Download hohology of cell complexes george e. cooke and ross l. 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
1967
JERSEY
l'orswo ra
These are notes based on an introductory course in
Copyright
®
196 7. by Princeton University Press
All Rights Reserved
algebraio topology given by Professor Norman Steenrod in
the fall of 1963.
The principal aim of these notes is to
develop efficient techniques for computing homology groups
of c omplexes.
The main object of study is a regular compl ex:
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 f ewe r cells than the number of simplices necessary
to realize the space as a simplicial complex.
Published in Japan exclusively
by the University of Tokyo Press;
in other parts of the world by
Princeton University Press
procedure I
orientation
~ chain
complex
And yet the
~homology
groups
is essentially as effective as in the case of a simplicial
comple x.
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 g i ve se veral examples of complexes, regular and irregular,
and comple te the chapter with a section on simplici al complexes.
In Chapter II we def ine orientation of a regular complex,
and chain complex and homology groups of an oriented regular
Prir,ted in the United Stat e s of America
The defi nition of orient at ion ot a regular complex
this diff icul ty we al ter the topology on the product.
The pro­
requires ce r t a in properties of regular complexes wh ich we
per notion is that of a compactly generated topology.
In order
cal l redundan t r es tri c t ions.
to prov i de a proper point of view for this question we
complex.
We assume that all regul ar
comple xes sa tisfy these restric tio ns, and we prove in a
include in Chapter IV a d iscus sion of categorie s and func­
leter chapte r ( VIII) that the restrictions are indeed re dun­
tors.
dant .
The main results o f the res t of Chap ter II are : a
In Chapter V we prove the Kunneth theorem.
We also
proof that d iffe r ent orientat ions on a given re gular complex
compu te the homolo gy of the join of two complexes, and we
yie ld isomorphic homology groups, and a proof of the
complete the chapter with a se ction on re lative homology.
un ive r sal coeffi oi en t t heo rem for regular complexes which
have fini tely many cells in each dimension.
s tate s that homeomorphic finite regular complexes have
I n Chapter III we define homol ogy groups of spaces
which are obta1 ned from r egular compl exe s by making cellular
identifications.
This te chnique simpl i fies the computation
of the homolo gy groups of many spaces by r educing the number
of cells required.
certain
~ -manifolds
In Chapter VI we prove the invariance theorem, which
isomorphic homo l ogy groups.
We also state and prove the
seven Eilenberg-Steenrod ax ioms for cellular homolo gy.
In Chapter VII we define singular homology.
We state
and prove axioms for singular homology theory, and show that
We compute th e homo lo gy of 2 -manifol ds,
if X is the underly ing to po logical space of a regular complex
called lens spaces, and r eal and complex
K, then the s i ngul ar homo logy groups of X are naturally
pro j ect ive spaces .
isomorphic to the cellular homo logy groups of K.
Chapter IV prov ide s background fo r the Kunneth theorem
In Chapter VIII we prove Borsuk's theorem on sets in
on the homol ogy of the product of two regular complexee.
Sn which separate Sn and Brouwer's theorem ( invariance of Given r egular compl exes K and L there 1s an obv i ou s way to
m
domain ) that Rand
Rn
are homeomorphic only if
define a cell structu re on
cells in X and in
L.
IKl)( lLl --simpl y take produ c ts of
But the product topo logy on
IKI X IL'
is in general t oo coarse to be the weak topo logy with
res pect to c lo sed cells.
Thus
KK L, with the pro duct
topology, i s not in general a regular complex .
To get a round
m· n.
We ahow that any regular complex satisfies the redundant restrict io ns stated in Chapter II, and settle a question raised i n Chapter I concerning quasi complexes. In Chap te r
IX we def ine skeletal decomposition of a space and homo l o gy groups of a skeletal decomposition.
We
prove that the homology groups of a skeletal decompo s i t ion
are isomorphic to the singul a r homo l ogy groups of t he
underlying space .
Barbara Duld, June Clausen, and Joanne Beale for typing
and correcting the manusc ript.
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
complex are i somorphi c to the singular homology groups of X.
We shoul d men tion that we sometimes refer to the
"homo l ogy" of a space wit hout noting which homolo gy theory
we are using.
This is be cause all of the different
definitions of t he homology groups that we give agree
on their common domains of def inition.
We a lso rem ar k t hat cohomology groups, which ws de f i ne
for a regular complex in Chapter II, are only to uched 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 gra titude to those who
haTe helped us in the preparation of t hese no t es.
First, we
tha.n k Martin Arkowit z for his efforts in our behalf--he
pa instakingly read the first draft and made many helpful
suggest ions for revision.
Secondly, we thank the National
Science Foundation fo r supporting the first-named a uthor
dur ing a protion of this work.
thank
And finally, we wish to
Elizabe th Epstein, Patricia Clark, Bonnie Kearns,
June 2, 1967
Table of Contents
Foreword
Table of Contents
Introduot ion i
I. Complexe s 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
IX. Skeletal Homology 207 239 INrRODUCTION
One of the ways to proceed from geometric to algebraic topology is
to. associate with each topological space
X a se quence of abelian groups
00
(H (X)}
q
,and to each co.ntinuous map
q=O
f
o.f a space
X into a space
Y a sequence of ho.momorphisms
f
one for each
q.
q*
H (X) ---;;;. H (Y) ,
q
q
:
The groups are called homology
as the case may be, and the morphisms
morphisms of
space
f:
X
X
~
---;;;.
---;;;.
X or of
of
Y,
are called the induced homo-
q*
f , or simply induced homomorphi sms.
e;eometry
map
f
grou~s
Schematically we have
ale;ebra homology groups
H (X) q
Y ---;;;. induced homomorphisms
f
q* The transition has these propertie s, among others: 1.
If
f:
X ~ X is the identity map, then
is the identity iso.morphism for each
2.
each
q
If
f:
X........,. Y
and
g:
f
q*
H (X) ---;;;. H (X)
q
q.
Y ---;;;. Z , then
(gf) q*
g
That is, if the diagram
Y
f
/
\,.
g
Z
X
gf
of spaces and mappings is co.mmutative, then so is the diagram
i
q
q*
• f
q*
for
If'
H (Y)
'1
I\
f ••
H'l (X)
f
exists, ,-re say t hat
lently, that
is an extension of
has been extended to a mapping
h
f
to
of
X, or, e'1uiva­
into
X
y.
There a r e famous sol utions of this problem in point-s et topology .
gq.
H (Z)
--..;;.>
h
f
The Urysohn Lemma is one .
'1
The hypotheses are:
but othenfise arbitrary; the subspace
(gf) '1-)(
X
a.re d isj oint, close d subsets of
of homology groups and induced homomorphisms , for each
'1.
In modern
interVal
parlance, homology theory is a func tor from the category of spaces and
to
[0 ,1)
and the map
Ao
the space
A
h:
A
X is normal,
the space
U
Al ,\-Ihere
Ao
and
Al
Y is the closed unit
---> Y carries
A
o
to
o
and
Al
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 a l ge ­
For example, a disc and a point have the same homology
braic objects.
groups.
So do a
o
I- sphere and a solid torus (the cartesian product of
a one-sphere and a diSC.)
Despite this, the methods of algebraic topol­
ogy may be applied to a broad class of problems, such as extens ion
Under these assumptions the Urysohn Lemma asserts that there exists a
mapping
f:
X ---> Y such that
problems.
Suppose we are given a space *
h
of
of
A into a space
A into
x,
If we let
Y
a subspace
i
A
of
x,
i/
and a map
denote the inc l usion mapping
(Ao U AI)
f
hi
X
"
l'
'.:::.I.
~
[0,1]
Y
X, then the extension problem is to decide whether there
The Tietze Extens i on Theorem is another solution.
exists a map
of
X into
Here,
X is
f , indicated by the dashed arr ow in the diagram below,
Y such that
h
normal, A
fi.
set
is closed in
X, a nd
h
is an arbitrary map of
A to the
Y of real numbers. X
i
A
"
/
h
,
f
'.>I
> Y
*Unless otherwi se stated, the \{Ord space will mean Hausdorff space.
The theorem asserts the existence of an extension
ii
iii
f:
X ---> Y of
h.
Path~~se
x
Let
connectivity can also be described in terms of extensions.
be the closed unit interval, and let
is pathwise connected if, given two points
mapping
h:
A~Y
extended to a mapping
defined by
f:
h(O)
= (O,l}
A
Yo
= Yo
and
and
A space
of
Y , the
= Yl
' can be
Yl
h(l)
HCJ.(X)
Y
i·l
"-
"­
¢
"­
"'­
CJ.
CJ.
X~Y
Given the groups, the homomo~hisms
homomo~hism
If
¢= f
o
f
cation with a prescribed two-sided unit in a space
an extension problem.
We seek a map
m(l,y)
Y X Y ~ Y and an element
m:
= m(y,l) = y
Y can be phrased as
for each
y
in
Y
ment that there be a two-sided unit determines a function
space
A
and
h*, does there exist a
--7>
H (Y)
is a solution of the algebraic problem for each
CJ..
On the
Cl.*
Furthermore, the problem of the existence of a continuous multipli­
Y such that
i*
such that i~
h*?
CJ.
is a solution of the geometric extension problem, then
¢: HCJ.(X)
¢
other hand, if
in
~
H (Y)
h*
H (A)
= (1 X Y) U (Y X 1) = YVY
Y X Y into
of
Y.
to the existence of a map
m:
does not exist either.
of algebraic topology.
on a subThe existence
Let
r etraction if
stances,
A map
A be a subspace of a space X
fx = x
for each point
in
x
A is called a retract of X
A •
Y X Y ~ Y such that mi
= h •
X can be retracted to
n
R
{n-l)-sphere in
A, then
A.
n-space
~
will denote the closed
Rn , and Sn-l will denote the unit .
