LECTURES ON HOMOTOPY THEORY
NORTHHOLLAND MATHEMATICS STUDIES 171 (Continuation of the Notas de Matematica)
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LECTURES ON HOMOTOPY THEORY
NORTHHOLLAND 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.
NORTHHOLLAND AMSTERDAM
LONDON
NEW YORK ' TOKYO
LECTURES ON H0MOTOPY TH E0RY
Renzo A. PlCClNlNl University of Milan Milan, Italy
1992
NORTHHOLLAND  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.  (NorthHolland 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 0444892389 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 9140793 514'.24dc20
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
This Page Intentionally Left Blank
Preface Men can do nothing without the makebelieve of a beginning. Even Science, the strict measurer, is obliged to start with a makebelieve 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 clockfinger 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 compactopen 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 Hspace and CoHspace 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 EckmannHilton 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 ndimensional 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 CWcomplexes, 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 CWcomplex 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 SeifertVan Kampen Theorem. The chapter is concluded with a section on some special CWcomplexes, the EilenbergMac 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 Hspaces and CoHspaces . . . . . 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 CWComplexes 6.1 CWcomplexes . . . . . . . . . . . . 6.2 Homotopy theory of CWcomplexes . 6.3 EilenbergMac Lane spaces . . . . . .
........ ........ ........
153 . . . 153 . . . 174 . . . 197
7 Fibrations Revisited 7.1 Sections of fibrations . . . . . . . . . . . . . . . . . . . . 7.2 3Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Universal Ffibrations . . . . . . . . . . . . . . . . .
.
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 compactopen 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 subbasis for the compactopen 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 compactopen 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 compactopen 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 subbasis 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 fl(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 ql(z) and
2
EX
and therefore, we can decompose the map k as k = (kql)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
CHAPTER 1. HOMOTOPY GROUPS
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 3 (Y,yo) take M , ( X , Y ) to be the space of all based maps f : (X,a,) with the compactopen topology.
Lemma 1.1.5 Let ( X ,zo),( Y , y o ) E Top, be given with X Huusdorfl. I f S is a subbasis 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 subbusis 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 subbasis 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

1.1. FUNCTION SPACES
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 as the onedimensional unit sphere S1 of R2 (centered taking (X,t,) 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 subbasis 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 fl(U) of EX. If q : S' x X + EX denotes the identification map,
K x
(32)
c ql(fl(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 .
1.1. FUNCTION SPACES
7
Now suppose that X is Hausdorff. According t o Lemma 1.1.5, it is with ) , K c X compact and U c CIY is of enough to study @  ' ( W K , ~ 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 subbasis 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 subbasis 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
CHAPTER 1. HOMOTOPY GROUPS
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 counterexample 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]
1.1. FUNCTION SPACES
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 and we are given three elements of Top,, say ( X , z , ) , (Y,yo) (2,2,). Define a function
by setting (a(f)(z))(y) = f ( z A y), for every f E M,(X A z E X and y E Y . Then prove the following statements:
Y,z),
(a) a is onetoone;
(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
Hspaces and CoHspaces
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. f1 and K : fi f2 be free Let fo, fi, f2 E M ( X ,Y ) and let H : fu 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 compactopen 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. HSPACES AND COHSPACES
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 Hspace if there exists a based map 1.1 : Y x Y + Y  called Hmultiplication  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 Hspace (Y,yo) with an Hmultiplication 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 Hspace.
1.2. HSPACES AND COHSPACES
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 Hmultiplication 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 pathconnected. Prove that if one fibre of ( E , p , B ) is pathconnected, 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 arrowmap. An arrowmap f : ( E , p , B ) ( E ' , p ' , B ) is a fibre homotopy equivalence over B if there exists an arrowmap 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,StS) 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
C H A P T E R 2. FIBRATIONS A N D COFIBRATIONS
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 arrowmaps (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 socalled 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
$(.)
= SUP I t t€ I
 +,t)
I
*
We begin by proving that q5 is welldefined and continuous. Take the map e x I. I , ( z , t )1 t  T I ( z , t ) I
:x
and note that, for every z E X , O({z} x I) is compact and thus, bounded; this implies that q5 is welldefined. Now fix a point z, E X ; suppose that 0 < q5(zo)< 1 and take E > 0 so that (q5(zO)~,q5(z0)+e). From [24, Lemma 3.5.81 (the “tube lemma”) we conclude that there exists an open set U1 C X such that
hence, for every z E Ul,$(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.
C H A P T E R 2. FIBRATIONS A N D 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
C H A P T E R 2. FIBRATIONS A N D COFIBRATIONS
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 , conclude that ( A , i ( f ) B , U j ( A x I))is a thus, writing i ( f ) = j i , . ~we
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 . the identity map 1~ : B + B shall also write z0 for the map j i ~ Take 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 selfmap 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 and at level 1, we have: H ( [ a , t ] 1) , = [u,O] and H ( [ b ] 1) , = [ b ] . On the 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 In1x (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.
CHAPTER 2. FIBRATIONS AND COFIBRATIONS
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 (nonclosed) 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 (nonclosed) 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.
CHAPTER 2. FIBRATIONS AND COFIBRATIONS
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 ) ~j is a homotopy that the map f factors out as f = ~ f i ( fwhere , ( f ) )is a cofibration. Using these statements equivalence and ( A , i ( f ) M 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.
CHAPTER 2. FIBRATIONS AND COFIBRATIONS
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.
2.4. APPLICATIONS OF T H E MAPPING CYLINDER
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 111Suppose 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 hyi = 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, = ?f’we obtain the equality T‘f‘ = J’gi“ and thus, from the equality 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
CHAPTER 2. FIBRATIONS AND COFIBRATIONS
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 hy 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 homotopy 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.
2.4. APPLICATIONS OF T H E MAPPING CYLINDER
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 DyerEilenberg 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 DyerEilenberg 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 .
This Page Intentionally Left Blank
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:
nnI


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 basepreserving 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:XPZ 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 ) the homotopy class of the constant function at the base that j l ( [ g ] is 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 pathcomponents 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).

a,) + 3  1 3 Let P : ( E ,e), + ( B ,b,) be a covering map; let fo, f l : (X, ( 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], C3n 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

where a* = p ( z ; ) , i = O,...,n; note that the coefficients ai 2 0 and 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+(lt)&) = 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
CHAPTER 4. SIMPLICIAL COMPLEXES
and
1
4 f I (P‘),I f I (a‘)) = Td ( 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
11
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 noncolinear 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:
4.1. SIMPLICIAL COMPLEXES
91
x(')
1) =T ; 2) T(')is the collection of all nonempty, 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
c71 C . * * C kr = u 1
92
CHAPTER 4 . SIMPLICIAL COMPLEXES
such that dim7j+' = 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.
4.1. SIMPLICIAL COMPLEXES
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 2,) E 'Y z$ set u = {zo,

