Homology of Cell Complexes

138 Pages·015.927 MB·English

by  Cooke, Finney

HOHOLOGY OF CELL COMPLEXES
BY
GEORGE E. COOKE AND ROSS L. FINNEY
(Based on Lectures by Norman E. Steenrod)
PRINCETON UNIVERSITY PRESS
AND THE
UNIVERSITY OF TOKYO PRESS
PRINCETON, N~d JERSEY
1967
l'orswora
These are notes based on an introductory course in
®
Copyright 1967. by Princeton University algebraio topology given by Professor Norman Steenrod in
Press
All Rights Reserved the fall of 1963. The principal aim of these notes is to
develop efficient techniques for computing homology groups
of complexes. The main object of study is a regular complex:
a C'-complex such that the attaching map for each cell is
an embedding of the boundary sphere. The structure of a
regular complex on a given space requires, in general,
far fewer cells than the number of simplices necessary
to realize the space as a simplicial complex. And yet the
Published in Japan exclusively
by the University of Tokyo Press; procedure I orientation ~chain complex ~homology groups
in other parts of the world by
Princeton University Press is essentially as effective as in the case of a simplicial
complex.
In Chapter I we define the notion of CW-complex, due
to J. H. C. Whitehead. (The letters CW stand for a) closure
f i n ite--the closure of each cell is contained in the union
of a finite number of (open) cells-- and b) weak topology-­
the topology on the underlying topological space is the weak
topology with respect to the closed cells of the complex.)
We gi ve several examples of complexes, regular and irregular,
and complete the chapter with a section on simplicial complexes.
In Chapter II we define orientation of a regular complex,
and chain complex and homology groups of an oriented regular
Prir,ted in the United States of America
complex. The definition of orientation ot a regular complex this difficulty we alter the topology on the product. The pro­
requires certa in properties of regular complexes which we per notion is that of a compactly generated topology. In order
call redundant restrict ions. We assume that all regular to provi de a proper point of view for this question we
complexes satisfy these restrictions, and we prove in a include in Chapter IV a discussion of categories and func­
leter chapter (VIII) that the restrictions are indeed redun­ tors.
dant. The main results of the rest of Chapter II are: a In Chapter V we prove the Kunneth theorem. We also
proof that differ ent orientations on a given regular complex compute the homology of the join of two complexes, and we
yield isomorphic homology groups, and a proof of the complete the chapter with a section on relative homology.
universal coeffioi ent theorem for regular complexes which In Chapter VI we prove the invariance theorem, which
have finitely many cells in each dimension. states that homeomorphic finite regular complexes have
In Chapter III we define homology groups of spaces isomorphic homology groups. We also state and prove the
which are obta1ned from regular complexes by making cellular seven Eilenberg-Steenrod axioms for cellular homology.
identifications. This technique simpl i fies the computation In Chapter VII we define singular homology. We state
of the homology groups of many spaces by reducing the number and prove axioms for singular homology theory, and show that
of cells required. We compute the homology of 2-manifolds, if X is the underlying topological space of a regular complex
certain ~-manifolds called lens spaces, and real and complex K, then the si ngular homology groups of X are naturally
projective spaces. isomorphic to the cellular homology groups of K.
Chapter IV provides background for the Kunneth theorem In Chapter VIII we prove Borsuk's theorem on sets in
on the homology of the product of two regular complexee. Sn which separate Sn and Brouwer's theorem (invariance of

Given regular compl exes K and L there 1s an obvious way to domain )
that Rma nd Rn
are homeomorphic only if m· n.
define a cell structure on IKl)( lLl--simply take products of We ahow that any regular complex satisfies the redundant

L. IKI IL'
cells in X and in But the product topology on X restrictions stated in Chapter II, and settle a question

is in general too coarse to be the weak topology with raised i n Chapter I concerning quasi complexes.

respect to closed cells. Thus KK L, with the product In Chapter IX we define skeletal decomposition of a

topology, i s not in general a regular complex. To get around space and homology groups of a skeletal decomposition. We
Barbara Duld, June Clausen, and Joanne Beale for typing
prove that the homology groups of a skeletal decompos i t ion
and correcting the manuscript.
are isomorphic to the singular homology groups of the
underlying space . Finally, we use skeletal homology to
George E. Cooke
show that the ho. ology groups we defined in Chapter III
Ross L. Finney
of a space X obta i ned by identification from a regular
June 2, 1967
complex are i somorphi c to the singular homology groups of X.
We shoul d mention that we sometimes refer to the
"homol ogy" of a space without noting which homology theory
we are using. This is because all of the different
definitions of t he homology groups that we give agree
on their common domains of definition.
We also remark that cohomology groups, which ws defi ne
for a regular complex in Chapter II, are only touched on
very lightly throughout these notes. We do not cover the
cup or cap products, and we do not define singular coho­
mology groups.
Finally, we wish to express our gratitude to those who
haTe helped us in the preparation of t hese not es. First, we
tha.nk Martin Arkowitz for his efforts in our behalf--he
painstakingly read the first draft and made many helpful
suggestions for revision. Secondly, we thank the National
Science Foundation for supporting the first-named author
dur ing a protion of this work. And finally, we wish to
thank Elizabeth Epstein, Patricia Clark, Bonnie Kearns,
Table of Contents
Foreword
Table of Contents
Introduot ion i

