.


71

2.4.5 Let (X,ao),(Y,yo) E Top, be given with the condition that i ( { y o } , i , Y ) and ( { z o ) , j , X )are cofibrations. Prove that

2.4.6 Let ( A , i , X )be a cofibration. Prove that if A is contractible, then the identification map q : X + X / A is a homotopy equivalence. (see Exercise 2.1.1); de2.4.7 Recall the notion of homotopy in Top‘ Let ( A , i , X ) and ( B , j , Y ) fine homotopy equivalence in Top’. be given cofibrations and let ( 9 ,h ) : i + j be a homotopy equivalence. Prove that if f : B + C is an arbitrary map, then C U j Y and C Ufs X have the same homotopy type.

2.4.8

*

Prove the following dual to gluing theorem: Let (Y,g,B) and (Y’,g‘, B’) be fibrations, let f : A + B and f’ : A‘ + B’ be maps and let hy : Y + Y’, h B : B + B’ and h.4 : A 4 A‘ be homotopy equivalences such that g‘hs = hgg and f‘h.4 = hBf

.

Then A, nj Y and ALf riff Y‘ have the same homotopy type. (See

2.4.9

PI.) * Prove the Dyer-Eilenberg adjunction theorem: Let ( A , i , X )be a cofibration and f : A --+ B be a map. If B and X are LEC spaces, then so is B U j X . (See [15, Corollary A.4.141.)

2.4.10

* Prove the following dual to the Dyer-Eilenberg adjunction the-

orem. Let ( A ,f,B ) ,( E , p ,B ) € Top’ with ( E , p ,B ) a fibration. Prove that if A, B and E are LEC spaces, so is the pullback space

A, Uf E .


Chapter 3

Exact Homotopy Sequences 3.1 Let

Exact sequence of a map: covariant case

f : ( Y , y o ) + (Z,z,) be a given map. By Lemma 2.2.5 the arrow

(PZ,e l , 2 ) is

a fibration; its fibre over z, is the loop space SlZ (loops based at z o ) . Now consider the pullback space L , = Y n PZ of el and f ; the arrow ( L f , c , Y )is also a fibration (see Lemma 2.2.2) and its fibre over yo is 02. In this way we obtain a gequence of topological spaces and maps -

RZ

f -LL”f-% Y + z

where j denotes the inclusion of 02 into L f (see Figure 3.1.1 ).

FIGURE 3.1.1

74

CHAPTER 3. E X A C T HOMOTOPY SEQUENCES

Actually, by loop iteration of these maps we obtain a sequence of spaces and maps which extends indefinitely to the left:

nn-I

-

-

Q"Z -2. . . i L -% Y f '

f

z.

This is the long sequence of spaces and maps associated to f.

Lemma 3.1.1 For every integer n

2 1, the

arrow (fInL,,fIn,,fInY)

is a fibration.

Proof - It is enough to prove the lemma for the case n = 1. We first observe that we can identify the space QPZ to PQZ simply by reversing the order of the parameters. Now take the arrows (Pa2= Q P Z , E= Qq,02) and (QY,Qf,QZ ),and construct a pullback diagram for them. We know that, up to homeomorphism, Lnf is a pullback space for these arrows; we must prove that so is Q L f . To this end, take the pullback diagram giving rise to L j and loop all spaces in sight to obtain a commutative diagram, having in the upper left corner the space Q L f ; then we use routine arguments to prove that such a diagram satisfies the universal property of pullbacks. 0 Let Set, be the category of based sets and base-preserving functions. Given arbitrarily ( A , a , ) , ( B , b , ) E Set, and f : ( A , a , ) -+ (B,b,), define the based sets im(f) = { b E B I (3a E A ) f ( a )= b} and ker(f) = ( a E A I f ( a ) = b,) (with base points b, and

a,,

( A ,a,)

respectively). We say that a sequence

( 4bo) 9-, (C,co)

is ezact (at ( B , b , ) )if im(f) = ker(g). In particular, our based spaces and functions could be groups and homomorphisms, respectively; in that case, the subsets im(f) and ker(g) are normal subgroups and evidently, we can continue talking about exact sequences, this time, of groups.

3.1. COVARIANT CASE

75

Theorem 3.1.2 For every ( X ,z,), (Y,yo), (2,2,) E Top, and every map f : (Y,yo) + (2,z,), the sequence of based sets and groups *

- [ X ,R"+12], ""j:[ X ,02"Lf]*R".l,

--f

"'

=

-%[ X , L j ] *

-

f* [X, O"Y]*R" +' - '

[ X , Y ] *A [ X , Z ] *

induced by the long sequence of spaces and maps associated to f, is exact.

Proof - Exactness at [ X , Y ] , : The contractibility of PZ implies that the composite map fq is homotopic to the constant map taking L f into z, E 2; thus, im(T;,) c ker(f,). Now let [g]E [ X ,Y ] *be such that f*( [ g ] )is the homotopy class of the constant function cz, : X + 2. Let H : X x I + 2 be a homotopy such that H ( - , 0) = ct0 and H ( -, 1) = fg; by the exponantial law, the homotopy H defines a map

H:X-PZ such that = fg and thus, by the universal property, there exists a map Ic : X L f and therefore, ker(f,) c im(q,). Exactness at [ X , L j ] , : Begin by observing that, on the one hand im(j,) C ker(5,) because q j takes RZ into yo. On the other hand, if [g] E [ X , L f ] ,is taken into the trivial class by q*,there is a homotopy ---f

H :X x I

+Y

, H ( x , O ) = Fig(.)

, H(z,l) = yo

and from it, because ( L f , q , Y )is a fibration, we can construct a homotopy G : X x I + L j of g such that FG = H . This last equality shows, in particular, that G(-, 1) is actually a map from X into RZ; furthermore, j * ( [G(-, l)])= [g]. Exactness at [ X ,RZ],: For every loop a E RY,j R f ( a ) = (yo,f a ) ; define the path fat in PZ by the formula fcrt(s) = f a ( s t ) and the homotopy H : OY x I + L j , H ( a , t ) = ( y o , f a t ) to see that j,Of* is the trivial function and therefore, im(Of,) ker( j * ).

c

76

CHAPTER 3. EXACT HOMOTOPY SEQUENCES

In order to prove the opposite inclusion, take [g] E [ X , n Z ] ,such that j l ( [ g ] is ) the homotopy class of the constant function at the base point (yo,cr,) of L,. Let

H :X x I

+L j

be the homotopy H(z,O) = jg(z) and H ( z , 1) = (yo,czo). Take the function h : X + fly, h ( z ) = ~ H ( Z , - ) and the homotopy

K : X x I + 02 , K ( a , t ) ( s )= f H ( z , t s ) ( t + s - t S ) for every 2 E X and t , s E I . Easy computations show that K is a homotopy between 02f(h)and g thus, proving that [g]= flf,([h]). Similar arguments apply to prove exactness at all other points; to check exactness at [X,flnL,]. we use the fact that ( P L , , WZi, W Y ) is a fibration (see Lemma 3.1.1). Note that in view of Theorem 1.2.5 the exact sequence of the previous theorem is, after a certain point, an exact sequence of groups and group homomorphisms. This exact sequence assumes a particularly interesting aspect whenever the based map f : Y + 2 is a fibration; in this case, we revert to our older notation ( E , p , B ) and right away prove the following result:

Lemma 3.1.3 Let ( E ,p , B ) be a fibration with fibre F over b, E B; then the spaces L, and F have the same homotopy type. Proof - Consider the map h : F -+ L,

, h ( e ) = (e,cb,)

where cb, is the path of B which is constantly equal to b, (see Figure 3.1.2). We are going to prove that h is a homotopy equivalence. To this end, take the homotopy

H : L, x I

+B

, H ( ( e , a ) , t ) = a(1 - t )

and notice that, since H ( ( e , a ) , O )= p c ( e , a ) and ( E , p , B ) is a fibration, there exists a homotopy G : L, x I 4 E whose restriction to

77

3.1. COVARIANT CASE

F

E

P

-B

FIGURE 3.1.2

L, x (0) is just 6and pG = H . Since pG((e,a ) ,1) = H ( ( e ,a ) ,1) = b,, the restriction g = G(-, 1) maps L p into the fibre F . The objective is now to prove that the compositions gh and hg are homotopic to the appropriate identity maps: indeed, the map

K :F x I

--+

F

, K ( e , t ) = G((e,cb,),t)

is a homotopy from 1~ to gh and the map

L : L, x I

+

L,

, L ( ( e , a ) , t )= ((G(e,a),l - t ) , a , )

where at(s) = & ( s t ) ,is a homotopy between hg and 1 ~0 ~ . Notice that in the previous lemma, the homotopy equivalence h : F + L, is such that K h = i , the inclusion of F into E ; this fact and Corollary 1.2.13 allow us to give the following important reformulation of Theorem 3.1.2:

Theorem 3.1.4 Let ( E , p , B ) be a fibration with fibre F over b, E B ; let e, E F be viewed as base point for both F and E . Then, for every ( X ,a,) E Top*, the following sequence of based sets and groups is exact:

C H A P T E R 3. E X A C T HOMOTOPY SEQUENCES

78 a

-.3[ X ,F ] , 5[ X ,El, 2 [ X ,B ] , . 0

We now specialize the based space (X,s,) to be the based unit 0sphere (S',e,). Recall that ru(Y,y,) is the set of all path-components of Y . The previous theorem implies the following: Theorem 3.1.5 Let ( E , p ,B ) be a fibration with fibre F = p-'(b,); let e, E F be the base point of both F and E . Then the following sequence of groups and based sets is exact:

+ The exact sequence of Theorem 3.1.5 is the exact sequence of the fibration ( E , p ,B ) , with fibre F .

EXERCISES 3.1.1 Prove the exactness at each point of the sequence of groups and based sets described in Theorem 3.1.2. 3.1.2 Let (X,2,) and (Y,yo) be given; use the fibration prl : X x Y + X given by the projection on the first factor and Theorem 3.1.5 to prove that

for every n

2 1 (cfr. Exercise 1.3.1).

-

3 - 1 3 Let P : ( E ,e), + ( B ,b,) be a covering map; let fo, f l : (X, a,) -+ ( E ,eo) be maps such that p f o p f i . Prove that fu ,,, fl.

3.2. CONTRAVAMANT CASE

3.2

79

Exact sequence of a map: contravariant case

Let f : (Y,yo) + (Z,z,) be a given base point preserving map and let (Z,Z,Cj) be its mapping cone cofibration (see Section 2.3). Now take the constant map c,, : 2 (zo} and use the law of horizontal compositions and the relationship between CY and CY to conclude that

-

(20)

ucz0cf

{ 2,)

u c z 0J

ZY

CY

We then construct an infinite sequence of spaces and maps by successive iterations: (see Figure 3.2.1 ).

FIGURE 3.2.1 For a fixed (X, a,) E Top,, the based map preserving function

-

f* : [ Z , X ] * and a group homomorphism

(see Theorem 1.2.12).

f induces

[Y,X]*

a base point

C H A P T E R 3. E X A C T H O M O T O P Y SEQUENCES

80

Theorem 3.2.1 For every ( X , Z ~ ) , ( Y , ~ ~ ) , ( ZE, ZTop, , ) and every map f : (Y,y,) --+ (2, zU), the sequence of based sets and groups

-

. . . [C"+'Y,X], C-3n c * [C"Cf,X]*5 [ C " Z , X ] *Y f ' -

-

-

. . . Cn,: [C,,X]* 5[ Z , X ] * f '

- *

[Y,X]*

is exact.

Proof - The homomorphisms C n q * , En%*and C"f* are induced from the maps YC,,, En%and Enf obtained by successive iterations of the suspension of c,,, i and f, respectively. We shall prove only two parts of the result, leaving the remaining parts as exercises. I) im(C) C ker(f*): It is enough to prove that Zf is homotopic to the constant map to the base point [zO]of CJ; this is done via the homotopy H : Y x I 4 Cj , H ( y , t ) = J ( [ t , y ] ) . 2) ker(%*)C im( 0; 3) C X E X P ( 4 = C I E S ( P ) P ( 4 = 1. Given that p , q E I K I we define the distance X E X

and easily verify that this distance is a metric in I K regard I K I as a metric space. Suppose that p € 1 K 1 has support s ( p ) = {zo, 2 1 ,

* *

I.

Thus, we may

,zn}

--

and denote by 20, - ,xn the functions of V ( K )which satisfy the conditions: z;(z;) = 1 and zc;(z) = 0 for every z E X \ {z;},i = 0, * * ,n. Then we can write the function p as

-

note that the coefficients ai 2 0 and where a* = p ( z ; ) , i = O,...,n; that C I E X a;= 1. If K is a finite abstract simplicial complex, the geometric realization I K I is called a polyhedron. Let K = (X,Y) be a finite abstract 2,) be an element of T. Take simplicial complex and let u = {x", the abstract simplicial complex K' = (u,p ( u ) \ { O } ) , where @(a)stands for the set of all subsets of 6. Now let us identify 20, ,x, to points of an euclidean space R" such that the vectors z1 - zo,... ,z, - x o are linearly independent; then the geometric realization

-

a ,

--

8

is a closed convex subset of R': to prove its convexity, let

be two arbitrary points of 1 6 1; the claim now follows trivially in view of the fact that the points of the line segment from p to q can be written

4.1. SIMPLICIAL COMPLEXES

87

in the form n

T

=tp

+ ( 1 - t ) q = C(ta;+ ( 1 - t)P;)z; i=U

with t E I and Cr=o(tai+(l-t)&) = 1. Furthermore, because d ( p , q ) 5 for every p , q € 1 u 1, I u I is bounded and thus, is compact. We say that I u 1 is a simplex of K 1. Notice that a polyhedron I K I is a compact subspace of a convenient euclidean space; furthermore, it satisfies the following properties: 1) if 1 cr 1 is a simplex of I K 1 and T c u,then I r 1 is a simplex of

a,

I

I K 1;

I I

2) if I u I and I T I are simplexes of K then, either 1 u I n I T I= 8 or I u 1 n I r 1 is the geometric realization of a common face; 3) a set F CI K 1 is closed in the polyhedron 1 K I if and only if, for every simplex I u IC I K 1, F n I u I is closed in I u I. The first two properties follow from the definitions; as for the third, any I u I is closed in 1 K 1, as a compact subset of a Hausdorff space; hence, if F n I u I is closed in I u 1 it is also closed in I K 1 and thus, F = ulvlFn u I is closed, since I K I is a finite set of simplexes. Property 3) shows that I K I has the topology determined by the family of its simplexes I u I .l Let K = (X,T) and L = ( Y , O ) be abstract simplicial complexes; a simplzcialfunction f : K + L is a function f : X + Y which takes the simplexes of K into simplexes of L . A simplicial function induces a function

I

Lemma 4.1.1 Let K = ( X ,T) and L = (Y,0 ) be two abstract simplicial complexes and let f : K + L be a simplicial function. T h e n the induced geometric function 1 f I is continuous.

Proof - We are going to prove that, for every given p € 1 K 1, there is a constant c ( p ) # 0 which depends on p and such that, for every 'In general, a space X has the topology determined by a family of subsets say, C X is closed in X iff U n U, is closed in U, for all A. This topology is sometimes also called weak topology.

{U, c X I X E A} if U

CHAPTER 4. SIMPLICIAL COMPLEXES

88

Assume that

and that ~ ( z i= ) ai

9

i = O , - - - , nq, ( y j ) = p j , j = O , - * . , m .

We have two cases to consider. Case I ) : s ( p ) n s ( q ) = 0 - In this situation,

i=O

j=O

because a; = 1, the minimum of the function Cy='=o a: will be achieved only when a,= l / ( n l), for all i = O , . . . ,n.It follows that

+

-

d ( p , q ) L 1/m Because d ( Jf

I ( p ) , { f 1 (4))_
= 42(n 1). Case 2 s ( p ) n s ( q ) # 0 - Relabel the indices of the elements of s ( p ) and s ( q ) so that the common elements are:

+ +

Notice that s ( p ) U s ( q ) has precisely m r 1 elements. Consider the elements Xi7 O 0. Let s be the largest index for which y3 < 0; take a

(notice that - A = CE:;, y; sequences of real numbers:

> 0)

< 0 and

and construct the following two finite

We now observe that

belong to I K

1 since

and s(p’) c s ( p ) , s(q’)

now use the equalities

c s(q).

But s ( p ’ ) n s ( q ‘ ) = 0 and so, by Case I),

90


and

1

4 f I (P‘),I f I (a‘)) = T-d ( l f I (P),I f I ( a ) ) plus the fact that

-4

5

-4

to conclude that

+

and therefore, also in this case, c ( p ) = d 2 ( n 1). It is now easy to see that I f I is continuous at p : for every E > 0, take S = E / c ( ~ ) Since . p is taken arbitrarily, it follows that I f 1 is continuous. 0 The previous lemma can be formulated in a slightly more general form; in fact, we define a function

to be tinear; clearly, if f : K + L is a simplicial function, the map 1 f I is linear. We should notice at this point that linearity was indeed the key factor in the proof of Lemma 4.1.1; hence, we can state the following result:

Theorem 4.1.2 Every linear function F

:I

K

1-1

L I is continuous.

0

Next, we wish to discuss the idea of burycentric subdivision of an abstract simplicial complex; looming in the background is a geometric argument that actually comes from considering the convex hull of three non-colinear points in a plane (a triangle): if ABC is a triangle and D is its barycentre, by connecting D to the vertices A,B and C we obtain three triangles which, when considered together, produce the “same” space as A B C . Let K = ( X , T)be an abstract simplicial complex; the first barycentric subdivision of K is an abstract simplicial complex

K(1) = (X(]),J(1)) defined as follows:


91

x(')

1) =T ; 2) T(')is the collection of all non-empty, finite subsets of that (uao,- . ,a&} E (Tio c * * c bin

F) *

-

-

T such

.

The T~~ barycentric subdivision of K - denoted by K(') - is obtained by it eration.

Theorem 4.1.3 Let K = (X,T) be a n y finite abstract simplicia1 complex and let T be a n arbitrarily given positive integer. Then the polyhedra I K I and I K(') I are homeomorphic. Proof - It is enough to prove the theorem for T = 1. We begin with some notation. For every

I 1,

we denote by b(u) the burycentre of the geometric simplex u to say

that is

Let F be the function which takes any vertex a of K(') into b(a) (note that the vertices of K(') are the simplexes of K ) ; now extend F by linearity to a linear function (denoted by the same letter) n

n

This function is continuous by Theorem 4.1.2; we are going to prove a;& be an arbitrary point of I K ( ' ) 1; that it is a bijection. Let p = Cy=O notice that r " , - - - uare n simplexes of K such that a" c u 1 c . . . c a n .

+

Assume that dim& = r ; then we may suppose that dima' = r 1 (otherwise, we may take intermediary simplexes of K forming a chain u0

=7-

0

c7-1 C . * * C kr = u 1

92

CHAPTER 4 . SIMPLICIAL COMPLEXES

such that dim7-j+' = d i m d + 1 and to which we attribute a zero coefficient in the summation representing p ) . As a consequence of this, we can assume that go = (2")

, ,

c1= {xu,x1)

............ and therefore,

Now if

n

Q = CPixi

€1

K

1

i=U

coincides with F ( p ) it follows that

............ Pn

= an/(n

+ 1)

and

1 L Po

2 P I 2 ... L P n 2 0 .

On the other hand, given that

the numbers

satisfy the previous equalities. Thus, the numbers a;and Pi mutually determine each other in a unique fashion, proving the bijectivity of F.


93

To complete the proof of the theorem, we just notice that F is a continuous bijection from a compact to a Hausdorff space. 0 At this point we want to prove a lemma which will be useful later on; it gives a characterization of the simplexes of an abstract simplicia1 complex. However, we first observe that we can associate to each vertex z E K an open set O(z) K 1; the construction goes as follows: take

CI

and use the following argument to show that O(z) is open: the function

is continuous because, for every q

€1

K

I,

and then observe that O(z) = S;l(O,ca). Notice that the set {O(z) I z E X } is an open covering of 1 K 1. We are now ready for our lemma.

Lemma 4.1.4 Let K = ( X ,T)be given; for every set u = {zo, 2,) E 'Y z$

--

20,

- - ,2, a

E X , the

a ,

ri O(%) # 0 .

i=O

Proof -

+: If u E T,the barycentre n

+: If p

i

n

E ny=c,O(z:;), then p ( z ; ) > 0 for every i = O,... ,n;this

means that (20, *

*

- ,%} c s ( p ) E T -


94

EXERCISES 4.1.1 Define an appropriate notion of subcomplez of an abstract simplicial complex. Find an abstract simplicial complex K together with a subcomplex L such that 1 K [ Z B" and I L 12 S"-l. 4.1.2 Let K = ( X ,Y ) be a finite abstract simplicial complex of dimension n. Prove that K can be realized as a polyhedron contained in the Euclidean space R2n+l. 4.1.3 Let K = (X, Y) be a finite abstract simplicial complex; for a given v # X , define the abstract simplicial complex vK = (XU{v), T'), where Y' = {v} u T u { d u {v} I d E T} .

We call vK the abstract cone on K with vertex v. Now let M be a closed bounded convex set in R" and let K = ( X ,T)be a finite abstract simplicial complex such that 1 K 1 2 d M , the boundary of M in R" (assume, if you like, that the vertices of K lie in 6M). Prove that, for every v E M \ dM,I vK (2M . 4.1.4 Let K = (X,T) be a finite abstract simplicial complex whose vertices lie in R";let v E Rn+l be such that v # X. Prove that

(the space on the right hand side of the above homeomorphism is the unreduced cone of I K I).

4.1.5 Let K = (X, T)be an abstract simplicial complex; two simplexes d , E~ T are said to be contiguous if either Q = T or there is a finite sequence (61,

*

- .,d " }

(ii) rn = T and (iii), vertex of both cr; and C T ~ +i ~=, 1 , . -,n - 1. Prove

of simplexes of K such that: (i)

n

6; C T ; + ~is a

(T

=

dl,

-

that:

(1) Contiguity is an equivalence relation in T; (2) if T has only one contiguity class, 1 K

I is path-connected.

4.2. SIMPLICAL APPROXIMATION THEOREM

95

U = {UA 1 X E A} be an open covering of a space X ; assume that, for every X E A, UA # 0. Prove that K ( A ) = ( A , F ( A ) ) , where F(A) is the set of all finite subsets 2 of A such that

4.1.6 Let

nw0

XEX

is an abstract simplicial complex. K ( A ) is called nerve of the covering 24. 4.1.7 Let K = (X,T) be an abstract simplicial complex. Define the star St(a) of a simplex u of K to be the set of all simplexes of K having c as a face; then, we say that K is locally finite if, for every simplex u of K , St(u) is finite. Prove that K is locally finite if, and only if, I K I is locally compact.

4.1.8 Let K = (X,T) be an abstract simplicial complex; we have seen that if K is finite, then the metric topology of I K I coincides with the topology determined by the family {I u ( 1 u E T}. Prove that these two topologies on 1 K I also coincide if K is locally finite. Furthermore, prove that if K is not locally finite, the metric topology of 1 K I is strictly coarser than the topology determined by the I u 1’s.

4.2

Simplical approximation theorem

In the previous section we have seen that if f : K map, the induced function

-+

L is a simplicial

is continuous; in this section we want to prove a sort of homotopic inverse of that result, namely: if K and L are finite, every map from I K 1 to I L I is, up to homotopy, the geometric realization of a simplicial

C H A P T E R 4. SIMPLICIAL COMPLEXES

96

map from a convenient barycentric subdivision of K to L. This is the essence of the simplicial approximation theorem which will be made precise later on. All abstract simplicial complexes used in this section are finite; this condition is always to be assumed, even when not stated explicitly. In view of Theorem 4.1.3 we identify I K 1 with I K(') 1, for every r ; in particular, the identification between I K 1 and I K(') 1 is done by associating the vertices of K(') to the barycentres of the corresponding simplexes of K . We begin our work by making some observations about the geometry of the barycentric subdivisions. The length of a l-simplex of Kfr)is the distance between its vertices viewed as points of 1 K(') 1; the diameter of K(') is the maximum of the legths of all the l-simplexes of K(r). Notice that the diameter of K is equal to however, the diameter of K will decrease with each successive barycentric subdivision:

a;

Lemma 4.2.1 For every real number E > 0 there exists a positive integer T such that the diameter of K(') is smaller than E. Proof - Assume that dimK = n. Let {nO,c1}be a l-simplex of K ( l )such that go = { G o ,

*

- ,Xi,}

and QI

= { xi0 3 *

* *

xi,

3:j o

7 * * *

,zj,, }

+ +

with p q 2 5 n. In view of the identification l }given by length of the l-simplex { a o , ~ is 9

1

d(C k=O + T

x

i

k

)

k=O

p

+

{(x+ + 1

=

c

1

9

1

p

q

2

K ( ' ) 1, the

+ e=o c P + q + 2 zit) = P

+ 2X i k

I K 1-1

)2(4+ 1) + ( p

1

1 +

+

2 )"P

+ l)}l'z=

97

4.2. SIMPLICAL APPROXIMATION THEOREM since

p + q + l 0, is a simplicial approximation to f if for every p € 1 K 1, then I g I ( p ) belongs to the geometric realization of the support of f ( p ) . As we are going to see, the geometric realizations of all the possible simplicial approximations to f are homotopic; indeed, we prove the following: Lemma 4.2.2 Let f :I K (+( L I be a map and let g : K(') + L be a simplical approximation t o f . Then f and 1 g I are homotopic.

Proof - Once more we recall that we are dealing with finite simplicial complexes and that we are identifying I K(') I with K 1; we also recall from the previous section that the geometric realization of a simplex is a convex set. Hence, for every p € 1 K 1, the line segment from f ( p ) to I g 1 ( p ) is totally contained in I s ( f ( p ) ) Ic( L I. The map

I

I.

is a homotopy between f and I g 13 We now state and prove the main result of this section namely, the simplicia1 approximation theorem:

Theorem 4.2.3 Let f :I K 1-1 L 1 be a given map. Then there ezist a n integer T > 0 and a simplicial function g : K(') + L such that f and I g I are homotopic. Proof - Suppose that K = (X,T) and L = ( Y , O ) . Consider the (finite) open covering {O(y) 1 y E Y } of 1 L 1 (see the observations preceeding Lemma 4.1.4) and let E be the Lebesgue number of the open 1 y E Y};next, let T be a positive integer for which covering {f-'(Og) the diameter of K(') is less than ~ / 2(see Lemma 4.2.1). This implies that, for each given vertex u of IT('),there is a vertex y E Y such that O(4

cf-'(W>

CHAPTER 4. SIMPLICTAL COMPLEXES

98

and in this way we obtain a function g : K") + L

)

g(a) = y

.

Lemma 4.1.4 easily shows that g is a simplicial function. In fact, for a simplex of Idr), say {ao, ,a"}, we have

- .

iiO ( 4 # 0

;

a=O

- ,n,

but, for every i = 0,

O ( 4 c f-'(O(g(aW and so,

ri O(flZ)c f-Yfi

r=O

implying that

O(9(a2>>)

1=u

iiO ( g ( 4 ) # 0

1=0

--

and therefore, { g ( c " ) , ,g(a")} c 0. We now prove that g is a simplicial approximation to f . Let p E 1 K(') I be such that s ( p ) = {a",- .- a"} ; )

then p E O ( d ) , for every i = 0, -.. ,n, and thus,

f(P) E f ( O ( a 9 ) c 0 ( 9 ( 4 > showing that {g(a"), summation form

- - - ,g(an)}c s ( f ( p ) ) . Now, if we write p in its n

p =

Calmz 2=0

we obtain that

n

I9 I (PI = Caz9(4 1=u

I.

and therefore, that I g I ( p ) € 1 s ( f ( p ) ) Finally, we use Lemma 4.2.2 to conclude the proof. 0 As an application of the simplicial approximation theorem we prove the following:

4.2. SIMPLICAL APPROXIMATION THEOREM

Theorem 4.2.4 F o r every n

2 1 and

every 0

99

5 r 5 n - 1,

Proof - First notice that because S" is path-connected, no(Sn,eU)= 0. Let us now prove that within the homotopy class of a based map f : S' + S" there is always a based map which is not onto. In fact, let K = (X,T) and L = ( Y , O ) be two abstract simplicial complexes K I and S" Z of dimensions T and n respectively, such that S'

1 L I; by the simplicial approximation theorem, there is a convenient barycentric subdivision of I K I and a simplicial map g : K ( t ) -+ L such that f -1 g I. Because dim Idt)= dim K = r < n = dim L , the function I g 1 cannot be onto. We prove next that if a map f : S' + S" is not onto, then it is homotopic to a constant map. Suppose that p E S" \ f(S'). Let 6 : S"

\ ( p ) + R"

be the homeomorphism given by the stereographic projection from p ; form the map g : S'

and let co : S'

-+

R" be the constant map at 0 E R". The homotopy

H : s' x I shows that g

N

A S" \ ( p ) -% R"

R" , H ( z , t ) = (1 - t ) g ( ~+) ~ c " ( z )

co; but P 1 g

N

f since 19 is a homeomorphism.

0

EXERCISES 4.2.1 Prove the following relative version of the simplicial approximation theorem. Let K and L be finite abstract simplicial complexes and let M be a subcomplex of K ; now let f :I K I-+I L 1 be a map whose restriction to 1 M I is the geometric realization 19' 1 of a simplicial function 9' : 14 -+ L. Then there exist an integer r and a simplicial function g : K(') -+ L such that g 1 L(') = 9' and 1 g 1- f rel. I M I.


100

4.2.2 Let K and L two finite abstract simplicial complexes. Use the simplicial approximation theorem to prove that the set of homotopy classes of maps f :/K )+) L I is countable. 4.2.3 Let f,g : X -+ S" (with n 2 1) be maps for which there is no z E X such that f ( z ) = -g(x); prove that f g. N

4.2.4 Prove that any map f : B"

+ B" has a fixed point. (Hint: Show that if the statement is not true, there exists a retraction of B" to the sphere S"-l; then use the relative version of the simplicia1 approximation theorem to realize simplicially this retraction and study the inverse image of the barycentre of an ( n - 1)- simplex

of

4.3

27-1.)

Polyhedra

In this section we look more carefully into the nature of polyhedra; our first result deals with the notion of product of polyhedra.