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 
CHAPTER 4. SIMPLICIAL COMPLEXES
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 2 d M , the boundary abstract simplicial complex such that 1 K 1 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 pathconnected.
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 lsimplex 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 lsimplexes 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 lsimplex 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 lsimplex { 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 11
)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 11 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 fYfi
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 pathconnected, 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.
CHAPTER 4. SIMPLICIAL COMPLEXES
100
4.2.2 Let K and L two finite abstract simplicial complexes. Use the simplicial approximation theorem to prove that the set of homoL I is countable. topy classes of maps f :/K )+) 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
271.)
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
CHAPTER 4. SIMPLICIAL COMPLEXES
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 ) zr1
= (~a~yyar)
zr = ( z s , ~ , )
... ... ...
c r  1  cr2 Cr  Cr1
... ... ...
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 Cr1; moreover, according to the definition of z,] and z,, we conclude and similarly, that c,.+ = a,. Hence, that c ,  ~= 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 rskeleton K'  here r is a nonnegative integer  is the set of all simplexes of K of dimension 5 r ; then the subpolyhedron 1 K r 1 is the rskeleton 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.

CHAPTER 4. SIMPLICIAL COMPLEXES
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 2simplex; place it on the ) on the origin (O,O,O) (z,y)plane of R3with its barycentre b ( ~sitting 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 nskeleton 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 nsimplex
~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 Mn1.Let T n : Mn + Mn1 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 ndimensional cube; (ii) 1 K 1 is a 2dimensional 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 pathconnected if there exists a path X : I t X such that X(0) = z and X(1) = y. Prove that pathconnectivity is an equivalence relation in X . The equivalence classes in X defined by pathconnectivity are called path components. 4.3.4 A space X is pathconnected if every two points z,y E X can be connected by a path in X . Prove that every pathconnected space is connected (give an example to show that the converse is not necessarily true).
4.3.5 Prove that the pathcomponents of a polyhedron polyhedra of I K (.
I K I are sub
106
C H A P T E R 4. SIMPLICIAL COMPLEXES
4.3.6 Prove that a connected polyhedron is pathconnected (i.e,, for polyhedra the notions of connectivity and pathconnectivity 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 counterexample: 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 pathconnected polyhedron; then g(l K I) must be contained in a pathcomponent 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 pathconnected,
argue using its pathcomponents (note that there are finately many pathcomponents 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, consider the map g : X E given by g ( x ) = (O,x), for every x E X , and the homotopy H : X x I + B defined by

o l t l ; H ( x , t )=
2t1,
; < 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
C H A P T E R 4. SIMPLICIAL COMPLEXES
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 nsimplexes 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:
C H A P T E R 4. SIMPLICIAL COMPLEXES
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 0simplexes (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 E and k :I K I+ based map; decompose g into the maps g :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 0sphere 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:
C H A P T E R 4. SIMPLICIAL COMPLEXES
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 0simplexes
of
t o = 0 < tl < ..' < t , = 1 and 1simplexes {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 1simplex {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 .
4.4. FIBRATIONS AND POLYHEDRA
113
The first problem is to contruct
Go :I K(') lo X I+ E . Let I x 1 be a 0simplex 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
CHAPTER 4. SIMPLICIAL COMPLEXES
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 nsimplex. 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.
4.4. FIBRATIONS AND POLYHEDRA
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 ndimensional 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
C H A P T E R 4. SIMPLICIAL COMPLEXES
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 pathconnected. 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 (Y,yo), from g to f is a pair of based maps ( a , b ) with a : (Z,z,) 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) arrowmap. If ( a , b ) and (a‘,b’) are two arrowmaps 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 arrowmap 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 arrowmaps is an equivalence relation. The homotopy class of an arrowmap ( 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 arrowmaps from g to f. We now extend the definitions of CoHspaces and CoHgroups 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 CoHarrow if (2,I , ) and ( W,w,) are CoHspaces with CoHmultiplications uz :
and UII,’:
z +z v z
w +w v w
such that (uz, v1.1,) is an arrowmap, 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 CoHarrow g : (2,zo) , (W,wo)of Top,' is associative if the CoHmultiplications 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 arrowmap (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 CoHarrow g : (Z,z,) (W,w,) is said to be a CoHarrowgroupif 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 CoHspaces (2,zo) and (W,ID,), respectively; see Exercise 1.2.7); then the arrowmaps
(4% v 1z)vz;
(+LJo
v 1r1)vrr)
and (c(12 v cz,)vz; a(111 v cw,)vrl,)
are homotopic to the arrowmap
(111.,
1~).
2. 2 and W have coinverses (i.e., Z and W are CoHgroups; see Exercise 1.2.8 for the definition) $z : 2 + 2 and $11, : W + W such that (+z,$ql.): g g is an arrowmap and, denoting the based homotopies of the coinverses by
Lz : a ( $ z v 1z)v.Z

cx,
9
120
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
The arrowmap (c,,, two) given by the commutative diagram of Figure 5.1.3 is called a counit of g.
FIGURE 5.1.3 The arrowmap ($z,q!q,.) : g + g is a coinverse of 9. Finally, a CoHarrowgroup 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 arrowmap .) 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 CoHarrowgroup for every n 2 1, which is commutative if n 2 2. To reach this goal we need to understand very well the CoHspace structure of S" (see Lemma 1.2.6) and that of Bn+'together with the interplay between these structures;
5.1. HOMOTOPY GROUPS OF MAPS
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,Cnl(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 CoHmultiplications 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
5.1. HOMOTOPY GROUPS OF MAPS
and