:to
,.
Y
Corresponding to each Beometric extension problem is a homology
extension problem, described for each
is a
Under these circum-
X is a normal space which contains an arc
unit ball in Euclidean
m
h
X ~ A
The Tietze Extension Theorem
Throughout this and subseCJ.uent chapters,
Y VY
f:
t,m-sided unit is now eCJ.uivalent
~
YXY
'1
This
latter implication is the principal tool in the proofs of many theorems
implies that if
of a continuous multiplication with
f
1
The reCJ.uire­
h
does not exi,s t, then
CJ.
by the followinB diagram:
THEORD1:
Sn-l
~ •
is not a retract of
Suppose to the contrary that there exists a map f:
such that
fx
the identity map
x
for each
x
sn-l in
h:
iv
v
Then
f
~ ~ Sn-l
is an extension of
En
r
i
This l as t diagram shows once more that the transition from geometry
"'"f
n-l
S
If
1, the fact that
n
shows that no such
f
h
to algebra may be crude.
..
n-l
S
~
homeomorphism, yet
n-l
H _ (S
)
n 1
El
is connected, while
SO
is not,
n > 2 , a different argument is required.
homology groups of
En
and of
Sn-l
are, as
Assume that the
'1
{
0
Hq (8n-l ) ==
:
'12: 1
If
f
exists, then for each
'1
{z
'1
does not map
X ~ X is a pOint
maps
n-l
S
x
in
fx = x.
e
H (Sn-l)
'1
Using the fact that
Sn-l
is not a retract of
onto
En X
for En
we
o
O<'1<n-l
~ mapping
En ~ En
f:
has.!:: ~ pOint .
Suppose to the contrary that there exists a mapping
f:
r
~ En
which has no fixed point.
Then we can define a retrs,ction g : En ~ Sn-l
For each pOint
we let Rx
x
of
En
denote the directed line segment
the following diagram is commutative :
f """"':""'/ q
f:
i
~
is a
can prove the Brouwer Fixed-point Theorem:
0, n - 1
H (En)
For
even though
A fixed point of a mapping
THEOREi,l:
e
onto
Furthermore
r
will later show them
.~
to be,
VEi')
is not one-to -one.
n
H _ (E)
n 1
Sn-l ~
i:
can exist.
which
If
i*
The inclusion mapping
which starts at
fx
f or each
En, since
If
f q.
h* >
H (Sn-l)
identity
q
homomorphism
tha t
x
x
in
lies in
g
and passes through
Sn-l, then
f
x.
Note that
has no fixed pOint.
gx = (x) n Sn-l = x.
Let
FLx
is defined
gx = Rx n(Sn.l_{fx)
The verification
is continuous is left as an exercise.
n - 1 , however, the diagram looks like this:
4f~~
z
h*
>
identity
tit
f n _1 ,
*
Z
Since the only homomorphism of the zero group into
morphism, there is no
¢
such that
after all.
¢i*
=
h-~
Z is the zero homo­
Hence
f
cannot exist
Thus
g
is a retraction of
En
onto
Sn-l.
This contradicts the preceding
t heorem, and completes the proof of this theorem.
vi
vii
Chapter I
COMPLElCES
Complexes
1.
If
A is a subspace of
f:
(X,A) ~ (Y,B)
of
X into
of pairs.
X and
of the pair
Y that carries
B a subspace of
(X,A)
A into
to the pair
B
Y, then a
(Y,B)
~
is a map
One can compose mappings
For each pair there is an identity mapping of the pair onto
itself.
A homeomorphism of
X onto
Y which carries
(X,A)
onto
A onto
B.
(Y,B)
is a homeomorphism of
A more important concept is
that of a relative homeomorphism.
A mapping
fj (X-A)
maps
f:
X-A
(X,A)
f:
Let
(X,A) ~ (Y,B)
to-one onto
Y-B.
(Y,B)
is a relative homeomorphism if
homeomorphically onto
morphism need not map
Exercise.
~
A onto
Y-B.
A relative homeo­
B •
X be a compact space, and suppose that
is a mapping such that
Show that if
fl (X-A)
maps
X-A
Y is a Hausdorff space then
onef
is
a r e lative homeomorphism.
Exercise.
space then
1.1.
f
Find an example to show that if
Y is not a Hausdorff
may not be a relative homeomorphism.
The definition of a cOSRlex.
A complex
K
IKI
consists of a Hausdorff space
of' subspaces, called skeletons, denoted by
which satisfy the following conditions.
1
IKn
1
n
and a se~uence
-1, 0, 1, ••• ,
1.
IK_1 1 is the empty set, and
I~I
2.
Each
3.
IKI '" UIKn I •
4.
For
is closed in
IK_11
C
IKol
c
IKll • . •
c
IKbI
c
If there exists an integer
is called the dimension of
If
n
n
n
al , a2 ,···, ai' ..•
n > 0 , the components
j~1 .
are open sets in the relative topology of'
the
n-cells of
of'
1Kn l
-
I'Kn_l l
though
They are referred to as
IK
/K'
n
n-cell
a.
of
1
=
Ca~1 - a~)
1
.
ln
For each
•
-n
a i denote that subspace of
K let
n
a
i
which is the closure of
rP1
IKI
a~1
though it need not be homeomorphic to
The topology of
cells of'
section
7.
K:
A
there exists a relative homeo­
IKol
is a discrete subspace of
q _< n
are called vertices of
K
The relative homeomorphisms
f
IKOI
set of
fls
0 E
r
K , even
IKI .
Its components,
IK I
n
IKn I
coincide:
IK
n
I
and the weak tOPOlogy defined
i f and only i f
and for each
A of
A
IK I
n
is closed
n a~1 is closed in
The structure of
~
are not part of
n,l
The cells
o
K.
To show
fls
is
replaced by another.
n-cell
O~
of
1
actually home.omorphic to the
not be homeomorphic to
mo rp1.l~
wise,
f
.
n,l
of
En
K is irregular.
relative homeomorphisms
a subset
.
IK I , by (4).
A complex is to remain unchanged when one set of
n
E
- S
n-l
-n
ai' then
a~
If
1
.
n,l
K
1
need
1
there exists a homeo­
- --- - ­
is called regular.
is a cell of
rP
Its boundary
•
a~
If, for each
onto
f
K, with the relative topology, is
n-cell
Sn-l
other-
K and if nOne of the
is a homeomorphism, then
a~
is called
irregular.
In order to be sure that the topologies in (6) and (7) are well-
i
defined, one must know that
1.2.
The least such
that a candidate for a complex satisfies (5) one need only exhibit some
1
The relative topology on
for each
of
By (5), an
is the ,~ak topology defined by the closed
IKI
in the relative topology of
1
En.
IKI is closed if and only if each inter­
is closed in -n
o.
by the closed cells of'
a~
--
A of
a subset
n -n
0 i
for each
q
K is not defined as a collection of cells.
The subspace
,rith the relative topology, n
n-l)
(-n.n
n-l
morphism
f'n,l.: ( E, S
~ a.,
a.)
which carries S
onto
1
1
'n
-n
a
For convenience, o. is called a closed
i
n-cell of K , even
1
6.
IK I
K.
For each
and let 11
K.
which are pOints by (5), are also open sets of
5.
q+
K, we will sometimes write
is a cell of
0
such that
K is said to be of finite dimension.
q?: r , then
IKI
r
IKI '"
u a~ and that
q, i
complex.
1
IK I '"
u
n .1; q _< n
a~
1
These two facts follow, as they should, from conditions (1) through (5).
A complex
K is said to be a complex~
jKI ,and
the underlying~, or the geometric rea lization of
complex is one with finitely many cells.
K
IKI
is called
He leave their verification to the reader.
A finite
A complex is called infinite
Exercise.
s trQcture of
if it is not finite.
2
Establish the following elementary facts about the
K.
3
.
1.
IKI
2. I f
3.
=
n
-n
U . cr i
n,l U cr .
1
n,i
q
cr
is a
I~I
q-cell of
cr~
u
1
i; q:: n =
q-cell
cr
q
of
1
;q
q
=a n
(While the first three parts of this
.
5. Let X be a space that is the union of finitely many disjoint
let
cr
oi
i
- cr
•
i
cr.
1
denote the closure of
Suppose that for each
k
i)
for some
the dimension of
cr.
2.
Cr
1
cr
i
X
cr.
1
a
k
k-l
(E , S
)
and
onto
k-cell and call
sq
spheres,
is
lies in a union of cells of dimension lower than that
only if
7.
f
/0
If
from
/KI
-q
1n cr.
X is continuous if and
into a space
is continuous for each cell
cr of
K.
K is of . finite dimension, then condition (7) implies
S
q
( q-l)-skeleton
The
sq , and
cr~ , whose union
and
sq _ Sq-l
1
f
, each
f
q,l
E
q
-q
~ cr
is the homeomorphism defined by vertical
1
projection:
.
X .
n
consists of a
fl(x...,
q, . 1. ... ,X)
q
are the cells of a complex on
A function
SO
1
SSe.·· e s
into two (open) he.mi­
cri
The map
.)
0e
q-sphere
U sq •
Sn
Sq- l is the equator in
1
6.
q>n
sn , and
pair of points.
of cr. •
Then the cr
i
Sn
a Hausdorff space,
1S
O-skeleton
di vides
(we call
k.
in
i
1. There exists a relative homeomorphism of
(ai'
cr
i
S
is closed in
The
Let
q::n
IKI '" Sn , and
Let
K be the
q l
sq '" {sn n R +
n. Certarnly
to require some moments of reflection.)
cr , cr , .. ·,cr •
l
2
m
q-ske.leton of
IKq_ll
exercise are easily seen to be true, the reader may find this fourth part
subsets
let the
-q
cr. •
U
i; q:: n
K.