I. Complexes 1

II. Homology Groups for Regular Complexes 28

III. Regular Complexes with Identifications 66

IV. Compactly Generated Spaoes and

Product Complexes 86

V. Homology of Produots and Joins

Relative Homology 106

VI. The Invariance Theorem 138

VII. Singular Homology 177

VIII. Introductory Homotopy Theory and the

Proofs of the Redundant Restrictions 207

IX. Skeletal Homology 239
INrRODUCTION
One of the ways to proceed from geometric to algebraic topology is
to. associate with each topological space X a sequence of abelian groups
00
(H (X)} ,and to each co.ntinuous map f o.f a space X into a space
q q=O
Y a sequence of ho.momorphisms
f : H (X) ---;;;. H (Y) ,
q* q q
one for each q. The groups are called homology grou~s of X or of Y,
as the case may be, and the morphisms f are called the induced homo-
q*
morphisms of f, or simply induced homomorphisms. Schematically we have
e;eometry ---;;;. ale;ebra

space X ---;;;. homology groups H (X)

q
map f: X ~ Y ---;;;. induced homomorphisms f
q*

The transition has these properties, among others:

1. If f: X ~ X is the identity map, then f H (X) ---;;;. H (X)
q* q q
is the identity iso.morphism for each q.
2. If f: X........,. Y and g: Y ---;;;. Z , then (gf)q* g • f for

q* q*
each q That is, if the diagram
\ g
Y
f /
,.

X Z
gf
of spaces and mappings is co.mmutative, then so is the diagram
i
If' f exists, ,-re say that f is an extension of h to X, or, e'1uiva­
H (Y)
I \
'1
f of X into y.
lently, that h has been extended to a mapping
f •• gq. There are famous solutions of this problem in point-set topology.
H'l (X) --..;;.> H (Z) The Urysohn Lemma is one . The hypotheses are: the space X is normal,
'1
(gf) '1-)( but othenfise arbitrary; the subspace A Ao U Al ,\-Ihere Ao and Al
a.re disjoint, closed subsets of X the space Y is the closed unit
of homology groups and induced homomorphisms , for each '1. In modern interVal [0,1) and the map h: A ---> Y carries A o to o and Al
parlance, homology theory is a functor from the category of spaces and
to 1.
mappings to the category of abelian groups and morphisms.
The correspondence here between geometry and algebra is often cru.de.
Topologically distinct spaces may be made to correspond to the same al ge­
braic objects. For example, a disc and a point have the same homology
groups. So do a I-sphere and a solid torus (the cartesian product of o
a one-sphere and a diSC.) Despite this, the methods of algebraic topol­
Under these assumptions the Urysohn Lemma asserts that there exists a
ogy may be applied to a broad class of problems, such as extension
mapping f: X ---> Y such that f hi
problems.
X
Suppose we are given a space* x , a subspace A of x , and a map i/ " l'
'.:::.I.
h of A into a space Y If we let i denote the inclusion mapping (Ao U AI) [0,1] Y
~
of A into X, then the extension problem is to decide whether there
The Tietze Extens i on Theorem is another solution. Here, X is
exists a map f, indicated by the dashed arr ow in the diagram below,
normal,
A is closed in X, and h is an arbitrary map of A to the
of X into Y such that h fi.
set Y of real numbers.