vn
 Bn+1 ,

Bntl
123
v Bn+l
are associative CoHmulti~~ications on Sn and Bn+' respectively; moreover, the object in : ( S " , e , ) + ( B n + l , e o ) of Top,'
is a CoHarrowgroup 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
vpt1
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,knl((l  t ” ) t ‘ , y ) ) , kn(2t, kn“t’t’’,y))),
osts;
 1,kn1 (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 arrowmap and (@svn,dBcn)
N
kn
and
knl
(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
5.1. HOMOTOPY GROUPS OF MAPS
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 CoHarrowgroup. 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 welldefined. To check associativity, take arbitrarily [ a ,b],[a', b'] and [a", b"] E n ( g , f ) and note that
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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 arrowmap from g x 11 to f . The identity element of n ( g , f ) is given by the arrowmap
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 CoHarrowgroup. 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)thhomotopy 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 ~ ) homotopy classes of arrownote that the elements of ~ , , + ~ ( f , yare maps 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' ,
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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.
5.1. HOMOTOPY GROUPS OF MAPS
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 arrowmap 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
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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
% rnl(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
sn1
:I
srr2
:I x
Sn2+ Bnl
+
factors through the map Cr1q
giving rise to a map bn1
such that
bn1cn2
.~ n  I
sn1
= knl (see Figure 5.1.9
).
2
9 t
+
B"
b
X
FIGURE 5.1.9
21ncidentally, notice that b,,  1 induces a homeomorphism Bnl/Snd2 Z Snl.
5.1. HOMOTOPY GROUPS
OF MAPS
131
for every [a,b] E 7rn(f,yo), where cz,, is the constant map taking Sn2 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 arrowmap ( 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 (cn1 (t,Y)) =
{
gb‘(cnl(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,bl(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
5.1. HOMOTOPY GROUPS OF MAPS
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+ ) rnl(B,yo)
is defined as ( l ~ , c l . and ) ~ so, takes the homotopy class of an arrowmap (a,b) : in1 +i into the based homotopy class [a]E r n  l ( B , y o ) ; the function j * ( n  1 ) : r n  ~ ( Y , y o )+ rnl(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 arrowmap ( a , c ) : i,l c (where c is the contant map from B" into yo)  into the homotopy class of the arrowmap (cy,,
abnl) : 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))
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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
5.1. HOMOTOPY GROUPS OF MAPS
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 arrowmaps
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
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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 : Snl + B" is the inclusion. Let ( a , b ) : i,l arrowmap. 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 Snl,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 arrowmaps connected by a homotopy ( H , K ) : in1 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 arrowmap ( 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 arrowmap ( u , b ) the arrowmap ( h ,c) can be used again, this time associating to (8,I?) the arrowmap ( & c ) : i,l + c i i . Applying the exponential law to i, we obtain a homotopy of arrowmaps ( h , c ) (h',c). + be a given arrowmap; 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 an arrowmap ( a , b ) : inl i. This shows, ultimately, that tJis a 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
C H A P T E R 5. RELATIVE HOMOTOPY GROUPS
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 nthhomotopy group of the triad ( X ;A , B ; zo) by
rta(X;A,B;2 0 ) = TnI(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 CoHarrowgroup, 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 , c., is the constant map groups w,(X,z,) and T ( ~ , , , C ~ ~ ) where from X to {zo). 5.1.5 Prove that, for every n 2 1, there is an isomorphism between {xo}; zo) and ~ ( i , c, ,r o ) . Deduce from this and the groups 7rrL(X, 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 semidirect 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,Yl 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 arrowmap. 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 Snl to Y so that a+i+ = ai+
, a+i
a2 = aa
, aa+ = 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 welldefined 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 Hspace, [[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 Hspace if [ L , , L , ] = 0.
4. Let L,
The function [
5.2
, ] is the socalled
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 arrowmap. The homotopy
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
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 pathconnected 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 pathcomponents 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 and y2 true (see Exercise 5.2.1). We now define two based spaces 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 pathconnected 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 pathconnected 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 CS" be two copies of the cone CS" over the sphere S" with C+S" n CS" = S"; then the space CtS" U CS" 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", CS"; e,) are trivial: Theorem 5.3.1 For ,every integer
T
such that 2
5 T 5 2n,
rr(Sntl;CtSn, CS"; e,) = o
.
Proof  Assume that [f]E 7rT,(Sn+' ; C+S", CS"; e,) is represented such that by a map f : Ir' x I + Sntl
f : a(Ir') x I f : I"' x (1) f : Ir' x (0) u {io} x I
)
C+S"
+
cS"
t
e,
CHAPTER 5. RELATIVE HOMOTOPY GROUPS
148
(see Exercise 5.1.13). Consider the sets
and
E = { [ t , ~E ]CS" 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 antiimage 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 sequence of the triad CsS", C+S" \ { q } ; e,). Hence [f]= 0. 0
((7's";
150
C H A P T E R 5. RELATIVE HOMOTOPY GROUPS
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+', CS"; e,)
5 T 5 2n  1
and an epimorphism for
Proof  Take the exact sequence of the triad (S"+'; C+S", CS"; 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"+',CS") and notice, in view of the contractibility of the cones C+S" and CS", that they imply the isomorphisms
7rn+1(C+Sn,Sn;e,) 2 7rn(Sn,e,) and
(Sn+',e,) 2 7r,+] (Sn+*, CS"; e,) ; now use the isomorphism
X ~ + ~ ( C + S"; S " ,e,) 2 T,,+~(S"+', CS"; 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 socalled suspension homomorphism.) 5.3.3 Prove that if n # m, then R" and R" are not homeomorphic. S"l.) (Hint: Prove that R" \ (0) +
This Page Intentionally Left Blank
Chapter 6 Homotopy Theory of CWcomplexes 6.1
C Wcomplexes
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 CWcomplezes. 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 nballs. 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 CWCOMPLEXES 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. CWCOMPLEXES
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 nceZls of X ; the restriction is a homeomorphism onto ex, an open of f to an open ball B," \ S;' ncell, 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 ncells. The following result will be used in the next section.
156 C H A P T E R 6. HOMOTOPY THEORY OF CWCOMPLEXES
Lemma 6.1.2 Let ( X , A ) be a n adjunction of the ncells
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:YxIY by the formulae:
This ends the proof. 0 A pair ( X , A ) is called a relative CWcomplex if there exists a sequence of spaces
Xl
A
c Xo c X 1 c X 2c ... c XIa c ...
6.1. CWCOMPLEXES
157
such that: 1) X" is obtained from A by adjunction of 0cells (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 ncells; 3) X is the union space of the sequence
that is to say,
x = uxt 121
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 CWcomplex ( 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 nskeleton of the relative CWcomplex (X, A). Because compositions of cofibrations are cofibrations (see Exercise 2.3.6), a finite induction shows that if ( X , A ) is a relative CWcomplex 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 CWcomplex 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 CWCOMPLEXES defines a homotopy F : X x I + 2 such that F
FI(AxI)=G.
I
X = f and
0
The pair (Sn,Snl) with n 2 1 is a relative CWcomplex: it is created by taking the attaching map
which is the identity map on each component Sr' and with charac: Br + S" (these were defined in Section 5.1). teristic maps i+,L
Theorem 6.1.4 Let (X, A ) be a relative CWcomplex. 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 : xnlu ( 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 Xl = A = 0 and Xu is a discrete space, X is called a CWcomplex. For every integer n 2 0, the space X" is called nskeleton of X . Note that from a settheoretical point of view, a CWcomplex 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 CWcomplexes are Hausdorff and normal. As was the case for relative CWcomplexes, we define a CWcomplex X to
159
6.1. CWCOMPLEXES
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 CWcomplex with finitely many cells is said to be a finite CWcomplex; such a CWcomplex is clearly a compact space. The following are examples of CWcomplexes: 1) X = B"+l, with the skeleta: X o = X1 = *  . = XnI = {e,}, xn = s", x n + 1 = f p + 1 . 2) In this example we construct a CWcomplex X whose skeleta, up to and including X"', are constituted by a unique 0cell 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 nsphere S", the wedge product Vx Sy is also called a bouquet of nspheres. 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 CWcomplex with skeleta X" = X I =  = Xn* { ( e , ) } and one ncell for each A. 3) Let F = R , C or H be the field of real, complex or quaternionic numbers, respectively. Since H is noncommutative, 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 CWCOMPLEXES
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 nspace 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 ( Snkl, &  l , Bnk)and ( Snkl, fnl, FP"' ) (here k is the dimension of F as a vector space over R, inkl is the inclusion and fnl 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 gnl and j,I are defined as follows:
6.1. CW COMPLEXES
161
*I
I
znk 1
Znk1
FIGURE 6.1.3
 FPn'
fn1
Snk1
znk1
Jn1
 FP"
T
B"k
gn1
FIGURE 6.1.4 and jn1
([(yo, . . ' 9
) = [(yo, * * * 7 Yn1, O)].
~nl)]
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 gnl : Bnk\ Snk'
f
FP" \ FP"'
162 C H A P T E R 6. HOMOTOPY THEORY OF CWCOMPLEXES
 FPn'
fn1
,pk 1
\
I \ bak1
btk1
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
\ Snk1
   ,yn)] E FP" \ FP"'
where jj,, is the conjugate of yn and I n
Consequently, gnlijnl = 1 and ijnlgnl = 1. As in the previous example, the CWcomplex FP" has a unique Ocell. We prove next two theorems for CWcomplexes which, with the appropriate modifications of the statements, can be adapted to relative CWcomplexes.
Theorem 6.1.5 The topology of a CWcomplex is determined by the f a m i l y of its closed cells.
6.1. CWCOMPLEXES
163
Proof  Let X be a CWcomplex 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 SXnl
~ 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 ncells 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 CWcomplex 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 ncell 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 CWcomplex 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 CWcomplexes are compactly generated spaces.
0
Let il be a set of open cells of a CWcomplex 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 CWcomplex X are subcomplexes of X.
Theorem 6.1.8 Let X be a CWcomplex, 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 CWcomplex 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 ncells 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',An2)is an adjunction of (n  1)cells and An' is closed in X n  l . Let e be an ncell 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 ncells; 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 ncell 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 Xn1, 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 CWcomplex 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 ncells 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 Xnl= 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 CWCOMPLEXES
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 CWcomplexes in general? It is clear that since CWcomplexes are compactly generated spaces (see Corollary 6.1.7), the Cartesian product of two CWcomplexes 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 CWcomplexes and still obtain a CWcomplex. 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 CWcomplexes 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. CWCOMPLEXES
167
i T
FIGURE 6.1.8
Theorem 6.1.11 Let X and Y be CWcomplexes with skeleta X" and Y", respectively, n 2 0. T h e n there ezists a CWcomplex 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'
uxpl
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 nonnegative 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)nl.
168 C H A P T E R 6. HOMOTOPY T H E O R Y OF CWCOMPLEXES One can now see that ( X @ Y)" is obtained from ( X @ Y)"l by adjunction of ncells 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 CWcomplexes. If X is locally compact', then X @ Y = X x Y. Proof  Use Theorem B.6. Adjunctions of CWcomplexes are CWcomplexes 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 nskeleton A" of A into the nskeleton W" of W , for every n. We have:
Theorem 6.1.13 Let A be a subcomplex of a CWcomplex Y and let f : A + W be a cellular map. T h e n X = W Uf Y is a CWcomplex containing W as a subcomplex. lSee Exercise 6.1.6 for a characterization of locally compact CWcomplexes.
6.1. CWCOMPLEXES
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 ncells 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 ncells 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 +Xnl &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 CWCOMPLEXES
BAr are the topological sums of nballs 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 Xnl = m and ~
( j Ug 6)= I
.
We define the space 2, by 2n  x "  ' u W " . Since (Wn,Wn') is an adjunction of ncells, the law of horizontal compositions implies that (Zn,Xn') is an adjunction of ncells. 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 Ynl + 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 ncells (see proof of Corollary 6.1.9), it follows that (P, 2,) is an adjunction of ncells. Clearly, the attaching map for the ncells 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. CWCOMPLEXES
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 CWcomplexes and let f : X + Y be a cellular map; then the mapping cylinder M(f) is a CWcomplex. 0 We complete this section with another consequence of Theorem 6.1.13.
Corollary 6.1.15 Let A be a subcomplex of a CWcomplex X . Then X I A is a CWcomplex.
Proof  The constant map A {a,} into a 0cell {a,} is cellular. Now take Exercise 2.1.7 and the previous theorem. 0 It is possible to find counterexamples to the (false) statement: the quotient of two Hausdorff spaces is a Hausdorff space (exhibit a specific counterexample); the previous corollary shows that we should not look for counter examples in the category of CWcomplexes. In the next section we shall see that in the category of CWcomplexes, 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 nballs B,” give rise to a closed cofibration
172 C H A P T E R 6. HOMOTOPY T H E O R Y OF CWCOMPLEXES 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 CWcomplex. 6.1.4 Let X be a based CWcomplex. 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 CWcomplex 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 CWcomplex 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 endpoints 81, = {(O,O), (1, i)}, n E N \ (0) (see Figure 6.1.10). Show that X can be constructed as a CWcomplex,
FIGURE 6.1.10
6.1. CWCOMPLEXES
173
denoted Xcw. Prove that X with the topology induced from the euclidean topology of R2 call this topological space X ,  is not a CWcomplex. Prove that the spaces X,, and X , have the same homotopy type. 6.1.8
* Prove that CWcomplexes are LEC spaces. (Hint: Use Exercise 2.4.9.)
6.1.9
Let z, be a 0cell of a CWcomplex X . Prove that ( { z o ) , i , X )is a cofibration.
6.1.10 Let X be a CWcomplex. Prove that the unreduced cone cX and the unreduced suspension n X are CWcomplexes. If X has a base point z, which is a 0cell, the corresponding cone C X and the suspension EX are CWcomplexes; moreover, CX C X and a x EX.