°
q ~
f or each
then aqe IK I
q
K
4. Also, although this won't be used immediately,
for each
of Rn +l
t (~"", xn+l)I~+2 = ... '" xn+l = OJ
•
(x...1.' ••• , x , (1 - q2)1/2
)
q
1L: x.1
A reversal of sign gives the homeomorphism for
verification that the
cr~ -q
cr
The
2
are the cells of a regular complex on
sn
1
is now trivial by Exercise 5.
2.2. An irregular complex ~
Sn •
condition (6).
Let
and let the
2. Examples
2.1. ~
regular complex
For each
k,
~
°< k < n
IKI = Sn , let
n
, identify
4
k l
R+
denote the point
(0, ••• ,0,-1)
q-skeleton
IK I
S
crO
q
= {
if
O::q::n-l
if
q~n
00
Sn
with the subspace
5
in
Sn,
Conditions (1) through
(3)
satisfied vacuously for
q
are satisfied, and conditions
<
For the one
n
now define a relative homeomorphism
stages.
P
First, let
=
f:
(4)
and
on
Sn, and let
coordinate
set
S
=
S
~
xn+l
n-l
E
0.
(En, Sn-l) ---;;. (Sn, aO )
in two
The mapping
f
gh
then satisfies condition
(5).
As in Kxample 2.1,
(6) and (7) are satisfied because the complex is finite .
2.3.
A regular complex
~
n
E
(0, ... ,0,1) , the point diametrically opposit.e
be the closed hemisphere of points of
3n
..ole start wi. th the regular complex for
with
contains
E
sn
g:
IKq I = sq
Take
2.1.
Then
of pOints of
a relative homeomorphism
are
(sn _ aO ) , we
n-cell,
O
a
(5)
P, and has as boundary the
xn+l
O
(E,S) ---;;. (Sn, a
0.
~le
°< q < n
-
,
IKn I
described in Example
= En , and
IKq I = W
To show that these are the skeletons of a regular complex
q>n
with coordinate
for
sn-l
for
OD
define
En
it remains only to check the properties required of the one
n-cell
by doubling angles.
)
ED _ Sn-l.
It is certainly an open, connected subset of
and the homeomorphism
1>
f:
n
n-l
n
n-l
(E, S
) ---;;. (E ,S
)
IKn 1= W '
may be taken to
be the identity map.
E'
2.4.
An irregular complex ~
En.
We augment Example 2.2 by taking
and by letting
2. 5.
More precisely, for each pOint
x
in
E,
gx
is that point of
which is located on the great circular arc that starts at
through
x
to
O
a
sr (p,
P and passes
2 .
Lf.. (p,
e
and
¢
R3
whose coordinates
origin, x) •
The result of Exercise 2.1 can be used to show that
g ,
which is clearly
continuous, is a relative homeomorphism.
To complete the definition of
morphism
h:
f , compose
(En, Sn-l) ---;;. (E,S)
h(x l ,·· .,xn )
g
with the homeo­
defined by
(~, ... ,xn'
6
(1 _
~ x~//2)
1
l
7
q > n ,
­
T.
a > b , and let
(x,y,z)
for
be the identity mapping.
lie in the closed interval
be positive real numbers with
points of
in such a way that
origin, gx)
b
(En, Sn-l) ---;;. (En, gn-l)
An irregular complex on the torus
Let
Sn
f:
IKq I = IKn I = ~
[0, 2rr] ,let
T
a
and
denote the set of
satisfy the parametric equations
(a + b cos ¢) sin
e
y = (a + b cos ¢) cos
e
x
z
=
=b
Thus
0
= f(O,O)
1
= f«
a
sin ¢
a2
O, 2'IT) X 0)
1
a = f(O X ( 0 , 2'IT))
l
of = f«O,
The mappings
'(
f
f
.
n,l
2'IT) X ( 0 , 2'IT)) may be taken to be the appropriate restrictions of
.
x
2.6. A regular complex ~
T .
Subdivide as pictured below:
To obtain an irregular complex
K on
T, subdivide
T
into the cells
shown in the following picture.
~
l1r .....-~----t
,..
f
o
211"
I
81
(0,0)
,. f
2-c.ell
1"
~ l-
0"1.
~.
t 0'0. fl
~O'J.
"­
ZlT
(Y2.~
The cells can be described precisely by using the parametrization of
which defines a mapping
e ¢-plane
21r
This subdivision of
1"
induces a subdivision of
T
into the cells of
I
a regular complex:
four vertices, eight edges (l-cells), and four
2-cells, for which
f
2.,(.
the
".
onto
f
of the
2-cell
T .
1"
[0,
2'lTJ X [0, 2'lTl of
!::!l
irregular complex ~ ~ projective
A point of
T,
~+l
R
=
induces the required homeomorphisms.
pn
is an equivalence class of points of
(n+l
. .
.
R
- the oTlgm) under the relatlon
9
8
n-space,
?
(~" " ,xn+l) ~ (rxl,···,rxn + l ) ,
where
r
i s a real number different from
The t opology of
0
in the quotient space topology obtained from
?+l
R
~
t opology is defined as follows:
Let
n l
+
p:
map
T:
of
pn
is closed if and only i f
Sn ~ pn
15 = plsn:
p-l(A)
f7\
is
-pr.:.~ Recall that this
pn
formation which sends each point onto its class in
A
pn
be the trans­
pn
Then a subset
"n+l
R
is closed in
The
Remark.
- k
a
is a two-fold covering of
pn.
pl
is the covering transformation of'
For eac h
>
k
with the mapping
pn.
define the
po
(1),
to real projective
k-skeleton of
C Sl C
uO
po
P
1
Sl.
by a ttaching its boundary
pP
to be
To obtain
sl
Attach a closed
to
pl
po
pk
- P
k-l
to
f :
k
Ek
"P. •
a
onto the closed hemisphere
g
onto
pk _ pk-l
Pn-l , one constructs
Give n
-n
(J
to
p
n -l
Pn by attaching the boundary Sn-l
by the double covering map
2 . 8 . A r egular comp l ex
~
of
g •
pn
u
We start with the regular complex on the boundary
11
10
twice.
0 '· "'0 "­
denote the usual
-k
pl(= sl)
'P'
is the union of two open
~
around
'P'
is a cell.
k
Sl
is homeomorph ic
denote the upper cell, i.e., the one whic h has its last non­
Let
(;2
Q)
C pn
k-cells, each of which is mapped homeomorphically by
zero coordinate positive.
2-ce11
with the double covering map.
•
k-space, and we can exhibit relative homeomorphisms
k
l-cell
The resulting space is
p2 , adjoin a
&' ~
(2), and (3) are satisfied,
P
by attaching
g(Sk) .
pk
+
g -l( Pk - Pk-l) = S k - S k-l
E
by adjoining its end-points to
Observe that
C Sn
C pl C
to show that each difference
k
po
k-l
This suggests an indu ctive construc­
15
start with a point
P
g •
• •
projection of
.
has been obtained from
This attaching operation amounts to wrapping
0
SO
Conditions
to
pl
(-Xl' . .• ,- xn+l )
T(xl , ·· . ,xn+ l )
uk
k
defined by
(;1
Let
P
k-l
P
The involution
tion for
Sn ~ Sn
The set
to
Note that
i n+ l
of the unit
(n+l)-cube in
n l
R +
r-dimensional faces
The cells of the complex Qn
i n +l
of
(O.::;r'::;n)
On
3. L
'n+l
I
i n+ l
i n+l
(~, ~, ••• ,~) •
into a regular complex on
identifying diametrically opposite cells .
Let
we identify pOints
that are diametrically opposite with respect to the center
We thus transform the complex on
A com;Elex
~at
locally-f~i t.e.
is not
are the open
pn
by
e
i
be the union of countably many regular, closed
joined to a common vertex
vertex of
e'g
To illustrate:
IKII
and
e.
IKI
Give to
l
v's
The
vo
e's
For each
i
let
v.
l
l-cells
be the other
the weak topology with respect to the
and
v's
determine a regular complex on
IKI
that is not locally-finite.
Each neighborhood of
2.9.
A regular complex ~
meet each
n
R
whose vertices are the points of
Rn
with integral coordinates.
The
n
R •
v
E.~ercise.
The reticulation of
n
R
described 2.9 is locally-finite, although the complex is infinite. n
R
for any
, and this subset is never empty
4.1-
DEFHITTION.
(in symbols
(2)
of
ILq I
K.
of 3.1 cannot be embedded in
n
A complex
L C K) i f
IL l n IKq I
(1)
Subcomplexes
L
ILl
for each
is a subcomplex of a complex
is a closed subspace of
q,and(3)
13
12
i
IKI
Show that the space
4.
is contained in the union of a finite number of cells of the complex . Such a complex is locally compact.
e
it must
o
Locally-finite Complexes
A complex is locally-finite if each point has a neighborhood that
in Example
l
e's:
n-dimensional cubes
cubes, with all their lOvrer dimensional faces, are the cells of
3.
meets infinitely many of the
in an open subset of
e.
be cause it cuntains
We obtain this complex by considering unit
vo
each cell
IKI
of L
K
is a cell
The proposition below gives a combinatorial cnaracterization of
subcomplexes.
In particular, it shows that the collection of cells of
sub complex is denoted by
its underlying space is
4.2.
PROPOSITION.
collection
C of
K ; it is called the
q
(Characterization of subcomplexes.)
~ ~
in
0'
~
of
~
C, ~ boundary
in
The cells of a
sub complex
a
11 I •
(1 )
111
L of
K if
(2)
11q_ll
I1 I
11 q l
The proof of this proposition occupies the remainder of this section.