Theorem 4.3.1 Let K = (X,T) and L = ( Y , O ) be two abstract simplicial complexes with X and Y finite. Then there exists an abstract sirnplical complex K x L such that

Proof - We define K x L by taking the following steps: 1) order both sets X and Y; 2) for X x Y , take the dictionary order relation induced by the orders of X and Y ; 3) require that {(zip, yio), . - .,(zim,yjn)}be a simplex of K x L if and only if the next three conditions hold true:

101

4.3. POLYHEDRA

(b) {

~ ~ o , ~ ~ ~ , ~ zE, }

(c) {Yj,,*..tYj,} E

-

@

;

.

These requirements readily imply that the projections prl : X x

Y

, pr2 : X x Y ---+Y

X

on the first and second components respectively, are simplicial functions between the appropriate abstract simplicial complexes and therefore, they determine a map

4 =I

PTl

1 x I PT2 Id K

x L

I+(

K Ix IL

I

*

We are going to prove that 4 is a homeomorphism. Let p E JK I and q € 1 L 1 be given by

ccY;xa, m

p = and

m

, CLY;= 1 , XO < ". < x m

a;> 0

n

n

j=O

j=O

Define the real numbers

and

t

j=O

-

next, we order the set (0, ao, - . ,a,-l, b",.

.-

- - ,b,

= 1) so that

and, for every r = O , l , . ,m+n, we define the elements z, = (xi,yj) E K x L by requiring that the indices i (respectively, j ) be equal to the number of the real numbers a, (respectively, b,) contained in the set { C ~ , C ~ , ~ ~ ~ , C , . -Observe ~}. that


102 and that, if or to

the element z,+1 is equal either to at any rate, z, < zrtl and so,

Z, = (xi,yj),

(zi,yjtl);

zu = (xU,yu) < 21

Because

c

= Crizi

*

i=o

Let z, < zrtl < < z,+t be the elements z; of $ ( p , q ) whose first coordinate is equal to 2,; the situation is described in the following array: vertices coefficients in $ ( p , q ) zr-1

= (~a-~yyar)

zr = ( z s , ~ , )

... ... ...

c r - 1 - cr-2 Cr - Cr-1

... ... ...

C,+t - C r t t - 1 = b 8 , Ya+,) zr+t+l = (z,+, 7 ~ a + t ) C r + t + l - Cr+t

Zr+t

Observe that the coefficient rs in I p q I $ ~ ( p , q is ) just equal to C,+t Cr-1; moreover, according to the definition of z,-] and z,, we conclude that c , - ~= and similarly, that c,.+ = a,. Hence, y8 = c,+i - c , - 1

= a, - a,-1 = a,

4.3. POLYHEDRA

103

and ultimately, 1 p r l I $ ( p , q ) = p . Similarly, we prove that I pr2 $ ( p , q ) = Q; this and the previous fact imply that &/J = lpqxlq. Now let us take an element

cr

-

I

.

, (respectively, with C:=o = 1 and zo < z1 < -.< zy. Let zo,. ,z yo, * ,yn) be the distict vertices which occur in the first (respectively, second) coordinate of zo,. ,z,; then

.

a

-

and furthermore, zu = (zo,yo) and z, = (zrn,yn).Because of the definitions we can write that

cr

where a,is the sum of all coefficients of the elements zr whose first coordinate is equal to z;;one can easily see that CzOai= 1. Similar observations can be made regarding I prz I (u).It is now easy to see that $4 = 1IKKxLI. The compactness of I K x L I implies that 4 is a homeomorphism. 0

Let K = (X,T) and L = ( Y , O ) be abstract simplicial complexes; we say that L is a simplicial subcomplex of K whenever Y c X and 0 C 'Y. (Have you solved Exercise 4.1.1 ?) If L is a subcomplex of a finite abstract simplicial complex K , we say that I L I is a subpolyhedron of I K 1; in this case, the pair of polyhedra (I K I, 1 L I) is called a polyhedral pair. Clearly, I L I is a closed subspace of I K I. The skeleta of a polyhedron I K I constitute a special class of subpolyhedra: given that K = ( X ,Y) is a (finite) abstract simplicial complex, its r-skeleton K' - here r is a non-negative integer - is the set of all simplexes of K of dimension 5 r ; then the subpolyhedron 1 K r 1 is the r-skeleton of 1 K 1.

I)

Theorem 4.3.2 Let (1 K 1, I L be a polyhedral pair. T h e n the triple (I K I,i, 1 L I) where i denotes the inclusion m a p - is a cofibration.

-


104

Proof - According to Theorem 2.3.1 it is enough to prove that there is a retraction

---

,zrt}be a simplex of K and let I u I be the geometric Let Q = (zo, realization of (Q, p ( u ) \ (8)). Next, let da be the set of all faces of Q, except u itself; the geometric realization I da 1 is a subpolyhedron of I Q 1, the boundary of I u I. We are going t o prove that is a strong deformation retract of 1 Q I X I .It is not hard to see the retraction “geometrically”: suppose 1 Q I is a 2-simplex; place it on the (z,y)-plane of R3with its barycentre b ( ~sitting ) on the origin (O,O,O) and then project I u I X Ionto I i? I from the point (0,0,2);the reader interested in handling actual formulae may proceed as follows (see [27, Lemma 3.2.31): first introduce a new notation: given an arbitrary point p € 1 Q I and a real number t E I , let [ p ,t ]denote the point t b ( a ) + ( l - t ) p of the line segment from the barycentre of 1 Q I to p ; now define the homotopy H, :I Q I X I x I Q I X I

-+

by the equations:

For every integer n 2 -1, define

M,

=I

K

I x(0)u I K” u L 1

X I

where K” is the n-skeleton of K and with the proviso that K-’ = then IM-, =I K I x(0)u I L I X I . Now define

H,, : M, x I

+M ,

0;

105

4.3. POLYHEDRA such that, for any n-simplex

~7E

Y’\ 0 ,

and, for every ( z , t ) E x I , H n ( z , t )= z. In this way we obtain a strong deformation retraction of Mn onto Mn-1.Let T n : Mn + Mn-1 be the retraction defined by T , = Hn(-, 1). If dim K = m it follows that M, =I K 1 X I ; the map T~ T , is a retraction of I K I XIonto

IZI. 0

---

This theorem proves, in particular, that the inclusion of a skeleton of a polyhedron into the polyhedron (or a higher dimensional skeleton) is a cofibration.

EXERCISES 4.3.1 Construct finite simplicial complexes K = (X,T) such that: (i) 1 K I is an n-dimensional cube; (ii) 1 K 1 is a 2-dimensional torus. 4.3.2 Let L be a simplicial subcomplex of an abstract simplicial complex K . Prove that (I K(‘) I, I L(‘) I) is a polyhedral pair, for every T 2 1. The following exercises deal with the “connectivity” of polyhedra.

4.3.3 Two points z and y of X E Top are path-connected if there exists a path X : I --t X such that X(0) = z and X(1) = y. Prove that path-connectivity is an equivalence relation in X . The equivalence classes in X defined by path-connectivity are called path components. 4.3.4 A space X is path-connected if every two points z,y E X can be connected by a path in X . Prove that every path-connected space is connected (give an example to show that the converse is not necessarily true).

4.3.5 Prove that the path-components of a polyhedron polyhedra of I K (.

I K I are sub-

106


4.3.6 Prove that a connected polyhedron is path-connected (i.e,, for polyhedra the notions of connectivity and path-connectivity coincide).

4.4

Fibrations and polyhedra

A Serre fibration is a arrow ( E , p , B ) E Top' satisfying the covering homotopy property for polyhedra; more precisely, given an arbitrary arrow (I K 1,g, E ) with 1 K I a polyhedron, for every homotopy

such that H ( - , 0) = p g , there is a homotopy G :I K I X I + E which extends p g and such that pG = H . Clearly, every fibration is a Serre fibration; however, the converse is not true as demonstrated by the following counter-example: let

E = U(1x { l f i }u { ( t ,-t) 1 t E I } ) 221

with the topology induced from R2;now take B = I and form the arrow ( E , p , B ) ,where p is the projection on the first factor. Let 1 K be a path-connected polyhedron; then g(l K I) must be contained in a path-component of E , that is to say, either g(l K I) c I x {l/i} (for some i 2 1) or g(l K I) c {(I!,-t)1 t E I } . Define

I

G : I'l x I -+ E

4.4. FIBRATIONS A N D POLYHEDRA

If I K

107

I is not path-connected,

argue using its path-components (note that there are finately many path-components because 1 K 1 is compact). This shows that ( E , p , B ) is a Serre fibration. We now prove that the arrow under consideration is not a fibration. Take X = {l/i I i 2 1 ) U (0) with the topology induced from R. Next, E given by g ( x ) = (O,x), for every x E X , consider the map g : X and the homotopy H : X x I -+ B defined by

-

o l t l ; H ( x , t )=

2t-1,

; < t i 1

which, as one can easily check, extends pg. However, there can be no homotopy G : X x I -+ E such that G(-,0)= g and p G = H .

Theorem 4.4.1 Let (E,p,B ) be a Serre fibration and let be a polyhedral pair. Given a map g

:I K I

I

-

x(0)U L X I

(I K ),I L I)

E

and an extension of p g , say

1 X I - +B

H : (K

,

there is a homotopy G:lKIxI--+E

which extends g and such that pG = H (see Figure 4.4.1).

Proof - Let u be a simplex of K = ( X ,'Y) and consider the polyhedral pair (I cr I, I acr I) (refer to Theorem 4.3.2 for the notation). Observe that 15 I=/ u I x{O)U then, if we are given a map

and

a

homotopy

H,

:I I c 7

X I

+B


108

FIGURE 4.4.1 extending pg,, because ( E , p ,B ) is a Serre fibration there is a map

G,

:I u 1 X I-+

E

extending g, and such that pG, = H,. For every integer n >_ -1, define the space

(cf. the proof of Theorem 4.3.2) and consider the map

Our objective is to construct a sequence of maps

G, : M, -+E such that

G,

1

= G,-l

pG, = H I M,,

, n _> 0 ,

, n 2 -1

because once this is achieved, we simply define G as the map satisfying the condition G / M,, = G,. Our construction is easily done by induction: we already have G - l ; suppose we have defined the maps Gp,p < n; to define G,, we consider the n-simplexes u E Y \ 0 (we

4.4. FIBRATIONS A N D POLYHEDRA

109

assume that L = (Y, 0))and then, construct G, as in the first part of the proof, taking g, as the restriction of G,-1 to 1 3 1 and H , as the restriction of H to 1 CT 1 XI. 0 Let ( E , p , B ) be a Serre fibration with fibre F over b, E B. Select a base point e, E F c E and construct the space L, = E fl P B ; let Cb, E PB denote the constant path at bv and take (e,,cb,) as the base point of L,. Let h : F --+ L, , h(e) = ( e ,cb,) be the map obtained from the universal property of pullbacks (see Section 3.1).

Theorem 4.4.2 Let ( E , p ,B ) be a Serre fibration with fibre F over b, E B and let I K 1 be a polyhedron with base point x,, Then the function h* : [I K I, FI* [I K I, 41* induced by h is a bijection.

-

Proof - We first prove that h, is injective. Let fl,f 2 E Me( I K be such that h f , hf2 and let

1,

F)

N

H

:I

K

I X I + L,

be the based homotopy connecting these two maps: for every

2

€1 K 1

H ( z , 1) = hf2(34 = (f2(4, C b , ) and for every t E I ,

H(z,,t) = ( e v , C b , )

*

Decompose H into two maps 9 : )K

I

X I + E , k : )K

1

XI

-----$

PB

such that FH = g and jjH = k. Use the exponential law to define a map k:lKIxlxI--+B which is easily seen to have the following properties:


110

1 ) E(z,O,s) = c&) = b, , 2) q z , 1,s) = Cb,(9) = b, , 3) k(z,t,O) = k(z,t)(O)= b, and, 4 ) & ( 2 , , t , S ) = c&) = b, . Now regard I as a polyhedron with two 0-simplexes (0) and { l } , and form the polyhedral pair

(I

K

I X I , I K I x ~ 0 ) UI K I X U ) u (4x I ) ;

next, consider the map

ij

:I K I X Ix

(0) U (I K

such that

I x ( 0 ) u I K I x ( 1 ) U {zo} x I) x I

s I (I K I XI x (0)) = 9 s I (I K I x I ) = fl s I (I K I X W x I ) = sI x I x I) =

+E

9

X{O}

7

f2

and

((zo)

Ceo

*

Notice that the map

i:1K(xIxI+B defined by & ( z , t , s ) = k(z,t,1 - s) extends the map pij and therefore, by Theorem 4.4.1 there exists a map

G:IKIxlxI+E

-

extending ij and such that pG = 6 . Now take

J

:I

K

I XI

E

, J(z,t)= G(z,t,l)

and observe that because i ( z , t , 1) = b,, the domain of the map J turns out to be the fibre F . Furthermore, J(-,0) = f ~ J(-,1) , = f2 and J(z,, t ) = e,. Now we prove that h, is surjective. Let S K 1L, be a given based map; decompose g into the maps g :I K I+ E and k :I K I+ PB so that g = +j and k = p g . Let

:I

H:IKIxI+B

4.4. FIBRATIONS AND POLYHEDRA

111

be the map obtained from k via the exponential law. Note that H(a,,t) = 6, and H(a,O) = b,, for every z € 1 K I. Consider the map

a :I K 1 XI + B , B ( z , t ) = H(a, 1 - t ) and observe that f i ( a , O ) = pg( z ) ; then, because ( E , p , B ) is a Serre fibration, there exists a map

G:I K I X I + E such that pG = H and G restricted to I K I x(0) is equal to g. Now take the map f =G(-,l):IKIxI+F and define the homotopy

J :I K 1 X I --+ L, , J ( z , t ) = ( G ( a , t ) , I c ( ~ ) t ) where k ( a ) t ( s )= Ic(z)((l- t ) s ) . This is a based homotopy between hf and 3. 0 The previous result has a particularly important consequence:

Corollary 4.4.3 Let ( E , p , B ) be a Serre fibration with fibre F over b, E B; let e, E F be viewed as base point for both F and E . Then, for every polyhedron I K I and every base point x, € 1 K 1, the following sequence of based sets and groups is ezact:

* * *

A [I K

I, J ' ] *

[I K

/ , E l * % [I K I,B], .

Proof - Use Theorem 3.1.2. 0 In particular, if I K I is the 0-sphere So (with base point e,)) we obtain the following: Corollary 4.4.4 Let ( E , p , B ) be a Serre jibration with fibre F over b, and let e, E F be the base point of both F and E . Then the following sequence of groups and based sets is exact:


112

We now describe an important class of Serre fibrations. We say that a arrow ( E , p ,B ) is locally trivial with fibre F if B has an open covering {Ux I X E A} together with a family of homeomorphisms

such that pq$, = prl, the projection on the first factor. We also say that the set (Ux,& I X E A} determines the locally trivial structure of

(E,?J,B ) . Theorem 4.4.5 Every locally trivial arrow is a Serre fibration. Proof - Suppose that ( E , p ,B ) has a locally trivial structure determined by (Ux,$x}; let I K I be a polyhedron, g

:I K I x{O}

+E

be a map and

H:)KIxl+B be a homotopy extending p g . Let open covering

E

W1(Ux)

I

be the Lebesgue number of the E

A}

I K I X I .Regard I as the geometric realization of a simplicial complex L with 0-simplexes

of

t o = 0 < tl < ..' < t , = 1 and 1-simplexes {t;,ti+l},i = O,"-,m - 1; now take a barycentric subdivision K(') of K so that, for every simplex I u I of 1 K(') and every 1-simplex {ti,t;+l}, i = O , . . . ,m - 1, there exists a X E A such that H( 1 Q 1 ~ [ t i , t i +c ~ ]VA )

I

-

Using induction on the skeleta of construct a homotopy

whose restriction to 1 K

I x(0)

I K(')

121

K

1,

we are going to

is g and such that pG = H .


113

The first problem is to contruct

Go :I K(') lo X I-+ E . Let I x 1 be a 0-simplex of I K(') 1 and take UA,so that

Next, form the commutative diagram of Figure 4.4.2 and, for every

t E [O,tl],define

G,,(Suppose that H(I x I 2

IX

W

7

t ) = 4 A L ( H (- 1 t ) ,Pr24x;doI 0))

1

x[tl,t2])

c UA2;then

9

form the commutative

-P - " h 1

1

T

FIGURE 4.4.2

diagram of Figure 4.4.3 and define

FIGURE 4.4.3

"'

'

-UA,

x

F

114


for every t E [tl,t 2 3 . Proceeding in this way we define

and, ultimately,

-

Go :I K(') 1' X I

E

satisfying pGo = H 1 (I K('f 1" X I ) This . is a homotopy of g restricted to I K(') 10 x(0). The induction step follows a similar line: suppose that

has been defined; for a simplex I u I of dimension n+1 take Ux such that H ( / u 1 x[O,t,]) c Ux, form the commutative diagram of Figure 4.4.4 (recall that the boundary I du I of I u I is contained in I K(') I") and

P

//

FIGURE 4.4.4 finally, define a map from 1 u I x [0,t l ]to Ux x F which extends g U G, and which gives H 1 (I u 1 x [0, t 1 3 ) when composed with prl. This map is then used to construct the extension of g U G, to 1 u 1 x [ t l , t 2 ] and so on, ultimately giving a map from I u I X I into E . The map

Gnfl :[ K(') I n +*

XI

__t

E

+

is constructed recalling that the intersection of two (n 1)-simplexes is either empty or is an n-simplex. 0 We have already seen an example of a Serre fibration at the beginning of this section; we now use Theorem 4.4.5 to give other examples.


115

To begin with, note that the arrow (R,p,S1)where p ( t ) = elnit - see Section 1.3 - is a Serre fibration with fibre Z. Let C"" be the product of the complex line C with itself (n 1) times, with the topology given by R2n+2 . We consider two spaces associated to C"+l: firstly, the sphere

+

and secondly, the n-dimensional complex projective space CP", defined as follows: consider the equivalence relation in (2"'' \ (O,O, ,0) determined by

--

and set CP" = (C"+l \ (O,O,---,O))/ CP" by [ z o , z 1 , - . ' , z,], define the map

=. We denote the elements of

and consider the arrow (S2"+',p, CP"). Now, for each integer j such that 0 5 j 5 n,the set

is open in CP"; furthermore, {Uj I 0 CP". Finally, notice that the maps

5 j 5 n} is an open covering of

are homeomorphisms such that p 4 j = p q . Hence, ( S Z n + l , p ,CP") is a Serre fibration. The Serre fibrations (S2"+l , p , CP") for n 2 1 are called Hopffibrutions, in honour of Heinz Hopf; all these fibrations have fibre S1. If n = 1, then CP' = S2;the Hopf fibration ( S 3 , p , S 2 )was first discussed by Hopf in 1931 in the celebrated paper [17] and has considerable historical significance. By applying Corollary 4.4.4 to the Hopf fibration ( S 3 , p , S2)we conclude, in particular, that the following sequence of groups is exact:

116


but nz(S3,eo)2 nl(S2,e,) 2 0 (see Theorem 4.2.4) and thus, using Theorem 1.3.6,we conclude that

nz(S2,e,)2 nl(S1,eo) zz

.

If we apply Corollary 4.4.4 to the Sene fibration (R,p,S') with fibre 2, the discreteness of the fibre and the contractibility of R imply that n,( S', e,) = 0 for every n 2 2.

EXERCISES 4.4.1 Prove that pullbacks of Serre fibrations are Serre fibrations. 4.4.2 Let ( E , p ,B ) be a Serre fibration. Prove that p is onto whenever B is path-connected. 4.4.3 Let ( E , p , B ) be a Serre fibration. Let

I K 1 be a contractible

polyhedron. Prove that any map from I K

1 to B can be lifted t o

E. 4.4.4 Let H be the field of quaternions. Define the quaternionic projectiwe spaces HP" and show that, for every n 1, there exists a Serre fibration ( S4n+3, p , HP") with fibre S3.

>

4.4.5

* The intent of this (difficult) exercise is to show a generalization of Theorem 4.4.5 for arrows ( E , p ,B ) over certain spaces B . We begin by saying that a covering (not necessarily open) {Vx 1 A E A) of B is numerable if it admits a refinement by a locally finite partition of the unity (paracompact spaces have this property; see [24]for the definitions). Prove that ( E , p , B ) E Top' is a fibration iff B has a numerable covering {I4 I A E A) such that (p-'(Vx),px,VA) is a fibration, for every X E A. (See [8] or [lo].)

Chapter 5

Relative Homotopy Groups 5.1

Homotopy groups of maps

We are now going to work with the category Top,’ of arrows over Top, (cf. Section 1.2). The objects of Top,’ are the morphisms of Top,, that is to say, based maps f : (Y,yo) (X,zo). Given that f : (Y,yo) -+ ( X , z o )and g : (Z,t,) -----f (W,wo) are objects of Top,’, a morphism from g to f is a pair of based maps ( a , b ) with a : (Z,z,) (Y,yo), b : (W,wo)+ ( X , z o )and f a = bg; in other words, the diagram below is commutative:

-

-

As in Section 1.2, we write ( a , b) : g -+ f and call ( a ,b) a (based) arrow-map. If ( a , b ) and (a‘,b’) are two arrow-maps from g to f we say that ( a ,b ) is homotopic to (a’,b‘) - notation: (a,b) (a’, b’) - if there exists

118

CHAPTER 5. RELATIVE HOMOTOPY GROUPS

an arrow-map defined by based homotopies ( H ,K ) : f x 11 -+ g such that

H(-,0) = a, H ( - , l )

= a‘

,K(-,0) = b, K(-,1)

= b’

.

-

(Note explicitly that H(e,, t ) = yo and K ( e , , t ) = z,, for every t E I.) We use the notation ( H , K ) : ( a , b ) (a’,b’) to indicate that ( H , K ) is the homotopy connecting ( a , b) and (a’, b’). The homotopy relation between arrow-maps is an equivalence relation. The homotopy class of an arrow-map ( a , b ) is denoted by [a,b]. In the sequel, for every pair of objects g, f E Topi-‘, we shall indicate by r ( g , f ) the set of all homotopy classes of arrow-maps from g to f. We now extend the definitions of CoH-spaces and CoH-groups introduced in the category Top, to objects of the category Top,‘. The extension of these notions is done on a routine basis; however, for the sake of completeness, we shall write down all of the definitions. To begin with, we say that a based map g : ( Z , z , ) + (W,w,) is a CoH-arrow if (2,I , ) and ( W,w,) are CoH-spaces with CoH-multiplications uz :

and UII,’:

z +z v z

w +w v w

such that (uz, v1.1,) is an arrow-map, that is to say, such that the diagram in Figure 5.1.1 is commutative.

z

VZ

-zvz

9

FIGURE 5.1.1

5.1. HOMOTOPY GROUPS OF MAPS

119

The CoH-arrow g : (2,zo) --, (W,wo)of Top,' is associative if the CoH-multiplications vz and v11- are associative and the homotopies

(vzv 1z)vz

Hz : ( l z v vz)vz and

Hw : ( l w v Y+M.

-

(v11-

v lll+.ql~

form an arrow-map (Hz,Hw), i.e., such that the diagram of Figure 5.1.2 is commutative.

ZXI

WXI

HZ

CZVZVZ

-wvwvw

HII-

FIGURE 5.1.2

-

An associative CoH-arrow g : (Z,z,) (W,w,) is said to be a CoH-arrow-groupif the following conditions hold true:

1. Let czo : 2 + {z,} and cw, : W + { w , } be the constant maps (recall that these are the counits of the CoH-spaces (2,zo) and (W,ID,), respectively; see Exercise 1.2.7); then the arrow-maps

(4% v 1z)vz;

(+LJo

v 1r1-)vrr)

and (c(12 v cz,)vz; a(111- v cw,)vrl,)

are homotopic to the arrow-map

(111.,

1~).

2. 2 and W have coinverses (i.e., Z and W are CoH-groups; see Exercise 1.2.8 for the definition) $z : 2 + 2 and $11, : W + W such that (+z,$ql.): g g is an arrow-map and, denoting the based homotopies of the coinverses by

Lz : a ( $ z v 1z)v.Z

-

cx,

9

120


The arrow-map (c,,, two) given by the commutative diagram of Figure 5.1.3 is called a counit of g.

FIGURE 5.1.3 The arrow-map ($z,q!q,.) : g + g is a coinverse of 9. Finally, a CoH-arrow-group g : (Z,z,) -+ (W,wo) is said to be commutative if the maps

and

eI1-: wv w -+wvw , (w,wi)(wr,w)

are such that ( B Z v z , 8 ~ ~ ~ isv ~ anl arrow-map .) homotopic to (vz,~ 1 1 . ) . We are going to show that the closed cofibration

in : ( ~ " , e , )

----f

(~"+l,e,)

(given by the canonical inclusion) is a CoH-arrow-group for every n 2 1, which is commutative if n 2 2. To reach this goal we need to understand very well the CoH-space structure of S" (see Lemma 1.2.6) and that of Bn+'together with the interplay between these structures;


121

thus, we embark on a discussion of some special maps on or into spheres and balls. For every n 2 0, define C,

: I x S" + Bn+l

, c,(t,

z)= (1 - t ) e ,

+ tx

and notice that c, induces a homeomorphism (also denoted by cn)

c, : ( I x S " ) / ( I x e, U (0) x Sn)= I A S" Z Bn+' . Next, define the following embeddings of the ball on the sphere

i+

: B"

--t

S"

and

i-

: B" ---+S"

to construct, for every n

, i+(z) = ( x , d l - 11 z ]I2)

, L ( z )= (

x , - J r n )

2 1, the map

k, : I x S"-' + S"

for every t E I and every x E 5'"-' phism

kn

:

and thus, it induces a homeomor-

zy-'2 S"

(cf. Lemma 1.3.1). We extend k,, to a map

k, : I x B"

-+

Bnfl, n 2 1 ,

as follows: firstly, define Ic,,(t,e,) = e, for every t E I ; secondly, if z E B" is not equal to the base point e,, there exists a unique pair (t',y) E I x S"-l such that ~ , , - ~ ( t ' , y= ) z and so, we set kn(t,X)

= kn(t,Cn-l(tl,y)) = c n ( t ' , & t ( t , y ) ) .

122

CHAPTER 5. RELATIVE H O M O T O P Y GROUPS

Because k n ( O , z ) = kn(l,z) = kn(t,eo) = e,, for every z E B" and every t E I, we obtain a homeomorphism kn : ZB" =" Bn+'. We now use the maps k, and k, to define CoH-multiplications in spheres and balls: for every n 2 1, let u, : S" + S"

v S"

be the map defined by

(kn(2t,z),eo), Fn(kn(t, 2)) =

o5ts;

(e,,kn(2t - I,z)), f 5 t 5 1 .

The definitions just given show that the diagram of Figure 5.1.4 commutes:

S"

un

-

S"

v S" in

in 1

I

FIGURE 5.1.4

Theorem 5.1.1 For every n _> 1, the maps

v in


and

-

vn

- Bn+1 ,

-

Bntl

123

v Bn+l

are associative CoH-multi~~ications on Sn and Bn+' respectively; moreover, the object in : ( S " , e , ) + ( B n + l , e o ) of Top,'

is a CoH-arrow-group f o r every n 2 1 (commutative, i f n 2

2).

Proof - We first restrict our discussion to the case of the unit ball Eln+'. The based homotopy

H : Bn+' x I + Brit' x Brit' defined by H ( e , , s ) = (e,,e,) and, for every a E Bn+' \ ( e , ) with 2 = kn(t,y), H ( k n ( t ,Y), 8) =

{

(kn(t(2 (kn(ts

- s),

Y), kn(ts, Y)),

+ 1 - s,y),kn(2t

-

o_ 2, we proceed as follows. Firstly, take the map

dB :p

- 1

vp

t l

--+

p + l

vpt-1

which switches components around and prove that to fnusing the based homotopy

BBcn

is homotopic

HB : B”+1 x I + B”+1 x B”+l which takes any (kn(t,l ~ ~ - ~y)), ( t t”) ’ , into

(kn(2t,kn-l((l - t ” ) t ‘ , y ) ) , kn(2t, kn-“t’t’’,y))),

osts;

- 1,kn-1 (t’t’’,y)), kn(2t - 1,I&-]((

f 5t 51

(kn(2t

1 - t”)t’,y))),

n

Secondly, take the corresponding “switching map” 8s for S” and the corresponding homotopy

H~

:

esvn

un

;

since H s is constructed with the aid of the auxiliary maps it follows that (Bsvn,OBcn)is an arrow-map and (@svn,dBcn)

N

kn

and

kn-l

(Yn, fin) - 0

We know that the constant map ce, : B”+’ --+ B”+’ is a counit (see Exercise 1.2.7); the following homotopy shows explicitly that ceo is a (right) counit (i.e., c ( l g n + l V Ce,)vn 1B”ti): N

H : B”+l x I + B”+l


125

takes (eo,s) into the base point and, on each Ic,(t,y) E Bn+',

kn(t(1

H(k(t,Y),8) = kn(s

A similar formula shows that

+ 4,Y),

+ (1

ceo is

-

osti;

s)t,y),

f 5tL1

*

a (left) counit.

Theorem 5.1.2 Suppose that we are given g , f E Top,' with g a CoH-arrow-group. Then x ( g , f) is a group and in particular, i f g is commutative, then n ( g , f) is commutative.