6.1.11 Let X and Y be CWcomplexes with 0cells 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 CWcomplex. 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 CWcomplex (note that the fibre is a CWcomplex as a discrete space), then E is a CWcomplex; it seems natural to ask if the total space E of a fibration p : E + B is a CWcomplex whenever B and F are CWcomplexes. The answer is no, in general; however, one can prove the following: If p : E + B is a fibration over a connected CWcomplex B and with fibre F, a CWcomplex, then E has the homotopy type of a CWcomplex. (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 CWCOMPLEXES 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 CWcomplexes
We begin this section with a discussion about the important concept of nconnectivity for a pair of spaces. A pair ( X , A ) is said to be nconnected if every pathcomponent 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 nconnectivity:
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 nconnected iff the pathcomponents of X intersect A and, for every 1 5 T 5 n and every arrowmap ( 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 ncells 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 arrowmap (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 CWCOMPLEXES
ax
F
 Ux B;
b
Y
FIGURE 6.2.1 of ncells ( 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 Sil
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 CWCOMPLEXES 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 0cells and (Y,B ) is a pair such that all pathcomponents of Y intersect B then, for every arrowmap ( 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 CWcomplex of dimension 5 n and let (Y,B ) be an nconnected pair. Then, for every arrowmap ( 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' XIY
such that Hkl(,O) = jbk,, Hkl(,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
Hk1
:x"' x I + Y
given by f i k  l ( z , t ) = H k  l ( z , 1t)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 CWCOMPLEXES in other words, HL is a homotopy relative to A. Now take and the arrowmap (bL1, g) : Lk1 j