The cells of
K
q
1
C, and that
111
is a complex whose
is closed in
IKI
C
11 q l , and each
is closed in
11 q l
n IKq I , the intersection of
ILl
111
IKI
because
with a closed subspace of
IKI
(~closure of
dim a •
COROLLARY.
consist of the space
1
is a Hausdorff space because it is a subspace of
n (uIKql)
(3)
111
(4)
11q I - 11q- 11
=
ILl
are those of a sub complex of
-----
­
K .
of
IL q I - 11q- 11
Each
pOint.
=
n IKql)
u(ILI
= (111 n
whi ch is the union of the
4.3.
Let
We will show· that
q
cells are precisely the cells of
a) lies in the union of cells of C of dimension
IKI , minus
less than
K are
111 n IK q I .
K , and
q-skeletoIl of
IK q I .
~ only if, ~ each ~
0'
This
=
11 q I
and the subspaces
111
< q of a complex K determines a sub complex of K
dimension
IKI , and let
from
IKq I) - (ILl
q-cells of
ul1 q I •
n IKq- 1 1) = ILl n (1' Kq I - IKq­ 11) ,
Each
C
q-cell is a component
because each is connected and no two have a common
q.-cell is open in the relative topology of
This
11 q l
follows from the fact that
This corollary is an immediate consequence of 4.2.
We now begin the proof of the proposition.
of
C are those of a subcomplex
1
of
K, let
11q I - a
Suppose that the cells
a
be a
q-cell of
C , and let
which is closed in
closed in
~
(the closure of
0'
in
of dimension
We know from Exercise 2.3 that
union of cells of
that
a
= cr
1
~C 11q_ll
of dimension less than
and that
q.
11
q-
11
is the
It remains to show
But this follows from the fact that 111
is closed in
IKI
Now suppose that each cell
cr
0'
of
C satisfies the condition that
(4)
C of dimension less than
dim
0' •
that
11 q l
ILl
denote the union of the cells of
14
n (IKq I -
11q l , because
0')
IKq I -
,
0'
is
is the union of the cells of. C
i
that
11
-q ,
I1 I
\~
q-
11
is the union of cells of
,and -q
a i
is the closure of
f
.:
q,l
C of dimension
O'~
l
in
111 .
q
(E , sq) ~ (IT~, o~)
l
be taken as the one required for .L •
Let
1111
< q .
relative homeomorphism
lies in the union of cells of
a =
O'~ in IKI is
, the closure of
(5) For each q-cell O'~
l
l
q
a U cr~
Now o~ C 11 11 by hypothesis together with the fact
l
l
q-
IT ~l
ai
111 , and hence in
n
IKI •
It follows from
111) - a •
111 f1 IK q I - ILl
=
C with the subspace topology
15
l
< q.
Therefore,
Accordingly, the
given for
K may
80 far we have been able to avoid the ques"t1on of whether
and
ILl
are closed in
is closed in
lies in
IKI
I t has been sufficient to know that
ILl , and that if
e
is a cell of
0
IL q I
its closure in
ties ( 6) and (7).
To settle it, and to sho., several related fact s , \'Ie
only finitely
4.5.
K be a complex.
~
cells of
Each compact subset of
-o ~ IKI
f rom
meets
IKI
.
If
ea ch
K.
be a compact subset of
D
IKI •
He sh01f first that
be a collec tion of cells of
e,
e
the cells of
............... is
A
~
~~th
complex
of
0
K
e .
such that
Let
the relative topology inherited
A
such that
~ ~losed in
A
~
for each cell
2! lei ,
subset
e, then
of
0
n0
is closed in In particula~>
IK I •
meets
D
cells of only finitely many different dimensions. 8upposetb..at this is
q.
not so. Let {o 1) be an infinite sequence of cells of strictly ascendq.
ing dimension, and let X. be a point of (0 1 n D) for each i
A
\.le prove the le=a by showing that if
An ..
is closed.
He assume that
By the preceding lemma,
..
..
is a cell of
An ~ t ¢ ,
K, then
and ~~ argue as follows .
meets only finitely many cells of
1
given
IK I
finite.
the
contains
q
(Here ,Ie us e
xi .)
If
(8 no)
then
is closed in
{x.}
1
~
qi
C ({x.) n IKq I)
IKI , and in D
1
D, because
{Xi)' and if
If
0
as well.
8
=
D meets infinitely many
an infinite sequence of points
{x.}, and if
1
K , therefore,
S
is closed in
{~)
is a
{x . ) - {a limit point)
1
(Xi)
o
D
IKq I
16
of
q-cell of
{x.)
1
K,
Thus
whe re the first
n
of these are the ones that lie in
8
~
n+h
C U
-',,-e
Pi'
1
can have
would still
e.
knov that
n+h
D.
.. = ..
n (U
Pi)'
1
and theref'ore,
1
x.
is a cell of
IKq I
IKI
Pl ,P 2 ,···, Pn , Pn+l" " , Pn+h
x2 , ... )
{o~) , then there exists
q-cells
with
is
meets only finitely many
is either empty or consists of a single point.
clos ed in the weak topol ogy of
1
It follows that
But this contradicts the compactness of
is a subset of
n
{x.)
is finite, and therefore closed.
'de show next that in each dimension
cells.
q , so t hat
to denote the entire set
is a subset of
B
no limit point in
be closed .
only if
Xi
1
in
(D n o~)
.
1
dim < q , then
It follovs that
If
S
S
n
D.
~
n lei = ~ n ( U
It follows,
Pi)
n lei
=
~
Since
Pie Ie l ,
for
n
1
Pi)
<i <n
.. n (u P. ) C ~ n IeI
1
n (u
1
1
0
is
n
n+h
8
By property (7) of the complex
, and hence in
Ie I
IKI.
~~ in
Let
e
Let
LEMMA .
the union of
lei -be -----_
Let
LEMMA .
D, contradicting the
The le=a is now proved.
o ~.E: ~ \lJl.ion. of ~ of
prove two lenmtas .
4.4.
can have no limit point in
1
D.
compactness of
IKI
The question must be settled bef ore proceeding to proper­
ILl
{x.)
as before, that
ILql
= ~
n
n (u
1
17
1
n
P.. )
1
C ~ n (u P.) ,
1
1
Since
K,
so that
4.6. Exercises.
n
1.
Tn (UP.) =Tn Icl .
Let
cr be a cell of a complex.
Show that
a
lies in the union
~
1
of f i nitely many cells of dimension less than
dim cr •
Accordingly,
AnT n Icl
AnT
Since each
n
An
P.
AnT n (u
1
is closed in
~
P.) = T
~
2.
n
n [U (A n
1
P. ) 1
of
~
3.
P.~ , their union (finite) is closed
We now return to the proof of Proposition 4.2, and finish showing
that
cells of
is closed in
in
a
IL I)
a
is closed in
On
IKI
Then
cr
is closed in
and that
An
C.
J
Hence
'c . Now suppose that A n
a
A n
a
cells of
is closed
An application of 4.5 shows that A
AC IL I
q
a
a
K
K lies in the union
K.
a
for any cell of
IL q I
This will be seen
Show that any union of closed
IKJ •
is a closed subset of
An immediate consequence of Exercise 3 above is that if
IKI
is
K is locally finite.
L.
Now suppose that
crC IL I
q
IKI .
By applying
Hence
AC IL
q
K.
implies that complexes are closure-finite.
defined are the
The result of Exercise 3
The complexes that we have
CW-complexes of J. H. C. Whitehead [1].
I
4.5 with
5.
A is closed in
ILql .
A final application of Lemma
K is regular.
of the cells of a finite subcomplex of
ILl) is closed in that subset. Therefore,
Icl = ILq I , we see that A is closed in
the relative topology of
dim a.
A complex is closure-fi.n,ite if each closed cell lies in the union
is closed in the relative topology of
is closed for each
K
~~enever
locally compact then
ILl
ILq I the weak topology and the relative topology cOincide.
A n (any subset of
An cr
of
IL
later to be true
ILl . Then A n
A is closed in
is closed in that subset of'
and hence in
First suppose that
Show that each compact subset of a complex
number of cells of dimension less than
is the weak topology with respect to the
for any cell of
for each cell
(7)
IL I
First suppose that
(any subset of
H lies in the union of finitely many cells of
is actually equal to (not merely contained in) the union of a finite
The topology of
ILl
Show that
H is a compact subset
4. Assume that K is a complex for which each cell boundary
L is a complex.
(6)
K be a complex and suppose that
of the cells of a finite subcomplex of
IKI , so that AnT is closed.
in
IKI
Let
The Weak Topology for Skeletons
The proposition that we established in the preceding section shows
4.5 shows that
and the proof of Proposition 4.2 is now complete.
ILl
is closed in
IKI
1
that the
q-skeleton of a complex
K.
In exam­
ining the proof of the proposition to see how the proposition follows
from known properties of
18
K is a subcomplex of
K, one sees that property (7) of the defini­
tion of a complex is used in what may be an essential way in the proof
19
of
4 . )-\- .
I t is us ed t here to show that a set
S
because i t is closed in the veak topology of
IKI
is c lo sed in
liK
q
I.
circles
In this section
1' i
conve r ge to
-1
i
a
The
°2
increases , so that as sets the
1:.
decrease s as
0
°2
in the topol0t:iy of
l
at
°
l
d.iameters of the
-1
"re consider an examp l e ,,,.hich may t hr o"\.J some light on the extent to
.2
1
-1 , all of them tangent to the equator
1:.
-2
cr 1
See the d iagram belm?
which property (7) is really needed .
A quasi comple x
1%1
subspaces
of
Q consi sts of a space
IQI
definition of a comp lex.
eve~J
and a sequence of
satisfying cond i tions (1) t hrougb
5.6)
'<T€
of t he
qu~si
Every complex is a quas i
complex is a complex.
presented belovr does not satisfy condition (7) .