Proof - Suppose that g and f are the based maps

f : (Y,YO)

-

( X ,4;

define the group structure on n(g, f) as follows. Given [a,b],[a', b'] E n ( g , f ) take the commutative diagram of Figure 5.1.5 and set

W

V1V

*wvw

bvb'

xvx

Q

* x

FIGURE 5.1.5

One can easily see that this operation is well-defined. To check associativity, take arbitrarily [ a ,b],[a', b'] and [a", b"] E n ( g , f ) and note that


126

-

~ ( 1 1 1V. b ) ( ( b V b') V ~ " ) ( vVI ,l -j \ r ) ~ j , *

(likewise for a, a', a" and v z ) ; then observe that these homotopies form an arrow-map from g x 11 to f . The identity element of n ( g , f ) is given by the arrow-map

where

cvo : (4 ZO) Czo

: (Wwo)

-- -(Y,YO)

>

( X , % ) ,w

Yo

7

20

-

The (left) inverse of [a,b] is given by [a$z, b$,1-], where ( $ z , $ w ) is the coinverse for g. This claim and the preceeding one follow directly from the definitions of the counit and coinverse of a CoH-arrow-group. The proof of the commutativity of n(g,f) when g is commutative is straight forward. 0

Corollary 5.1.3 For every f : (Y,yo) + ( X , z o )E Top,' and every integer n 2 1, n(i,,,f) is a group, which is commutative if n 2 2. 0 For every f : (Y,yo) -+( X ,2,) E Top,', the group n(in,f ) will also be denoted by n,+l(f,yo). The group n,+l(f,y,) is the ( n l)th-homotopy group of f (see Exercise 5.1.7 for an interpretation of 7rn+l(f,yo) in terms of the mapping cylinder M ( f ) ) . Once again we note that the elements of ~ , , + ~ ( f , yare ~ ) homotopy classes of arrowmaps in + f given by commutative diagrams as in Figure 5.1.6.

+

FIGURE 5.1.6

There are three particular cases of importance:

5.1. HOMOTOPY GROUPS O F MAPS

1. f = i : (A,.,) + (X,.,) E Top*' is customary to use the notation

127

is an inclusion; in this case it

and call this group the ( n t 1)"-relative homotopy group of the pair (X,A)'

2. f = cyo is the constant map from Y into the base point yo: in this case r,,+~(cyo,yo) coincides - as a group - with the homotopy group n,( Y,y o ) (Exercise 5.1.4).

3.

= i : ( { z ~ ) , z ~-+ ) (X,.,) E Top,' is the inclusion of the base point in X ; in this case r,ttl(i,zo)coincides - as a group - with r n ( X , Z o ) , n 2 1. (Exercise 5.1.6).

f

Let f : (Y, yo) + ( X ,zo)E Top*' be given and let cyo and czo represent the constant maps of S"-' to yo E Y and of Bn to zo E X , respectively. We have seen that if n 2 2 ,

is a group; if n = 1, we regard Tl(f,YO)

= r(i0,f)

as a set with base point [cyo,c,,]. The following result gives a nice and important characterization of the identity element 0 = [cy,, czo] of nn(f ,Yo).

Lemma 5.1.4 Let [u,b] E r n ( f , y o ) . T h e n [a,b] = 0 iff there exists a based extension b' : Bn + Y of a (see Figure 5.1.7) and a homotopy rel. S"-' HI : B" x I --+ X such that (1) H'(-,0) = b

and, f o r every (ac,t)E S"-l x I ,

, H 1 ( - , l ) = fb' ,


128

B"

-X

b

FIGURE 5.1.7

Proof - =+: Let ( H , K ) : ( u , b ) B" + Y by the formulae b'(x) =

{

-

( C ~ , , C , ~ ) be

given. Define b' :

0 LJJ x I]_< H(&2

The homotopy H' : B" x

H'(x,t) =

-2

I +X

It 2 II),

1

2

;

511 2 115 1 -

defined by

K(&",t),

01 1 1 1 1 11 - 6

fb'(4,

1- f 5 11 2 115 1

i

is such that H'( -, 0) = b and H'( -, 1) = f b'. e=: Define the homotopy A : Sn-' x I + Y by A(a,t) = a(a), for every ( x , t ) E S"-' x I. Then ( A , H ' ) : ( a , b ) ( a , f b ' ) , since H' is a homotopy relative to Sn-'. Now define N

H : B"

x I -+ Y

, H ( z , t ) = b'((1 - t ) z + te,)

and next define

K:B"xI+X', and

H : S"-l x I

-+ Y

K=fH,

, H(z,t)= H ( z , t ) ;

'If A is a subspace of X ,we call (X, A ) a pair of spaces or simply, a pair.


129

finally, notice that ( H , K ) : ( a , fb’) (cy,,cxo). 0 The previous lemma shows that a based map (P1, e,) + (Y, yo) is homotopic to the constant map if and only if it can be extended to the unit ball B”.

Lemma 5.1.5 Let ( c , d ) : f

then, for every n

2 2,

--f

f’ be an arrow-map with

( c ,d ) induces a group homomorphism

This operation is easily seen to be independent of the representative chosen for the class [a,b]and moreover, is a group homomorphism. 0

Theorem 5.1.6 Let g : ( Z , z o )-, ((Y,yo), f : (Y,yo) + (X,zo) and h : (Z,z,) + (X,zo) be objects ofTop*’ such that fg = h (that is to say, such that the diagram of Figure 5.1.8 commutes). Then, for every

FIGURE 5.1.8 n

2 2, there is a base preserving function

130


which is a group homomorphism for every n 2 3. Furthermore, the following sequence of groups and based sets is exact:

'..

-

r n ( g r Z 0 ) (+ '

Z d n

rn(h,Zo)

(SJX)n

+ rn(f,Yo)

z

52

f)l

-% rn-l(g,zo)***

(gJx)1

. . - r 2 ( . f ,--+ ~ ~ri(g,z0) ) ('A ri(h,Zo)

+

rl(f,yo)

Proof - We first observe that the map kn- 1

sn-1

:I

srr-2

:I x

Sn-2+ Bn-l

+

factors through the map Cr1-q

giving rise to a map bn-1

such that

bn-1cn-2

.~ n - I

sn-1

= kn-l (see Figure 5.1.9

).

2

9 t

+

B"

b

-X

FIGURE 5.1.9

21ncidentally, notice that b,, - 1 induces a homeomorphism Bn-l/Snd2 Z Sn-l.

5.1. HOMOTOPY GROUPS

OF MAPS

131

for every [a,b] E 7rn(f,yo), where cz,, is the constant map taking Sn-2 onto the base point z, of 2. The function 6, is a homomorphism because

We are only going to prove the exactness at 7rn(g,zo) leaving the other cases as exercises. We begin by showing that (lz,f),S,+1 = 0. Let [a,b] E ~ , + ~ yo) ( f ,be given arbitrarily; consider the commutative diagram of Figure 5.1.10 and extend the constant map c,, to the con-

B"

bn

-

S"

*Y

a

f

*X

FIGURE 5.1.10

stant map Ez0 : B"

2. The based homotopy

is a homotopy between hEzo and fab,, relative to S"-l; hence, because of Lemma 5.1.4,the arrow-map ( lzc,,, fab,) is homotopic to the arrowmap (czo,czo): indl+ h defined by the constant maps. We now prove that ker(lz,f), c im6,+1. Let [a,b] E r,(g,zo) be such that ( a , f b ) (cz,,cI,); then there exists a map b' : B" + 2 extending a and a homotopy relative to S"-'

-

such that H(-,0) = f b and If(-, 1) = fgb' = hb'. Notice that g b ' ( z ) = b ( z ) = g a ( z ) for every z E S"-'; hence we can construct a map u" : S" + Y by setting

aI' z+ = g b I

,a

z- = b

I' '

C H A P T E R 5. RELATIVE HOMOTOPY GROUPS

132

where it and i- are the northern and southern embeddings of B” into S”,respectively. Next, we construct a map b“ : BnS1-+ X by factoring the homotopy H via the map

e : B” x I

,

-+ ~ n + l e ( Z , t ) = tit(z)

+ (1 - t)i-(.)

(thus, b“6’ = H ) . The constructions of u“ and b“ show that, for every E S”, fu”(z) = b”(z) and therefore, [u”,b”] E 7rnS1(f,yo).It remains to prove that 6ntl([u”,b”]) = [u,b]. To see this, first observe that 6 n + l ( [ a N , b N ]= ) [c,,,u’%,] and that 2

a”bn (cn-1 (t,Y)) =

{

gb‘(cn-l(2t,y)),

0I tI

b ( c n 4 ( 2- 2t,y)),

;5 t 5 1 .

Next, construct the based homotopy

K : B” x I --+ Y gb/(c,b-l(l- s

K ( c n - l ( t , y ) ,S) = b(cfl- 1 (

+ 2t,y)),

2 Y 11,

0I t5

;

’2 < - t j l

7

which implies that u“b, b. 0 We now describe an important particular case of the previous theorem: assume that 2 = B is a subspace of Y , z, = yo = b,, g = i : B C Y is the inclusion map, X = (9,) and the maps N

f = ‘1’

: (‘7YO)

-

({yO},YO)

7

= cs

’

(B,?/O)

.----)

({YO},YO)

are constant maps; in other words, take the commutative diagram of Figure 5.1.11. Then, the exact sequence of 5.1.6 takes on the following format:

133


FIGURE 5.1.11 .-n1(Y,yo)

J+(llr1(Y,B;y0)% r"(B,yo) 3~cI(Y,Yo);

this sequence is the ezact sequence of the pair (Y,B ) . Notice that the base preserving functions i,(n) are induced by the inclusion i, while the other two kinds of functions defining the previous exact sequence are slightly more complicated: the function an

: r n ( Y , B ; ~ o+ ) rn-l(B,yo)

is defined as ( l ~ , c l . -and ) ~ so, takes the homotopy class of an arrowmap (a,b) : in-1 +i into the based homotopy class [a]E r n - l ( B , y o ) ; the function j * ( n - 1 ) : r n - ~ ( Y , y o )+ rn-l(Y,B;yo)

-

is given by 6, : rn(cy,yo) +~ " - ~ ( i , yand ~ ) so, takes a homotopy class [a]E r n - l ( Y , y o ) - or equivalently, the homotopy class of the arrow-map ( a , c ) : i,-l c (where c is the contant map from B" into yo) - into the homotopy class of the arrow-map (cy,,

abn-l) : i,-Z

+i

.

As an application of the exact sequence of a pair, we study the homotopy groups of the wedge of two based spaces; the result is the following: Theorem 5.1.7 Given that (X, x o ) ,(Y, yo) E Top,, there exists an isomorp hism r n ( -X V

Y,( a o ,

yo))

r n

(X, z o )

r n (TIT, yo)

CB r n + 1 ( X x Y ,X

v Y ;( 2 0 ,yo))


134 for every n

2 2.3

Proof - Let i : X V Y i * ( n ) : nn(x V

-+X

Y,( z o ,

x

Y be the inclusion map

yo))

4

and let

rn(X x Y , (ro,yo))

be the induced homomorphism; we are going to prove that i,(n) is surjective. Consider the maps

i, a2

x v Y , i&) = (x,yo) , : Y + x v Y , iz(y) = (zo,y)

:

x

. )

and define the homomorphism en

Y,( z o , y o ) )

:rn(X x

-

rn(X V

(zo,yo))

+

(il)*(~ ) ( P T I)*(n)(a) (h)* (n)(prz)* ( n ) (a) In Exercise 1.3.1 we have seen that the homomorphism e n (a)=

4&4

= ((PTl>*(n)(a>, (prd*(n)(4)

is actually an isomorphism. Define +$n

Let us determine its inverse explicitly.

-

: r n ( x , x o ) @ rn(Y,Yo)

rn(x x

Y,(xo,~o))

+

= @I)*(+) (iiz)*(n)(P)* The obvious properties of the compositions of projections and inclusions into products show that $n$n is the identity isomorphism and so, because 4,, is known to be an isomorphism, $n is the inverse of $n. An easy computation now shows that $Jn(Q,P)

for every a E n,(X x Y,(ro,yO)).Therefore, the sequence

3The case n = 1 does not always work: see Corollary 6.2.9.

135


rn(X v Y,( z o , y o > >

'3 rn(x x Y,

(zoyyo))

is exact and splits, showing the desired result. 0 When working with the relative homotopy groups of pairs ( X ,A ) it is sometimes better to deal with hypercubes and their boundaries rather than with balls and spheres; thus, we reformulate the definition of the relative homotopy groups of (X, A ) in terms of maps from hypercubes. The transition from one formulation to the other is very simple: for every integer n 2 1, regard the hypercube I" as a polyhedron with base point io = (O,O,..-,O) and notice that I" 2 B"; moreover, the boundary a(1") of I" is homeomorphic to S"-'; hence, we can view r n ( X ,A; 2,) as the set of all the homotopy classes of arrow-maps

where in-' is regarded as the inclusion of a(I") into I" and i, as the inclusion of A into X ;we shall also use the notation

( b ,4 : ( I " ,

-

wn>>(X,A )

to indicate that we have a map b : I" -+ X whose restriction to a(In) maps a(I") into A. Another formulation for r,(X, A; zo) is the following: Consider the hypercube I" and its subspace

J"-' The space

= a(I"-') x

I

u In-'

x (0)

J"-' is contractible to the point io

.

= (0, -

- - ,O};

let

H : J"-' x I -+ J"-'

-

be a homotopy H : 15,-I ci,; since (a(I"),i,I")and (J"-',j,a(I")) are cofibrations (see Exercise 2.3.4), the homotopy H gives rise to homotopies K : a ( I " ) x I 4 O ( P ) and G : I f Zx I 4 I" which show that the inclusion

( I n ,a(In),i,)c ( I " ,a(In),P-')

is a homotopy equivalence and so, r n ( X ,A; q,)can be viewed as the set of all homotopy classes of maps from ( I " ,a ( I f l ) P , - l ) into (X, A , zo). The next result shows that it is possible to regard a relative homotopy group as a homotopy group of a space.

136


Theorem 5.1.8 Let ( X , A ) be a given pair of spaces with base point a, E A c X and let

PAAX= {A E X I 1 X(0) = z,,X(l) E A } Then r,(X, A;a,) E T ~ - ~ ( P > c,,), A X , for every n

.

2 1.

Proof - Let i : A 4 X be the inclusion map. Take the space PX = (A E X' 1 X(0) = z,} and construct the pullback diagram of the arrows ( P X , e l , X ) and ( A , i , X ) , where el is the evaluation at t = 1 (see Figure 5.1.12 and the beginning of Chapter 3).

A

i

-X

FIGURE 5.1.12 Let czo E P X be the constant path at zo and let (z,,czo) be the base point of L;. We are going to prove that

for every n 2 1. Let cLi : L, + {(zo,czo)} be the constant map; we must prove that there exists a bijection

where i,-l : Sn-l + B" is the inclusion. Let ( a , b ) : i,-l arrow-map. Now define the homotopy

H' : SrL-' x I

-+

X

---f

i be an

, ( a , t )H b ( c n - l ( t , z ) )

and observe that H'(a,O) = z,, for every z E S"-'; thus, its adjoint H' is a map from 5'"-* to P X . Because q H ' ( z,t ) = b( z)= ;a( a ) for every

5.1. HOMOTOPY GROUPS O F MAPS

137

x E Sn-l,there exists a unique map h : S"-l --+ Li such that slh = a and zh = H . Define e([a,b])= [h,c],where c : B" -+ {(zo,c~,)} is the constant map. In order to show that B is independent of the choice of represe.ntative within the class [ u , b ] ,we proceed as follows. Let (a,b) (u', b') : i,-l + i be two arrow-maps connected by a homotopy ( H , K ) : in-1 x 11 + i. Using the exponential law, we construct a commutative diagram . , - as in Figure 5.1.13, that is to say, we construct an arrow-map ( H , K ) : i,-l + .'i Because L! is a pullback space

-

FIGURE 5.1.13

of the arrows ((PX)' = P(X'),e;,X') and (A',i',X') (see Exercise 2.1.4), the technique used before to associate to the arrow-map ( u , b ) the arrow-map ( h ,c) can be used again, this time associating to (8,I?) the arrow-map ( & c ) : i,-l + c i i . Applying the exponential law to i, we obtain a homotopy of arrow-maps ( h , c ) (h',c). + be a given arrow-map; consider the Now let (h,c) : pullback diagram defining L; and form the maps

-

a = Elh : Sn-' + A

and

a' = ih : Sn-'-3 PX

.

Because qa' = ia and P X is contractible, the map i a can be extended to a map b : B" --+ X (see Exercise 2.3.13) that is to say, we obtain i. This shows, ultimately, that tJis a an arrow-map ( a , b ) : in-l bijection. Finally, we claim that Li 2 P..lS: the homeomorphism is given by the function P:,X + L; , x H ( A ( l ) , X ) . 0 --f


138

The following consequence of this theorem will be used in Section 6.2. A triad ( X ;A , B ; zo) is a space X together with subspaces A and B such that X = A U B and a base point z, f A n B. Define the nth-homotopy group of the triad ( X ;A , B ; zo) by

rta(X;A,B;2 0 ) = Tn-I(PBX,PAnBA;Cz,) where, as usual, czo denotes the constant map (path) at

2,.

Corollary 5.1.9 Let ( X ; A , B ; a , ) be a triad. Then there exists a n exact sequence of groups and based sets

... --+

-

rntl(A,AnB;a,) --+ T n t 1 ( X , B ; 2 , )

T ~ + ~ ( X ; A , B ;+ Z , r) , ( A , A n Biz,)

+. * .

Proof - Take the exact sequence of the pair ( P B X ,PAnBA) and use Theorem 5.1.8. 0 The exact sequence of Corollary 5.1.9 is called exact sequence of the triad (X; A , B; zo).

EXERCISES 5.1.1 Prove that the relation ( a ,b) (a', b') in the class of all objects of Top,' is an equivalence relation. N

5.1.2 Prove that if g E Top,' is a commutative CoH-arrow-group, then, for every f E Top,', r ( g , f) is a commutative group. 5.1.3 Prove that if f : ( Y , y , ) -+ (X,ao)is a homotopy equivalence, then r,(f,y o ) = 0 , for every n 2 1. 5.1.4 Prove that, for every n 2 1, there is an isomorphism between the groups w,(X,z,) and T ( ~ , , , C ~ ~ ) where , c., is the constant map from X to {zo). 5.1.5 Prove that, for every n 2 1, there is an isomorphism between the groups 7rrL(X, {xo}; zo) and ~ ( i , c, ,r o ) . Deduce from this and the previous exercise that r n ( X ,zo) Z T,(X, {zo};zo), for every n 2 1.

5.1. HOMOTOPY

GROUPS OF M A P S

139

5.1.6

Assume that f = i : ({z,}, zo) + ( X ,2,) E Top,' is the inclusion of the base point in X . Prove that r,,+l(i,2,) 2 r , ( X , z,), n 2 1.

5.1.7

Let f : (Y,yo) + ( X , Z , )be a given map; let (Y,i(f),M(f))be the cofibration associated to f (see Theorem 2.3.9). Prove that for every n 2 1

5.1.8

Prove the exactness of the sequence of Theorem 5.1.6.

5.1.9

Let ( X , A )be a pair of spaces with base point if A contracts to 2, over X, then

2,

E A. Prove that

for every n 2 3. If n = 2, we still have an isomorphism, but the right hand side might be a semi-direct product. 5.1.10 Let ( X , A )be a pair with base point retract of X , then

2,

E

A . Prove that if A is a

for every n 2 2.

5.1.11 Prove that for every pair ( X , A )with base point z,, the arrow

is a fibration; however, the arrow ( P , q X , q , X ) is not necessarily a fibration. 5.1.12 Prove that for every sequence of spaces A z, E A , there is an exact sequence

called the ezact sequence of the sequence A

cBc

C and every

c B c C.

C H A P T E R 5. RELATIVE H O M O T O P Y GROUPS

140

5.1.13 Use the exponential law of maps to prove that the elements of r n ( X ;A , B ;2.) are in a bijective correspondence with the homotopy classes of maps from

(In-' x I , ~ ( I " - ' )x 1,Y-l x {l},In-'x {0} u {io} x I ) into ( X ,A , B , z0). 5.1.14 H o m o t o p y addition theorem Let n 2 3 and let ( a , b ) : in-' + f, f : (Y,yo) + ( X ,g o ) , be a given arrow-map. Define the map c : B"-l -+ Y

such that c

I SnP2= a I S"-2, fc = f 1 B"-l .

Next, define the maps a+ and a- from Sn-l to Y so that a+i+ = ai+

, a+i-

a-2- = aa-

, a-a+ = c .

=c

,

+

Prove that [a,b]= [a+,&+] [.-,&-I. 5.1.15 For every (Y,y o ) E Top, and any p , q 2 1, define a relation

c : sp+9-'

2

a ( P x P)= ( I P x

a(P)u a ( P ) x P)

_+

Y

is given by

i

a(s), s E 1 p

c(s,t) =

qt),

s E

, t E a(I9)

a(P), t E 1 9

(arrange matters so that c(e,) = y o ) . Prove that the following statements are true:

141

5.2. QUASIFIBRATIONS

1. [

, ] is a well-defined function.

2. I f f

:(

K Y O )

--+

(Z,Zo)

f * ( P + 4 - 1",

PI11 = [ f * ( p ) ( [ a l f)*,(s)([bI)l

>

for every [a]E r,(Y,y,) and every [b] E r,(Y,yo).

3. If ( Y , y o ) is an H-space, [[a],[b]] = 0, for every [a]E r,(Y,y,) and every [b]E r,(Y,yo). = [Is.] be the homotopy class of the identity map 1s. : S" -, 5'"; then (Sn,e,) is an H-space if [ L , , L , ] = 0.

4. Let L,

The function [

5.2

, ] is the so-called

Whitehead product

.

Quasifibrations

The relative homotopy groups of a pair defined by the total space of a Serre fibration and a fibre are isomorphic to the homotopy groups of the base space; more precisely:

Theorem 5.2.1 Let ( E , p , B ) be a Serrefibration and let F = p l - ( b o ) for a base point b, E B; select a base point e, E F c E . Then, for every n 2 1, the map p induces a bijection

(isomorphism, in case both based sets are groups).

Proof - Let

be a given arrow-map. The homotopy


142

is a strong deformation retraction of I" onto {i,}) with i, = (0,. Now take the maps

H : I" x I +

B

. . ,O).

, H(z,t)= bD(z,t)

and

K : I" x (0) U {i,} x I

--$

E

the constant map to e,; notice that H extends p K . Thus, according to Theorem 4.4.1 there exists a map )

G :I" x I

-+

E

extending K and such that pG = H . Let b' : I" + E be defined by b' = G(-, 1). Then the restriction a' of b' to 8 ( P ) maps O(I") onto e, E F and moreover, p,(n)( [a', b']) = [a, b]. Now suppose that [a,b] E r,(E,F,e,) is such that p,(n)([a,b])= [Pu,pb] = 0. Then, there exist a map b' : I" + {b,} extending p a and a homotopy rel. a(I " )

H : I" x I

B

such that H ( - , 0 ) = pb and H(-,1) = ib' = Define the map g : I" x (0)

U a(F) x I

-+

cb,

(see Lemma 5.1.4).

E

by the conditions

O(I") x I ) , g(a,t) = a(a) . H(-,O)and therefore, because of Theorem 4.4.1,

(V(z,t) E

Notice that p g = there exists a homotopy

G : I" x I

-+

E

which extends g and such that pG = H . Let

b" : I" + E

, b"

= G(-,1).

5.2.

Q UASIFIBRATIONS

143

Because p b " ( z ) = pG(z,l) = H ( z , l ) = b, for every z E I", b" is actually a map from I" to F . Its restriction to a(I") coincides with the b rel. a(I"). Theorem 5.1.4 now proves that map a and finally, ib" [u,b]= 0. 13 It is clear that the previous theorem can be stated for a fibration rather than for just the more restrictive type of Serre fibration. Motivated by Theorem 5.2.1 we give the following definition: ( E , p , B ) E Top*' is a quasajbration if 1)p is onto, 2) for every b, E B and every e, E p-'( b,) = F c E ,

-

and 3) the sequence of based sets

is exact. (This definition was first given in [9].) From Theorems 2.2.6 and 5.2.1 it follows that any fibration with path-connected base space is a quasifibration; if ( E , p ,B ) is a Serre fibration and B is connected, then again p is onto (this comes trivially from lifting paths of B ) and we use Corollary 4.4.4 to show that ( E , p ,B ) is a quasifibration. We now give an example of a quasifibration which is not a fibration or a Serre fibration. Take the set

with the topology induced by R2.Next, take B = [-1,1] and let p : E + B be the projection on the first factor. The arrow ( E , p , B ) is a quasifibration because the spaces E , B and the fibres F are contractible. Let us prove that ( E , p , B ) is not a Serre fibration and thus, not a fibration, either. Consider the cube '1 = {*} and the maps g : I"

E

--i

H : 1" x I --+ B

, g(*) = (-1,l)

, H ( * , t ) = 2t - 1 ;

144

C H A P T E R 5, RELATIVE H O M O T O P Y GROUPS

notice that pg = H(-,0) and there is no G : '1 x I + E such that pG= H. The fibres of a Serre fibration or a quasifibration over a pathconnected space are not necessarily of the same homotopy type, as it is the case for fibrations (see Theorem 2.2.6); however, we still have an interesting situation which we describe anon. We begin with the following definition: A map h : Y + 2 is said to be a weak h o m o t o p y equivalence if it induces a bijective correspondence between the sets of path-components of Y and 2 and if, for every n 2 1 and every yo E Y ,

h*(n): %3(Y,Yo)

-

%l(Z,f(Y0))

is an isomorphism. In view of Section 5.1, we can formulate an alternative definition: A map h : Y + 2 is a weak homotopy equivalence iff, for every yo E Y ,

f*(O) : T"(Y,Y o )

-

m(X, f ( 4 )

is a bijection and ~ , ( f , y , ) = 0, for every n 2 1. Clearly, every homotopy equivalence is a weak homotopy equivalence; the converse is not true (see Exercise 5.2.1). We now define two based spaces and y2 to have the same weak homotopy type if there are a space X and weak homotopy equivalences fi:K+Xand

fi:Y2---'X.

Theorem 5.2.2 Let ( E , p , B ) be a Serre fibration o r a quasifibration; suppose that B is path-connected and, f o r s o m e b E B , F = p - l ( b ) is such that F,e,) = 7r1 ( F ,e), = 0, e, E F . T h e n all t h e fibres of p have the s a m e weak: homotopy type and p is onto. Proof - First factor p via its mapping track, i.e., write p as the composition p'u, where u : E ------t T ( p ) is a homotopy equivalence and p' : T ( p ) + B is a fibration (see Theorem 2.2.7). Because p = p ' r , the restriction of r to the fibre F takes F into p'-'(b). Now compare the exact sequence of ( E , p , B ) to the exact sequence of the fibration ( T ( p ) , p ' , B ) , via the functions induced by the maps l ~u ,and u I F ; the "five lemma" implies that, for every n 2 1, (c 1 F ) * ( n ): .rrn(F,e)

-

7rrL(P/-yb),U(e))

5.2. Q UASIFIBRATIONS

145

is an isomorphism, that is to say, a I F is a weak homotopy equivalence. For another fibre F' = p-'(b') we conclude, as before, that the map c I F" is a weak homotopy equivalence. But from Theorem 2.2.6 we know that there exists a homotopy equivalence

f : p'-l(b/) the composite map f(a I

F') is

p'-'(b) ;

4

a weak homotopy equivalence. 0

EXERCISES 5.2.1 Let X be the subspace of R2 constructed as follows: for every n 2 1, take the line segments An and B,, where A, has vertices ( - l , O ) , (O,l/n) and Bn has vertices ( 0 , - l / n ) , (1,O). Let C be the segment with delimited by the points (-l,O), ( 1 , O ) and set

Prove that the constant map c : X + ((0,O)) is a weak homotopy equivalence which is not a homotopy equivalence.

5.2.2 Let ( E , p , B ) be a Serre fibration and let b, E B, c B ; moreover, let E, = p - l ( B , ) and e, E p-'(b,). Prove that p induces a bijection

rn(E, Eo; eo)

B o ;b o )

r7h(B,

-

5.2.3 Let ( E , p ,B ) be the quasifibration ( E , p ,B ) given by

E = [-l,O]

x (1)

u (0)

x [O,11 u [O,11 x (0)

(with the topology induced by R2), B = [-1,1] and p equal to the projection on the first factor. Now take the map

h : I = [O,13 given by

h(t) =

{

B = [-1,1]

isin(l/t) t

>o

t=O

146

CHAPTER 5. RELATIVE HOMOTOPY GROUPS Prove that the pullback of ( E , p ,B ) and (I,h, B ) is not a quasifibration. (This contrasts quasifibrations from fibrations and Sene fibrations, because pullbacks of these last two structures maintain the structure.)

5.2.4 Prove that n,(SJ,e,)

~ , - ~ ( S ~ , e , ) $ n , ( S ~ , efor , ) ,everyn 2 1.

5.2.5

* Let ( E , p ,B ) be a Serre fibration with fibre F over b,

5.2.6

* This exercise is the “local” counterpart to Exercise 4.4.5. Let p : E -+ B be an onto map and let U = {Ux I X E A) be an open covering of B such that:

E B. Let ( P E , e l , E ) be the fibration defined by the evaluation at t = 1 (see Lemma 2.2.5). Take a pullback space F nPE of ( F ,i, E ) and (PE, € 1 , E ) . Show that there exists a weak homotopy equivalence

(a) for every x E Uxi n Uxj, there exists an open set such that

zE

U A E~U

uXkc uxin uxj;

(b) for every X E A, the restriction

(;.em,the pullback of p over the inclusion fibration. Then ( E , p , B )is a quasifibration. (See [9].)

Ux c B ) is a quasi-

5.3. SOME HOMOTOPY GROUPS OF SPHERES

5.3

147

Some homotopy groups of spheres

We have seen that the fundamental group of S' is the infinite cyclic group Z (see Theorem 1.3.6); moreover, as a consequence of the simplicial approximation theorem, we proved that 7r,(S", e,) = 0 for every 0 5 T 5 n - 1 and n 2 2 (see Theorem 4.2.4). Next, the exact sequence of the homotopy groups of the Hopf fibration ( S 3 , p , S2)shows that that 7r2(S2,eo)E Z. We have also seen that by applying Corollary 4.4.4 to the Serre fibration (R,p,S') given by p ( t ) = e l n i t , we obtain that 7rn( S1,e,) = 0 for every n >_ 2. Finally, we notice that, for every n 2 1, the spheres S" are path-connected and therefore, 7ro(Sn, e,) = 0. In this short section we are going to prove that for every n 2 1,7rn(Sn,e,) 2 Z and that the higher homotopy groups of the spheres do not need to be trivial, For a given integer n 2 1, let C+S" and C-S" be two copies of the cone CS" over the sphere S" with C+S" n C-S" = S"; then the space CtS" U C-S" is homeomorphic to the sphere Sntl, Note that

0 -

O+

0-

(similarly, for C S"); the spaces C S" and C S" are the open (n+l)cells of this (particular) decomposition of Sntl. The crucial result for the computation of rn(S", e,) is the following theorem which shows that the lower homotopy groups of the triad (Sntl;C+S", C-S"; e,) are trivial: Theorem 5.3.1 For ,every integer

T

such that 2

5 T 5 2n,

rr(Sntl;CtSn, C-S"; e,) = o

.