177
IT;(, 1) = 6
(here ~ k denotes  ~ the inclusion of X"' into X k ) ;because ( X k , X k  ' ) is an adjunction of kcells, 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 GWcomplex 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 mcell of X \ A with attaching map f : Sm' + Xml. The hypothesis on Y implies that jmlf factors
with m  1
through B" (see Exercise 2.3.13) and thus, extends to a map from Xml Uf B" to Y . Of course, this can be done with every finite adjunction of mcells 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 mcells to X"l and 9m1 a :x,"lty
is an extension of jml. 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:il)} is an increasing chain of 0 we define g':
uxy a

k'
by g' I X r  ' = ijzl. 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 CWCOMPLEXES 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 ncells are (n  1)connected is true, as we shall prove anon.
Theorem 6.2.5 Let ( X , A ) be an adjunction of ncells with n 2 1; then ( X , A ) is (n  1)connected. Proof  Suppose that
clearly, all the connected components of X meet A. B') (with T 5 n Now take the arrows (Srl,ir,F and let (a,b ) : i,l z
 1) and
( A , i ,X)
f
be an arbitrary arrowmap. 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 ) = bl(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
6.2. H O M O T O P Y T H E O R Y O F CWCOMPLEXES
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 CWcomplex 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 pathcomponent 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 CWCOMPLEXES 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) = iy6, 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'irl = rbirl = 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 CWcomplex and let 0 5 n < dimX; then (X, X n ) is nconnected. Proof  We first prove that for every m > n, ( X m ,Xn) is nconnected. The previous theorem shows that ( X " + ' , X " ) is nconnected; suppose that m 1 > n and, by induction, that (Xm', Xn)is nconnected. Now observe that (S", X m  ' ) is (m1)connected (again, by Theorem
6.2. HOMOTOPY THEORY OF CWCOMPLEXES
181
6.2.5) and that the pathcomponents of X" intersect X":in fact, the ( m 1)connectivity of (X'", X"') implies that the pathcomponents of X" intersect Xm' and the nconnectivity of (Xm',Xn) implies that the pathcomponents 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 pathcomponents 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
and hence, ( X m X , " ) is nconnected. To prove that (X,X") is nconnected 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 arrowmap 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 nconnected. 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 nspheres, 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 CWCOMPLEXES
Proof  Regard S” as a CWcomplex with just one 0cell and one 1cell. Since the rskeleton of the relative CWcomplex
remains unchanged up to and including the dimension 2n  1 (see Theorem 6.1.11), it follows that k
(JJ j=1
s;, v s>; k
j=1
=
(ns;, (n k
k
j=l
j=1
qSnl)
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 nonabelian; 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 pathconnected. 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 yo). classes relative to 81 of maps ( I ,01)+ (Y, 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 CWCOMPLEXES
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

fn1.
Next, for each i = 0,    ,n, choose paths
184 CHAPTER 6. HOMOTOPY THEORY OF CWCOMPLEXES 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
fn1
* *
'
XfllflllX;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,)
* * *
[XnlfnlXil]

* 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 welldefined, 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:.ItV = [f]where f is (both taking 8I into a0). Then 4(w) = [ir;f][i~.g] defined by composition of paths:
O n 2 1; moreover, assume that ( B ,A n B ) is a relative CWcomplex 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 CWcomplexes
. 

A; = ( A n B ) U E l U *   UE j
, B; = B U A,
and form the sequences of CWcomplexes
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,AiI)+ (Bi,Bii)
. ,Ic,
6.2. HOMOTOPY THEORY OF CWCOMPLEXES
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 c B;. Now we use the "five lemma" to prove the sequence Bo c 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 CWcomplexes
..
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 CWCOMPLEXES 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 arrowmap 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 sub. gives rise to a commutative diagram of homotopy space K(B‘ X I )This
6.2. HOMOTOPY THEORY O F CWCOMPLEXES
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 onetoone 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 CWcomplexes is homotopic to a cellular map:
Theorem 6.2.11 Let X and Y be CWcomplexes 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 0cell, or f(z) is connected to a 0cell of Y by a path since every pathcomponent of a CWcomplex contains at least a 0cell). Next, define
F” : KC’ x I
+Y
192 C H A P T E R 6. HOMOTOPY T H E O R Y OF CWCOMPLEXES 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
Fnl
: K"'
I (X"x (1)) c Y o .
xI+Y
such that
F"1
1 (Xnlx (0) u L
and
x I)=F
I (Xn1
x (0) uL x I )
F"l(x"'x (1)) c Y"l .
Let e be an ncell 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 Fnl

:
x I + X x (0)
u K"l
x x (0) u KrLIx I
+
x I
Y.
6.2. H O M O T O P Y T H E O R Y O F CWCOMPLEXES
193
Because the restriction of be to Sf'' x { 1) maps that space into Y"l C Y" and the pair (Y,Y")is nconnected (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 Fnl.) The crucial property of the restriction F," 1 (P' U e) x (1) is that
in1
x 11
Y FIGURE 6.2.4
194 C H A P T E R 6. HOMOTOPY THEORY OF CWCOMPLEXES such a map is cellular. Repeating this process for all ncells of X we obtain an extension of FnI 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 socalled 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 CWcomplexes 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 CWcomplex (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 Theo~ , = 0 for every rem 2.3.9); because the homotopy groups T , ~ ( T [f(zCo)]) 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 nconnected, 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 CWCOMPLEXES
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 nconnectivity 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) CWcomplexes ( 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 nequivalence 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
nconnect ed .
+
Y is an nequivalence iff ( M ( f ) X, ) is
2. Prove that if f : S + Y is an nequivalence and K is a CWcomplex, 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 CWCOMPLEXES 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 CWcomplex with n 2 1, and ( B ,A n B ) is an mconnected relative CWcomplex. 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 nconnected CWcomplex, 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. EILENBERGMAC L A N E SPACES
197
FIGURE 6.2.5
6.3
EilenbergMac Lane spaces
In this section we shall study a very important class of CWcomplexes, first investigated extensively by S.Eilenberg and S. Mac Lane (see [13] and [14]), known as EzlenbergMac Lane spaces. Formally, we shall say that a pathconnected CWcomplex X is an EilenbergMac 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 CWcomplexes with a unique 0cell to which all the other cells are attached; this will be achieved with constructions similar to our general construction of CWcomplexes 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 CWcomples with a 0cell 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 CWCOMPLEXES 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 CWcomplex 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
6.3. EILENBERGMAC L A N E SPACES
199
Remark  The lemma is still valid if we assume X to be a union space of a sequence of based CWcomplexes
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, pathconnected CWcomplex M ( T ,n ) such that
Proof  Consider the CWcomplex 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 CWCOMPLEXES
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 , follows that [g~,] = [fZ], and hence, gZ = gLx fx, for classes [ L ~ ] it 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).