(Theorem
(6)
There are re gular quasi cO]llplexes , irreGular
qua si complexe s , finite one s , and so forth .
compl e x, but not
IQI
The
qu~ si
complex
cr.­
2
In Chapter VIII
prove that each regular quas i complex satisfies con­
dition (7) (i.e., each regular quas i comp lex i s a complex).
We sta rt vith the r egular complex
Ie
on
E3
described in 2.1,
and pictured be lovr .
Subdiv i sion o f
erG
Each pair
0-;0
2
I
8.
-1
a
( 1: i ' 02 )
1:
i
(E\ So )
i s cons idered to be the image of
r elative homeomorph ism.
and the set s
~
The
= (\ - o~)
•
l-cells of
1'111 - IQol
IQ11
The space
ar e
1
c'l'
under
1
°2 ' is the union of the
cl os ures of a ll these l-c e lls , "rLth the Euclidean subspace topology obt a ined from
The qua s i comple x
Q vrill have
IQI
IKI, but
Q '.'ill have infinitely
E3 .
\,]e then let
being the components of
I ~ I = IK21
The
2·cell s of
Q, I ~ I - 1'11 1 , are ?
many cells.
Q:
IQo I
=
In dimension zero , there is no difference between
o~
by subdividing
U
Ie
and
cr~
In dimensions one and t"l-I'O, ne'" cells are defined
cri
The cell
cr~
will be inc luded in
I~I
1)
the crescent- shaped domains of
2)
the single cell
unchanged .
['he cons truction of
1:
i ) , and 2
°2
Q is completed by letting
The subdivision is canied out by constructing an infinite sequence of
21
20
cri - (U
I~I
equal
I~I , for
n > 3
each
There is only one
3-cell in
I~I
definition, the topology on each
IQI
topology, inherited from
IQI ,
In
IKI
=
J3
the cell
Q:
A
is the Euclidean subspace
the Euclidean topology and the weak topology agree.
IQ11
nor in
Euclidean topology agree with the weak topology.
I~I
It
does the
This, in fact, is why
J3.
the eYBmple must be so elaborate as to contain the
3-cell
ca=ot stop our construction with
because neither
~
is a quasi complex.
Q"'J.
q-dimensional face
s'
of an
s
n~simplex
is a
simplex whose vertices are included among those of
= E3 •
is easily seen, however, that in neither
or
then topologized by this metric.
By
or with
~
s'
is to 'be identified ,,'ith the pOint of
on the vertices of
zero ).
s'
Ql
having the same coordinates
This identification is an isometry of
O-face of
Each point of
(the other coordinates of such a pOint of
with which i t is identified.
We
s
s.
q-dimensional
Each vertex
s , and we identify
s'
A of
s
are
with the subset of
s
determines a unique
s
A with the point of that
O-face.
The identification of vert ices with functions allows us to regard
A quasi complex that fails to be a complex
does so because one of its skeletons, as a subspace with the relative
a simplex as a single topological space, namely, the collection of func­
topology, fails to have the weak topology with respect to its own closed
tions
cells.
a simplex as a subspace of that simplex.
Such a skeleton, by itself, must violate condition
(6).
a
topologized by the metric
d.
in common they have points in common:
Exercise.
IQI
Verify that the Euclidean topology given for
IKI,
If two simplexes have vertices
the points of the common face that
is the simplex determined by the common vertices.
in the quasi complex just described, agrees with the weak topology defined
In
Rn+l, let
A. = ( 0 , ••• , 0 ,1, 0 , ••• ,0) , with the
1
Q
by the closed cells of
We may also regard a face of
place, let
s
be the simplex defined by the objects
1
in the
.th
1
Ai' and, using
vector addition, let
6. Simplicial Complexes
a=
n-simplex
An
s
consists of a collection
{A)
of
called vertices, together wIth the set of all functions
such that
L a(A) = 1.
A
n+l
a:
The functions are called points of
of
s .
1
n+I
111
objects,
(A) ~ [O,lJ
s , and the
values of a function are called barycentric coordinates of the pOint.
The point whose value is
n+l
[ a(A.)·A.
for each
o~
a
in
s.
Because of the restrictions
a(A ) < 1 , the collection
i
~ = (a)
convex closure of the vertices
A.
defines an isometry of
s
s
and
1
of points of
Rn +l
The correspondence
1
The space
. 1 ex III
.
SlIDP
Rn+l
Be1
ow ·1S a
. t
P1C
ure f or
1
d(a,~)
1
s
on each vertex is called the barycenter
We define
[>-:; (a(A)
A
as the distance between the pOints
22
a
~(A))2 f
and
~.
The points of
s
are
n+l
[a(A.) = 1
23
n
2
and
is the
a <~
a
is called the standard
i s called a simp li Cial complex .
Since e ach subset o f a li s ted set is li s t ed , each face of a simp l ex
of
i s a simplex of
K
K
The int ersecti on of two s implices of
if not empty, is a simplex of
secting s implices .
Hext we ShOlf that an
Use the i sometry to get
xn+l
of
n-s:iJ:nplex is homeomorphic to the
s
Ef
is a simp licial complex .
En
< n +l vertices .
n l
R + , project on to the hyperp lane
into
0 , and , \li th a s:iJ:nilarity transformat t on, move
=
n - bal l
Ra dia l projection from an interior poin t of
s
En
for
thus
Kn
which is a fa ce of each of the inter­
of simplices of
K
<n
of dimension
Take al l s implice s on the original list with
This new' li s ting specifi es
K
n
A weak top o l ogy ,,'i, th respect t o the s implices of
t o the interior
onto
s
The set
K
K,
jKj ) the lUlion of t he simplices of
K
can be defined
K, because each simp lex has a
top·o logy .
yie lds the des ired h omeomorphism .
A metric topol ogy for
Extend each point
a
of each simp lex
the set of all vertices of
on v ertices of
can a l so be de fined i n a natural ,ray .
jKj
K
0
of
K
[0,1 ) which
into
2 ) is
sand
s
to that mapping
a
of
1) agrees with ex
on the other vertices of
K
Le t
K
be the collecti on of such extended funct i ons, ,nth the metric
p(a,(3 )
If
a
and
~
are distinct points of an
line segment containinG
(to + (l-t) ~ j O
of p oints
to + (l- t~ )
the function
~
OL
s
ex and
. t0
In
n 1
"R
+
5t 5
~
I) .
0f
d
G
It is easily seen that for each
s
Rn +l
on
IKj .
f'unc tion
If'
.
¢-lj s
K
¢(a) = a ) induc es a metric
IKI , defined by
For t',IO points
the s ame simplex,
t
I~ote that under the isometry
G 'lS an or d'l nary l'lne segrnen t ·In
¢: K ~
The bijection
s , the
i s defined to be the collect ion
is a point of
th e llllage
'
( n +l)- simp l e x
~(A)2)~
[ L: (a(A)
A
d(a , ~ )
a
and
p(a,~)
~
of
j KI , not necessarily from
FOT e a ch s implex
DEFnrrTION .
vertices
Suppose we a re given (1 ) a countab le collection of
{A}, and (2) a fa.rnily of fini te subsets of'
p r operty th'lt each subset o f a li sted sub se t i s listed.
subset de t e rmines a simp l ex, and the collection
24
K
{A}
,·n th
the
i s finite , t hat is i f
Each listed
of the se simp lices
that each s implex of
K
of
K, the
is an isometry.
K
has fin itely many simplices , the
weak topology and the metric top o logy agree .
6.1.
s
K
The proof r e qu.ires sho'"ing
is closed. i n the metric topology of
IKI
•
If
is infini te , the two topologies may differ, as t he foll Owing exa.'l1p l e
sho''{s .
Start •.nth the vertices
Vi
and t h e
25
I - simplices
e.
1.
of 2 .12.
Here,
~I
IK I
n
denotes the union of the simplices of
K of dimension
<n
-Yo
~:::
Conditions (1) and (3) present no difficulty.
in
.
IKI
be the poin:t in
e
sets of
defined by
i
Since
d(a:.,
v )
l
0
topology for
(0:. }
l
IKI
and
o:.(v.)
l
l
l/i •
the
n-simplices of
Vo
in the metric
of
However, in the weak topology for
IKI , the set
n-simplex
= .J2/i
the
0:.
l
converge to
K to the
do not coincide.
open
An example of a finite simplicial complex is a simplex with all
IKn l
is closed.
its faces.
Thus the two topologies for
IKI
We will usually use a single symbol to denote both a simplex
If
the topological boundary of
as well as the simplicial complex con­
s
sisting of' the proper faces of
s
s
is a simplex, then
s
and the complex it determines.
will denote
(those faces of dimension less than
IKnl
~
s
n
n-
K:
~ ~
simplex, with
in
simplicial complex
I~I
0:
the~roperty
that each face of a simplex of
The interior
~
(A l ,··· ,An +l }
= (s - s)
of an
n-cell of
i'laPPiDBs
the standard
f:
o
s
Each
because its complement is closed in
5 lies in
of
It is easily seen that
simplices of
K is a collectiQn of faces of a sill81e
are precisely the interior points of
is connected, and is referred to as an open simplex.
Each point
is a closed
11
(0,1)
interval
IKn­ 11
is an open
Thus
n-cell of
s =s
n IKn- 11
K and that
s
K.
(En, sn-l) --;.. (s,s)
are provided by isometries with
n-simplex.
condition (5) and since
An alternative definition of finite simplicial co~lex.
is the union of
IKn Ins
functions from some listed set
n-simplex is open in
is closed
is the union of a finite number of closed
s
'K I - IK
Since each simplex of
dim s) •
6.2.
n
IKn I
IKI
The points of
o:.(v
) = 1 - l/i
l
0
K then
is a simplex of
s , so that
faces of
Let O:i
s
If
Also,
IKI
K is a closed cell of
K in the sense of
has the ,-reak topology with respect to the
K, condition (6) is satisfied.