Proof - Assume that [f]E 7rT,(Sn+' ; C+S", C-S"; e,) is represented by a map f : Ir-' x I + Sntl such that

f : a(Ir-') x I f : I"-' x (1) f : Ir-' x (0) u {io} x I

-----)

C+S"

---+

c-S"

t

e,


148

(see Exercise 5.1.13). Consider the sets

and

E- = { [ t , ~E ]C-S" 10 5 t 5 1/4} C6- S" and regard both I'-' x I and Sn+las geometric realizations of suitable abstract simplicial complexes L = (Y,0) and M, respectively. By the simplicial approximation theorem 4.2.3, there exist a barycentric subdivision L(') of L and a simplicial function g : L(') 4 M such that I g 1- f; furthermore, we may assume that the subdivision L(') is so fine that every u E 0(')such that f(l u I) n E+ # 0 has the property

I)

O +

that f ( l B CC S" (similarly, for the other hemisphere of S"+'). We now define the following two subsets of a('):

C1 = {u E 0(') I

f(l

I) f l E - # 0 , dimg(a) 5 n} Cz = (a E 0(') 1 f ( l a I) n E - # 8 , dimg(a) = n + 1) . a

Because E - can be viewed as the geometric realization of a simplicial complex of dimension n 1,

+

E-\(

u f(lfll)nE-)#0;

UECl

take p E E- \ (UaECIf( I a I) n E - ) and note that p # e, and that there must be a a E C2 such that p E f ( l a I). The anti-image f-'(p) is a polyhedron of Ir-'x I of dimension 5 T - (nt1);if r : Ir-'x I +I'-' is the projection on the first factor, K = r-'(r(f-'(p))) is a polyhedron contained in I'-' x I and dim K

induced by inclusion and assume that

+[f']= [f]. But

since C+S" is a strong deformation retract of Sn+' \ { p } and the last group is trivial, as one can conclude from an inspection of the exact CsS", C+S" \ { q } ; e,). Hence [f]= 0. 0 sequence of the triad

((7's";

150


Corollary 5.3.2 The inclusion

i : (C+S", S " ) --+ (S"+l,c - S " ) induces a homomorphism

i r ( ~: rr(C+Sn, ) S"; e,) whach is a n isomorphism f o r 2 T = 2n.

+ 7rr(Sn+', C-S"; e,)

5 T 5 2n - 1

and an epimorphism for

Proof - Take the exact sequence of the triad (S"+'; C+S", C-S"; e,) and use Theorem 5.3.1. 0

Theorem 5.3.3 For every integer n 2 1, Tn( S", e,) Z 2. Proof - We already know that sl(S1,e,) and 7r2(S2, e,) are isomorphic to the group of integers; we now prove that for every n 2 2, rn(Sn,eo) r,+l(Sn+',eo). Take the exact sequences of the pairs (C+S", S") and (S"+',C-S") and notice, in view of the contractibility of the cones C+S" and C-S", that they imply the isomorphisms

7rn+1(C+Sn,Sn;e,) 2 7rn(Sn,e,) and

(Sn+',e,) 2 7r,+] (Sn+*, C-S"; e,) ; now use the isomorphism

X ~ + ~ ( C + S"; S " ,e,) 2 T,,+~(S"+', C-S"; e,) given by the previous corollary to complete the proof of the statement. 0

The final result of this section is the following:

Theorem 5.3.4 7r3(S2,e,) 2 Z

.

Proof - The exact sequence of the Hopf fibration ( S3,p , S2)and the fact that ?r,(S',e,) = 0 for every n 2 2 prove that

s3(S2,e,)2 r3(S3,e,)2 Z

.U

The problem of finding all the homotopy groups of the spheres is very difficult and is still an open question.

5.3. SOME HOMOTOPY GROUPS OF SPHERES

151

EXERCISES 5.3.1 Prove that S"-' is not a retract of B". 5.3.2 We have seen in Theorem 5.3.3 that T n ( S n , e o ) ~,+1(S"+',e,,), for every n 2 2. This result can also be retrieved as a particular case of the Freudenthal suspension theorem (see Exercise 6.2.5). As preparation for that future starred exercise, prove that the correspondence

defined by E#(n)([f]) = [IS,A f] for every [f]E x , ( X , z , ) is a group homomorphism for every (X, zo) E Top, and every n 2 1. (This is the so-called suspension homomorphism.) 5.3.3 Prove that if n # m, then R" and R" are not homeomorphic. (Hint: Prove that R" \ (0) S"-l.) -+


Chapter 6 Homotopy Theory of CW-complexes 6.1

C W-complexes

The goal of this section is to introduce and discuss some of the main properties of a class of spaces which is crucial for the development of Homotopy Theory, namely the class of CW-complezes. For a given set A, let {S;-*

I X E A}

be a set of copies of the (n - 1)-dimensional sphere, and let

{Bx” I

E

A)

be the family of the corresponding n-balls. We know that, for every X E A, the inclusion

ix : s;-1 B,” gives a closed cofibration (SXn-’, ix, By). Taking topological sums, we obtain a closed cofibration

(Us y ,2, uB i ) . x

A

For a given map f : uxSi-’

+A ,

let

X=AUj(UBI;) x

154 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES be the adjunction of

uABg to A via f, Exercise 2.3.12 shows that X

\ A zz U(B; \ S,”-’) x

(the homeomorphism is given by the appropriate restriction of the map

f:UBj;+X x

created by the pushout construction); furthermore, from Lemma 2.3.6 we conclude that the arrow ( A , i , X ) is a closed cofibration.

Theorem 6.1.1 normal, so is X .

If A

is Huusdorfl, so is X = A U j

(1,B,”);if A

is

Proof - We first prove that X is Hausdorff if A is Hausdorff. Let B = uxB,” and S = uASi-’; given any two distinct points of X , say z and y, we must find open sets U,V c X such that z E U ,y E V and U n V = 0. We distinguish three cases: 1) z,y E B \ S: Because B \ S is Hausdorff and open in X, there are two open sets U , V c B \ S satisfying the required conditions. 2) z E B \ S and y E A: Because B is a regular space and S is closed in B , there exists a closed subset W of B which contains z in its interior and such that W n A = 0. Now W is closed in X since i-’(W) = 0; thus, V = X \ W is open in X , it contains y and has empty intersection with U =$, which contains 2 and is open in X . 3) z,y E A: Because A is Hausdorff, there exist two disjoint open sets of A , say U‘ and V‘ such that z E U’ and y E V‘. Clearly, f-’( U‘) and f-’(V‘) are disjoint and open in S; the trouble is, they might not be open in X but, as we shall see, they can be expanded to open sets of X by letting them grow within the balls used to construct X . Roughly speaking, this process of “growing” in B is done by attaching a

“collar” to the open sets f-’(U’) and f-’(V‘); the formal construction is as follows. Let V c A be open; the set

Cf(V) = v u

j({h 12 E

f-’(V),1/2 < t 5 1))

has the following properties:

Cf(V)nA f-’(cf(v))

=

=

V { t z I z E f-’(V),1/2 < t 5 1)

6.1. CW-COMPLEXES

155

(these two properties imply that C j ( V ) is open in X);furthermore, Cj(V) contains V as a strong deformation retract, as seen with the aid of the homotopy: H : Cf(V) x I + Cf(V)

H(x,s)=

{

x€V

x' - s ) t z f((1

+

SX),

z E f-'(V),i

are disjoint and contain, respectively, the points z and y. Let us prove next that X is normal if A is normal. In view of Tietze's extension theorem, it is enough to prove that for any given closed subset C c X , every map k : C + I can be extended to a map k' : X ---f I. The normality hypothesis on A implies that there is an extension g' : A 4 I of g = k I A n C. Let h : f-'(C)U S 4 I be the map given by h 1 $-'(C) = kfc (where fc : f-'(C)t C is the map induced by $) and h I S = g'f. Now the map h can be extended to a map h' : B -+ I because B is normal. The map k' is now given by the universal property of pushouts. 0 The open set Cj(V) of X associated to the open set V of A is a collar of V and the process used to obtain it is called collaring. For each X E A, f ( B 1 )= E x is a compact subspace of X (closed, if A is Hausdorff). The subspaces Ex are the n-ceZls of X ; the restriction of f to an open ball B," \ S;-' is a homeomorphism onto ex, an open n-cell, whose closure coincides with EX. The map

is a characteristic map for the cell ex; the map

which glues the cell ex to A is an attaching map for the cell ex. The pair ( X ,A ) is an adjunction of n-cells. The following result will be used in the next section.

156 C H A P T E R 6. HOMOTOPY THEORY OF CW-COMPLEXES

Lemma 6.1.2 Let ( X , A ) be a n adjunction of the n-cells

el,-.-,Ck.

Let y = A u (El \ { P l ) ) lJ . * ' u (Ek \ { P k H be the space obtained by removing a point p j from each open cell j = 1, * * ,k. Then A is a strong deformation retraction of Y .

ej,

Proof - The space X is defined by the pushout diagram of Figure 6.1.1. For each j = 1 , - -- ,k, let fj be the restriction of f to By. There

ulj=ls;-1

f

A

I

FIGURE 6.1.1 is no loss of generality in assuming that p j = f T i ( O ) , j = 1,.* - ,k. Now notice that each sphere ST-' is a deformation retract of Bj" \ (0) via the homotopy Hj : (By \ (0)) x I + BY \ (0) (Gt)

-

-t

2

I1 2 II

+ (1 - t ) x .

Define the homotopy

G:YxI-Y by the formulae:

This ends the proof. 0 A pair ( X , A ) is called a relative CW-complex if there exists a sequence of spaces

X-l

A

c Xo c X 1 c X 2c ... c XIa c ...

6.1. CW-COMPLEXES

157

such that: 1) X" is obtained from A by adjunction of 0-cells (i.e., Xo is the topological sum of A and a discrete space); 2) for every integer n >_ 1, the pair ( X " , X " - l ) is an adjunction of n-cells; 3) X is the union space of the sequence

that is to say,

x = uxt 12-1

with the topology determined by the family { X i I i 2 -l}, namely: C c X is closed in X if and only if, for every i 2 -1, C n Xz is closed in X i . If the sequence X - l c X" c - . C X" c - * is stationary at n, that is to say, if X"-l # X" and X' = X " for every T 2 n, we say that the relative CW-complex ( X , A ) has dimension n - notation: d i m X = n; otherwise, X has infinite dimension - notation: dimX = 00. The space X" is called the n-skeleton of the relative CW-complex (X, A). Because compositions of cofibrations are cofibrations (see Exercise 2.3.6), a finite induction shows that if ( X , A ) is a relative CW-complex of finite dimension, the arrow (A,;,X ) is a cofibration. The infinite dimensional case is slightly more delicate:

-

-

Lemma 6.1.3 Let ( X ,A ) be a relative CW-complex with dim X = 00. Then (A,%,X ) is a cofibration.

Proof - Let f : X x {0} -+ 2 and G : A x I --+ 2 be maps which coincide when restricted to A x (0). We want to prove that, for every n 2 0, there is a homotopy F, : X" x I --+ 2 such that (i) F, ( A x I ) = G , (ii)Fn(-,0)= f I S"and (iii)F,,+l I (S"x I ) = F,. Suppose that we have constructed such a F,, for some n (this can be done by observing that the inclusion of A into X " is a cofibration). To construct Fn+l,consider the restriction f XnS1,the homotopy F, and use the fact that ( X n , i n , S f l + l )is a cofibration. The sequence

I

I

{ F , : X" x I

--+ 2

In

2 0)

158 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES defines a homotopy F : X x I + 2 such that F

FI(AxI)=G.

I

X = f and

0

The pair (Sn,Sn-l) with n 2 1 is a relative CW-complex: it is created by taking the attaching map

which is the identity map on each component Sr-' and with characteristic maps i+,L : Br + S" (these were defined in Section 5.1).

Theorem 6.1.4 Let (X, A ) be a relative CW-complex. If A is normal, so is X . Proof - We use Tietze's extension theorem. Let C c X be closed and let k : C --+ I be a map. Using the normality of the spaces X" (use induction and Theorem 6,1.1), define inductively extensions Ic, : X" +I of the maps

IC' n : xn-lu ( C n xn) -+

I

given by the conditions

The set of maps {k, I n 2 0) now gives rise to a map Ic, : X + I which extends 8 . 0 Note that if A is 7'1, then as it is easily seen, X is 7'1 and so, if A is also normal, then X is Hausdorff. If X-l = A = 0 and Xu is a discrete space, X is called a CWcomplex. For every integer n 2 0, the space X" is called n-skeleton of X . Note that from a set-theoretical point of view, a CW-complex X is just the disjoint union of its open cells; furthermore, while the closed cells are closed (and compact) subsets of X, its open cells are not necessarily open subsets of X (indeed, an open cell of X is not open if it intersects the boundary of a cell of higher dimension). The previous theorem shows that CW-complexes are Hausdorff and normal. As was the case for relative CW-complexes, we define a CW-complex X to

159

6.1. CW-COMPLEXES

be finite dimensional of dimension n if its sequence of skeleta becomes stationary at n; moreover, the inclusion of any skeleton of X into a skeleton of higher dimension i s a cofibration and so is the inclusion of any skeleton X' into X . A CW-complex with finitely many cells is said to be a finite CW-complex; such a CW-complex is clearly a compact space. The following are examples of CW-complexes: 1) X = B"+l, with the skeleta: X o = X1 = * - . = Xn-I = {e,}, xn = s", x n + 1 = f p + 1 . 2) In this example we construct a CW-complex X whose skeleta, up to and including X"-', are constituted by a unique 0-cell e,; to this end we generalize the concept of wedge of two based spaces. For a given based space (X,a,) and a set A, let

I A E A)

OA = {(xA,zo)

be a family of based copies of X ; define the wedge product of the family U A to be the set

V X , = {(zx) E n X x 1 zx x

# z, for at most one X E A}

A

endowed with the final topology given by the canonical map 4 : uxx x

_f

vxx ; x

as base point of Vx X x , we take the element (2,) whose components are a l l equal to a,. If the space X is the n-sphere S", the wedge product Vx Sy is also called a bouquet of n-spheres. Now consider the pushout diagram of Figure 6.1.2 in which c is the

constant map to the point e,. Notice that

XrL

vs;

;

x

-

thus, X = X" is a CW-complex with skeleta X" = X I = - = Xn-* { ( e , ) } and one n-cell for each A. 3) Let F = R , C or H be the field of real, complex or quaternionic numbers, respectively. Since H is non-commutative, we will consider only multiplication on the left. We define on F"" \ ((0,. ' ,0))

-

160 C H A P T E R 6. H O M O T O P Y THEORY OF CW-COMPLEXES

FIGURE 6.1.2

the following equivalence relation: for all (YOjY1, ***, Yn) in Fnfl\ ((O,.. * 7 o)},

2

= (zo,z1, ...,;c,),

z w y w (3 X E F \ (0)) ('d i = 0, ...,n) 2; = Xyj

y =

.

We then define the Projective n-space to be the space

FP" = (Fntl \{(O,.**,O)})/

w

with the quotient topology. We wish to prove that FP" is a CWcomplex; to do so, we shall prove that FP" is a pushout space of the arrows ( Snk-l, & - l , Bnk)and ( Snk-l, fn-l, FP"-' ) (here k is the dimension of F as a vector space over R, ink-l is the inclusion and fn-l is defined by

the equivalence class of (zO,z1, union space of the sequence

FPu = { e , }

...,~ ~ - 1 )and )

then take FP"

as

the

c FP' c . . . c FP" = FPn . . .

Given the pushout diagram of Figure 6.1.3 construct the commutative diagram of Figure 6.1.4 where gn-l and j,-I are defined as follows:

6.1. CW- COMPLEXES

161

*I

I-

znk- 1

Znk-1

FIGURE 6.1.3

- FPn-'

fn-1

Snk-1

znk-1

Jn-1

- FP"

T

B"k

gn-1

FIGURE 6.1.4 and jn-1

([(yo, . . ' 9

) = [(yo, * * * 7 Yn-1, O)].

~n-l)]

By the universal property of pushouts, there exists a unique map w : FPn-' UjIt-[ Bnk+ FP" such that the diagram of Figure 6.1.5 commutes. We show that w is a bijection, for then, since F P - l and Bnkare compact, F P - ' Ujn-, Bnk is compact and as a continuous bijection from a compact space to a Hausdorff space, w is a homeomorphism. In order to show that w is a bijection, it is sufficient to prove that gn-l : Bnk\ Snk-'

----f

FP" \ FP"-'

162 C H A P T E R 6. HOMOTOPY THEORY OF CW-COMPLEXES

- FPn-'

fn-1

,pk- 1

-\

I \ bak-1

btk-1

FP" FIGURE 6.1.5

-

is a bijection. To do this, define a map

-

9fl- 1 : F

as follows: for all [(yo,

P \~F p - 1

Bnk

\ Snk-1

- - - ,yn)] E FP" \ FP"-'

where jj,, is the conjugate of yn and I n

Consequently, gn-lijn-l = 1 and ijn-lgn-l = 1. As in the previous example, the CW-complex FP" has a unique O-cell. We prove next two theorems for CW-complexes which, with the appropriate modifications of the statements, can be adapted to relative CW-complexes.

Theorem 6.1.5 The topology of a CW-complex is determined by the f a m i l y of its closed cells.

6.1. CW-COMPLEXES

163

Proof - Let X be a CW-complex and let U c X be a set whose intersection with any closed cell of X is closed; we want to prove that U n X" is closed, for every integer n 2 0. Because Xu is discrete, U n X o is closed. Assume that U n Xn-' is closed in X"-'. Recall that the skeleton X" is determined by a pushout diagram as in Figure 6.1.6 and therefore, we must prove that

f

ux SXn-l

~ n - 1

I

I

i

FIGURE 6.1.6

f"(U n X " ) is closed in B,". The map f induces a set of characteristic maps for the n-cells of X ; by hypothesis

ux

(fx I X E A}

f"l(un x")= fil(u n ex) is closed in By, for every

X

E A. Hence,

f-'(~n xn)= U !;*(vn xn) A€ A

is closed in

ux B,". 0

Theorem 6.1.6 Let K be a compact subset of a CW-complex X . Then K is contained in a finite u n i o n of open cells of X .

Proof - Let S c K be obtained by taking a point 2, E e n K from each open cell e which intersects K ; our objective is to prove that S is finite. We begin by observing that S n X" = K n Xu is a discrete, closed subset of K and thus, S n Xu is finite. Assume, by induction, that

164 CHAPTER 6. HOMOTOPY THEORY OF C W - C O M P L E X E S

S n X"-'is finite. For every closed n-cell E , S n e consists of at most 2, and the finitely many elements S n Xn-' and therefore, S n E is either empty or is a finite set, in any case, a closed subset of Z. But X" is itself a CW-complex and thus, according to Theorem 6.1.5, its topology is determined by the family of its closed cells; thus S n X" is a closed subset of X" which is discrete and contained in the compact space K and therefore, is a finite set. We have shown that, for every n 2 0, S n X " is a finite set and so, S is a discrete, closed subset of X and of K ; but a discrete, closed subset of a compact space is finite and so, S is finite. 0 As a consequence of the previous results, we have:

Corollary 6.1.7 CW-complexes are compactly generated spaces.

0

Let il be a set of open cells of a CW-complex X and let A be the union of all the cells of 0; we say that A is a subcomplex of X if, for every open cell e E R, e c A and A is given the topology determined by the closure of all cells in 0. Hence, arbitrary unions and intersections of subcomplexes of a CW-complex X are subcomplexes of X.

Theorem 6.1.8 Let X be a CW-complex, let il be a set of open cells of X and let a be the u n i o n of the cells in il. T h e following are equivalent: 1) A is a subcomplex of X . 2) A i s a CW-complex determined by the skeleta A" = A n X " , n 2 0.

Proof - 1) + 2): We must prove that, for every n 2 0, (A",An-') is an adjunction of n-cells and that A, is closed in X " . This is done by induction on n; the case n = 0 is clearly true. Now suppose that, for an integer n such that n - 1 2 0, (An-',An-2)is an adjunction of (n - 1)-cells and An-' is closed in X n - l . Let e be an n-cell of the set il and let : B," + X be a characteristic map for 5. Condition 1) implies that fe(B:) = e c A and so, the attaching map fe : S:-' X factors through An-' that is to say, e is attached to X via the pushout diagram of Figure 6.1.7 . This shows that as a set A" is an adjunction of n-cells; the question is: does A" have the topology determined by the adjunction process ? To study this fact, take U c A" satisfying the conditions: U n A"-' is closed in A"-l and, for every n-cell e E il with --f

165

6.1. C W - C O M P L E X E S

FIGURE 6.1.7

characteristic map fe, f;'(U) is closed in B,". Since A"-' is closed in Xn-1, it follows that U is closed in X " . In particular, taking U = A", we obtain that A" is closed in X " . Hence, U is closed in the subspace A" of X". 2) =+ 1): Trivial. 0

Corollary 6.1.9 Let A be a subcomplex of a CW-complex X . Then A is closed in X and the arrow ( A ,i, X ) determined by the inclusion m a p is a cofibration.

Proof - Since X is the union space of the sequence

xo c x'... c X " c

.

.

I

and A n X" = A" is closed in X " , then A is closed in X . We prove by induction that (A",i,,X") is a cofibration. This is clearly so for n = 0; assume it true for n - 1. The law of horizontal compositions implies that (Xn,X"-l U A") is an adjunction of n-cells and therefore, (X"-l U A n , j n , X n ) is a cofibration. On the other hand, because (An-',i,-l, X"-') is a cofibration and A" n Xn-l= A"-l, then (A",j:, ,P - 1 U A") is a cofibration. Define in = jnj; and use Exercise 2.3.6 to conclude that ( A " , i , , X " ) is a cofibration. To prove that ( A , i , X )is a cofibration, it is enough to show that X x I is a union space for the sequence

xo x I c x' x I c - c X " x I c -

But this is easy: let f : X x I

--f

9

*. *

.

2 be a map such that

f" = f 1 ( X ' , x I ) : X" x I

--t

2

166 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES

is continuous for every n 2 0. The exponential law implies that the adjoint function : X" 2'

p

__f

is continuous for every n and thus, f : X + 2' is continuous; again by the exponential law, f is continuous. 0 In Chapter 4 we have defined the product of two polyhedra; what can we say about CW-complexes in general? It is clear that since CW-complexes are compactly generated spaces (see Corollary 6.1.7), the Cartesian product of two CW-complexes may fail to be a CWcomplex because this product may fail to be a compactly generated space (see Appendix B). However, we shall see that we can always define a "product" - in the categorical sense - of two CW-complexes and still obtain a CW-complex. Let us be more precise. A product of two objects X and Y of a category C is an object P of C together with two morphisms 7rl : P -+ X and 7r2 : P --f Y such that if fl : Q t X and fz : Q -+ Y are two morphisms of C, then there exists a unique morphism f : Q --f P such that 7rlf = fi and 7r2f = f 2 . In Top, the product of two spaces X and Y is just the usual Cartesian product X x Y (endowed with the product topology) together with the projections on the first and second factors; in the category of compactly generated spaces CG, the product is given by X 8 Y = Ic(X x Y ) together with the projections on each factor. To define the product of two CW-complexes we need the following.

Lemma 6.1.10 Let (AI,i1,&) and (A2,i2,y Z ) be cofibrations, and let (Al ,f i ,B1) and (Az, f2, Bz) be given arrows; f o r m the adjunction spaces X I = B1 Uf, K and Xz = B2 Ufi y2. Then,

f2.

where g = f1 x fz Ufl x (If the cofibrations and arrows are defined by objects of the category CG, then the products indicated are t o be viewed as products in CG.)

Proof - We wish to prove that the diagram of Figure 6.1.8 is a pushout. This fact follows from the definitions. 0

6.1. CW-COMPLEXES

167

i T

FIGURE 6.1.8

Theorem 6.1.11 Let X and Y be CW-complexes with skeleta X" and Y", respectively, n 2 0. T h e n there ezists a CW-complex X 8 Y whose skeleta are given by the sets

(X@Y)"=

u

P+q=n

n 2 0, and with the topology determined by the f a m i l y of the products of the closed cells of X by the closed cells of Y .

Proof - Notice that ( X @ Y)"is discrete. The product BP 8 BQ Z BP x BQ (see Theorem B.6) is homeomorphic to BP+Qand its boundary d(BP x Bq) is actually given by

a(BP x BQ)= BP x SQ-'u

sp-'

x Bq

.

Thus the previous lemma shows that the pairs

(XP 8 Y9,XP 8 Yq-'

uxp-l

8 YQ)

+

are adjunctions of ( p q)-cells (products in CG); denote the attaching maps of these ( p q)-cells by fp.q. For a fixed integer n 2 1 and for any pair of non-negative integers p , q such that p q = n,

+

+

XP 8 y9- 1 u XP- 1 8 IY ' c ( X€3 . y

;

moreover, the corresponding attaching maps fp,q fit together to determine a map f n from their topological union to the space ( X @ Y)n-l.

168 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES One can now see that ( X @ Y)" is obtained from ( X @ Y)"-l by adjunction of n-cells via fnIt remains to prove that X @ Y is the union space of the sequence

( X @ Y)" c ( X @ Y)'c

- c ( X @ Y)"c ... . *

This is done in three steps: firstly, we notice that, for every p 2 0, XP @ Y is the union space of the sequence

secondly, X @ Y is the union space of the sequence

X " @ Y c x' @ Y

c ... c X" @ Y c ..* ;

thirdly, X @ Y is the union space of the family {XP @ Y QI p , q 2 0 ) (that is to say,

uP@YQ

X@Y=

P4Z"

with the topology determined by the subsets X P @ Y Q and ) thus, X @ Y is the union space of the sequence

( X @ Y ) " C( x @ Y ) lc

-*.

c ( X @ Y ) " c ..* . n

Corollary 6.1.12 Let X and Y be CW-complexes. If X is locally compact', then X @ Y = X x Y. Proof - Use Theorem B.6. Adjunctions of CW-complexes are CW-complexes as long as the attaching map is cellular : a map f : A + W between two CWcomplexes is said to be cellular if it takes the n-skeleton A" of A into the n-skeleton W" of W , for every n. We have:

Theorem 6.1.13 Let A be a subcomplex of a CW-complex Y and let f : A + W be a cellular map. T h e n X = W Uf Y is a CW-complex containing W as a subcomplex. lSee Exercise 6.1.6 for a characterization of locally compact CW-complexes.

6.1. CW-COMPLEXES

169

Proof - For every n 2 0, construct the space

X " = W" ujn Y" where fn : A" + W" is the restriction of f to A". Note that Xu is a discrete space. We are going to prove that, for every n 2 1 the pair ( X n , X n - l ) is an adjunction of n-cells and that X is a union space of

x" c . . . c X " c . . .,

The first of these assertions will be proved by constructing an intermediate space X"-' c 2, c X " such that ( X " , 2") and (&, X"-') are adjunctions of n-cells with the attaching map of (Xn,2") factoring through X"-'. Assume that we succeeded in constructing 2, with the aforementioned properties. Let g :

s,\= u s;-l +Xn-l &A

h : SAI =

u S:-'

4

2,

p€Al

be the attaching maps for h decomposing as

(Zn, X"-')

SAI

and ( X " , Z,,), respectively, with

% x " - ' --L . 2"

where i is the inclusion. Let j : 2, + X " be the inclusion map. We claim that the commutative diagram of Figure 6.1.9 (where BA and

FIGURE 6.1.9

170 CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES

BAr are the topological sums of n-balls corresponding to the topological , and the vertical arrows sums of (n- 1)-spheres SAand S A ~respectively, are inclusions) is a pushout. For this, take maps I : BA U B,jr -+ 2 and m : X"-l -+ Z giving rise to a commutative diagram when composed with the appropriate maps; then use the universal property of the pushout for (Zn,X"-l) relative to the maps I 1 BA and m to obtain a map k : 2, -3 2 which will be used in the pushout diagram of ( X " , Z , ) to generate a map P : X" -+ 2 such that P

1 Xn-l = m and ~

( j Ug 6)= I

.

We define the space 2, by 2n -- x " - ' u W " . Since (Wn,Wn-') is an adjunction of n-cells, the law of horizontal compositions implies that (Zn,Xn-') is an adjunction of n-cells. The same law also implies that 2,

W" Ufn (A" U Y"-')

.

The space W" U j n (A" U Y"-l) is a pushout space for the diagram determined by fn and the inclusion A" c A" U Y"-l; let

J : A" U Yn-l + W" U j n (A" U Y"-l) S 2, be

a

characteristic map. Taking the inclusion

A"

u Yn-'c Y" ,

viewing f as an attaching map and using the law of vertical compositions we conclude that

s't 2 z,, u j Y'&; since (Y",A" U Y"-') is an adjunction of n-cells (see proof of Corollary 6.1.9), it follows that (P, 2,) is an adjunction of n-cells. Clearly, the attaching map for the n-cells of this pair factors through A" UY"-l and thus, through Y"-'; but the induced map Y"-' -+ 2, factors through x n - 1 , which completes this part of the proof.

6.1. CW-COMPLEXES

171

It remains to prove that X is a union space for the spaces X ” . Let j , : X” + X be the canonical maps and let g : X --+2 be a map such that, for every n 2 0, g j , is continuous. These maps give rise to two sequences of maps {hn : W” + Z 1 n 2 0)

{k, : Y” -+ Z I n 2 0) which, by the universal property of adjunction spaces, produce a continuous function X -+ 2 that coincides with g . 0 The following result is an immediate consequence of the previous theorem:

Corollary 6.1.14 Let X , Y be CW-complexes and let f : X + Y be a cellular map; then the mapping cylinder M(f) is a CW-complex. 0 We complete this section with another consequence of Theorem 6.1.13.

Corollary 6.1.15 Let A be a subcomplex of a CW-complex X . Then X I A is a CW-complex.