X
/ " ,
i f
A h >' .>I Y
*Unless otherwise stated, the \{Ord space will mean Hausdorff space.
The theorem asserts the existence of an extension f: X ---> Y of h.
ii
iii
Path~~se connectivity can also be described in terms of extensions.
HCJ.(X)
Let x be the closed unit interval, and let A = (O,l} A space Y i·l "- ¢
"­
is pathwise connected if, given two points Yo and Yl of Y , the "­ "'­
mapping h: A~Y defined by h(O) = Yo and h(l) = Yl ' can be H (A) h* ~H (Y)
extended to a mapping f: CJ. CJ.
X~Y
Given the groups, the homomo~hisms i* and h*, does there exist a
homomo~hism ¢: H (X) --7> H (Y) such that i~ h*?
CJ. CJ.
If f is a solution of the geometric extension problem, then
o
¢ = f is a solution of the algebraic problem for each CJ.. On the
Cl.*
other hand, if ¢ does not exi,st, then f does not exist either. This
Furthermore, the problem of the existence of a continuous multipli­ latter implication is the principal tool in the proofs of many theorems
cation with a prescribed two-sided unit in a space Y can be phrased as of algebraic topology.
an extension problem. We seek a map m: Y X Y ~ Y and an element 1
Let A be a subspace of a space X A map f: X ~ A is a
= =
in Y such that m(l,y) m(y,l) y for each y in Y The reCJ.uire­
r etraction if fx = x for each point x in A • Under these circum-
ment that there be a two-sided unit determines a function h on a sub-
stances, A is called a retract of X The Tietze Extension Theorem
space A = (1 X Y) U (Y X 1) = YVY of Y X Y into Y. The existence
implies that if X is a normal space which contains an arc A, then
of a continuous multiplication with ~ t,m-sided unit is now eCJ.uivalent
X can be retracted to A.
to the existence of a map m: Y X Y ~ Y such that mi = h •
Throughout this and subseCJ.uent chapters, ~ will denote the closed
Y X Y
n
unit ball in Euclidean n-space R , and Sn-l will denote the unit

'1
m
n

{n-l)-sphere in R .
:,to.
h
YVY Y
THEORD1: Sn-l is not a retract of ~ •
Corresponding to each Beometric extension problem is a homology f: ~ ~ Sn-l
Suppose to the contrary that there exists a map

extension problem, described for each CJ. by the followinB diagram: such that fx x for each x in sn-l
Then f is an extension of
the identity map h:
iv
v
En
r This l ast diagram shows once more that the transition from geometry
r
i "'"f to algebra may be crude. The inclusion mapping i: Sn-l ~ is a
n-l h .. n-l
S ~ S homeomorphism, yet i* is not one-to-one. Furthermore ~ maps
H _ ( Sn -l ) onto H _ (En) even though i does not map Sn -l onto En

n 1 n 1
If n 1, the fact that El
is connected, while SO is not, A fixed point of a mapping f: X ~ X is a pOint x in X for

shows that no such f can exist.
which fx = x. Using the fact that Sn-l is not a retract of En we
If n > 2 , a different argument is required.
Assume that the
can prove the Brouwer Fixed-point Theorem:
homology groups of En and of Sn-l
are, as .~ will later show them
to be, THEOREi,l: ~ mapping f: En ~ En has.!:: ~ pOint .
{z

'1 0 r
VEi') e { : Hq (8n-l ) == '1 0, n - 1 Suppose to the contrary that there exists a mapping f: ~ En
'12: 1 o O<'1<n-l which has no fixed point. Then we can define a retrs,ction g : En ~ Sn-l
For each pOint x of En we let Rx denote the directed line segment
If f exists, then for each
'1 the following diagram is commutative :
which starts at fx and passes through x. Note that FLx is defined
H (En)
f """"':""'/
f or each x in En, since f has no fixed pOint. Let gx = Rx n(Sn.l_{fx)
e f q. If x lies in Sn-l, then gx = (x) n Sn-l = x. The verification
H (Sn-l) h* > H (Sn-l)
that g is continuous is left as an exercise.
'1 identity q
homomorphism
For q n - 1 , however, the diagram looks like this:
tit
4f~~
*
fn_1,
z h* > Z
identity
Since the only homomorphism of the zero group into Z is the zero homo­
morphism, there is no ¢ such that ¢i* = h-~ Hence f cannot exist Thus g is a retraction of En onto Sn-l. This contradicts the preceding
after all.
t heorem, and completes the proof of this theorem.
vi
vii
Chapter I
COMPLElCES
1. Complexes
If A is a subspace of X and B a subspace of Y, then a ~
f: (X,A) ~ (Y,B) of the pair (X,A) to the pair (Y,B) is a map
of X into Y that carries A into B One can compose mappings
of pairs. For each pair there is an identity mapping of the pair onto
itself. A homeomorphism of (X,A) onto (Y,B) is a homeomorphism of
X onto Y which carries A onto B. A more important concept is
that of a relative homeomorphism.
A mapping f: (X,A) ~ (Y,B) is a relative homeomorphism if
fj (X-A) maps X-A homeomorphically onto Y-B. A relative homeo­
morphism need not map A onto B •
Exercise. Let X be a compact space, and suppose that
f: (X,A) ~ (Y,B) is a mapping such that fl (X-A) maps X-A one-
to-one onto Y-B. Show that if Y is a Hausdorff space then f is
a relative homeomorphism.
Exercise. Find an example to show that if Y is not a Hausdorff
space then f may not be a relative homeomorphism.
1.1. The definition of a cOSRlex.
K IKI
A complex consists of a Hausdorff space and a se~uence
IK
of' subspaces, called skeletons, denoted by n 1 n -1, 0, 1, ••• ,
which satisfy the following conditions.
1