The proof of (7) is left
to the reader.
K is likewise
K.
REFERENCES
Exercise.
Show that 6.2 agrees ~~th 6.1.
1. J. H. C. Whitehead, Combinatorial homotopy I, Bull. Am. rvlath. Soc. 6.3.
THEOREM.
Let
K ~ ~ simplicial complex.
the weak topology, together with the subspaces
--- ----
----- ---
The ~
IKI
with
vol. 55 (1949), pp. 213-245. IK I , determines a
n
regular complex.
26
27
In this section the ring
Chapter II.
i ntegers.
HONOLOGY GROUPS FOR REGUIAR COl<lPLEXES
1.
Redundant Restrictions on Regular Complexes.
over
Homology Groups.
R
He will sometimes l"Yite
Z ".
C
If'
11
will allmys be the ring
chain comple:0' for
is a chain complex over
11
Z
of
cha in complex
Z, its chain modules are
abelian groups.
In this chapter we describe the process of associating with any Given a regular complex
K, we will define homology groups
given regular complex an infinite sequence of groups called the homology f or
groups of the complex.
K by first givillg
K
an orientation.
Any such orientation
Fundamental to this process are the concepts of gives rise to a chain complex, and then the homolor;y groups for
K
chain complex and homology of a chain complex: will be the homology groups of this chain complex.
1.1.
DEFINITION.
A chain complex
C
over a ground ring
R
(R assumed to be commutative lvith unit) is a collection of unitary R-modules (e :qEZ
Q
dq-l dq
o
For every q, Cq
or the qth chain module, and
of C.
He will often ,,'rite
the boundary operator of C.
1.2.
DEFINITION.
If
are zero.
d for the collection
1-:e then write
q
},o)
dq
Z
Bq
q
B
C Zq
q -
DEFnU:L'ION.
( {C }, d) •
and
Bq
groups of
UZq},O)
B( C) , is the
.
K
the R-modules of q-cycles
d
qO
q+ l O ,
•
K, many different ways of
Thus it ,rill appear that the homology
K ..reI'e not wel-l-defined.
h omoloGY groups of
)-le will show, however, that the
are independent of the orientation used to
de f ine them.
As 'ile have just remarked, an orientation on a regular complex
He will see that only the boundary
operators of the chain complexes so induced may vary as we v-a:ry
ori entat ions.
The chain groups, however, are defined independent of
ori e ntation in the following way.
1. 4.
of
Hote that since
K.
will induce a chain complex.
q
and whose boundary operators
1m dq + l
defining an orientation of
and call d
is a chain complex, then the
and q-boundaries, respectively, of C.
1.3.
as
{c\}
Z(C) , is the chain complex
= Ker
((Bq},O), where
For each q, we call
we have
C
The complex of boundaries of C, l-.'ritten
chain complex
q,
is called the qth boundary operator
q
complex of cycles of C, 1"Yitten
Z
q
such that for every
is called the R-module of q-chains of C, d
C = ({C
whose qth chain module is
C ~ C 1
q
q-
d:
q
together with R-homomorphisms
that there may be, for a given complex
DEFIlHTION.
K, written
Given a regular complex
K, the group of q-chains
Cq(K), is the free abelian group on the q-cells of
K .
Thus for
q ::: -1,
C (K)
q
=
0 •
In order to define a boundary
operator: ve will assume that the regular complex
Given a chain complex
module of C is the factor R-module
The homology chain complex of
C
Z
q
IEq
C =
Uc
q
},o) , the qth homology
and is denoted by
is the chain complex 1,Titten
It will be clear
tain restrictions.
K
satisfies cer­
At this time ve will state these restrictions
--~----=~
H (C) •
q
H*( C) •
and assmne that they are satisfied by all regular complexes.
1·1e
wi ll prove later that this is the case by using the homology theory
of simplicial complexes.
Thus we will show now that the redundant
28.
29.
restrictions for regular complexes are satisfied by the subclass of
x
simplicial complexes.
hypotheSiS,
Three Redundant Restrictions on a Regular Complex
R.R . l:
If
r < q , and
~, then
point of the r-cell
taking
.
~C ~
note that a q-cell of
By
~C
R.R.l,
By
the inductive
0
K is simplicial, we
~.
S2
The only vertices of the complex
~.
are the two vertices of
This complex is of
K is then the interior of a simplex of K.
"r
If a point of the interior of an r-simplex lies in a closed q-cell,
R.R. 2:
that is, in a q-simplex, then every vertex of the r-simplex is a
If
~
and
p
the dimensions of
vertex of the q-simplex, and the desired relation holds.
p
In a regular complex
K, if
is a cell whose
0
and
closure contains the cell
~,then
~
is a face of
~
is called a proper face of
0,
~
we write
<
is called a face of
~
If
0 .
~< 0
f
~
and
then
0
p
If
0.
< O2 <
~
K such that
are two cells of
and
~
are
q
there are precisely two (q+l)-cells
DEFINITION.
0
and the conclusion follows.
course irregular.
l.5.
Since
to be a 2-cell whose boundary is collapsed to a point of
0
some l-cell
To see that R.R. l is satisfied if
contains a vertex.
~C 0 ,
r < q.
with
An example of a complex not satisfying R.R.l is obtained by
is a q-cell whose closure contains a
0
~
is closed,
K
~
is a point of some r-cell
and
0
q+2
and
l
p
<
~
and
respectively, then
O
2
such that
p
< 0l <~
•
If the dimension
is zero, this means that a vertex of a
q
2-cell is a vertex of exactly two l-cells contained in the boundary
0.
of the 2-cell:
l. 6.
PROPOSITION.
If
K
is a regular complex and
then the collection of faces of
finite by Lemma 1.4.4) for
of
is a sub complex (necessarily
0
o , and the collection of proper faces
is a finite sub complex for
0
Proof:
0.
PROPOSITION.
then
a
Proof:
If
0
R.R.2 states that an edge
lying in the boundary of
p
3-cell is the common face of exactly two 2-cells which also lie
in the boundary of the 3-cell:
is a cell of the regular complex
'r~
K,
conta~s at least one vertex.
The proof is by induction on the dimension of
proposition is obviously true if
0
true for cells of dimension less than
;
q = l,
If
a
This proposition follows from Proposition 1.4.2.
l. 7.
Then
a cell of K,
0
C
K l
- q-
and
0
is nonempty.
30.
is a vertex.
q, and let
If
x
0
.
R.R.2 is satisfied if
The
Suppose it to be
and
~
be a q-cell.
and
dim ~
and
~
0
is a point of
0
,
then
K is simplicial.
to be open simplices such that
q+2.
Then
p
p
<
~
Indeed, take
,with
is a simplex on vertices
is a simplex on the vertices of
3l.
p
dim p
p
q
AO,Al,···,A q ,
and two other vertices
A
q+l
and
It is clear that we may take
Aq+2 •
plex on the vertices of
vertices of
1.8.
p
and
0:
cells a .
on
1"
Aq+l , and
and
a
1
to be the sim­
the simplex on the
2
[a:1")
We extend
If
a
C (K)
q
K, an incidence
1.10. LElIji,lA.
?P?p
q q+l
For all integers q,
of
by line­
°.
K is a :function assigning to each ordered pair of
K an integer :f-.com the set
{-1,0,+1}
denoted by
Proof: Let
T be a senerator of
is an edge 1fith vertices
+
Cq+l(K) •
Then, for
q>l,we
h ave
0:
0:
] )
o0:
q 0 q+ IT = 0 q (2:[T:O a
is nonzero if and only if
[a:A]
1"
is a face of
A
[a:B] =
and
(by def.)
a of one
B, then
°
2: h:o]oO:o
a
q
(by linearity)
2: [T:O](2:[o:p]p)
a
p
(by def.)
2: (2:[T:O][O:p])p
p a
< 1" and dim p = dim 1"-2 , and a l and a2 are the
two cells with p < 0i < 1" for i = 1,2 , given by R.R.2,
(iii) If
over all of
arity.
101fer dimension.
( ii) cPq
To justify this de f .i nit ion, we have the folloving lenn:na:
[a:1")o: (or [a:1"J for short) such that:
(i) 1.3. 4 .
terms by l€mma
Aq+2
Given the regular complex
DEFIIIJITIOE.
£'unction
p
0
P
(by Def. 2.18 (iii))
°
Thus, by linearity, the result follows for all (q+l)~chains,
q ~ 1 •
then
h: a l ]· [al:p] + h: 02 ][02: P ]
[O:1"J
=
is called the incidence number of the pair
K is oriented by the incidence fUnction
orientation on
°.
If
00:
q < 1 , then the map
q
This completes the
is the zero map.
proof.
0,1".
0: , or that
0:
He say that
yields an
1.11. DEFUrITION.
If
c
is a O-chain, the index of c, written
In c , is the sum of the coefficients of c. K.
Remark.
Remark on condition (ii) above.
It follows easily from the
definition of a regular complex that every I-cell has precisely two
Using methods similar to the proof of the above lemma , it is easy to see that for
a
E
Cl(K),
In
(o~a) =
° , as
a
consequence of Definition 11.1.8 (ii). vertices.
1.l2. DEFIl'TITIOJlT.
1.9.
DEFTIIJITION.
If
0:
is an incidence fUnction on
associated boundar"J operator
defined as follows.
cPo
q
2: [0:1"]0:1"
For
a
?p,
with
K, the
2P:
C (K) ~ C l(K) , is
q
q
q-
a generator of
C (K) ,
q
Note that this sum has finitely many nonzero
TEK
K
is a regular complex oriented by the in­
cidence :function 0:, then the chain co~lex induced by
CO: (K) , is the chain complex
0:, written The grOUPS of cycles and
({Cq(K) ,00:)
K in dimension q are the SToups of cycles and boundaries
0: Ct
of the cha in complex CCt(K) , and are written Zq (K) and Bq (K) ,
boundaries of
r e spectively.