Proof - The constant map A {a,} into a 0-cell {a,} is cellular. Now take Exercise 2.1.7 and the previous theorem. 0 It is possible to find counter-examples to the (false) statement: the quotient of two Hausdorff spaces is a Hausdorff space (exhibit a specific counter-example); the previous corollary shows that we should not look for counter examples in the category of CW-complexes. In the next section we shall see that in the category of CW-complexes, all maps are homotopic to cellular maps (see Theorem 6.2.11). .--)

EXERCISES 6.1.1 Prove that the inclusions of the spheres S;-’ into the n-balls B,” give rise to a closed cofibration

172 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES 6.1.2 Let X be a space obtained as a pushout space of the arrows ( A , f , B )and ( A , g , C ) ,with A , B and G compact. Prove that X is compact. 6.1.3 Show that the geometric realization of a finite abstract simplicia1 complex (i.e., a polyhedron) is a CW-complex. 6.1.4 Let X be a based CW-complex. Prove that a function f : X --+ Y is continuous H for every closed cell E x of X , the restriction f I E x is continuous (j for every characteristic map fx of X , ffx is continuous. 6.1.5 Let e be an open cell of a CW-complex X . Prove that the set X ( e ) defined as the intersection of all subcomplexes of X which contain e is a finite subcomplex (hence, a compact subspace) of

X. 6.1.6

* Prove that

a CW-complex X is locally compact iff every open cell of X meets only finitely many closed cells of X . (See [15].)

6.1.7 Let X be the subset of R2obtained by taking all the segments I,, with end-points 81, = {(O,O), (1, i)}, n E N \ (0) (see Figure 6.1.10). Show that X can be constructed as a CW-complex,

FIGURE 6.1.10

6.1. CW-COMPLEXES

173

denoted Xcw. Prove that X with the topology induced from the euclidean topology of R2- call this topological space X , - is not a CW-complex. Prove that the spaces X,, and X , have the same homotopy type. 6.1.8

* Prove that CW-complexes are LEC spaces. (Hint: Use Exercise 2.4.9.)

6.1.9

Let z, be a 0-cell of a CW-complex X . Prove that ( { z o ) , i , X )is a cofibration.

6.1.10 Let X be a CW-complex. Prove that the unreduced cone cX and the unreduced suspension n X are CW-complexes. If X has a base point z, which is a 0-cell, the corresponding cone C X and the suspension EX are CW-complexes; moreover, CX C X and a x EX.

-

-

6.1.11 Let X and Y be CW-complexes with 0-cells zo E X and yo E Y viewed as base points. Suppose that X is finite. Prove that the smash product X A Y is a CW-complex. 6.1.12 Let p : E -+ B be a covering map. Prove that if B is a CWcomplex, so is E . 6.1.13

* Exercise 2.2.3 shows that a covering map p : E + B is a fibration. The previous exercise shows that if B is a CW-complex (note that the fibre is a CW-complex as a discrete space), then E is a CW-complex; it seems natural to ask if the total space E of a fibration p : E + B is a CW-complex whenever B and F are CW-complexes. The answer is no, in general; however, one can prove the following: If p : E -+ B is a fibration over a connected CW-complex B and with fibre F, a CW-complex, then E has the homotopy type of a CW-complex. (See [15, Section 5.41.)

6.1.14 Let $4 : CP' x

be defined by

CP' 4 C P 2

174 CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES Show that

4 factors through a map

4 : (CP1x CP2)/Z2+ C P 2 where (CP1 x CP2)/Zzis the orbit space of the obvious action of 22 on CP' x C P 2 (not a fixed point free action) and is a homeomorphism.

4

6.2

Homotopy theory of CW-complexes

We begin this section with a discussion about the important concept of n-connectivity for a pair of spaces. A pair ( X , A ) is said to be nconnected if every path-component of X intersects A and, for every base point z, E A c X and every 1 5 T 5 n, 7r,.(X,A; 2,) = 0. The following is an interesting characterization of n-connectivity:

Lemma 6.2.1 Let A be a subspace of X and, for an arbitrary choice of base point 2, E A c X , let i : ( A ,2,) + ( X ,z,) be the based inclusion map. Then ( X ,A ) is n-connected iff the path-components of X intersect A and, for every 1 5 T 5 n and every arrow-map ( a , b ) : i,-l + i, there exists an extension b' : B' + A of a such that ib' and b are homotopic rel. S'-l.

Proof - See Lemma 5.1.4. 0 Lemma 6.2.2 Let ( X , A ) be an adjunction of n-cells with n 2 1. Let (Y,B) be a pair such that 7r,(Y,B;y,) = 0 f o r all yo E B c Y . Let Z : A + X and j : B Y be the inclusion maps; then, for any arrow-map (a, b ) : 6 -+ j, there exists a map b' : X + B extending a and such that jb' b rel. A .

-

-

Proof - Form the commutative diagram of Figure 6.2.1 in which the middle square is the pushout diagram giving rise to the adjunction

175

6.2. H O M O T O P Y T H E O R Y O F CW-COMPLEXES

-ax

F

- Ux B;

b

-Y

FIGURE 6.2.1 of n-cells ( X ,A ) and i ~ z;\ , are the canonical inclusions, for each A E A. Because rn(Y,B;yo) = 0, there exists a map

whose restriction t o S:-’ to s,”-1

is afix and there exists a homotopy relative

HA : B; x I

+Y

such that H A ( - , 0) = jb’, and H ( - , 1) = b f ; ~ .These homotopies fit together to give rise to a homotopy relative to UA Si-l

H

:

u B,”

xI

----t

Y

x

from j ( u Ab‘,) to bf. At this point, define b’ : X pushouts and the maps a : A

_t

B using the universal property of

B and

notice that b‘ extends a. The homotopy

G:SxI-+Y

176 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES is relative to A and relates jb' to b.' 0 Note that if n = 0 the previous lemma is interpreted as follows: if (X, A ) is an adjunction of 0-cells and (Y,B ) is a pair such that all pathcomponents of Y intersect B then, for every arrow-map ( a ,b ) : ;i + j there is a map b' : X -+ B extending a and such that jb' b rel. A . N

Theorem 6.2.3 Let ( X , A ) be a relative CW-complex of dimension 5 n and let (Y,B ) be an n-connected pair. Then, for every arrow-map ( a , b ) : i + j there exists a map b' : X --+ B extending a and such that jb' b rel. A . N

Proof - The result is true for the pair ( X - ' , A ) ; now, suppose by induction that for a given k - 1 < n, we have constructed a map

bi-' : X"' and a homotopy

I&-'

-B

:xk-' XI-Y

such that Hk-l(-,O) = jbk-,, Hk-l(-,l) = b I X"' and H h - l ( z , t ) = j a ( z ) , for every z E A. In order to construct the next tier in our set of maps from the skeleta of ( X , A ) to B and the corresponding homotopies, we first have to modify conveniently the map b I X k to this effect. We start by using the fact that the inclusion of X"' into X k is a cofibration: take the homotopy

Hk-1

:x"' x I + Y

given by f i k - l ( z , t ) = H k - l ( z , 1-t)for every ( x , t ) E X"* X Ibecause ; this homotopy coincides with b I X"' when restricted to X"' x {0}, there exists a homotopy

such that HL( -, 0) = b I Sk'and which is equal to f i k - 1 when restricted to Xk-'x I . Notice that, for every ( z , t )E A x I , HL(z,t) = j u ( z ) or, 'The homotopy G could also be defined using Exercise 2.1.5, the universal property of pushouts and the maps ja and H .

6.2. H O M O T O P Y THEORY OF CW-COMPLEXES in other words, HL is a homotopy relative to A. Now take and the arrow-map (bL-1, g) : Lk-1 j

-

177

IT;(-, 1) = 6

(here ~ k denotes - ~ the inclusion of X"-' into X k ) ;because ( X k , X k - ' ) is an adjunction of k-cells, Lemma 6.2.2 shows that there exists a map bi : X k + B which extends bi-, and such that j b i 6 rel. X"'. In particular, j b i b 1 X k rel. A. 0 The following result will be used in Section 6.3.

-

N

Lemma 6.2.4 Let ( X ,A ) be a relative GW-complex whose cells in X \ A are all of dimensions 2 n + 1. Let (Y,y o ) E Top, be such that n,(Y,y,) = 0 f o r every r > n. Then any based m a p g : A + Y extends to a based m a p j :X --f Y. Proof - By induction on the relative skeleta X". Suppose that g has been extended to a map

> n. Let E be an m-cell of X \ A with attaching map f : Sm-' -+ Xm-l. The hypothesis on Y implies that jm-lf factors

with m - 1

extends to a map through B" (see Exercise 2.3.13) and thus, from Xm-l Uf B" to Y . Of course, this can be done with every finite adjunction of m-cells to X"-'. For the general case, we use Zorn's Lemma. Let U be the class of all pairs ( X r - l , i j z - l ) where X r - l is obtained by the adjunction of finitely many m-cells to X"-l and 9m-1 -a :x,"-l--ty

is an extension of jm-l. Partially order the set U by inclusion of the sets and restriction of the maps. We observe that increasing chains of U have upper bounds; in fact, if { (Xr-', ij:i-l)} is an increasing chain of 0 we define g':

uxy a

-

k'

by g' I X r - ' = ijz-l. Thus, U has a maximal element ( X ' , g z ) ; a contradiction argument and Exercise 2.3.13 show that X ' = X " . 0

178 C H A P T E R 6. HOMOTOPY THEORY OF CW-COMPLEXES We have seen that the pair (Sn,Sn-')is an adjunction of two ncells; furthermore, its exact sequence, Theorem 4.2.4 and Theorem 5.3.3 show that (Sn,Sn-')is (n - 1)-connected. Indeed, the more general statement that adjunctions of n-cells are (n - 1)-connected is true, as we shall prove anon.

Theorem 6.2.5 Let ( X , A ) be an adjunction of n-cells with n 2 1; then ( X , A ) is (n - 1)-connected. Proof - Suppose that

clearly, all the connected components of X meet A. Now take the arrows (Sr-l,ir-,F B') (with T 5 n and let (a,b ) : i,-l z

- 1) and

( A , i ,X)

---f

be an arbitrary arrow-map. Because b(B') is a compact subset of X, there are finitely many cells of the adjunction, say el, ,e k , such that

b(B') c A u .El u *

- u *

El,

(see Theorem 6.1.6 ). Since there are no cells of dimension higher than n in X, the open cells e j =_ ej \ ( E j n A ) , j = 1, ,k, are open in X . For every X E A, let px = fx(O), 0 E B,";consider the set

---

y=

\(Pl))U.**U(~k\{Pk))

and observe that b-'(Y) is open in X: in fact,

=A

uf (UXEA(BY \ (0)))

is open in X and b - ' ( Z ) = b-l(Y). In this way we obtain an open covering { b-'( Y), b - l ( el ), - . ,b-' ( e k ) } of B'. Consider ( B " ,9"'') as the geometric realization of a pair ( K ,L) formed by an abstract simplicia1 complex K = ( X ( K ) , T )and a subcomplex L. Lemma 4.2.1 shows that there is a suitable barycentric subdivision K(")of K - and consequently, of L (see Exercise 4.2.1 ) - such


179

that, for every simplex u of K("),either b(l u I) c Y or b(l u for some j = 1, * ,k. Take the following subcomplexes of K :

-

and, for every j =

1 , e . e

1) c

ej,

,k,

At this point we should observe that

and that

Br=IBIUIB1pJ...Up,,j . For each j = 1,.. . ,k, define

I

aBj = Bj n B = {u E T(') b(l u

I) c e j \ { p j } }

;

notice that the pairs (Bj,aBj) are disjoint and regard (I Bj I,/ aBj I) as a relative CW-complex of dimension 5 P 5 n - 1. We now claim that the pair ( e j , e j \ { p i } ) is (n - 1)-connected, for every j = 1,. . . ,k. In fact, the only path-component of ej meets ej \ { p i } ; moreover,

and since the sphere Sycl of radius f is a strong deformation retract of (Bj" \ Sj"-') \ {0}, it follows that ( e j , e j \ { p j } ) has the same relative homotopy groups as (B?,Sy-'). The previous observations and Theorem 6.2.3 allow us to conclude that, for every j = 1,.. . ,Ic, there is a map b> : B3 + e j \ { p j } which extends the restriction of b to exists a homotopy Hj : Bj x I

I

8Bj

----f

ej

I

and furthermore, there

180 C H A P T E R 6. HOMOTOPY THEORY OF CW-COMPLEXES from ijbi (here ij is the inclusion map) to b /I Bj 1, relative to I aBj I. Let be the restriction of b to the space I B \(Us,1 1 Bj I); note that

I

P :I

B I \(U:=l

I Bj

I)

+A

*

The disjointness of these subcomplexes of K ) = B' shows that the maps b', b:, j = 1,.. . ,k fit together to define a map

?,:B'+Y such that

& I S'-*

is the composition

moreover, the homotopies H i , j = 1,.. . ,k and the constant homotopy a(a) for every ( z , t ) E (I B 1 \(Us=, 1 Bj I)) x I also fit together to give rise to a homotopy H :B' x I +

x

-

which is rel. Sr-' and such that 8(-,0) = iy-6, I?(-,l) = b (here i l ~: Y X is the inclusion map). But A is a strong deformation retract of Y with retraction T : Y + A ; thus, define

b' = r6 : B' and notice that

--f

A

-

b'ir-l = rbir-l = r i A a = a

ib' = irb = iyi,4rb , i y i A r b i l - b re1.A ,

,

-

and il,-i b rel. S'-'. N

This concludes the proof. 0

Corollary 0.2.6 Let (X, A) be a relative CW-complex and let 0 5 n < dimX; then (X, X n ) is n-connected. Proof - We first prove that for every m > n, ( X m ,Xn) is n-connected. The previous theorem shows that ( X " + ' , X " ) is n-connected; suppose that m- 1 > n and, by induction, that (Xm-', Xn)is n-connected. Now observe that (S", X m - ' ) is (m-1)-connected (again, by Theorem

6.2. HOMOTOPY THEORY OF CW-COMPLEXES

181

6.2.5) and that the path-components of X" intersect X":in fact, the ( m- 1)-connectivity of (X'", X"-') implies that the path-components of X" intersect Xm-' and the n-connectivity of (Xm-',Xn) implies that the path-components of Xm-' intersect X" or, in other words, the following two functions induced by the inclusion maps are onto:

then, r o ( X " ) + r o ( X m )is onto and so, the path-components of X" intersect X". Now for any z, E X", the exact sequence of the spaces X" c X"-' c X" (see Exercise 5.1.12) shows that

, " ) is n-connected. and hence, ( X m X To prove that (X,X") is n-connected we proceed as follows. For any m > n, let in," : X" 4 Xnl be the inclusion map; also, denote by i the inclusion of X" into X . Now, for any 1 5 T 5 n, take the : s'-* -+ B' and an arrow-map inclusion ( a , b ) : i,-l --+

i

.

Since b(B') is compact, there is an m > n such that b(B') c X" (see Theorem 6.1.6). Because of Lemma 6.2.1 there exist a map b' : B' + X" extending a and a homotopy relative to Sr-' of in,"b' to b; but then ib' is homotopic rel. S'-' to b. Lemma 6.2.1 now proves that ( X , X " ) is n-connected. 0 As an application of the last corollary we prove the following:

vf=l

Theorem 6.2.7 Let Sj" be a finite wedge product of n-spheres, with n 2 2; f o r every j = 1, * ,Ic, let

be the canonical inclusion m a p . T h e n T , , ( V ~ =Sj", ~ ( e , ) ) is the free abelian group generated by the classes [ ~ j ] j, = 1,- - ,k.

182 C H A P T E R 6. HOMOTOPY T H E O R Y O F CW-COMPLEXES

Proof - Regard S” as a CW-complex with just one 0-cell and one 1-cell. Since the r-skeleton of the relative CW-complex

remains unchanged up to and including the dimension 2n - 1 (see Theorem 6.1.11), it follows that k

(J-J j=1

s;, v s>; k

j=1

=

(ns;, (n k

k

j=l

j=1

qSn-l)

and thus, by Corollary 6.2.6

for 1 5 T 5 2n - 1. This and a finite induction argument on Theorem 5.1.7 prove our result. 0 The previous theorem holds true also in the case n = 1, with the appropriate modifications as we are dealing with groups which are possibly non-abelian; however, its proof requires a more general result discovered independently by H.Seifert and E.R. Van Kampen.

Theorem 6.2.8 Let U and V be subsets of a space X ; suppose that the following conditions are satisfied: 1) U and V are open in X ; 2) X = U U V ; 3) U n V # 0; 4) U ,V , U fl V and X are path-connected. T h e n , f o r every base point x , E U n V , the commutative diagram of groups and homomorphisms (induced by the appropriate inclusion m a p s ) depicted in Figure 6.2.2i s a pushout in the category of g ~ o u p s . ~

Proof - For the proof of this theorem we shall regard the fundamental group of a based space (Y,y,,) as the group defined by the homotopy classes relative to 81 of maps ( I ,01)+ (Y, yo). We must prove that given any group G and homomorphisms

3See Appendix A.

183

6.2. H O M O T O P Y T H E O R Y OF CW-COMPLEXES

I

FIGURE 6.2.2

such that u((il)*(l))= v((Z&(l)), there exists a unique group homomorphism 4 : T ~ ( X , Z+ ~ )G such that +(iu)*(l) = u and + ( i l F ) . ( l ) = 21. Let a E r1(X,x0)be represented by a map f : I + X taking aI into 2., Consider the open covering { f-* (U), f-'( V ) }of I ; because of Lemma 4.2.1,we can subdivide I by

-

so finely that, for every i = 1, * ,n, f( [t;-l,t i ] )is contained either in U or in V . We can also assume that f ( t ; )E U n V for every i = 0, ,n: in fact, f ( t 0 ) = f ( t n ) = 2, E UnV ;if for a given i # O , 1 , f ( t ; )E U\ V , then both f( [ t ; - l ,t ; ] )and f( [ t i , t i + , ] ) must be contained in U and hence, we simply could have eliminated the point t; (same argument applies if f ( t i ) E V \ U). For each i = 0, - ,n - 1, define the map

--

-

Notice that f can be written as a composition of the paths f;,namely:

f = f0fi

---

fn-1.

Next, for each i = 0, - - - ,n, choose paths

184 CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES such that X;(O) = z,, Xi(1) = f ( t i ) and with the proviso that A" = An = cz,, the constant path at a,. Then

f = f"f1

--

X"fOX,'X,fl

fn-1

* *

'

Xfl-lfll-lX;l

and since each class [XifiX;,',] (i = O,... , n - 1) belongs to either nl(U, a,) or to 7r1 (V,a,), it follows that

w = [X"fOA,'J[X,f1A,'] is a word of the free product

7rl (U, 2,)

* * *

[Xn-lfn-lXil]

-

* nl (V,a,)

4 : .lrl(U,.O) * nl(V,ao)

and a = +(w), where

7r,(X,aC,)

is the homomorphism induced by (iu)*(l) and (i\-)*(l) (Appendix A). By combining consecutive elements of w which belong to the same component group of the free product, we can assume that the word w is written as

-

w = alp1 - * a,pn

with

cti

E nl(U,z,) and

p; E nI(V,a,),

+(a)=

for i = l , - . . , n . Now we define

U(~I)V(PI) .*.~(an)v(Pn)

*

+

We do not know yet if is well-defined, that is to say, if it is independent of the representation of the elements a;if it is, then It, is a homomorphism and satisfies the required conditions. To show the independence of on the representation of a,it is enough to prove that if a word w = alP1representing a is such that

+

4(w) = ( i ~ ) * ( l ) ( a l ) ( i ~ , )=* (1l ) ( ~ ~ ) then $(a)= U ( Q l ) V ( P * ) = 1 *

Suppose that a1 and

/31

are represented respectively by the maps

f:I-+U,g:.I---tV (both taking 8I into a0). Then 4(w) = [ir;f][i~.g] = [f]where f is defined by composition of paths:

O n 2 1; moreover, assume that ( B ,A n B ) is a relative CW-complex with cells of dimension > m only. Then the homomorphism

+

is an isomorphism for 1 5 r 5 m n - 1 and is surjective i f T = m t n.

Proof - Case I : Suppose that A = ( A n B ) U En+' and that B = ( A n B ) U Em+'. In this case, we proceed exactly as for Theorem 5.3.1 to prove that

.rr,(X;A,B;z,)=O, 2 < r < m t n and then use the exact sequence of the triad ( X ;A , B ; zo)(see Corollary 5.1.9). Note that

because n 2 1 and in view of Theorem 6.2.5. Case 2 Suppose that B = ( An B ) U Em and that A = ( A n B ) U El U U E k , where the cells E l U * U Ek have dimension > n. For every i = 1,' ,k define the CW-complexes

--. --

-

A; = ( A n B ) U E l U * - - UE j

, B; = B U A,

and form the sequences of CW-complexes

and

B(,= B

c B* c . - * c Bk

=

x.

Using the result of Case 1 above, we conclude that for every i = 1, the inclusion j(i) : (Ai,Ai-I)+ (Bi,Bi-i)

. ,Ic,

6.2. HOMOTOPY THEORY OF CW-COMPLEXES

189

induces a homomorphism of the homotopy groups which is an isomorphism for 1 5 r 5 m -t n - 1 and is onto if r = m n. We are going to prove by induction that, for every i = 0,. ,k, the inclusion

-.

+

e(i) : (Aj7A n B ) -+ (Bi,B ) induces a homomorphism .!(i),(~) of homotopy groups which is an isomorphism for 1 5 r 5 m n - 1 and is onto if T = m n. The assertion is clearly correct for i = 0; suppose it is proved for i - 1. The maps j(i) and l ( i ) induce a morphism from the exact sequence of A0 c A;-1 c Ai (see Exercise 5.1.12) into the exact sequence of the sequence Bo c c B;. Now we use the "five lemma" to prove the required statement (note that for r = 2,

+

+

since n 2 1). Case 9: Assume that A = ( A n B ) U El U . - .U e k , where the cells El U U t?k have dimension > n; furthermore, suppose that B = ( A n B ) U E ; U . . . U Z ; , wherethecells E T , i = l,...,.!,havedimensions > m. For every 1 5 i 5 1, define the CW-complexes

..-

and

Xi = A U Bi; then form the sequences

and

c Xe = X . Xu = A c Xi c Factor the inclusion map i : (A, A n B ) (X, B ) as ---f

ze,(Xl, Bc) = (X, B) and note that, because of Case 2 above, each map

190 CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES induces, for each positive integer

T,

a homomorphism

which is an isomorphism for 1 5 T 5 m -t n - 1 and an epimorphism for P = m n. Hence i * ( r ) has the same properties. General case: Let [b,a] E r,(X, B;zo)be given; since b(B‘) is compact there exists a finite subcomplex M of X containing it, according to Theorem 6.1.6. Form the commutative diagram of homotopy groups as in Figure 6.2.3 and observe that if 2 5 T 5 m -t n, (i I M ) * ( T )is

+

r , ( A , A n B ; z0)

i*(T)

Ij.B;

*%(X,

a,)

FIGURE 6.2.3 onto (use the previous cases). But [b,a]is in the image of j , and so,

+

is onto, for every 1 5 T 5 m n. On the other hand, let [b,a],[b’,a’] E r , ( A , A n B ; ao) be such that i * ( ~[b, ) (a ] ) = i*(~)( [b’,a’]). Then there exists an arrow-map of based homotopies ( K ,H ) : i,-l x 11 -+ ia (where iB : B such that

--f

X and i,-l

K(-,0)= b , K ( - J )

:

Sr-’

-+

B‘ are the inclusion maps)

= b’,H(-,0) = a,H(-,l) = a‘.

Let M be a finite subcomplex of X which contains the compact subspace K(B‘ X I )This . gives rise to a commutative diagram of homotopy

6.2. HOMOTOPY THEORY O F CW-COMPLEXES

191

groups just like the previous one. Consider, in particular, the homomorphism ( j ’ ) * ( r :) R,(M n A , M n ( A n B ) ;z0)

-

T,(M, M n B ;z0)

induced by inclusion and take the elements [b,, all, [bi,a:] of R,(M n A , M n ( A n B ) ;z0) SO that

But (i 1 M ) , ( T )is one-to-one for 1 5 r 5 rn

+ n - 1 and

+

proving that [b,a]= [b’,a’] for 1 5 r 5 rn n - 1. 0 Corollary 6.2.6 is also used to prove that any map between two CW-complexes is homotopic to a cellular map:

Theorem 6.2.11 Let X and Y be CW-complexes and let L c X be a subcomplex; also, let f : X + Y be a map whose restriction to L is cellular. Then there exists a cellular map g : X + Y such that g f rel. L.

-

Proof - For every integer n such that 0 5 n 5 dimX, take K” = X ” U L and define the map

F :X x F(z,t)=

{

(0)U L x I + Y

z E X and t = 0, f(4, f ( x ) 7 (0) E L x I.

Now, for each z E Xu \ L , choose a path A, : I + Y such that X,(O) = f(z) and X,(l) E Y o (this can be done because, either f ( z ) is a 0-cell, or f(z) is connected to a 0-cell of Y by a path since every path-component of a CW-complex contains at least a 0-cell). Next, define

F” : KC’ x I

+Y

192 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES Note that

F" I ( X " x (0) u L x I ) = F 1 ( X " x (0) u L x I ) and Fo I (KO x (1)) is cellular; in particular, Fo Suppose that we have defined a map

Fn-l

: K"-'

I (X"x (1)) c Y o .

xI+Y

such that

F"-1

1 (Xn-lx (0) u L

and

x I)=F

I (Xn-1

x (0) uL x I )

F"-l(x"-'x (1)) c Y"-l .

Let e be an n-cell of X \ L with characteristic map

and attaching map c, : S'+l T,

given by

: B" x

-

--t

I

X"-'

. Observe that there is a retraction

B" x (0) U S"-lx I

t )=

T " ( 2,

(The reader might have encountered this retraction while proving that the arrow formed by the inclusion of a sphere into the corresponding ball is a cofibration: see Exercise 2.3.2.) Now define the map

b,. : B" x I as the composition of

ze x 11 : B"

T~~ with

x (0)

E'

the maps

u S"-l

and

F 1 ( X x (0)) u Fn-l

-

:

x I + X x (0)

u K"-l

x x (0) u KrL--Ix I

--+

x I

Y.


193

Because the restriction of be to Sf'-' x { 1) maps that space into Y"-l C Y" and the pair (Y,Y")is n-connected (see Corollary 6.2.6),

thus, be I (S"-l x (1)) extends to a map &e

: B" x (1) + Y"

and, denoting the inclusion of Y" into Y by j,, there is a homotopy

HZ : B" x I

+Y

relative to S"-l x (1) between jnbe and be I (B" x (1)) (see Lemma 5.1.4). Now we can construct a commutative diagram (see Figure 6.2.4) whose square is a pushout and thus, giving rise to a map

F," : X x (0) u (K7'-lu e) x I

-+ Y

.

(The map F' of Figure 6.2.4 is the restriction F 1 ( X x (0)) U Fn-l.) The crucial property of the restriction F," 1 (P-' U e) x (1) is that

in-1

x 11

Y FIGURE 6.2.4

194 C H A P T E R 6. HOMOTOPY THEORY OF CW-COMPLEXES such a map is cellular. Repeating this process for all n-cells of X we obtain an extension of Fn-I to a map

\ L,

F, : K" x I + Y such that F n ( X n x (1)) c Y" and

F, I ( X " x (0) U L x I ) = F I ( X " x (0) U L x I ) . The union space of the sequence the maps F, produce a function

{K" I n 2 0) coincides with X ;

G:XxI+Y which is continuous and is a homotopy rel. L between f and g = G(-,l), a cellular map. 0 Theorem 6.2.11 above is the so-called cellular approximation theorem; it should be juxtaposed to the simplicia1 approximation theorem. We conclude this section by proving the Whitehead realizability theorem:

Theorem 6.2.12 Let X , Y be CW-complexes and let f : X + Y be a weak homotopy equivalence. Then f is a homotopy equivalence. Proof - Because of the cellular approximation theorem we can assume that f is cellular and hence, that the mapping cylinder M ( f ) is a CW-complex (see Corollary 6.1.14). Moreover, by the definition of weak homotopy equivalence, r,(f,zC,) = 0, for every n 2 1. Now recall that the map f can be written as f = r f i ( f ) ,where r f is a homotopy equivalence and ( X ,i(f),M ( f ) ) is a cofibration (see Theorem 2.3.9); because the homotopy groups T , ~ ( T [f(zCo)]) ~ , = 0 for every n 2 1, Theorem 5.1.6 applied to the commutative triangle determined by the decomposition f = ?ti( f), shows that 7rn(i(f),z,)= 0 for every n 2 1, that is to say, that the pair ( M ( f ) , X )is n-connected, for every n 2 1. In order to prove that f is a homotopy equivalence, we are going to prove the equivalent statement that X is a strong deformation retract

6.2. H O M O T O P Y T H E O R Y OF CW-COMPLEXES

195

of M ( f ) (see Corollary 2.4.2).4 To show that X is a strong deformation retract of M ( f ) we need to deform the identity map

into a retraction

T

:M ( f )

-, X

-

via a strong deformation retraction

G : M ( f )X I

X

.

The construction of G follows exactly the same steps as the construction of the homotopy G in Theorem 6.2.11; in the present case we use the nconnectivity of ( M ( f ) ,X ) for every n 2 1 (instead of the n-connectivity of ( X , X " ) as in the cellular approximation theorem). 0

EXERCISES 6.2.1 Let f : (Y,yo)--$ (X,z,) be a base preserving map. Prove that f is a weak homotopy equivalence if, and only if,

f*(O) : .I,"(Y,YO)

-

.I,o(X,Z,)

is onto and, for every arbitrarily given pair of (based) CW-complexes ( K , L ) - note that L is a subcomplex of K - and base preserving maps b : K --+X , a : L --f Y such that f a = b I L , there exists an extension b' : K --+ Y of a such that fb' N a,rel. L.

6.2.2 A map f : X --+ Y is an n-equivalence if f*(O) : 7ro(X,zo) + 7ro(Y,f(z0)) is a bijection and 7rr(f,zo)= 0, for every a, E X and every 1 5 r 5 n. 1. Prove that f : X

n-connect ed .