6.3. EILENBERGMAC L A N E SPACES
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 pathconnected CWcomplex M(.rr,n) with a unique 0cell 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
lR+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 CWcomplex 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 CWcomplex 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 CWCOMPLEXES
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 CWcomplex 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. EILENBERGMAC LANE SPACES
203
for every T 2 1 and therefore,
(the pathconnectivity 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 EilenbergMac 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 CWcomplex X = K ( n ,n ) with only one 0cell 2, such that T, r = n r r ( X i ~ o ) 0, r # n . Proof  We first construct the CWcomplex 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
204 C H A P T E R 6. HOMOTOPY T H E O R Y OF CWCOMPLEXES
i
v,
B,:+2
f

xn+2
FIGURE 6.3.3 and in this way, we construct a relative CWcomplex ( X , + Z ,M ( T ,n)) having only one 0cell, 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 constant map (see Exercise 2.3.13); this implies that 7rn+I(Xn+2,zo) Z
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 pathconnected, based CWcomplex 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
6.3. EILENBERGMAC LANE SPACES
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 CWCOMPLEXES
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 EilenbergMac 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. EILENBERGMAC 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,")
208 C H A P T E R 6. HOMOTOPY T H E O R Y OF CWCOMPLEXES
FIGURE 6.3.7
is nconnected; 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
6.3. EILENBERGMAC L A N E SPACES
.
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 CWCOMPLEXES 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 EilenbergMac 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 EilenbergMac 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 EilenbergMac 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 pathconnected space and let n 2 1 be an integer. Then there exists a based pathconnected space X ( n ) such that: 1. ( X ( n ) , X ) is a relative CWcomplex 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
6.3. EILENBERGMAC LANE SPACES
f
X
211
Y
FIGURE 6.3.8
Lemma 6.3.12 Let ( X , z o ) , ( Y , y o )E Top, be pathconnected 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 CWcomplex ( 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 pathconnected space ( X , z , ) there is a tower of spaces and maps as in Figure 6.3.9, where the maps pn,nl 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 CWCOMPLEXES
x FIGURE 6.3.9
EXERCISES 6.3.1 Prove that for every space X there exists a CWcomplex 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 Hspace. (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 CWcomplex ( X ,zo)and every integer n
1, the set
Byx,7r) = [ X ,K ( a ,n)]*
has an abelian group structure.
2
6.3. EILENBERGMAC L A N E SPACES
213
(b) A based map f : (Y,yo) + ( X )z,) of CWcomplexes 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 CWcomplex ( X ,z,) and every integer n 2 1,
B"X,
R) %
H"+'(CX, R ) .
(e) If ( X ,a,) is based CWcomplex 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 CWcomplexe with 0cells and 1cells only. Prove that R ~ ( Xzo) , is a finitely generated free group.
This Page Intentionally Left Blank
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 compactopen 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 subbasis 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 81 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 compactopen 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.
CHAPTER 7. FIBRATIONS REVISITED
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 lefthand 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*

C H A P T E R 7. FIBRATIONS REVISITED
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,1t); 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 p. such that pG = H . It is easy to see that ii = G(, 1) E sec(el,p(el)) Now we must prove that A is welldefined. 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:BxIE,
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’
,
7.1. SECTIONS OF FIBRATIONS
(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 endpoints 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 Gl 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
CHAPTER 7. FIBRATIONS REVISITED
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. endpoints, 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*
7.1. SECTIONS OF FIBRATIONS
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 Kl : 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 endpoints 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
C H A P T E R 7. FIBRATIONS REVISITED
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 pathcomponent 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 pathconnected,
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 pathconnected fibre F over p ( e ) ; moreover, we assume that the arrow defined b y the inclup # 0. Then there sion of { p ( e ) } in B is a cofibration and that sec(e,p(e)) is a bijection
Corollary 7.1.13 Let ( X , z o ) (Y,yo) , E Top, be based spaces such that Y is pathconnected 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)
CHAPTER 7. FIBRATIONS REVISITED
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 arrowmaps. 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 arrowmap 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, Snl)is a cofibration, A can be extended to a homotopy : sn1 X I X I + Y .
At
Now take the homotopies A1 1 7
A' A; : 29
y  1
XIY
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 Snl,
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 arrowmap 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 pathcomponent of yo, there is a function A : W Y 0 , Z O ) x Tl(f,9") 7rn(f,zo) ' *