32·
If
The qth homology STOUP of
K
is the STOUP
ZO:~)jBO:(K)
, and is denoted by rf(K).
q
q
q
The collection of homology
groups and the zero boundary operator form the homology chain com­
plex of
i;(K).
K, -written
Notation and minor definitions:
A
q-chain of a regular complex is sometimes referred to as a chain on
K
if it is zero off of the cells of some subcomplex
L, or to be a Chain on
is often said to lie on
zl
and
z2
we -write
z
2
O.
[O : T)
those of
< a as follows.
If
T
~
< q-l , we
is of dimension
is of dimension
with one vertex,
0
If
q-l, its vertices are
A.,
say, omitted.
1
[a:T)
We set
(_l)i.
Two q-cycles
that the function so defined is an incidence function for
We leave as an exercise to the reader the verification
K.
There is a practical procedure for defining an incidence
f unction on a finite regular complex of dimension at most 3:
The collection of q-cycles homologous to a
is called a homology class and is denoted by
We shall also say that two chains
cl - c
set
~
for
AO, ••. ,Aq , we define incidence numbers
L of K it
-- that is, if their difference is a boundary -- and
zl - z2
given cycle
[a: ~)
are said to be homologous if they lie in the same co­
B~(K)
set of
L.
by the (ordered) vertices
{z} •
c 'c
are homologous, and -write
l 2
(i)
Associate an arrow with each I-cell
the value
+1
to
[o:A)
arrow, and the value
-1
for
0
of
K, and assign
A the vertex at the head of the
to
[o:B)
B
>o
for
B the vertex at the tail
' if their difference is a boundary.
of the arrow.
Next we note that for a finite regular complex K, oriented
by the incidence function
the qth homology group
0:,
~(K)
'A
is a
(11)
Associate a circular indicatrix with each 2-cell
o.
For
factor group of two finitely generated abelian groups and so is
each I-cell
finitely generated.
T < 0 , assign the value
~(K)
ated abelian groups,
tions of the indi.c atrix on
R.R. 3:
The finite summa.nd is called the
q
if the torsion subgroup of
-1
if
(iii)
R~(K) •
[0: T )
2
[O:T ) = +1
assign
+1
0
To
-1
if not.
It is satisfied if
If
K is simplicial.
0
to
[O:T)
T
< o.
We
if the orientation induced by the corkscrew
agrees with the circular indicatrix attached to
is a simplex spanned
[a:~) =
34.
3
0; the corkscrew
induces an orientation (indicatrix) on each 2-cell
on
K.
-1
Associate a corkscrew with each 3-cell
This restriction allows one to define homology groups for
see this, order the vertices of
[O:T )
"t'2,
There is an incidence function for any regular complex K.
all regular complexes.
agree,
l
~ is non­
The rank of the free summa.nd is called the qth Betti
K and is -written
T
't3
'tI
~(K) , and K is said to have non-trivial
torsion in dimension
number of
if the direc­
[O:T)
a and the arrow on
not.
trivial.
to
splits into a direct sum of a free
abelian group and a finite group.
torsion subgroup of
+1
By the fundamental theorem on finitely gener­
+1
~
in (11),
a on a regular complex
Once we have an incidence fUnction
K , we may compute the homology of
It appears at this point that the homology so obtained
l-ce·l ls.
a on K.
a
K, the homology groups of
C~(K).
C
(K)
In fact, the chain complexes
same structure.
Let
It
will turn out that this is not the case: given two orientations
Sl
q
-f
0 •
eo
and two I-cells
and
TeO
Ca(K) and C~(K) have the
[el:e ]
O
We have presented a procedure for constructing homology
Given a topological space
X is the space of some regular complex
del
generated by
1
X, such
HO(S )
K, we may define the
We must of course verif'y that if X
so
We
as follows:
+1
=
d Tel = e O - TeO '
ZO(Sl)
1
= coiBo "'" z .
~(Sl) ~ Z.
zl(S) "'" Z , generated by
Hq(Sl)
k
>
=
0
for
S2k+2.lS
0 ,
0
q
CO(Sl) "'" z EEl z ,
eo - TeO' s o
e 1 + Tel' ~(Sl) =0,
>1
bt·
alne d fr om
S2k+1
by a d
···
JOlnlllg
can be realized
two
homology groups.
Sl
(See diagram.)
eO' Teo,Bo(sl) "'" Z , generated by
(2k+2) cells
e
L, then K and L have isomorphic
as the space of another complex
Tel'
ITe :Te ] = -1
1
O
+1
=
Te O- eO'
For
K.
and
l
[Te :e J
1 O
-1
to be the homology groups of the
X
inter changing antipodal
n-spher~.
Then
homology groups of the space
e
construct an incidence fUnction on
[e :Te ]
1
O
groups of a regular complex.
Sl
is the space of a regular complex with two O-cells
are isomorphic to those of
Computation of the Homology Groups of the
--7
a,~,
The precise sense in which this is true will be
Example:
Sl
T be the mapping
points; then
explained in section 5.
cOI@lex
for
K using the chain ccJmplex
might depend upon the particular incidence fUnction
that
Hq(SO) = 0
The I-sphere is obtained from the O-sphere by adjoi ning two
Ca(K).
on
0
HO ( S ) "'" Z EEl Z •
0
BO(S ) = 0 , so
In other words, we must demonstrate that the se­
+ ,
2k 2
Te
+ , with the following orientation:
2k 2
[e2k+2:e2k+1J = [e2k+2:Te2k+lJ = [Te2k+2:e2k+l J = [Te2k+2:Te2k+1J
=
+1 •
quence of homology groups is a topological invariant for the cate­
gory of spaces which realize regular complexes.
in a later chapter.
AssUIIJ.ing topological invariance for now, we
compute the homology of the space
of a regular complex
This will be done
other incidence numbers
K and by defining an orientation on K.
demonstrate this technique with the n-sphere
We
For
Sn.
cells
The O-sphere is the space of a regular complex with two
O-cells; orientation is possible in only one way.
cept when
q
0, and
CO(SO) "'" Z EEl Z •
36.
Then
Cq (SO) '" 0
o
ZO(S ) =
ex­
0
CO(S )
S2k+ 1
are as in
X by giving X the structure
and
k
>
S2k+3
0
e 2k+ ,
3
Te
is obtained from
S2k+2
by adjoining two (2k+3 )
+ ' with the following orientation:
2k 3
[e2k+3:e2k+2J = - 1
[Te2k+s:e2k+2J
[e2k+3:Te2k+sJ
[Te2k+3:Te2k+2J
=
+1
37.
=
+1
-1
The reader
~
check that the functions given satisfy the require­
ments for incidence functions.
Sn
n+1 .
q-cells of
III dilJlens ions
Z2k+l (S
2k+2
)
n,
Z2k+l (S
Proof:
H (Sn+l) ~ H (Sn)
r
r
Z (~k+l) = Z (S2k) ~ Z
2k
2k
2k
K
THEOREM. I f
is a finite complex and
for
n
­
> 1,
rank of
X(K) = rfiimOK (-l)%:
q=
q
K , then
~ Z
Te
2k
- e
with generator
2k e 2k+ l + Te 2k+ l
B
q-
is free since it is a subgroup of the free abelian group
This implies that
D
= rBq-l
°
- P
q
Te 2k - e 2k S=y:
1
q ~C q ~Bq- 1~0
Thus there is a homomorphis~
11c(Sk+l)
For
=
0,
n
>
oe 2k+ 2
e 2k+1 + Te 2k+ 1
q
{z
r: B
q-
~
g
~
B 1
q-
is the quotient of
¥ , compute
where
dim K.
n
b)
z,
H (¥) "" Z EEl
l
z, ~(¥) ~
Z
Using a regular complex whose space is the projective plane
2
,compute
HQ(JP 2 ) ~ Z,
2
Hl(JP ) "" Z2'
~(]p
2)
= 0 •
But
C\ +°2
2.1.
DEFINITION.
complex
K
The EuJ.er Characteristic
The Euler character ist ic of a finite regular
is defined by:
X(K) = z:dim K( -l)qR (K) •
q=0
q
-°
Z
q
by
B
q
rank D
q
, so
n-
l+P -0 ) + (_l)np ,
n
n
n
3
+ •.. + (-l)nO:n •
Once we have proved that the homology groups are topologi­
cally invariant, we will know that
We can compute
sition of
X(K)
X(K) , as defined above, is
B,y Theorem 2.2, so is
IKI •
For example, Z:(-l)%:q •
using the theorem and any regular decompo­
X(Sn)
n
X(E )
x
38.
B 1
q-
where
Po = ° 0 ' and the proof is complete.
also a topological invariant.
2.
Z ffiD
q
q
= 1 .
Everything drops out except
Using a regular complex whose space is the
HQ(¥) ~
Cq
0qr
The rest is aritl1metic:
+ (_l)n-l(p
PO-
torus
such that
we have that rank
f
a)
q
C
q-l
Z:(-l)qRq(K) = (PO+ P -0: ) - (P +P -( ) + .••
2
2
l
1
l
for q = 0 or n
o otherwise
Z EEl Z for q = 0
Hq(SO) =Lo otherwise
Exercise.
1 ~ C
is a direct summand and
q
Since
H (K)
q
q
Z
Rq(K) = Pq - (Oq+l - P q+ ) .
l
(k+l
-1\.+1 S
) ~ Z
R
1, H (Sn) =
q
is of course the
°q
Consider the short exact sequence
O---;..Z
(2k+l
2k+l S
) = 0
Thus
q •
o
with generator
2k+l)
Z , for each
q
Pq '" rank of
Cq
n+l:
(S2k) = 0
oe 2k+ 1 let
r of n,
B
]P
is the nurriber of
°q
so obtained is the same as in example 2.1, chapter 1.