--+

Y is an n-equivalence iff ( M ( f ) X, ) is

2. Prove that if f : S + Y is an n-equivalence and K is a CW-complex, the induced function

'If you have not read Section 2.4, try Exercise 6.2.3.

196 CHAPTER 6. H O M O T O P Y THEORY OF CW-COMPLEXES is a surjection if dim K 5 n and is an injection if dim K 5 n - 1. (Hint: Use Theorem 6.2.3 with X = K , A = 0 for the first part and with X = K x I , A = K x 6’I for the second one.)

6.2.3 Use the previous exercise to give another proof for Whitehead’s realizability theorem. (Hint: Assume K to be X and then Y . ) 6.2.4

Let ( X ; A , B ; z , )be a triad such that ( A , A n B ) is an nconnected relative CW-complex with n 2 1, and ( B ,A n B ) is an m-connected relative CW-complex. Prove that the inclusion map ( A ,A n B ) -+ ( X ,B ) induces a homomorphism

*

+

which is an isomorphism for 1 5 T 5 n m and an epimorphism for T = n+m. (This is the homotopy excision theorem. A possible source of good help in solving this - and the next - problem is [33].)

6.2.5

* Let X

be an n-connected CW-complex, n 2 0. Prove that the suspension homomorphism

is an isomorphism for 1 5 T 5 2n and an epimorphism for 2n 1. (This is the fieudenthal suspension theorem.)

+

T

=

6.2.6 Let L : X -+ QCX be the adjoint (under the exponential law) of the identification map l ~ .Prove ~ . that the diagram of Figure 6.2.5 is commutative. (This shows that ~ , ( n is ) an isomorphism iff &(n) is an isomorphism.) Clearly, the suspension homomorphisms are isomorphisms if X is contractible; the following is an example in which these homomorphisms are not isomorphisms: take X = S-‘ and n = 7 ; it is known that n7(S4) 2 Z @ Z I 2 and that 7r8(S5)E Z,, - see [34].)

6.3. EILENBERG-MAC L A N E SPACES

197

FIGURE 6.2.5

6.3

Eilenberg-Mac Lane spaces

In this section we shall study a very important class of CW-complexes, first investigated extensively by S.Eilenberg and S. Mac Lane (see [13] and [14]), known as Ezlenberg-Mac Lane spaces. Formally, we shall say that a path-connected CW-complex X is an Eilenberg-Mac Lane space of type (?r,n)- or, X is a K ( T , ~for ) short - if X has only one nontrivial homotopy group, namely x,(X,x,) = T. An obvious example is the sphere S' which is a K(2,l). We are going to prove that we can construct K ( r , n ) ' s for every integer n 2 1 and every group ?r (abelian, if n > 1). Our EilenbergMac Lane spaces will be contructed as CW-complexes with a unique 0-cell to which all the other cells are attached; this will be achieved with constructions similar to our general construction of CW-complexes but using wedge products of spheres and balls instead of topological unions of such spaces. We begin with a technical lemma.

Lemma 6.3.1 Let X be a CW-comples with a 0-cell xo acting as a base point and let (Xm)be the set of all finite subcomplexes of X with base point a,. Then, for every n 2 1,

198 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES Proof - The set { X a } can be viewed as a direct system of spaces indexed by its own elements (by inclusion) and X is its direct limit (see Appendix A). For X , G X p , denote by $,,p the inclusion map and form the direct system of groups

Next, for each a E A, take the homomorphism

induced by the inclusion map h, : X, + X . Our result will be proved if we show that the hypotheses of Theorem A.3 are valid in the present context. For every [f]E r n ( X , z o ) ,there exists an index a E A such that f(Sn)C X , (see Theorem 6.1.6). Now regard the map f as the composition f = h,f, where f : S" --f X,; hence,

and condition 1) of Theorem A.3 holds true. Next suppose that [g]E r,(X,,zo)is such that

where c,, : 5'" -+ X is the constant map at 2,. This fact, together with an appropriate reformulation of Lemma 5.1.4, shows that hag can be extended to a map g' : B"+' + X . But g'(B"+') is a compact subset of the CW-complex X and thus, by Theorem 6.1.6, it is contained in some finite subcomplex Xp of X and so, 9' decomposes as hog', with

Arrange matters so that X u C XR and so, (4,,a)g extends to

showing that (4a,p)*(n)([9]) = 0. This is condition 2) of Theorem A.3. 0


199

Remark - The lemma is still valid if we assume X to be a union space of a sequence of based CW-complexes

The previous lemma allows us to enlarge the scope of Theorem 6.2.7 and Corollary 6.2.9:

Theorem 6.3.2 Let v, Sl be a wedge product of spheres indexed by a set A; for each a E A, let L,

:

s:

4

v s,n a:

be the canonical inclusion map. If n > 1, n,(v, S ,: ( e , ) ) is a f r e e abelian group generated by the hornotopy classes [L,], If n = 1, the group R ~ ( VSA, , ( e , ) ) is free with generators [ha]. Proof - The theorem is an immediate consequence of Lemma 6.3.1, Theorem 6.2.7 and Corollary 6.2.9. 0

Corollary 6.3.3 Let n 2 1 be a given integer and let T be a free group (abelian if n > 1) with generating set A. Then there exists a based, path-connected CW-complex M ( T ,n ) such that

Proof - Consider the CW-complex M(.rr,n) =

v SE ,Ell

with base point 2, = (e, ); the previous theorem shows that the nth homotopy group of M ( T , ~is) the free (abelian, if n > 1) group with generating set 0 = {[ha] I a E A} . The function q5 : 0

t

A

, q 5 ( [ ~ ~= ] ) CY

induces an isomorphism r , ( M ( T , n ) , z , ) S T . 0

200

C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES

Lemma 6.3.4 Let n 2 1 be an integer and let R and F be f r e e groups (abelian if n > 1) with generating sets I? and @,respectively. For every homomorphism 4 : R -+ F there exists a m a p f : M ( R , n ) + M ( F , n ) , unique up to homotopy, such that

f*b):

4 w , + o )

coincides with

4,

-

nn(M(F,n),yo)

Proof - Recall that

v SI

M ( R , n )=

XEr

and that

M(F,n)=

v S; . U€@

For every a: E I? take represented by a map

4([~,]) E 7rn(VvEaS:,yo) fZ

: S" +

and assume it to be

v s; YE@

which, because of the homeomorphism S" S S:, can be viewed as a map from S:, namely fx :

s,. + v s; . I/€@

These maps fit together to produce a map

f:VS,.+ XEr

vs; I/€ @

such that f*(n)([bx]) = 4([Lx])* To prove uniqueness, assume that there exists another map

g:vs:-vsyn TEr

ll€@

-

with the required properties. Since T,,( M ( R, n ) ,2,) is generated by the classes [ L ~ ] it , follows that [g~,] = [fZ], and hence, gZ = gLx fx, for every a: E I?. Thus, g f. 0 The next theorem shows that we can construct spaces M(.rr,n) as in Corollary 6.3.3 for any group (not necessarily just free groups).

-


20 1

Theorem 6.3r5 Let n >_ 1 be an integer and let x be a group (abelian i f n > 1). Then there exists a path-connected CW-complex M(.rr,n) with a unique 0-cell and cells of dimensions n and n 1 only, such that

+

0, O < ~ < n - l

T,(M(x,n),z,)

x, ~

=

n

Proof - As we have seen in the previous corollary, the theorem is true if x is free. Let dJ

l-R+F+x+l be a free presentation of x . Let f : M ( R , n ) -+ M ( F , n ) be a map such that f,(n) = q5 (see Lemma 6.3.4). Case 1 - n > 1 : Theorem 5.1.6 and the properties of the spaces M ( R , n ) and M ( F , n ) now imply the exactness of the following sequence: R 4 F -+x n ( f , z , )+ 0 ----f

and so, n,(f,z,) S T . The question is now how to realize xn(f , z,) geometrically. To this end, we proceed by taking the following steps: Firstly, we construct the (based) mapping cylinder o f f , namely the space obtained by adjunction of the cylinder M ( R , n ) x I to M ( F , n ) via the cellular map

f : M ( R , n ) x { 0 } u {z,} x I

. f

M(F,n)

taking (2,) x I into 8,; this is done with the aid of the pushout diagram of Figure 6.3.1. Theorem 6.1.13 shows that M ( f ) is a CW-complex with M ( R ,n ) as a subcomplex. Secondly, we recall Exercise 5.1.7 to conclude that

Thirdly, we construct the cone C M ( R , n ) of the based space M ( R , n ) (see Section 2.3) which is also a CW-complex in view of Corollary 6.1.15. Take the map

i :M ( R , n )

-

M(f),z

H

(z,l)

202 CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES

I

I M(R,n)x I

M(f) f

=

M ( F , n ) Uf ( M ( R , n )x I )

FIGURE 6.3.1

FIGURE 6.3.2

and attach the cone C M ( R , n ) to the mapping cylinder M ( f ) to produce a CW-complex M ( f ) U C M ( R , n ) (see Figure 6.3.2 and use Theorem 6.1.13). Our fourth step is to use Theorem 6.2.10 in conjunction with the triad

(M(f)U C M ( & n)i M ( f ) ,CM(& n);s o ) to show that

is an isomorphism, for every 1 5 C M ( R , n )implies that

T

5 n. Finally, the contractibility of

6.3. EILENBERG-MAC LANE SPACES

203

for every T 2 1 and therefore,

(the path-connectivity of M ( f ) U C M ( R , n ) follows by construction). Case 2 - n = 1 : Take M ( f ) U C M ( R , l ) and its open, pathconnected subsets

u = (Wf) u C M ( 4 1))\ C W R ,1 ) and

v = (M(f) u CM(R11)) \ M ( F , 1 ) ;

notice that U has the homotopy type of M ( F , 1 ) and V is contractible. Now use Theorem 6.2.8 to prove that Tl(f,

z,)

= q ( M ( f )u C M ( R ,l),

2,)

2

F/RE II

thus concluding the proof of the theorem. 0 We are now ready to construct the Eilenberg-Mac Lane spaces of type (n,n)(with x abelian if n > 1 ) .

Theorem 6.3.6 For every integer n 2 1 and every group n (abelian if n > l), there exists a CW-complex X = K ( n ,n ) with only one 0-cell 2, such that T, r = n r r ( X i ~ o ) 0, r # n . Proof - We first construct the CW-complex M(.lr,n) as in the pre-

-

vious theorem. Next, take a free presentation

0

--+

R

F 9, T,+1(M(x,n),z0) -+ 0

where F has a generating set represented by a map

a.

fv : Sn+l

Suppose that for each y E

M(n,n) ;

altogether these maps determine an attaching map

f:

V S,”+l--+

v€@

M(a,n)

a, q ( y ) is


i

v,

B,:+2

f

-

xn+2

FIGURE 6.3.3 and in this way, we construct a relative CW-complex ( X , + Z ,M ( T ,n)) having only one 0-cell, namely z, and cells of dimensions n, n 1 and n -t 2 (see Figure 6.3.3). Theorem 6.2.5 implies that (Xn+2,M(7r,n)) is (n+l)-connected and thus, rr(Xn+2, $ 0 ) 2 nr(M(r,n),zo)

+

for every T 5 n. Denote the inclusion of M ( r , n ) into each generator y ,

+

i*(n i1 M Y ) ) = i*(n 1)([fUl)

=

[%I

Xn+2

by i. For

=0

because each i f v factors through B;+’ and therefore, is homotopic to a Z constant map (see Exercise 2.3.13); this implies that 7rn+I(Xn+2,zo)

0. Continuing in this way, we construct a sequence of based CWcomplexes M ( n ,n ) c Xn+2 c Xn+, c . * * whose union set is the path-connected, based CW-complex X we are searching for; of course, we must use the Remark after Lemma 6.3.1. 0

Lemma 6.3.4 has its counterpart for arbitrary groups

7r

and

K’ :

Lemma 6.3.7 Let n 2 1 be a n integer, K and 7r’ groups ( d e l i a n zf n > 1) and let p : 7r 7r‘ be a homomorphism. For every n > 1 there exists a m a p (unique up t o homotopy) _t


205

such that g,(n) = p .

Proof - Suppose that x and x' have, respectively, the following free presentations: 4 l+R+F& n+l

The homomorphism p : n --+ x' induces group homomorphisms a : R + R' and /3 : F --+ F' such that /3c$ = #a and pq = q'p. In view of Lemma 6.3.4 we can construct a homotopy commutative diagram of based spaces and base preserving maps as in Figure 6.3.4 such that a,(n) = a,b,(n) = p , f*(n)= 4 and fi(n)= #. Now we use

b T

FIGURE 6.3.4 this diagram, the homotopy equivalences M ( f ) M ( F , n ) , M ( f ' ) M ( F ' , n ) , the pushout giving rise to M ( n ' , n ) = M ( f ' ) U C M ( R ' , n )and the fact that the arrow ( M ( R , n ) , i , M ( f ) )is a cofibration, to define a map ij : M(f) + M ( x ' ,n ) N

N

such that ji is equal to a map which factors through C M ( R ' , n ) and so, is homotopic to the appropriate constant map. Thus, we can factor ji through C M ( R , n ) (see Exercise 2.3.13) and hence, by the universal property of the pushout diagram which defines M ( x , n ) , we obtain a map g : M ( x ,n ) + M ( n ' , n ) and indeed, the homotopy commutative diagram of Figure 6.3.5 which

CHAPTER 6. HOMOTOPY THEORY OF CW-COMPLEXES

206

b

a T

9 T

T

FIGURE 6.3.5 shows that g has the required property. Now we extend the previous lemma for Eilenberg-Mac Lane spaces.

Theorem 6.3.8 Let n 2 1 be an integer and let 7r and (abelzun .if n > 1). For every group homomorphism p : T exists a map (unique up to homotopy)

be groups + r' there

T'

r : K ( n , n ) -+ K ( n ' , n )

such that T , ( n ) = p.

Proof - We first define a map i; : M ( K ,n )

such that +*(n) = p. Suppose that

with

7r'

K ( d ,n )

has a free presentation

F' determined by a generating set a'. By construction,

Let

and

i

:

v s,. Y€@'

+M ( n ' , n )

6.3. EILENBERG-MAC LANE SPACES be the inclusion maps. For each y E

far : s;

-f

207

a' let K(r,n)

be a representative map of i * ( n ) [ ~ = ~[i~,] ] E r'. Define

F :

v Sc

-+

K(?r',n)

,€@'

by setting F 1 S: = fi/.Recall that ( [ L , ] } is a generating set for the group ?r,(V, S:, 2,) (see Theorem 6.3.2); since

F*(4([Larl)= [ F L U 1 = [far1= ; * ( 7 4 ( [ L Y l )

s,",~,),

then, for every Q! E nn(Var F*(n)(a) =L(~)(cY). Suppose that f : S" + VUEa,S: is an attaching map for an ( n + l ) cell E of M ( d ,n ) ;then the diagram of Figure 6.3.6 shows that if factors S"

f

=v

FIGURE 0.3.0

through B"+' and so, i*(n)([f]) = 0. But then F,(n)([f]) = 0 implying that F f : S" + K ( r ' , n ) factors through B"+' and so, F extends over Z; in this way we obtain a map j : M(7r',n) --+ K(7r',n)

which extends F . Now consider the commutative diagram of Figure 6.3.7 and show that the homomorphisms i,(n) and F,(n) are onto. The first of these homomorphisms is onto because the pair ( M ( r ' ,n ) ,V, S,")


FIGURE 6.3.7

is n-connected; the second, because of the definition of F and the fact that the generating sets of T’ and K,(V, Si,zo)are in a bijective correspondence, In particular, j,(n) is onto. Notice also that if Q E nfl(M(n’,n),z,) is such that j * ( n ) ( a ) = 0, then there exists a p E n,(V, Sy”,2,) such that i.(n)(P)= a = 0; thus, j,(n) is an isomorphism. We now consider the group homomorphism

j*(n)-lp: 7r

_f

K

I

.

By Lemma 6.3.7 , there exists a map

k : M(n,n)

---)

M(n‘,n)

such that k,(n) = j * ( n ) - * p ; now define ? to be the composition ? = jk : M(7r,n) --f K(n’,n)

.

We wish to extend? to amap K ( K , ~--+) K ( d , n ) ;this can be done trivially with the aid of Lemma 6.2.4 but then we have to contend with , the isomorphism i,(n) induced by the inclusion of M ( K ,n ) into K ( K n). The trick is to consider the group homomorphism pi,(n) : K construct a map T’ : M(7r,n) K(r‘,n)

-

+ K’,

as before so that rL(n) = pi,(n) and then, extend r’ to K ( n , n ) via

Lemma 6.2.4. Then, r*(n)i,(n)= rL(n>= pi*(..)

209


.

and so, ~ , ( n= )p Finally, we must prove that the map T is unique up to homotopy. Suppose that TO,T1 : K ( n ,n) + K(n',n) are two maps satisfying the condition (~o)*(n) = ( q ) * ( n )This . implies, in particular, that

for every y E homotopy

a'.

Hence,

H

and therefore, there exists a

[ T ~ L = ~ ] [TIL~]

:TO

1 V S;

TI

Y

I V S; . Y

Define the map

G : M(n,n) x (0) U

v Si

x I U M ( n , n ) x (1)

Y

-

K(?r',n)

and extend G to a homotopy

K :M(n,n) x I

-+

K(a',n)

via Lemma 6.2.4. The next step is to define a map

L : K(7r,n) x (0)

u M ( n , n ) x I u K ( n , n ) x (1) + K (n ' , n )

by the conditions

L 1 (K('.,n) x (0)) = Tu L I ( M ( n , n )x I ) = K L 1 ( K ( n , n )x (1)) = TI

210 C H A P T E R 6. HOMOTOPY T H E O R Y OF CW-COMPLEXES and then use again Lemma 6.2.4 t o extend L to a homotopy

M :K(n,n) x I -

K(d,n)

connecting TO and T ~ 0 . The proof of the previous theorem can be repeated "ipsis litteris" if, instead of K ( d , n ) we take an Eilenberg-Mac Lane space X of type ( ~ ' , nthat ) is to say, X is a based space satisfying the conditions

More precisely, we have the following result:

Theorem 6.3.9 Suppose that we are give an integer n 2 1, the groups A and T' (abelian if n > 1) and an Eilenberg-Mac Lane space of type (T',n). Then for evey group homomorphism p : T ---f A' there exists a based map P : K ( T , ~+ ) X such that r,(n) = p. 0 Corollary 6.3.10 Eilenberg-Mac Lane spaces of type (;rr,n)are unique up to homotopy type.

Proof - Just take K' = A and p = 1, in the previous theorem. 0 The technique used in the proof of Theorem 6.3.6 can be used to easily prove the following result:

Theorem 6.3.11 Let ( X ,xo) be a based path-connected space and let n 2 1 be an integer. Then there exists a based path-connected space X ( n ) such that: 1. ( X ( n ) , X ) is a relative CW-complex with cells of dimension 2 n 2;

+

2. r,(X(n ), e , ) = 0, for every

3. if a; : X

+X(n)

T

_> n + 1;

is the inclusion map,

is an isomorphism, for every 0 5 r 5 n. 0


f

X

211

-Y

FIGURE 6.3.8

Lemma 6.3.12 Let ( X , z o ) , ( Y , y o )E Top, be path-connected and let f : ( X ,xo) 4 (Y,y o ) be a bused map; finally, let 1 5 m 5 n be integers. Then there exists a based map fn,n : X ( n ) + Y ( m ) such that the diagram of Figure 6.3.8 commutes,

Proof - By construction, the relative CW-complex ( X ( n ) , X )has cells of dimension 2 n 2 only; also by construction, the homotopy groups ?r,(Y(m),y,) = 0 for all T 2 M 1 and so, in particular, these groups are trivial for all T > n t 1. Now apply Lemma 6.2.4. 0 Lemma 6.3.12, coupled with Theorem 6.3.11, shows that for every based path-connected space ( X , z , ) there is a tower of spaces and maps as in Figure 6.3.9, where the maps pn,n-l are induced by the identity map l,y : X + X , for every n > 1. Furthermore, the arrows ( X ,i;, X ( n ) ) satisfy the properties indicated in Theorem 6.3.11. Finally, notice that decomposing the map

+

Pn+l ,n : X ( n t

+

1) + X ( n )

into a fibration and a homotopy equivalence as in Theorem 2.2.7, we obtain a fibration whose fibre is a K ( K , , + ~ (z,), S , n 1);this statement is proved by inspection of the exact sequence of the fibration (see Theorem 3.1.5) and the homotopy groups of the spaces involved. The tower of spaces and maps described in Figure 6.3.9 is called the Postnikov tower of the space X .

+

212 CHAPTER 6 . HOMOTOPY T H E O R Y OF CW-COMPLEXES

x FIGURE 6.3.9

EXERCISES 6.3.1 Prove that for every space X there exists a CW-complex X' and a map f : X ' --+ X which is a weak homotopy equivalence. 6.3.2 Show that for every abelian group 7r and every integer 7t 2 1, K(7r,n) has the homotopy type of an H-space. (Hint: Use the fibration of Lemma 2.2.5.) 6.3.3 Let

7r

be an abelian group. Prove the following statements:

(a) For every based CW-complex ( X ,zo)and every integer n

1, the set

Byx,7r) = [ X ,K ( a ,n)]*

has an abelian group structure.

2


213

(b) A based map f : (Y,yo) + ( X )z,) of CW-complexes induces a group homomorphism

q f: )

Hn(Y,R)4 H"(X)R)

for every n 2 1. (c) If f above is a homotopy equivalence, then the homomorphisms H"(f) are isomorphisms. (d) For every based CW-complex ( X ,z,) and every integer n 2 1,

B"X,

R) %

H"+'(CX, R ) .

(e) If ( X ,a,) is based CW-complex and ( A ,2,) is a subcomplex of X , the sequence

B"(X/A,n) +~ " ( X , R 4) B " ( A , R ) is exact at

P(x,R).

The group a " ( X ,R) is the nth (reduced) cohomology group of X .

6.3.4 Let X be a finite, connected CW-complexe with 0-cells and 1-cells , is a finitely generated free group. only. Prove that R ~ ( Xzo)


Chapter 7 Fibrations revisited 7.1

Sections of fibrations

In Section 1.2 we have hinted that the sets [ X , Y ]and [ X , Y ] ,are related; in this section we wish to study this relationship. To this end, we shall view [ X , Y ](respectively, [ X , Y ] , )as the set of homotopy classes (respectively, based homotopy classes) of sections of the trivial fibration ( X x Y , p r l , X ) - where prl is the projection on the first factor - and proceed to work at the level of sections. A section of prl is just a map s : X --+ X x Y such that prls = 1s. Let s e c p l be the set of all sections of prl with the topology induced from the compact-open topology of M ( X , X x Y ) .

Lemma 7.1.1 secpq

S

M(X,Y).

Proof - The function

6' : M ( X , Y ) -+ secprl defined by 6'(f)(z) = (z,f(z)), for every f E M ( X , Y ) and 2 E X , is a bijection. To show that 6' is continuous at fo E M ( X , Y ) , take WK,u in the sub-basis of open sets of secprl such that B ( f o ) E W ~ C Now ,~.

implies, by a generalization of the tube lemma (see [24,Lemma 3.5.8 and Exercise 3.5.10]), that there exist open sets U.y and Ul.of X and

CHAPTER 7. FIBRATIONS REVISITED

216

Y , respectively, such that

then f, E W K , ~and , . 9(W~,u,.)c W K , ~The . inverse map 8-1 is also continuous. 0 For two given based spaces (X, z,) and (Y,yo), let sec(,o,vo)p q be the set of all base preserving sections of the projection X x Y + X with the topology induced from the compact-open topology of M,(X, X x Y ) . As before, there is a homeomorphism 8, : sec(,o,v,)p q 2 M , ( X , Y ) . Now let us bring in the homotopy classes. Given the homotopy H : X x I + Y , construct the homotopy

fi : x x I

-

X x Y

, (z,t)

H

(z,H(z,t))

and observe that for each fixed t E I , ( - , H ( - , t ) ) = 8 H ( - , t ) is a section of p r l . Then, the sections f i ( - , O ) and fi(-,l) are homotopic but by a homotopy which is a section at each level t E I; such a homotopy (called a vertical homotopy), is an equivalence relation in secprl and so, partitions secprl into disjoint classes. Let [secpq] be the set of these equivalence classes; we use the notation [sec(,,,y,) p q ] , in the based case. Since 9 and 9, are homeomorphisms, there are bijections [secprl] S [ X ,Y ] and [set(,,,,,) pl],2 [ X ,Y ] , . Hence, we are going to study the relationship between [ X , Y ] and [ X , Y ] ,by studying the relationship between [secprl] and [sec(,,,y,)prl],. We shall do this by analyzing the sections of a general fibration; however, we first prove a lemma which will play a very important role in the development of this section.

Lemma 7.1.2 Let ( A , i , X ) be a cofibration and let ( E , p ,B ) be a f i bration. Let g : X x (0) U A x I E be a map and let H : X x I + B be a homotopy such that the composite map --f

s x (01u A x I -, x x I 5 B equals pg. Then there exists a map G : X x I -+ E whose restriction to X x (0) U A x I is g and such that pG = H (see Figure 7.1.1).

217

7.1. SECTIONS OF FIBRATIONS

X x (0) U A x I

-E 9

i B

X X I

FIGURE 7.1.1

Proof - The hypothesis that ( A , i , X ) is a cofibration implies, on the one hand, the existence of a map 4 : X -+I such that +-'(O) =A (see Theorem 2.3.3); the map

?I, : x x I

+I

, (qt)

tqq.)

F+

is such that $-'(O) = X x ( 0 ) U A x I . On the other hand, ( A , i , X ) is a cofibration iff X x (0) U A x I is a strong deformation retract of X x I ; let K be such a strong deformation retraction. Define the homotopy

k :( X x I ) x I

+X

x I

by the formulae

and consider the commutative diagram of Figure 7.1.2 in which the map r is just the retraction K ( ( - , -), 0). Because ( E , p ,B) is a fibration, there exists a homotopy

G : (Sx I ) x I

-

E

whose restriction to (X x I ) x ( 0 ) is gr and such that p G = HK. The map

G :X x I

---+ E

( z 7 t )H G ( ( x 7 t ) , $ ( z , t ) )

satisfies the conditions spelled out in the statement of the lemma.

0

C H A P T E R 7. FIBRATIONS REVISITED

218

(X x I)x I

K

-XxI

-B

FIGURE 7.1.2 Note that if the map

g :

x x {O}

uA x I

E

of the lemma is such that g I A x {t} = g I A x { 0 } , for every t E I , then the homotopy G obtained is rel. A . Let (E,p, B) be a fibration. A section of (E,p, B ) - or simply, of p is a map s : B + E such that ps = lg. If the spaces E and B are based and p : ( E ,e,) + ( B ,b,) is a based map, we say that the fibration p is based; a based section of the based fibration p is a base preserving map s : (B,b,) + (E,e,) such that p s = l g . We adopt the notation secp and sec(,,,bo)p for the sets of all sections of p and all based sections of p, respectively. Two sections so and s1 of p are said to be vertically homotopic if there exists a map

H:BxI-+E such that H(-,0) = so, H(-,1) = s1 and, for every t E I , H(-,t) E secp; in the based case, the homotopy must also be rel. { b o } .

Theorem 7.1.3 Two sections of the fibration p are homotopic tically homotopic.

zfl

ver-

Proof - Clearly, if two sections of a fibration ( E ,p, B ) are vertically homotopic, they are homotopic. Conversely, let H : B x I + E be a homotopy between the sections so = H ( -, 0) and s1 = H(-, 1). Define

7.1. SECTIONS

OF FIBRATIONS

the homotopy h : B x I

-+

219

E by

for every ( z , t )E B x I . The homotopy ph : B x I -, B coincides with the projection on the first factor p q : B x I -+ B whenever t = 0 or t = 1 and furthermore, for every t E I and every 2 E B ,

These properties imply that ph is homotopic rel. B x d I to pl:in fact, to begin with define a homotopy

K :( Bx I ) x I

---f

B

,

p h ( z , t ( l - t‘))= 2 ,

05t5

p h ( z , ( l - t ) ( l - t’)),

:5 t 5 1 .

K ( ( 2 ,t ) ,t‘) = between ph and prl; next, define a map

Ic : ( B x I ) x (0) u ( B x 81)x I

+E

w ,t ) ,t’) = h b , t ) and consider the commutative diagram of Figure 7.1.3 where L is a homotopy rel. B x d I obtained via Lemma 7.1.2. The homotopy

v = L((-,-),l)

:B x

I

+E

is the desired vertical homotopy between so and 31. Thus, according to the preceeding theorem, [secp] is the set of the usual homotopy classes of sections of p . For the based case we need an extra condition on the base point of B :

Theorem 7.1.4 Let ( E , p ,€3) be a fibration and Eet e E E be such that the arrow (p(e),i,B)is a cofibration. Then 8 , s’ E sec(,,,(,))p are bused homotopic iff they are based vertically homotopic.


220

( B x I ) x (0)

u(Bx I ) x I

Ic

-E

P

( Bx I ) x I

K

*B

FIGURE 7.1.3

Proof - Corollary 2.3.5 applied to the cofibration (p(e), i, B ) shows that ( ( B x 01) u ({p(e)I x I),L , B x 1) is a cofibration. Now follow the same line of proof as in Theorem 7.1.3, but taking care to consider based maps and based homotopies and replacing the left-hand vertical arrow of Figure 7.1.3 by the inclusion

( B x I ) x (0) u ( B x dI

u {p(e)} x I ) x I c ( B x I ) x I .

This theorem shows that if ( p ( e ) ,i, B ) is a cofibration, [sec(e,p(e)) p] is the set of all usual based homotopy classes of based sections of p . The next result characterizes based vertically homotopy classes of

sec(,,p(,))p,under the now familiar condition on p(e). Lemma 7.1.5 Let ( E , p ,B ) be a fibration, e E E and (p(e),i, B ) be a cofibration. Then s , s' E set(,,,(,)) p are vertically and based homotopic iff there exists a homotopy K : s s' whose restriction t o { p ( e ) } x I is homotopic rel. { ( p ( e ) , O ) ,( p ( e ) ,1)) t o the constant path at e.