C H A P T E R 7. FIBRATIONS REVISITED
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
sn1,
.
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 arrowmap
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 arrowmap
and homotopies A:) and B:. Next, construct the homotopies
A : S"' x I
+
B : B" x I + X
Y
,
7.1. SECTIONS OF FIBRATIONS
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 arrowmap (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
C H A P T E R 7. FIBRATIONS REVISITED
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)
SnI
v snI ,
= bi]
.
Consider the CoHmultiplications vn1
. s n  1
$
Ynl : B" + B" V B" and extend them to v,~ x 11 and
&l
x 11 in the obvious way. Then
is an unbased arrowmap 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)vnr A I {e,) x I = A(t) , B I B" x (01 = r(b, v bb>Vn1 , 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) : in1 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
CHAPTER 7. FIBRATIONS REVISITED
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.
be a based space such that ( ( z , } , i , X ) is a cofibration. 7.1.2 Let (X,z,) 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 3sphere 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
C H A P T E R 7. FIBRATIONS REVISITED
FFibrations
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 nonempty 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 Tarrow if p is onto and, for every b E B , p  ' ( b ) = Eb is an object of F. An arrowmap ( g , h ) : ( D , q , A ) + ( E , p , B ) is an Farrowmap 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 Farrowmap (9,h ) : (D, q, A ) 3 ( E , p , B ) simply by (9, h ) : q p. The category of Farrows and Tarrowmaps is denoted by TF(CG)+. We shall indicate that an arrow ( E , p , B ) is an object of T'(CS;)simply by writing ( E , p , B ) E TF(CG)' or p E TF(CG)+; if an arrowmap ( 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 Farrowmap. Moreover, i f ( D ,q, A ) E TF(CG)*and (9,f) : q p is an Farrowmap, the arrow ( l ,1,i) : f
7.2. FFIBRATIONS
23 7
3 ji  where,L is the unique map determined by the universal property of pullbacks  is an Farrowmap. 0
q
In particular, if A is a subspace of B and f is the inclusion map, the 3arrow 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 Fhomotopic 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 3arrowmap; we use the notation (9,h ) 2 (g', h') to indicate Fhomotopy. If A = B , h = h' = 1~ and H = prl is the projection on the first factor, (G,prl) is an Fhomotopy over B . We use the notation (g,lg) (g',lg)  or simply g g' to indicate that (g,lB) and (g',lg) are 3homotopic over B. An Farrowmap (g,l~): (0, q, B) + (E,p, B)is an Fhornotopy equivalence over B if there exists an Farrowmap (g',lg) : ( E , p , B ) + ( D , q , B )such that gg' and g'g are Fhomotopic over B to the respective identity maps. We want to study these Fhomotopy equivalences over a space more closely (see Theorem 7.2.4), but first, we analyse just onehalf of that concept: more precisely, we say that an Farrowmap (g', 1g) : p +q is a right Fhomotopy inverse of (g,ls): q + p if gg' wg 1 ~Let .
;
2
( H , P R ) :P x
wg
be the Fhomotopy 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))}
.
C H A P T E R 7. FIBRATIONS REVISITED
238
These observations prompt us to define an Fsection of (g,lB) as a map u : E + G such that
is an Farrowmap 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 Farrowmap (g,lB) : q topy inverse iff it has an 3section. 0
+
p has a right Fhomo
We now turn our attention to Fhomotopy 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 Fhomotopy 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 Fsection u : El.
+ Gv =
{(d,X) E G I q(d) E V }
of ( g l r , l \ r ) , there exists an Fsection 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 3section r : E
+
G defined by
r ( e ) = (g'(e), H ( e , ))
, Ve E E .
Define a homotopy
J : G XI &
E'
239
7.2. TFIBRATIONS 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 Fsection 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 Fhomotopy equivalence over B and U c B is an arbitrary subspace of B , the restriction
is an Fhomotopy 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 Farrowmap (g,lB) : q + p which is an Fhomotopy equivalence over each element of a special covering of B , then ( g , l ~ is) an Fhomotopy 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. 3FIBRATIONS
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 (DoldMay) Let ( g , l ~ ): ( D , q ,B ) + ( E , p ,B ) be a given 3arrowmap. Let C = {U} be a numerable covering of B such that, for every U E C, (gu,1u) : qu 4 pu is an 3homotopy equivalence over U. Then (g,lB) is an 3homotopy equivalence over B . Proof  It is enough to prove that the hypotheses allow us to conclude that (g,lB) has a right Fhomotopy inverse (g',lg) over B: in fact, let (ju,1u) be an 3homotopy inverse of (gu,1u) over U ,for each U E C; then 3 1 .% &.fgUgb gU
;
and thus, (gt/,lu)is an 3homotopy equivalence over U , for every U E C; but then (gl,1 ~ has ) itself a right 3homotopy 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 Fhomotopy inverse of (g,1 B ) over B and hence, that ( g , l ~ )is an Fhomotopy 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. .
CHAPTER 7. FIBRATIONS REVISITED
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 7section 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 Tsection
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 Fsection p : Eu, + GLI,such that p = r1over #J*(l) n U. Now define
243
7.2. 3FIBRATIONS
Then (W,q.) > (V, a contradiction. 0 The following space generalizes the notion of function space: for any two Farrows ( 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 necessarily a fibration) associated to q,p E T7(CG)); such an arrow is 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 3arrow 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
CHAPTER 7. FIBRATIONS REVISITED
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 Farrowmap (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. FFIBRATIONS
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 Farrows which are fibrations. An object ( E , p , B ) E T’(CB)^ is an FJibration if it satisfies the Fcovering homotopy property with respect to all Farrows that is to say, if for every Tarrow ( D , q , A ) ,every Farrowmap ( 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 Farrowmap (see Figure 7.2.3).
D x I
G
*E
P 1
AxI
H
*B
FIGURE 7.2.3: Fcovering homotopy property
246
C H A P T E R 7. FIBRATIONS REVISITED
Note that if F E F,( F x B , p r z , B ) E this is the trivial Ffibration.
TF(CG)’ is an Ffibration;
Lemma 7.2.7 Ffibrations are fibrations.
Proof  Let ( E , p ,B ) be an Ffibration; we are now given an arrow ( D , q , A ) ,an arrowmap (9, h ) : q + p and a homotopy H : A x I + B . The arrow ( A , nh E , p , A ) obtained by pullback is an Farrow; hence, there is a homotopy GI : ( A , nh E ) x I + E such that (GI, H ) : jj x 11 + p is an Farrowmap. 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 arrowmap. 0 Clearly, an arbitrary fibration might fail to be an Ffibration because F may not have enough morphisms. The following lemma gives a simple characterization of Farrows which are also Ffibrations.
Lemma 7.2.8 A n Farrow ( E , p ,B ) is an Ffibration if f o r every map f : W + B , any Farrowmap (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 Farrowmap.
Proof 
+ Follows easily from the fact that p is an Ffibration.
+ Take an Farrow ( D , q , A ) ,an Farrowmap ( 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 Farrowmap. 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 Exercise 2.1.3). Since p ( g p r l ) = hprl(q x l ~ )the , universal property of pullbacks gives rise to a unique map
9 : D x I + ( A P n h E )x I
7.2. FFIBRATIONS
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:DxIE,
it is easy t o check that G(,0) = g and that ( G , H ) : q + p is an Farrowmap. 0 As we have already seen, the functional arrow associated to two Farrows is not necessarily an Farrow; thus, the functional arrow associated to two Ffibrations does not have to be an Ffibration; however, we have the following result:
Theorem 7.2.9 The functional arrow associated to two FJibrations is a fibration.
Proof  We are given two 3arrows ( 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 arrowmap (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 Farrowmap 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 arrowmap 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 Tfibration and Qo is an 3arrow, there exists a homotopy
K’ : ( ( W x (0)) n 0)x I
+
E
such that K’(, 0) = and (K’,k‘) is an Farrowmap. Next, consider the Tarrowmap (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 Ffibration, we obtain a hornotopy H’ : x I ) , nh 0)x I  D
((w
such that (H’,h’) is an 3arrowmap. Now form the commutative diagram of Figure 7.2.6, where each square is a pullback and define the map
7.2. FFIBRATIONS
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 Ffibrations in terms of functional arrows and fibrations:
Theorem 7.2.10 An Farrow ( E , p ,B ) is an 3fibration 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 Farrowmap. Define the map
CHAPTER 7. FIBRATIONS REVISITED
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 Farrowmap. Theorem 7.2.8 now proves that ( E , p , B ) is an Ffibration. 0 As it is the case for fibrations, Ffibrations 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 Flifting function if the arrowmap ( r " , w ) (see Figure 7.2.7) is an 3arrowmap.
then
Theorem 7.2.11 An 3  a r r o w is an Ffibration ifl it has a n Flifting 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. FFIBRATIONS
FIGURE 7.2.7 of the evaluation map io such that (CZ,w) : p
x I +p
is an Farrowmap. Let I? : B' fl E + E' be the adjoint of CZ; then r is an Flifting function for p . + Let ( E , p , B ) be an 3arrow with a lifting function I'. Also let (D, q, A ) be an Farrow, let (g,h ) : q +p be an 3arrowmap 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 3arrowmap. 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 Fhomotopic 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 Ffibration. Then there exists a functor Fp : IIB H 3

from the fundamental groupoid of B t o the homotopy category of F .
252
CHAPTER 7. FIBRATIONS REVISITED
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 Ffibrations. We begin by observing that Lemma 7.2.1 readily implies that pullbacks of Tfibrations are again 7fibrations; what is nice (and not so trivial) is that pullbacks of 3fibrations are wellbehaved 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 3fibrations 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.
FFIBRATIONS
253
FIGURE 7.2.8 and observe that L ( ( a , r ) , s , O ) = ( a , r ) . Since fi is an Ffibration, 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 Farrowmap. We define next the map
M : H*E x I
N : H*E x I
+ H*E

H*E
CHAPTER 7. FIBRATIONS REVISITED
254 3. pN
( p ~ l ) ( xp 11) .
1
Hence, N is an 3homotopy 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 3homotopy
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 Fhomotopic 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 Ffibration; if B contracts to bo E B , then ( E , p , B ) is 3homotopy equivalent over B to the trivial 7fibration (p'(bu) x B , p 2 , B ) .0
EXERCISES 7.2.1 Prove that Fhomotopy 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. UJ E , p , f , A ) and (D,U, E , p g , A ) . Take the pullback arrows (Dp Prove that the space of all Farrowmaps ( 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 TFIBRATIONS
255
7.2.3 Let ( D , q ,A ) , ( E , p , B ) E TF(CG)+and let
be two Tarrowmaps. Prove that (9,h ) and (g‘, h’) are Thomotopic iff their corresponding sections to q *1 p are 3homotopic over A .
7.3
Universal Ffibrations
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 Ffibrations 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 3fibrations is to classify any Tfibration 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 3fibrations ( E , p , B ) E T ~ ( c 6 ) we consider in this section (with the possible exception of the universal 3fibrations): the base spaces B will always be pathconnected CWcomplexes. Because 3fibrations are indeed fibrations (see Lemma 7.2.7 ), the fibres of each of our 3fibrations 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
CHAPTER 7. FIBRATIONS REVISITED
( F l ) Every morphism f E F ( X ,Y )is an Fhomotopy 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 Ffibrations relative to this category of fibres are called Gbundles with fibre F . We now define an Ffibration (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 CWcomplex nor that it be pathconnected. The latter requirement will be seen to be true later on (Theorem 7.3.1). An Ffibration (E,,p,,B,) is extension universal if, for every (D, q, A ) E T’(CG),for every subcomplex L c A and every Farrow‘Because this category of Tfibrations 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 3FIBRATIONS
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 Ffibration, we give some notation. For a given pathconnected CWcomplex X , let & ( X ) be the set (view this class as a set!) of all 3homotopy equivalence classes over X of 7fibrations 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 pathconnected 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 pathconnected CWcomplexes). Besides, that theorem also shows that an 3fibration (E,,pa, B,) defines a natural transformation
a. a

4() : [,B3(;]
&()
*
An 3fibration (E,,p,, B,) is free universal if the natural transformation $() is an equivalence. Theorem 7.3.1 If (E,,p,,B,) E TF(CG)’ fibration, then B, is pathconnected.
is a f r e e universal
F
CHAPTER 7. FIBRATXONS REVISITED
258
Proof  Let zu,zl be arbitrary points of B,; by regarding them as functions z0,21 :
* + B,
and pulling back p , over * via the functions zo and z1we obtain two Ffibrations over * which are Fhomotopy equivalent over to the Ffibration ( F ,ck, *), where CF is the constant function. Thus, the maps 20 and a1 are homotopic, that is to say, there exists a homotopy H from x 1 to B, such that H(,0) = zo and H(,1) = a l ; but H(*,t) is a path in B, connecting xu to z l . 0
*
*
Theorem 7.3.2 A n Ffibration (Em, p,, B,) iff it is extension universal.
is aspherical universal
Proof  + : Give an Ffibration (D, q, A ) together with a subcomplex L c A and an Farrowmap ( g L , h L ) : q t + p,. Let S L : L + DL* E, be the section of q~ p , which corresponds to (gL,hL) by Theorem 7.2.5 . Let j be the inclusion map of DL*E, in D*E,; then (4 *1 p , ) j ~=~i : L
A
9
.
Now, because A is pathconnected and 7r,(F*E,,q$) = 0 for every q$ E F E, and every n 2 0, p l p , is an nequivalence for every n 2 1 and thus, the pair ( M ( qk1 p , ) , D E,) is nconnected 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 FFIBRATIONS
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
Dk 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 Farrowmap ( 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 conby Corollary 7.2.6 there struct the pullback %arrow (S" x F, CF, exist two Farrowmaps
for a certain b E B,.
(k',Ic") : C F

s");
p,
(c',cI') : cr.. +p ,
C H A P T E R 7. FIBRATIONS REVISITED
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 CWcomplexes is important for the proof of Theorem 7.3.4.
Lemma 7.3.3 Let X be a pathconnected CWcomplex. 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 FFIBRATIONS
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 CWcomplexes are paracompact [15, Theorem 1.3.51 and locally contractible [15, Theorem 1.3.21. For every n E N, let X" denote the nskeleton of X . Suppose that Xn+l is obtained from X " by the adjunction of a family of (n 41)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)
xn+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'"\
lylll 7
ui+1 = Hri+Jtl
(7
0)l(E)
;
. by induction on j,
CHAPTER 7. FIBRATIONS REVISITED
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
=
Hni1
2 , for
I u:
every
2
E
17;. This is done
x [0,11
It L j
g;i, (ptj  t 1( a , O ) , t ) ,
0
Hn+jtl (
3' < t