-

Proof - + This is clear from the definition of based vertical homot0PY. + Let G : ( { p ( e ) ) x I ) x I --$ E be such that

7. I. SECTIONS OF FIBRATIONS

221

4,

1. G((p(e),t ) ,0) = K ( p ( e ) ,

2- G((p(e),O),t’) = G ( ( P ( 4 I), , t’) = e , 3. W ( e ) , t ) ,1) = e for every t , t‘ E I . Because ( { p ( e ) } x I , L , B x I ) is a cofibration the inclusion map), there is a homotopy

H : ( B x I) x I

(L

is

+E

whose restrictions to ({p(e)}x I)x I and ( Bx I)x (0) are, respectively, G and K. Now we define a homotopy L : B x I -P E by

L(z,t)=

i

H ( z , 3t

-

1, 1),

H ( z , 1 , 3 - 3t),

;5 t 5 p p 5 t 5 1.

Now L is a based homotopy from s to s’ and thus, using Theorem 7.1.4 we conclude that our based sections are indeed based and vertically homotopic. 0 Let X be a space and let IIX be the category whose objects are the points of X and whose morphisms, say from zo to z1, are the homotopy classes rel. 01 of all paths from zu to zl.We denote the set of all morphisms of IIX from zu to z1by rIX(zo,zl). Observe that if 20 = z1, DX(zu,zo) = T * ( X , Z O ) * The category IIX is the fundamental groupoid of X . We also consider the full subcategory II*X consisting of all points z E X such that the arrows ( { z ) , i , X ) - where i is the inclusion map - are cofibrations. In the presence of a fibration ( E , p ,B ) we shall normally be interested in the fundamental groupoid ITE and the full subcategory II#E of points e E E such that ( p ( e ) ,i, B ) are cofibrations.

Theorem 7.1.6 Let ( E , p ,B ) be a fibration. For every eo, el E II#E, there exists a function

A : II#E(eo, e l ) x

b ( e O * p ( e ~ , )PI)

-

bec(PL,p(e,)) PI*

-


222

Proof - Let [A] E n#E(eo,el) and [s] E [sec(eo,p(e,,))p]* be given. Choose representatives A E (A] and s E [s]. The assumption that (p(el),z,B) is a cofibration implies the existence of a homotopy fi : B x I + B such that H ( - , 0 ) = 1 B and, for every t E I , H ( p ( e l ) , t ) = pX(1 - t ) . Now consider the maps g : B x { 0 } U {p(el)} x I

--+

E

given by g = (sH(-, 1))U A and

H:BxI+B defined by H ( z , t ) = H(Z,1-t); because p g is equal to the restriction of H to B x {O}U{p(el)} x I , it follows by Lemma 7.1.2 that there exists a homotopy G : B x I + E whose restriction to B x { O } U p ( e l ) x I is g and such that pG = H . It is easy to see that ii = G(-, 1) E sec(el,p(el)) p. Now we must prove that A is well-defined. To this end, take arbitrarily s’ E [s]and A’ E [A]; the sections 5 and s‘ are related by a based vertical homotopy

K:BxI-E,

H(-,O)=s, H(-,l)=s’

and the paths X and A‘, by a homotopy rel. aI

L:IxI+E, As for

8

L(-,O)=X, L(-,l)=X’.

and A, the section s‘ and the path A‘ give rise to homotopies

H:BXI+B and G ’ : B x I -+ E such that: f i ’ ( - , O ) = l ~fi’(p(el),t) , = pX’(1 - t ) , for every t E I , G’(-,0) = s’fi’(--,l), and GI(--, 1) = if‘ E sec(e,,p(eL))p. Observe also that by setting H I ( - , t ) = 1 - t ) , we obtain a homotopy s‘H‘ : s’&’( -, 1) s‘. In this way, we have the following string of homotopies

-

G-’ : S -

a’(-,

sa(-,l),sH : s f i - s , K : s -

9’

,


(s’H’)--I : 9’

a‘@’(

-, 1) , G‘ : s’B’( -, 1)

223 I’

which, when composed in the order presented above, give rise to a homotopy M : ii Z’. Our objective is now to show that the restriction of M to (@(el)} x I is homotopic rel. the end-points of { p ( e l ) } x I to the constant map, so that 3 and S‘ result to be based and vertically homotopic by Lemma 7.1.5. Let ( p ( e l ) , t )be an arbitrary point of the unit segment {p(el)} x I; when t varies from 0 to 1, this point traces, under the action of M, a path which can be broken up into five different paths, each produced by the action of the individual five previous homotopies: N

1. under G-l our wandering point describes a path from el to eU along the existing path A-’;

2. under sH we have a path from eo (as s E sec(,o,p(,o)p)to s ( p ( e 1 ) ) along SPA;

3. under K , a path from s(p(el)) to s ’ ( p ( e l ) ) along K ( p ( e l ) , t ) ;

4. under ( d H ‘ ) - ’ , our point moves from s‘(p(e1))to eU along s’pA’-*; finally, 5. under G’, a path from eo to el along A‘.

Let

N :I

x

I 3 E be defined by N ( t , t ’ ) = K(pL(t,t’),t’) .

Then, if * E I x I is a point moving on three edges of the square I x I, from (0,O)to ( l , O ) , then from ( 1 , O ) to (1,l) and finally, from (1,l)to (0, l), N(*)describes the three paths given in 2., 3. and 4. above. Now connect the point (0, f ) to the vertices (1,O) and (1,l)and take the geometric figure formed by the following union of three segments:

with t E I . When t = 1 this figure is exactly the union of the three edges connecting (0,O) to ( l , O ) , then (1,O) to (1,l) and finally, (1,l) to (0,l); for t = 0, this figure is the segment ( ( O , O ) , ( O , l ) ) . In this


224

way we obtain a deformation of the boundary of I x I minus the edge ( ( O , O ) , ( 0 , l ) ) onto this later edge, showing that the path formed by steps 2.,3. and 4. can be deformed into the path obtained applying N to a point moving along the edge (O,O), (0,1), from (0,O) to ( 0 , l ) ; thus, the path given by M ( p ( e l ) t, ) , t E I shrinks to just the path determined by steps 1. and 5. above. Since X A' rel. end-points, the path given by 1. and 5. is homotopic to the constant path at el. 0 Notice that if eo = el, we have a function N

moreover, if F is the fibre of p over p ( e 0 ) and sec(,o,p(,o))p# 0, the exact sequence of the fibration ( E ,p , B ) (see Theorem 3.1.5) shows that nl(F, eo) is a subgroup of nl(E , eo) and therefore, we have a function

A :~ I ( F eo), x [~ec(eo,p(eo)) PI*

4

[Sec(eo,p(eo))

PI*

*

Let G be a group with identity element 1 and let X be a set; a function

A:GxX--+X is a left action of G on X if the following two conditions hold true: 1. For every 2. for every

a: E 2

X , A ( l , z ) = t and,

E X and g,gf E G,

A(s, Ng', 4) = k?', 4' If, for every g E G and a: E X , A(g,a:) = 2, the left ction A is said t be trivial. The relation a: x' iff 3g E G such that 2' = A ( g , a ) is an equivalence relation; the set of the equivalence classes determined by it is denoted by X / G . N

Theorem 7.1.7 Let ( E , p , B ) be afibration, let e E l l # E be such that sec(,,p(,))p# 8 and let F be the fibre of p over p ( e ) . Then the function

A

:

F, e) x

PI*

[~ec(v,p(e))

4

[Sec(,,p(c))

PI*


225

Proof - We first observe that if (A] E r I ( F , e ) , for every [s] E [sec(e,p(e))p].,we can take the homotopy H defined in the construction of h ( [ X ] , [ s ] ) to be the projection on the first factor; moreover, the homotopy G : B x I t E of s to G( -, 1) extends s U X and is a vertical homotopy, because pG = p r l . It is easy to check that the identity element of rl(F, e ) , namely the class [ce] of the constant map at e, is such that A([c~],[s]) = [s], for every [sl E [Sec(e,p(e)) * Now take arbitrarily [A], [i] E r l ( F ,e) and [s] E [sec(e,p(e))p]..We first extend s U X via a vertical homotopy G : B x I + E such that G(-,0) = s, G(-,1)= s' and G(p(e),t) = A ( t ) ; next, we extend s' U 1via a vertical (but not necessarily based) homotopy G' such that GI(-,0)= s', G'(-,l) = 5 and G'(p(e),t)= X(t). On the other hand, we take

PI -

X i : I d E

and extend s U X i via a homotopy K : B x I --+ E such that K ( -, 0) = s, K(-,1) = I' and K ( p ( e ) , t )= X i ( t ) . By putting together the homos, G : s s' and G' : s' s we obtain a homotopy topies K-l : 8' M : B x I + E from 8' to 8 whose restriction to ( p ( e ) } x I is the loop (Xx)-'Xx which is homotopic rel. the set of the end-points of { p ( e ) } x 1 to the constant loop at e; now use Lemma 7.1.5. 0

-

-

-

Corollary 7.1.8 For every (Y,yo) E Top, and every n Zeft action of (Y,yo) on nn(Y,yo).

2 1, there i s a

Proof - Take the fibration (S" x Y,p q , S") and select (eo,yo) as base point of S" x Y.Since the inclusion of ell in S" produces a cofibration (see Exercise 2.3.2) and the fibre of p r l over efJ is Y ,there is a left P T ~ ] *= r n ( Y , yo). 0 action of rl(Y,yo) on the set [Sec((eo,gO),en) For the moment we shall be content to study some consequences of the action of the fundamental group of Y on the set nn(Y,yo);later on, we shall see that the action indeed takes into consideration the fact that .n(Y,yo) is a group (see Corollary 7.1.17 ). By associating to each


226

class [s]E [~ec(,,~(,))p]. the free homotopy class [s]f E [secp], we obtain a function

# : [sec(e,p(e))PI*

4

[set PI

*

We wish to study this function more closely whenever we have a fibration ( E , p , B ) with fibre F over p ( e ) , where e E E is such that sec(,,p(,.)p 0 and ( { p ( e ) ) , i , B ) is a cofibration; these conditions will be tacitly assumed in the next two theorems.

+

Theorem 7.1.9 T h e function

induced by

cp

is injective.

Proof - We prove this result by showing that two sections 8,s’ E sec(,,(,))p are free homotopic iff there exists [A] E n,(F,e) such that

A([XI, [sl) = is‘]; j If 8,s are free homotopic they are also vertically homotopic (see Theorem 7.1.3); thus, there exists a homotopy K : B x I -+ E such that pK = p q . Take the path X = K ( p ( e ) , - ) : I+

E

and notice that because p X ( t ) = p(e) for every t E I , X is actually a loop in F . The map s u X :B x { 0 }

u {p(e)} x I + E

is extended by K and the definition of A shows that A([X], [s])= [s’]. += Suppose that X : I 4 F is a loop such that A([X], [s])= [s’];this means that there exists a homotopy K : B x I --t E extending s U A, and such that p K = p q , K ( - , 0 ) = 9 and K(-,1) = s‘. 0 As for the surjectivity of # we first give the following result:

Lemma 7.1.10 For every section s of p there i s a based section ii of p which is vertically homotopic t o s iff s ( p ( e ) ) and e are in t h e s a m e path-component of F .

7.1. SECTIONS

OF FIBRATIONS

227

Proof - =+ If H : s S is a vertical homotopy, H ( p ( e ) ,-) is a path in F connecting s ( p ( e ) )to e. + Let X : I + F be such that A(0) = s ( p ( e ) )and X(1) = e. Take N

g = s U X :B

and H = p r l : B x I homotopy G : s S . 0

+

x {0} U{p(e)} x I

+E

B to obtain from Lemma 7.1.2

a

vertical

N

Theorem 7.1.11 If F is path-connected,

4 is surjective.

Proof - An immediate consequence of the previous Lemma. 0 Theorems 7.1.9 and 7.1.11 together prove the following result: Theorem 7.1.12 Let ( E , p , B ) be a fibration with path-connected fibre F over p ( e ) ; moreover, we assume that the arrow defined b y the inclusion of { p ( e ) } in B is a cofibration and that sec(e,p(e)) p # 0. Then there is a bijection

Corollary 7.1.13 Let ( X , z o ) (Y,yo) , E Top, be based spaces such that Y is path-connected and the arrow ( {xo}, i, X) is a cofibration. then there exists a bijection

-

~ ~ , y l * l ~ l ( y , [X,Y] ~o) * Proof - Apply the previous ideas to the fibration (X x Y , p q , X ) . 0

In particular, if Y is simply connected, [ X , Y ] 2 , [X,Y]. We have seen in Corollary 7.1.8 that the group rl(Y,yo)acts on the underlying set of r,(Y,y,); this fact was established by studying the set of homotopy classes of sections of the fibration (S" x Y ,p r l , Y ) but, as a matter of fact, we can define the action of the fundamental group on r,(Y,y,) by a direct method, establishing at the same time that the left - action

-

A : m(Y,yo) x ~n(Y,Yo) %(Y,YO)

-

is such that, for every [A] E 7r1(Y,yo),

A"XI,-)

: .rrn(Y,Yo)

.n(Y,Yo)


228

is a group automorphism. We shall prove this assertion as a particular case of a more general situation described in Theorem 7.1.16. We begin with the following.

Lemma 7.1.14 Let f : ( Y , y o ) + ( X , f ( y , ) ) be a n object ofTop*+ and let ( a ,b ) , (a', b') : i n - 1 f

-

be two arrow-maps. Suppose that

is a n unbased homotopy f r o m ( a , b) t o (a', b') such that A(e,, t ) = A ( t ) is homotopic rel. OI t o the constant m a p cyo. Then [ a ,b] = [a', b'], that is t o say, ( a , b ) and (a', b') are homotopic by a based arrow-map hornotopy (cfr. L e m m a 7.1.5).

Proof - Let H : I x I

+Y

be a homotopy such that

H ( - , 0 ) = A , H ( 0 , - ) = H(1,-) = IT(-,1) = cyo . Define the map

A : sn-'x

I x (0) u { e , ) x I x I

-+Y

by the conditions

A I Sn-'x I x ( 0 ) = A , A I {e,) x I x I = H ; since ( {e,}, i, Sn-l)is a cofibration, A can be extended to a homotopy : sn-1 X I X I + Y .

At

Now take the homotopies A1 1 7

A' A; : 29

y - 1

XI-Y

A;(z,t) = A'(Z,O,t) A k ( Z , t ) = A'(z,t,l)

, ,

.31,(Z,t) = A y e , 1,1 - t ) ,

7.1. SECTIONS O F FIBRATIONS

I

229

for every (z,t ) E S"-* x I , and define the product

A'(z,t) =

05t

4(2,2t), &(z,4t

-

2),

A$(z,4t - 3),

5

;

4 5t5 3

;5 t 5 1 .

Observe that, for every z E Sn-l,

A'(z,O) = a ( z ) , A ' ( z , l ) = d ( ~ )

fa

and, for every t E I , A'(e,,t) = yo. Now take and B to define a map B : B" x I x {O}US"-' x I x I + X which can be extended to a homotopy

B : B" x I x I

+

x

because B") is a cofibration. Similarly to gives rise to a homotopy

B ' : B" x I

-+

A', the map B'

X

such that, for every z E B",

B'(2,0) = b(z) , B ' ( z ,1 ) = b'(z) and, for every t E I , B'(e,, t ) = f ( y , ) . Moreover,

is a based arrow-map homotopy between ( a ,b ) and (a', b'). 0

Theorem 7.1.15 Let f : (Y,y,) -, (X,f(yo))E Top, be given arbitrarily. For every z, € Y in the same path-component of yo, there is a function A : W Y 0 , Z O ) x Tl(f,9") 7rn(f,zo) ' -*

-


230

Proof - Let X : I -+ Y be a path with X(0) = yo, X(1) = toand let be an arbitrary element. Define a function

--

[a,, b,] E n,(f, yo) =

ii : Sn-'x {0} u { e , } x I = a,(z), ?i(e,,t) = X(t),

2

Z1(5c,O)

E

Y

sn-1,

.

tEI

We now proceed in two ways: on the one hand, we consider the cofibration ( { e , } ,i, S"-' ) to extend ii to a homotopy

A, : S"-' x I --+ Y and on the other hand, we use

6, : B"

-

fA, and b, to define a map

x (0)

u S"-'

xI

X

and extend it to a homotopy

B,: Bn x I The two maps a1 = A,( -, 1) and

bl

+X

= B,(

.

-, 1) define an arrow-map

and thus, an element

We must prove that [al,bl] depends solely on the classes [A] and [ao,b,]. Let A' E [A] be given. Proceed as before, but using the path A' to obtain an arrow-map

and homotopies A:) and B:. Next, construct the homotopies

A : S"-' x I

-+

B : B" x I + X

Y

,


231

by setting

A,(z,l

- 2t),

05t 5

f,

A ( x , t )= AL(2,2t - l ) , f 5 t 5 1 and

B ( x , t )=

B , ( z , l - 2t), 0 5 t 5

{

BL(2,2t - l ) ,

$,

5 t 5 1,

to obtain an unbased homotopy

( A , B ) :in-* x 11 + f from ( a l , b l ) to (ui,bi). But A ( e , , t ) = X - ' X ' ( t ) and thus, A(e,,t) is homotopic rel. dI to cz,; from the previous lemma we conclude that [al,bll = [a:,b:l. Now take (ub,bb) E [a,,b,] arbitrarily. Let

f

(A',B') : i,&-lx 11

be a based homotopy connecting (u,, b,) to (a:, bb). Use the path X and the arrow-map (a,,b,) to construct the homotopy ( A o , & ) ; next, use the same path and (uL,b;) to construct (AL,BL).Finally, define

A : S"-' x I +Y and

B:B"xI+X by the formulae:

!

A , ( z , l - 2t), 0 5 t 5

A ( z , t )=

A'(z,4t

-

2), 1 2 -< t 5

4 ( 2 , 4 t - 3),

and

$,

9,

3 5 t 5 1,

Bo(2,1 - 2 t ) , 0 5 t 5

;,

3 B'(z,4t - 2), f 5 t 5 1,

BL(2,4t - 3 ) , f 5 t 5 1

.

232


Note that

A(-,O) = a17 +,O) = bl, A(e,,t) = X-'C,~X(~),

A(-,1) = a:, B(-, 1) = b i , t E I,

and thus, by Lemma 7.1.14, [ a l , b l ] = [a;,bi]. 0

Theorem 7.1.16 Let f : (Y,y,) ( X , f ( y o ) )and X : I with X(0) = yo, X(1) = zo. The function --f

+Y

be given,

is a group isomorphism, f o r every n 2 2.

Proof - We first prove that A([X], -) is a group homomorphism. Take [ao,bo],[ab,bb]E rn(f,yo) and let ( A , B ) , (A',B') be the arrowmap homotopies constructed to produce

A([X],[ab,bi]) = [A'(-,l)

=

, B'(-,I)

Sn-I

v sn-I ,

= bi]

.

Consider the CoH-multiplications vn-1

.- s n - 1

--$

Yn-l : B" + B" V B" and extend them to v,-~ x 11 and

&-l

x 11 in the obvious way. Then

is an unbased arrow-map homotopy whose value at t = 0 is

A : Sf'-'

x I

-

Y

233

7.1. SECTIONS O F FIBRATIONS

defined by

A I sn-' x (01 = v ab)vn-r A I {e,) x I = A(t) , B I B" x (01 = r(b, v bb>Vn-1 , B I S"-' x I = fA ~ ( u O

and define A : Sn-' x I -+ Y , B : B" x I

A(x,t)=

by:

A l ( z , l - 2t), 0 5 t L

i

+,2t

B ( x , t )=

+X

{

Because A(e,,t) = A-'A(t),

-

l),

,

;5 t 5 1 ,

B l ( x , l - 2t), 0 5 t 5 B ( x , 2 t - l),

3

,

f 5 t 1.1 .

it follows that

[ A ( - , I),B ( - , 1)1= A(IA1, [ a l , b l l ) ~ ( [ 4 [a:, , b:l) Since [ A ( - , l ) , B ( - , l)]is also equal to A([A], [al,bl][a;,b{]), we conclude that the assertion made at the beginning of the proof is true. Next, we prove that A([A], -) is a bijection. Take [a,, b,] E n n ( f , y o ) and, using (ao,b,) and A, construct the unbased homotopy

(A,, Bo) : in-1 x 11 + f which defines A([X],[a,,b,]) = [al,bl]. Then take ( a l , b l )and A-' to obtain the unbased homotopy

( A i , B i ): i n - i x 1 1 + f which produces h ( [ A - ' ] , [al, b l ] ) = [a;,bb]. Now define the homotopies A : Sn-' x I -+ Y and B : B" x I --+ S as

A&, 1 - 2 t ) , 0 5 t 5 A ( z , t )= A,(2,2t- 1),

+,

f 5t 41 ,

234


and B,(z,l

-

2t), 0

5 t 5 ;, 11

B(x,t)=

B ' ( Z , Z t - l), 51 5 t 5 1, and use Lemma 7.1.14 to see that [ub,bb] = [a,,b,]; this shows that the composition of homomorphisms A( [A-'I, -)A( [A], -) is the identity function. Similarly, we prove that composing these homomorphisms in the other order we obtain again the identity function. 0 Corollary 7.1.17 There is a n action

such that, for every [A] E

is

a

TI(~,Y,),

group automorphisrn, for every n 2 1.

Proof - State Theorem 7.1.15 for n use Theorem 7.1.16. 0

+ 1, yo = z , and f = car,;then

EXERCISES 7.1.1 Let ( E , p , B )be a fibration with fibre F over p(e,). Prove that if SeC(eo,p(eo) P # 0, then

Tn(E, eo)

TrI(F,eo)

a3 T l ( B , p ( e o ) )

for every n 2 2.

7.1.2 Let (X,z,) be a based space such that ( ( z , } , i , X ) is a cofibration. Let Y be a space together with a path X : I -+ Y such that X(0) = yo and X( 1) = yl. Prove that X induces a bijection

7.1.3 Prove that the group action of 7r1(Y,yo)on itself is given by inner automorphisms.

7.1. SECTIONS OF FIBRATIONS 7.1.4

235

* Regard the 3-sphere S3 as the set of all pairs of complex numbers (zO,zl) such that I zu l2 + 1 z1 12= 1. Let p , q E N \ (0) be relatively prime. Define the function

4 : S3 -, s3

mPaiq

(zO,zI) H (e

p

zu, e

p

21)

.

Prove the following statements: (a) q5 defines a (continuous) left action

2, x

s3 s3, ( C ( ~ 0 , Z I ) ) +

H df(ZU,Zl)

-

which is fixed point free; (b) Define (zo,zl) ( Z : , Z ~ ) in S3 iff there exists 7 E 2, such 0 ~= @(zo, z l ) and let L ( p , q ) = S3/ -. Then the that ( ~ ’2;) quotient map II : + L ( p , q ) is a covering projection map (with fibre Z,). (c) The space L ( p , q ) - called Eens space of type ( p , q ) is a CWcomplex.

s3

Remark: The lens spaces L ( p , q ) have fundamental group Z, and moreover, 7 r n ( L ( p , g ) , * ) 2 7r,(S3,e0),for every n 2 2. Thus, homotopy groups cannot make a distinction between L ( p , q ) and L(p,p’), q # 4’; however, lens spaces can be differentiated by cohomology (see [16, Section 5.101).

7.1.5 Let ( E , p ,B ) be a fibration and ( A ,i, X ) be an arrow such that A is a strong deformation retract of X . Suppose also that f : A -+ E and g : X -+ B are maps such that p f = g i . Then prove that there exists a map F : X + E such that F i = f and p F = g. 7.1.6 Let ( E , p , B ) be a fibration. Show that there exists a covariant functor S from II# to the category of sets Sets which assigns to each e E II# the set [sec(,,,(,))p], and to each morphism [A] of II#, the function A( [A], -).

236

7.2


F-Fibrations

In the previous section we proved that, for any two spaces X and Y ,M ( X , Y ) E secpq, the space of sections of the trivial fibration ( X x Y , p r l , X ) (and the corresponding based case); we also studied the sets of homotopy classes associated to such spaces. In the present section we shall generalize the previous situation to arrows determined by surjections and whose fibres are constrained to live in a certain fixed category; in order to generalize the basic theorem relating a function space to a space of sections, we need the more general format of the exponential law and its related results. Hence, from now on we shall work in the category CG of compactly generated spaces (see Appendix B for the relevant definitions and results); thus, all the spaces considered are tacitly assumed to be compactly generated and moreover, all the categorical constructions (as products, pullbacks, etc.) and function spaces (based and unbased) are supposed to be taken within Cp. Let 3 be a non-empty category whose objects are objects of CG and whose morphisms are such that, for every pair of objects X, Y E F, the set F ( X , Y )of all morphisms in F from X to Y ,is a subset of the set CG(X, Y )of all morphisms in CG from X to Y . An arrow ( E , p ,B ) is said to be an T-arrow if p is onto and, for every b E B , p - ' ( b ) = Eb is an object of F. An arrow-map ( g , h ) : ( D , q , A ) + ( E , p , B ) is an F-arrow-map if, for every a E A , the restriction

is a morphism of T ;notice that the map g completely determines the map h. As in Section 2.1, we sometimes indicate an F-arrow-map (9,h ) : (D, q, A ) 3 ( E , p , B ) simply by (9, h ) : q p. The category of F-arrows and T-arrow-maps is denoted by TF(CG)+. We shall indicate simply by writing that an arrow ( E , p , B ) is an object of T'(CS;)( E , p , B ) E TF(CG)' or p E TF(CG)+; if an arrow-map ( g , h ) is a morphism of TF(CG)' we shall write (9, h ) E Ty(CG)+. --+

Lemma 7.2.1 Let ( E , p , B ) E TF(CG)+ and f : A + B be given. Then ( A , nf E , p , A ) is an F-arrow-map. Moreover, i f ( D ,q, A ) E TF(CG)-*and (9,f) : q p is an F-arrow-map, the arrow ( l ,1,i) : --f

7.2. F-FIBRATIONS

23 7

3 ji - where,L is the unique map determined by the universal property of pullbacks - is an F-arrow-map. 0

q

In particular, if A is a subspace of B and f is the inclusion map, the 3-arrow map obtained by the pullback of p and i will be denoted by (JT.4, PA,A). Two morphisms

(9,h ) ,(g', h') : (44, A ) of

-

TF(CG)-+ are F-homotopic if

G :g

-

(J%P , B )

there exist homotopies H : h

N

h' and

g' such that

( G , H ) : 4 x 11 - P

is an 3-arrow-map; we use the notation (9,h ) 2 (g', h') to indicate F-homotopy. If A = B , h = h' = 1~ and H = prl is the projection on the first factor, (G,prl) is an F-homotopy over B . We use the notation (g,lg) (g',lg) - or simply g g' to indicate that (g,lB) and (g',lg) are 3-homotopic over B. An F-arrow-map (g,l~): (0, q, B) -+ (E,p, B)is an F-hornotopy equivalence over B if there exists an F-arrow-map (g',lg) : ( E , p , B ) -+ ( D , q , B )such that gg' and g'g are F-homotopic over B to the respective identity maps. We want to study these F-homotopy equivalences over a space more closely (see Theorem 7.2.4), but first, we analyse just one-half of that concept: more precisely, we say that an F-arrow-map (g', 1g) : p -+q is a right F-homotopy inverse of (g,ls): q + p if gg' wg 1 ~Let .

-;

-2

( H , P R ) :P x

wg

be the F-homotopy gg' the fibration €0

and the map g : D

--+

u :E

1 ~ . Let

:E ' +E

12

-

D fl E'

, €"(A)

P be the pullback space of

= X(0)

E , and define

------f

D n E' , e H ( g ' ( e ) , H ( e ,-)) .

Note that, for every ( e , t ) E E x I , p H ( e , t ) = p ( e ) = q(g'(e)) and thus, the image of c lies in the space

G = ((d,X) E D n E1 I X ( I ) C p-'(q(d))}

.


238

These observations prompt us to define an F-section of (g,lB) as a map u : E + G such that

is an F-arrow-map and

- where K ( ( d , X ) , t ) = X(t), for every ( ( d , X ) , t ) E G x I - is an Fhomotopy over B. These definitions show that

Lemma 7.2.2 An F-arrow-map (g,lB) : q topy inverse iff it has an 3-section. 0

-+

p has a right F-homo-

We now turn our attention to F-homotopy equivalences over B. Our next result will be needed for the proof of Theorem 7.2.4.

Theorem 7.2.3 Let ( D , q , B ) , ( E , p , B ) E TF(CG)-) be given and let ( g , l ~ :)q -+ p be an F-homotopy equivalence over B. For a given map t$ : B + I = [0,1] define the subspaces A = t$-'(l)and V = t$-'((O, 11) of B . Then, for every F-section u : El.

+ Gv =

{(d,X) E G I q(d) E V }

of ( g l r , l \ r ) , there exists an F-section p : E + G of (g,lB) such that P EA = Q I EA.

I

Proof - Let (g', 1 ~: p) -+ q be such that

Consider the 3-section r : E

-+

G defined by

r ( e ) = (g'(e), H ( e , -))

, Ve E E .

Define a homotopy

J : G XI &

E'

239

7.2. T-FIBRATIONS where

k :I x I

--+ (I

x (0)) U ((0) x I ) U ((1) x I )

is our favourite retraction and

for every (d, A) E G and s , t E I . Now define the homotopy L : aK (where K ( ( d , A ) , t )= A ( t ) ) by the formulae:

N

1~

Now define the F-section p : E + G of g by

This function is such that, for every e E E.4, i.e., q5(p(e))= 1,

Notice that if ( g , 1 ~ :)q + p is an F-homotopy equivalence over B and U c B is an arbitrary subspace of B , the restriction

is an F-homotopy equivalence over U (this can easily be seen by constructing the commutative diagram of Figure 7.2.1 - whose squares are pullbacks - and using the dual to Theorem 2.4.4 - see also Exercise 2.4.8). We are going to prove a sort of converse to this observation: if we are given an F-arrow-map (g,lB) : q + p which is an F-homotopy equivalence over each element of a special covering of B , then ( g , l ~ is) an F-homotopy equivalence over B. To make our statement precise, we

CHAPTER 7. FIB RATIONS RE VISITED

240

*D

/

\ i B

U

z

;-B

7.

FIGURE 7.2.1

give the following definition. A covering (not necessarily open) C = { U) of B is said to be a numerable covering of B if there exists a set

{$A

:

-+

[O,11 I

E

A}

of maps - called partition of unity - such that: 1. (VU E C)(3X E A)

U c $il((O,l]);

2. (Va E B ) $ x ( z )# 0 for at most a finite number of indices

3. (VZ E B ) Ex ~ x ( z=) 1.

X E A;

7.2. 3-FIBRATIONS

241

Conditions 2. and 3. show that the family {4X1((0,1]) 1 X E A} is locally finite that is to say, every z E B - indeed, a certain neighbourhood of 2 - meets only finitely many sets 4x1((o,I]).

Theorem 7.2.4 (Dold-May) Let ( g , l ~ ): ( D , q ,B ) + ( E , p ,B ) be a given 3-arrow-map. Let C = {U} be a numerable covering of B such that, for every U E C, (gu,1u) : qu 4 pu is an 3-homotopy equivalence over U. Then (g,lB) is an 3-homotopy equivalence over B . Proof - It is enough to prove that the hypotheses allow us to conclude that (g,lB) has a right F-homotopy inverse (g',lg) over B: in fact, let (ju,1u) be an 3-homotopy inverse of (gu,1u) over U ,for each U E C; then 3 1 .% &.fgUgb gU

-;

and thus, (gt/-,lu)is an 3-homotopy equivalence over U , for every U E C; but then (gl,1 ~ has ) itself a right 3-homotopy inverse (g", 1 B ) and therefore, 3

It

gd9" -B g 9'9

-; g'g"

7

1B

,

proving thereby that (g', 1 ~is)also a left F-homotopy inverse of (g,1 B ) over B and hence, that ( g , l ~ )is an F-homotopy equivalence over B. Let { 4 ~ : B --$ [0,1]I X E A} be a partition of unity associated to the numerable covering C; indeed, we may assume that C is given by the set C = {Ux = $ i l ( ( O , l ] ) I A E A}

.

If V is an arbitrary union of elements of C, say

we define

41- : v

Hence, V = {z E B matters so that

I

-

[0,1] ,

41-(z)

2

I-+

c

&(z)

.

Al€I\'

> 0). Furthermore, we can arrange

v c w e (ih, 5411. .


242

Let A be the set of all pairs (V,a\r) such that V is a union of elements of C and a\* : El. + Gl- is an 7-section of (gl7,ll.) : -+ El( A # 0 because we could take V = Ux E C, in which case (Ux,ru,) exists by the conditions of the theorem). Define a partial order in A by saying that (V,CTI..) < (W,ail-)if V c W and, for every e E p-'( V) such that # J r , ( p ( e ) )= #JJiy(p(e)),then ar.(e) = q l . ( e ) . (Notice that if alT(e) # r\l-(e), we conclude that there exists an element U AE C such that p ( e ) E Ux,UAc W and U, $ V.) Let

be a totally ordered subset of A. Form the set V' = UyErVy. Let e E El-! be given arbitrarily; let V ( e )be the union of all Ux E C such that p ( e ) E Ux c V'; since V ( e ) is a finite union (C is numerable) and d'is totally ordered, there exists a -ye E r such that V ( e )c Vye. Hence, for every -y 2 -ye (the total order of A' induces a total order on I?), a, = ayeover V ( e ) .This fact let us define an T-section

of (gl.1, 1\71). In other words, A' has an upper bound (V', a l ~ ) .By Zorn's Lemma, the partially ordered set d contains a maximal element (V,al-). We claim that V = B. Suppose not. Then there exists a X E A such that U, V ;define W = UA U V and

I

#J

#J(4=

: W + [O, 11

#JA(4 5 h ( 4 l' #,

h(4 2 4 1 w

Thus, # J ( x )> 0 iff # I Z ( X ) > 0 and Crl- is defined over q5-'((0,1]). By the previous lemma, there is an F-section p : Eu, + GLI,such that p = r1over #J-*(l) n U. Now define

243

7.2. 3-FIBRATIONS

Then (W,q.) > (V, a contradiction. 0 The following space generalizes the notion of function space: for any two F-arrows ( D , q , A ) and ( E , p , B ) , q r ) ,

topologized as a subspace of E D . We wish to prove that 3 ( q , p ) is homeomorphic to the space of sections of a certain arrow (not necsuch an arrow is essarily a fibration) associated to q,p E T7(CG)-); defined as follows. Firstly, consider the set

and the function

Secondly, we take the set E U ( 0 0 ) (where 00 is a point disjoint from E ) , topologize EU (00) by saying that K c E U (00) is closed iff either K = E U (00) or K is closed in E and define E+ = k ( E U (00)) (for the definition of k - see Appendix B). Thirdly, we define the function j :D * E

.--)

(E+)D

by the conditions

Finally, give D j , E the initial topology with respect to the functions j and q * p and retopologiae this space by taking its compactly generated topology. The arrow (D * E , q * p , A x B ) is the functional arrow associated to q and p ; notice that q * p is not necessarily an 3-arrow and so, we have a construction involving two objects of TF(CG)-’ but which might land us outside T;L.(CS)+. Now take the arrow (D* E , q k1 p , A ) obtained by composing q * p with prl : A x B + A and let sec(q*1 p ) be the space of all sections of the arrow q *1 p topologized as a subspace of (D * E).‘l.

244


Theorem 7.2.5 For every ( D , q , A ) , ( E , p , B )E T;F(CG)+,the spaces F(q, p ) and sec(q *1 p ) are homeomorphic. Proof - Let d : F(q,p) -+ sec(qkl p ) be the function which assigns to each F-arrow-map (9,h ) : q -+ p the section s of q *1 p defined by s ( a ) ( z )= g(z), for every a E A and every z E D,. We are going to prove that 6 is a homeomorphism. To prove that s is continuous, we must prove that the compositions ( q * p)s and j s are continuous; the former coincides with the map (144,h): A + A x B , while the latter corresponds, by the exponential law (see Theorem B.7), to the map

The injectivity of d is clear from its definition. To prove that 6 is surjective, let s E sec(q*lp) and let l : A x D -+ E+ be the unique map which corresponds to the map j s : A -+ (I?+)" under the exponential law; now define 9 :D

, 9 ( 4 = l(q(z),zC>

-+

for every z E D ;the map h : A -+B such that pg = hq is automatically defined by 9. Note that O(g, h ) = s. Before we show the continuity of both 6 and O-', let us observe that the topologies of F(q,p) and sec(q*l p ) can be given by the initial topologies with respect to the following functions, respectively:

and

j' : sec(qkl p )

-+

((E+)")." , j ' ( s ) = j s

.

These two facts and the exponential law now conclude the proof. 0

Corollary 7.2.6 Let (0, q, A ) , ( E , p , B ) E T;F(CG)+and let (W,T,A ) be an arrow; let (W,,n,.D , q, W ) be an arrow constructed as a pullback of q and T . Then the space F(q, p ) is homeomorphic to the space M ( W,D* E ;r ) of all maps @ : W D * E such that ( q *1 p ) @ = r . -+

7.2. F-FIBRATIONS

245

=D*E

W P

!l I

B-

i

h

W

T

r

A

FA

FIGURE 7.2.2

Proof - The situation we describe in the corollary is illustrated in Figure 7.2.2. From the theorem we know that F(q,p)% sec(q*1 p ) ; on the other hand, the universal property of pullbacks shows that (Wqn, 0)* E 2 W n, ( D t E ) and so, sec(tj*, p ) 2 s e c ( q ) . The function 4 : s e c ( m ) + M(W,D * E ; r ) defined by e ( 8 ) = F S is a homeomorphism. 0 We devote the rest of the section to the study of F-arrows which are fibrations. An object ( E , p , B ) E T’(CB)^ is an F-Jibration if it satisfies the F-covering homotopy property with respect to all Farrows that is to say, if for every T-arrow ( D , q , A ) ,every F-arrowmap ( g , h ) : q + p and every homotopy H : A x I -+ B of h, there is a homotopy G : D x I + E of g such that ( G , H ) : q x 11 -+ p is an F-arrow-map (see Figure 7.2.3).

D x I

G

*E

P 1

AxI

H

*B

FIGURE 7.2.3: F-covering homotopy property

246


Note that if F E F,( F x B , p r z , B ) E this is the trivial F-fibration.

TF(CG)’ is an F-fibration;

Lemma 7.2.7 F-fibrations are fibrations.

Proof - Let ( E , p ,B ) be an F-fibration; we are now given an arrow ( D , q , A ) ,an arrow-map (9, h ) : q -+ p and a homotopy H : A x I + B . The arrow ( A , nh E , p , A ) obtained by pullback is an F-arrow; hence, there is a homotopy GI : ( A , nh E ) x I + E such that (GI, H ) : jj x 11 -+ p is an F-arrow-map. Let $ : D -+ A, nh E be the unique map satisfying the conditions: p$ = q and & = 9. The homotopy G = Gl(4 x 11)is such that (G, H ) : q x 11 -+ p is an arrow-map. 0 Clearly, an arbitrary fibration might fail to be an F-fibration because F may not have enough morphisms. The following lemma gives a simple characterization of F-arrows which are also F-fibrations.

Lemma 7.2.8 A n F-arrow ( E , p ,B ) is an F-fibration if f o r every map f : W -+ B , any F-arrow-map (9,h ) : (WP n,

E,P,W )

-

(E,P, B)

and any homotopy H : W x I -+ B such that H ( - , 0 ) = h, there ezists a homotopy G : (Wpn, E ) x I + E such that G(-, 0 ) = g and ( G ,H ) : p x 11 -+ p is an F-arrow-map.

Proof -

+ Follows easily from the fact that p is an F-fibration.

+ Take an F-arrow ( D , q , A ) ,an F-arrow-map ( g , h ) : q -+ p and a homotopy H : A x I 4 B such that H ( - , 0 ) = h. The hypotheses imply that there exists a homotopy H : ( A P n hE ) x I

+E

such that H(-,O) = h = prz and ( H , H ) : ji x If + p i s an F-arrowmap. Each of the rectangles of the commutative diagram of Figure 7.2.4 is a pullback and thus, the outside diagram is a pullback (see , universal property of Exercise 2.1.3). Since p ( g p r l ) = hprl(q x l ~ )the pullbacks gives rise to a unique map

9 : D x I + ( A P n h E )x I

7.2. F-FIBRATIONS

247

FIGURE 7.2.4 such that h p q 8 = g p q and ( p x 11)8 = q x 11. Actually, the commutativity conditions and the uniqueness of B show that, for every ( z , t )E D x I , O(z,t)= ( ( q ( z ) , g ( z ) ) , tNow ) . take the map

G=HB;

G:DxI-E,

it is easy t o check that G(-,0) = g and that ( G , H ) : q -+ p is an F-arrow-map. 0 As we have already seen, the functional arrow associated to two Farrows is not necessarily an F-arrow; thus, the functional arrow associated to two F-fibrations does not have to be an F-fibration; however, we have the following result:

Theorem 7.2.9 The functional arrow associated to two F-Jibrations is a fibration.

Proof - We are given two 3-arrows ( D , q , A ) and ( E , p , B ) ,an arbitrary space W , a map g : W 3 D * E and a homotopy H : W x I + A x B such that H(-,0) = ( q * p ) g ; we wish to find a homotopy G :W xI D * E extending g and such that ( q * p)G = H . Now ---f

define the maps

h :W x I

--+ A

Ic:WxI-+B,

,

h = prl H

,

k=pr*H;

form the commutative diagram of Figure 7.2.5 in which the rectangles labelled 2 and 3 are pullbacks (so, the larger rectangle obtained by putting together 2 and 3 is also a pullback), the arrow-map (a07

ko) :

+

p

248

C H A P T E R 7. FIBRATIONS R E V I S I T E D

of rectangle 1 is an F-arrow-map with QIo((w,0 > , 4 = 9 ( w ) ( 4

and ko is the restriction of k to W x (0). (Note that (ao,ko) E 3(Qo,p) is the unique arrow-map corresponding to g E M ( W,D E ; H ( -, 0)) according to Corollary 7.2.6.) Define

*

I}) n D

E*((WX{

I

P

-(W

x I )nD

6 D -

1

FIGURE 7.2.5

k‘ : (W x (0))

x I -+

B

, k’(w,O,t)= k ( w , t )

and notice that, because p is an T-fibration and Qo is an 3-arrow, there exists a homotopy

K’ : ( ( W x (0)) n 0)x I

-+

E

such that K’(-, 0) = and (K’,k‘) is an F-arrow-map. Next, consider the T-arrow-map (L,h ) of pullback 3 in Figure 7.2.5 and the homotopy

Since h‘ is a homotopy of h and (D, q, A ) is an F-fibration, we obtain a hornotopy H’ : x I ) , nh 0)x I -- D

((w

such that (H’,h’) is an 3-arrow-map. Now form the commutative diagram of Figure 7.2.6, where each square is a pullback and define the map

7.2. F-FIBRATIONS

249

FIGURE 7.2.6 notice that, for every ( ( w , t ) , i z )E (W x I ) n D ,qp((w,t),iz>= h(w,O) and therefore, by the universal property of pullbacks, there exists a unique map

8 : (W x I ) n D --+ such that hioprlB = p and (iju x

7 : (W x I,)n D

+

(wx (o}nD) x I

11)O

= 7, where

(W x (0)) x I

, ((W,t))Z)

H((W)O),t)

.

Finally, we define the map

K : (W x

l)qnh

D

--+

E

)

K = K'B

)

observe that pK = kij and use Corollary 7.2.6 to complete the proof. 0

The following result gives a characterization of F-fibrations in terms of functional arrows and fibrations:

Theorem 7.2.10 An F-arrow ( E , p ,B ) is an 3-fibration if the functional arrow ( E * E , p * p , B x B ) is a fibration.

Proof - 3 This is an immediate consequence of Theorem 7.2.9. -+We use Lemma 7.2.8 and Corollary 7.2.6 to prove sufficiency. Let f : W -+B be a given map and let

be a given F-arrow-map. Define the map


250

for every (w,z)E W, n, E . The composition ( p * p ) g ' is the restriction to W x (0) of the map

(fpr,,H) :W x I

*

-+ B

x B

*

and so, because ( E E , p p , B x B ) is a fibration, there exists an extension of g', say G' : W x I 4 E*E, such that (p*p)G' = (fprl,H ) . From Corollary 7.2.6 we conclude that there exists a map

G : (W x I ) pnfprlE

n, E ) x I

E (Wp

--+E

such that (G, H ) : p x 11 -, p is an F-arrow-map. Theorem 7.2.8 now proves that ( E , p , B ) is an F-fibration. 0 As it is the case for fibrations, F-fibrations can be characterized by lifting functions. Of course, we first must adjust the definition of lifting function to conform to the fact that the fibres belong to the category F: suppose that the arrow ( E , p ,9)has a lifting function

r:B1nEhE1 (see Section 2.2 and in particular, Figure 2.2.1). Let

ri : (B' nE) x I be the adjoint of I' and let w :

B' x I

-

B

+E

, w ( X , t ) = X(t) ;

I' is an F-lifting function if the arrow-map ( r " , w ) (see Figure 7.2.7) is an 3-arrow-map.

then

Theorem 7.2.11 An 3 - a r r o w is an F-fibration ifl it has a n F-lifting function. Proof -

+ The map w : B' x

1

-

B

, (A,t)

H

A(t)

is a homotopy of the evaluation map E ~ ~By . Lemma 7.2.8 there exists a homotopy k : (B' n E ) x I E

-

251

7.2. F-FIBRATIONS

FIGURE 7.2.7 of the evaluation map io such that (CZ,w) : p

x I +p

is an F-arrow-map. Let I? : B' fl E + E' be the adjoint of CZ; then r is an F-lifting function for p . + Let ( E , p , B ) be an 3-arrow with a lifting function I'. Also let (D, q, A ) be an F-arrow, let (g,h ) : q -+p be an 3-arrow-map and let H : A x 1 + B be a homotopy of h. Take H' : A + B' to be the adjoint of H and define the map

6 :D

+ B'

n E , @(z)= ( H ' q ( z ) , g ( z ) )

for every x E D. Now define G : D x 1 ---t E to be the adjoint of I'd; the definitions guarantee that (G, H ) : q x 11 --f p is an 3-arrow-map. 0

The homotopy category of F - denoted HF - is the category with the same objects of T and whose morphisms are homotopy classes in

F

of morphisms of F ;more precisely, and using the language of Fhomotopy over B , we say that f o , f i E F ( X , Y ) are homotopic in 3 if they are F-homotopic over a singleton space * (this simply means that the functions obtained at the various stages t E I of the homotopy are morphisms of F).

Corollary 7.2.12 Let ( E , p , B ) be an F-fibration. Then there exists a functor Fp : IIB H 3

-

from the fundamental groupoid of B t o the homotopy category of F .

252


Proof - For every b E B , let F,(b) be the fibre Eb over b. Let be a path in B with X(0) = b and X(1) = b'. The restriction of I" to ({A} x &) x (1) defines a morphism fx E F(Eb,&). Now let

H:IxI+B be a homotopy rel. d I of X to a path A' in B , also from b to b'; take the adjoint map H' ; I + B' and define the composition

to obtain a homotopy in F from fh to fx~. 0 Next, we investigate the pullbacks of F-fibrations. We begin by observing that Lemma 7.2.1 readily implies that pullbacks of T-fibrations are again 7-fibrations; what is nice (and not so trivial) is that pullbacks of 3-fibrations are well-behaved with respect to homotopies:

Theorem 7.2.13 Let ( E , p ,B ) E TF(C~)' be a n 7 - f i b r u t i o n and Eet f",fi : A -+ B be homotopic maps. Then the 3-fibrations obtained by pullback: ( A , nfo E , Pfo 9 A ) and ( A , nf, E , P h 7 A ) are F - h o m o t o p y equivalent over A ,

Proof - Let H : A x I --+ B be a homotopy from fo to f i ; form the pullback diagrams of Figure 7.2.8 where H*E = ( A x I ) pflH E and (H*E)" = ( A x { s } ) ~nj H*E, s = 0 , l . Note that

L :( Ax I ) x I x I

--+

AxI

L ( ( a , r ) , s , t )= ( a , ( l - t ) T +is)

7.2.

F-FIBRATIONS

253

FIGURE 7.2.8 and observe that L ( ( a , r ) , s , O ) = ( a , r ) . Since fi is an F-fibration, there exists a map

K : H*E x I x I

--+ H*E

such that, for every ( ( a ,T ) , e) E H*E and every s E I ,

is an F-arrow-map. We define next the map

M : H*E x I

N : H*E x I

+ H*E

-

H*E


254 3. pN

( p ~ l ) ( xp 11) .

1

Hence, N is an 3-homotopy over A x I from IH*Eto the map

k : H*E

+ H*E

defined by w v > , e >= M ( ( ( v ) , e ) , 4= " ( b , T ) , The next step is to define

IC"

e),O

: (H*E)' -+(H*E)'

k"(((a,I), el) = M ( ( ( a ,I), 4 0)

9

and

k' : (H*E)" -+ (H*E)' w a , O),e)) =

W((%0 > , 4 1)

'

Consider the 3-homotopy

M ( M ( - , s ) , 1) : (H*E)' x I

+ (H*E)*;

then M(M(-,O),l) = k'k" and M ( M ( - , l ) , l ) = kk I ( H * E ) l ,which in turn is F-homotopic to l ( ~ * , q l and so, klk' 2 1(H*,q1. We take

M ( M ( - , 1 - s),O) : (H*E)" x I * (H*E)O to prove that k"kl

1(H*,q0

.0

Corollary 7.2.14 Let ( E , p , B ) E TF(CG)+ be an F-fibration; if B contracts to bo E B , then ( E , p , B ) is 3-homotopy equivalent over B to the trivial 7-fibration (p-'(bu) x B , p 2 , B ) .0

EXERCISES 7.2.1 Prove that F-homotopy is an equivalence relation in TF(CG)-'. 7.2.2 Let ( E , p , B ) E T;F(CG)' and let f , g : A --+ B two given maps. Take the pullback arrows (Dp UJ E , p , f , A ) and (D,U, E , p g , A ) . Prove that the space of all F-arrow-maps ( k , l , A ) : p j + pg is homeomorphic to the space of all maps 4 : A -+ E * E such that

(P* P I 4

= (f,s>.

7.3. UNIVERSAL T-FIBRATIONS

255

7.2.3 Let ( D , q ,A ) , ( E , p , B ) E TF(CG)+and let

be two T-arrow-maps. Prove that (9,h ) and (g‘, h’) are T-homotopic iff their corresponding sections to q *1 p are 3-homotopic over A .

7.3

Universal F-fibrations

In this section we shall be concerned with three types of universal Tfibrations and their relations to each other. We shall not deal with the question of the existence of such universal objects. The existence of universal F-fibrations is thoroughly examined in [22] (see in particular Theorem 9.2 of that monograph); the reader is also directed to [23, Section 51 , where the authors present a slightly altered version of May’s theorem with an alternative proof. The central idea of the theory of universal 3-fibrations is to classify any T-fibration via a map (or even better, a homotopy class of maps) from the base space of the fibration to the base space of the universal one (see the observations preceeding Theorem 7.3.1); actually, by perusing the literature, one finds several different kinds of universality, which in most cases are equivalent. In this section we begin such a study. To start with, we shall put a restriction on dl the 3-fibrations ( E , p , B ) E T ~ ( c 6 ) -we consider in this section (with the possible exception of the universal 3-fibrations): the base spaces B will always be path-connected CW-complexes. Because 3-fibrations are indeed fibrations (see Lemma 7.2.7 ), the fibres of each of our 3-fibrations will have the same homotopy type; we wish to have this fact reflected in the definition of our category of fibres F and accordingly, we impose the following conditions on 3:

256


( F l ) Every morphism f E F ( X ,Y )is an F-homotopy equivalence over a singleton space * . As we shall also want to work with a “distinguished” fibre F , we shall assume: (F,) There exists a fixed space F E 3 such that, for every object

x E F,F ( F , X ) # 0.

Finally, throughout the section we shall always identify the products and X x * with X . A simple example of a category of fibres F is obtained by taking a fixed space F , all the spaces with the type of F and all possible homotopy equivalences between such spaces; other examples can easily be constructed. Another - but less direct - example is the following. Let G be a topological group and let F be a space on which G acts eaectively, that is to say, if

*x X

A:GxF+F is a left action and if R ( g , z ) = z, for every z E X, then g is the identity element of G. The objects of F are the pairs ( X , 4 ) where X is a space on which G acts effectively on the left and Q : F t X is a homeomorphism which preserves the action of G on F and X . The set of all morphisms between ( X ,4 ) and ( X I ,4’) is given by

The F-fibrations relative to this category of fibres are called G-bundles with fibre F . We now define an F-fibration (E,, p , ,B,) to be aspherical universal if, for any choice of base point Q E F * E , and every integer n 2 0, r,(F*E,,$) = 0. At this point we are requiring neither that B , is a CW-complex nor that it be path-connected. The latter requirement will be seen to be true later on (Theorem 7.3.1). An F-fibration (E,,p,,B,) is extension universal if, for every (D, q, A ) E T’(CG)-,for every subcomplex L c A and every F-arrow‘Because this category of T-fibrations leads towards the “classica1”category of fibre bundles, we direct the reader to [19] and [28] for a complete treatment on the subject, including the question of the existence of universal bundles.

7.3. UNIVERSAL 3-FIBRATIONS

extending ( g L , hL) (i.e., g

257

I DI,= g L , h I L = hL: see Figure 7.3.1).

FIGURE 7.3.1 Before we introduce our third kind of universal F-fibration, we give some notation. For a given path-connected CW-complex X , let & ( X ) be the set (view this class as a set!) of all 3-homotopy equivalence classes over X of 7-fibrations over X . As a consequence of Theorem 7.2.13 , &(-) is a contravariant functor from the homotopy category H C W , associated to - the category of path-connected CWcomplexes - to the category of sets (here notice that H C W , has for objects the same objects as but its morphisms are the homotopy classes of maps between path-connected CW-complexes). Besides, that theorem also shows that an 3-fibration (E,,pa, B,) defines a natural transformation

a. a

-

4(-) : [-,B3(;]

&(-)

*

An 3-fibration (E,,p,, B,) is free universal if the natural transformation $(-) is an equivalence. Theorem 7.3.1 If (E,,p,,B,) E TF(CG)’ fibration, then B, is path-connected.

is a f r e e universal

F-

CHAPTER 7. FIBRATXONS REVISITED

258

Proof - Let zu,zl be arbitrary points of B,; by regarding them as functions z0,21 :

* + B,

and pulling back p , over * via the functions zo and z1we obtain two F-fibrations over * which are F-homotopy equivalent over to the Ffibration ( F ,ck-, *), where CF is the constant function. Thus, the maps 20 and a1 are homotopic, that is to say, there exists a homotopy H from x 1 to B, such that H(-,0) = zo and H(-,1) = a l ; but H(*,t) is a path in B, connecting xu to z l . 0

*

*

Theorem 7.3.2 A n F-fibration (Em, p,, B,) iff it is extension universal.

is aspherical universal

Proof - + : Give an F-fibration (D, q, A ) together with a subcomplex L c A and an F-arrow-map ( g L , h L ) : q t + p,. Let S L : L + DL* E, be the section of q~ p , which corresponds to (gL,hL) by Theorem 7.2.5 . Let j be the inclusion map of DL*E, in D*E,; then (4 *1 p , ) j ~=~i : L

A

9

.

Now, because A is path-connected and 7r,(F*E,,q$) = 0 for every q$ E F E, and every n 2 0, p l p , is an n-equivalence for every n 2 1 and thus, the pair ( M ( qk1 p , ) , D E,) is n-connected for any n (see Exercise 6.2.2). Next, take the commutative diagram of Figure 7.3.2 and use Theorem 6.2.3 on the square to obtain a map s’ : A + D E , such that S’ I L = j,, , i ( q *1 p,)a’ z‘, re1.L

*

*

*

.

N

Let H : A x I + A be the homotopy rel. L such that q

-

7

0) = (! *1IPoop’

H ( - , l ) = 1A (recall that T ~ , . , ~ ~=Z 1.4). ~ Take the commutative diagram of Figure 7.3.3 with g defined by g

IA

x (0) = s

,g I L x I

By Lemma 7.1.2 there exists a homotopy

=js,

.

7.3. UNIVERSAL F-FIBRATIONS

259

FIGURE 7.3.2

4*1 Pcc T

f

FIGURE 7.3.3 G :A x I

-+

extending g and such that ( q k l p,)G s :A

D* E, = H. Let

D-k E,

be defined by 8 = G(-, 1). Note that s 1 L = js, and that ( ~ * ~ p , )=s 1 ~Use . again Theorem 7.2.5 to obtain an F-arrow-map ( g , h ) : q + p which extends ( g ~h, ~ ) . + : Let [k]E .rr,(F* E m , $ ) be given arbitrarily with

k : S"

-+

F* E,

, k(e,) = 4 : F

+ pG'(b)

Take ( F ,CI;,*),( S " ,cS7*) E TF(CG)+ and construct the pullback %-arrow (S" x F, CF, by Corollary 7.2.6 there exist two F-arrow-maps

for a certain b E B,.

(k',Ic") : C F

-

s");

p,

(c',cI') : cr.. +p ,


260

corresponding (uniquely) to k and the constant map c b ( S " ) = 46 E F * E,, respectively. Now form the commutative diagram of Figure 7.3.4 , with

$ :F

x I

-

E,

&: { e , ) x I Since (E,,p,,

, $ ( z , t ) = +(z)

E pil(b) ,

+ B, , & ( e o , t ) = b

.

B,) is extension universal, there exists

FIGURE 7.3.4

( K ' ,K") : cr; x 11 +p ,

8).

extending ((k' U c') U 6, (k" U c") U Applying again Corollary 7.2.6 we see that ( K ' , K " ) gives rise to a unique map

H : S" x I -+ F * E,

*.

such that (CF p,)H is the constant map from S" x I to We can verify that H is a homotopy between k and the constant map cb, i.e., [k]= 0. The following result about CW-complexes is important for the proof of Theorem 7.3.4.

Lemma 7.3.3 Let X be a path-connected CW-complex. Then there exists a numerable covering {U, I n E N} of X such that, f o r every n E N, the inclusion m a p

in : U,, -iX is homotopic t o a constant m a p .

7.3. UNIVERSAL F-FIBRATIONS

26 1

Proof - We shall give here a direct proof of this lemma, following that by A.Dold in [8, Proposition 6.71; however, we note that this lemma is also a consequence of the facts that CW-complexes are paracompact [15, Theorem 1.3.51 and locally contractible [15, Theorem 1.3.21. For every n E N, let X" denote the n-skeleton of X . Suppose that Xn+l is obtained from X " by the adjunction of a family of (n 4-1)-cells as in the pushout diagram of Figure 7.3.5. Now remove the centre of

FIGURE 7.3.5

each ball

and form the adjunction space

(We have already encountered a similar construction during the proof of Theorem 6.2.5.) Then %+l is open in X"+l and moreover, X" is a strong deformation retract of F+', say, given by the deformation retraction fptl

. jp+' x 1

+

H"f'(-,O)(P+l)

x-n+l

c X" ,

H"+l(-, 1) = 1.y.,,+,

7

, Y(x,t)E X" x I For a fixed n E N, define the open sets UL c H"+l(z,t)= x

XI'+'

in the following way: (Jo

n

= x'"\

lyll-l 7

ui+1 = Hri+Jtl

(-7

0)-l(E)

;

. by induction on j,


262

next, define U, = u j c ~ U i Notice . that (U,, 1 n E N} is a covering of X because, for every n E N, X " \ Xn-' C U, and hence, X " C UnENUn* We now prove that for each n E N, the inclusion map in : U,, + X is homotopic to a constant map. By induction on j , define the maps g; : u;3'x [O,j]

-----f

u;

j E N, so that:

and gi(2,O) E

u,", 2 E u;z .

Note, in particular, that gff(z,j) = by taking g1 n

and gi+l(,,t) =

i

=

Hni-1

2 , for

I u:

every

2

E

17;. This is done

x [0,11

It L j

g;i, (p-tj - t 1( a , O ) , t ) ,

0

Hn+jtl (

3' < t

Lectures on Homotopy Theory (North-Holland Mathematics Studies)

Lectures on Homotopy Theory

Lectures on homotopy theory