I t is clear that
B
Note that the regular complex for
2.2.
(torus)
1 + (_l)n 1
= 0
y
X(JP 2) 1
X( JPn)
1/2 + 1/2 ( _l) n
2
x _ 7x + 6X(M)
is Symmetric about the .1 1ne
0:
> 7
+
x
7/2 ,
f49 - 24x(M)
o -
2
·
This last equality follows from the fact that the regular decompo­
sition of
SO
yields a regular decomposition of
JPn
The number on the right is called the 1:Ieawood number ·of the mani­
with every
fold
two cells of the same dimension identified.
M
and is written
must have at least
h(I"';).
h eM)
Any simplicial decomposition of M
vertices .
We will see later that the product of two finite regular
complexes
KX L
plex
of
K and L can be given the structure of a regular com­
and
K
K XL
A q-cell of
T
a cell of
L
such that
Show that
Exercise.
regular complexes
K
and
is a pair
L.
(cr,T)
with
dim cr + dim
X(KX L)
X(K) ' (L)
cr
a cell
n
3.
circles. each of Euler
Given a simplicial decomposition of a space
0:
1
In particular, if
0:1
1
S ~(aa-l)
K
S (~l)
20:1
F2
IKI, with
0:0
Homology groups of a space are constructed in the hope that
of the space.
'de find in this section that there is a characteri­
the Oth homology group .
3.1.
If' We triangulate a compact 2-manifold
'~,o
M
(with­
THEOREM.
components of
t hen
~Qr_ every
If
Etxo - tal + ta2
H (K) ~ the
direct sum of the H (L . )
Proof: Clearly
[cr :T]
Cq(K)
is zero if
t hat boundary maps
> tao - 0:0 (0:0 -1) •
cr
q
is the direct sum of the
and
l
C (L.) •
q
l
Since
T are not in the same component, we have
C (L.) ~ C l(L.)
q l
ql
for each
i,q
Thus the
b oundary preserves the direct sum and the conclusion follows.
0:0 :::: 3 , and the graph of
41.
40.
~ , ••• , Ln'
q,
q
Since
of
First we prove the following theorem.
are ~he spaces of sub complexes
IKI
Then
%- 70:0 + 6X(M) > o .
te~
K is a regular complex, and if the connected
2 - simplices,
tao - 20:1
Thus
Homology and Connectedness
zation of the number of connected components of a space in
i s a finite 2-dimensional simplicial complex,
6X(M) =
,the binomial coefficient .
out boundary) every I-simplex is the face of exactlY
so
h(JP2)
they will provide some information about the topological properties
edge s , etc . , we know that
O:k
Exhibit simplicial triangulations of these mani­
h(torus) = 7 .
for any two finite
characteristic zero, has Euler characteristic zero .
vertices,
Exercise .
This show's, for example, that the
n- dimensional torus, being the product of
6 .
h(S2)
folds with these least numbers of vertices.
q
T
4.
Examples .
3.2.
DEFINITION.
vertices of
K is a regular conwlex and
If
K, then an edge path from A to
sequence of vertices
o < i < n-l , such that A
as vertices for each
3.3.
THEOREJv!.
conwlex
AO'
B =
An'
and
B are
is a finite
B
0 < i < n , and edges
(Ai)'
A+
i l
A and
ai
that
i
Ai
The following three statements are equivalent for a
K
IKII
(iii)
There is an edge path between every two vertices of
is connected.
K.
\<le
> (iii).
A be any vertex of
K.
lies in
let
K.
IKII
be connected, and
U be the union of the cells of
K
if and only if one of its vertices does.
U
either contains
T
conwlex of
U is open and closed in
K,
logy property.
or is disjoint from
IKII
Since
T
is connected, and
IKII
So
Rl
U
is a sub­
by the weak topo­
contains
U
A I-cell
A and there­
IRlI.
> (i).
Proof that (iii)
dim K
let
of
> (ii).
first prove by induction on dimension that (i)
This is clearly true for
a
is connected for each cell
fore all of
Proof:
is connected since we have now sho.m
all edge paths connecting A with other vertices of
T
( ii)
IRlI
Proof that (ii)
let
IKI is connected.
lall
and
i
(i)
K contains cells of arbitrarily high dimension the same
argument proves that
(a ) ,
has
If
Since the union of the cells of
Suppose it is true for
1
a path of edges is a connected set, (iii) implies at once that
dim K < n
let
K be a regular conwlex of dimens ion
n+ 1
is connected.
that
IKI
is connected.
Suppose
IRlI
Q,
= P U
Since every closed cell of
We show that
union of all cells of
Q must be enwty.
let
pI
Q'
sets (finite chain theorem in Kelley's General Topology) to deduce
that
similarly.
QI
is empty, so is
Q.
If
a
a
is a subconwlex of
tion hypothesis,
lie either in
a
1;1 1
K of dimension
(and
P'
and
P or in Q.
QI
P'
are disjoint and exhaust
or in
K.
Q'.
Thus
It follows
Moreover,
P'
Q') meets each closed cell a in the null set or in a and
IKI,
Q'
is empty.
Then so is
B,y the connectedness
Q, and so
IKII
IKI
is connected.
LE~~.
3.4.
in
If
K. and
Proof:
If
c
c
K is a connected regular conwlex,
a O-chain on
K, then
c
~
(In c)A
is a multiple of a vertex, say
c
=
AO
mAl ' then there
AO
B,y R.R. 3 we may choose an orientation on
LBia i
I-chain with the following properties:
of the edge path from
=
Al -AO.
AO
Then
to
AI'
K •
let
(i) the a i
(ii)
mAO ~ mAl ,so
a
i
=
a vertex
O
ex i sts, by the previous theorem, an edge path from
d(Eaia i )
42.
is a connected space.
B,y the tpduc­
(It is non-enwty by Prop. 1.7.)
so is closed by the weak topology property.
of
< n-l.
K is a connected conwlex
K,
(the I-skeleton of ~) is connected and must
and all its faces must lie either in
that
is connected.
In what follows we will say that
is a cell of
if
then
IKI
P contains a vertex of a I-cell it contains that
If
I-cell, so if
IRlI , we use an elementary theorem on connected
be the
P, and define
K having a vertex in
K is connected and con­
open, closed,
P
tains points of
and non-enwty.
IKII
such
to
AI.
be the
are the I-cells
.:t 1 , and (iii)
c ~ (In c) AO
If
so that
c
=
Lfl=O
Ai - AO
m.A.
l
join edge paths from each
l
bounds.
Then
I;(m.A. - m.A ) = c - (In c)A
l
C
'V
l
l
o
O
'
A.
to
l
4· . 2 .
AO
The latter bounds. so b-y definition
'
(In c )AO •
3.5.
THEOREN.
K be a connected. regQLar compl ex .
K
belongs geneTates
for some l-chain
= m(mP.) =
Thus, the coset of
A
HO(K).
c.
rnA
bounds fOT some
to '-rhich
m, then
A
rnA = dc
Thus
K
Let
Any cycle on
mus t lie in some
Lo;'
Thus
is a tree
K
in
T
ve rtex of
HO(K) "" Z
T
{Lo;'
The union
and we may apply 3.3.-
We claim that
By the remark after Definition loll,
O .
In(dc)
If
BO(K)
He must show that the set of
connected, because any two vert ices of'
be any
A
Then, by the preceding l emma, any O-cycle is homo l o-
gous to a multiple of
m
Let
K has a maximal
K , ordered by inclusion, has t he property that any totally
ordered set of trees in
Let
vertex of
The p roof uses Zorn" s Lemma.
ordered subset has an upper bound.
K is co=ected if and only if
HO (K) "" Z •
Proof:
Proof:
trees in
A regQLar complex
Each co=ected reguJac compl ex
tree .
miAi - m/· O b01mds, and
'
PROPOS ITION.
L
L
0;
E A}
be a totally
of the tre es
L
Lo;
is
must lie in some
Lo;'
L , being a finite complex,
is a tree .
By Zorn's Lemma, there
\'rhich is maximal i-Tith respect to inclusion.
is maximal in the sense of 4.1.
T, and l et
B
be any vertex of'
Let
A
Since
K .
be a
K
is con­
nected, "e may apply 3. 3 to sho1{ that there exist s a path of edges
If
K
is not co=ected, yre may apply the fir s t theorem of
A
this section, after noting that by Prop . 1.4. 2 a conne cted component
= Al ,
e ,A ,e , ••. ,e , An+l
l 2 2
n
< u+ 1
larges t integeT
of a regular compl ex is a subcomplex.
then
3. 6.
COROLLARY.
is isomorphic to
HO(K)
is the number of components of
3.7.
OOROLIoARY.
Ho (K)
IKI
RO(K)
of
Z
I{here k
no. of components of K.
That is , if
is homeomorphic to
K
ILl, ,t hen
and
L
are complexes such
Ho(K) ~ HO(L) .
maxil!lal tree in
T
Computation of Homology Groups
Thus
out effectively .rhenever
K
DEFllUTIOH.
A
has a finite or
Ide order them in a silIIple sequence
union of
L _
n l
{Ln}
inductively by
Let
j
be the
If
j
< n+ 1 ,
A.
J
T, Which contradicts
n+l
B
and is a
~
countab~
e number of
{e } , and define a
i
e
l
, and
L
n
is the
and the closure of the first edge in the ordering
with p recisely one vertex in
If.1.
contains
contains
T
B.
to
The construction of a maximal t ree can be carried
sequence of trees
4.
T
A
K
Remark .
e dge s.
such that
~rom
is a tree strictly containing
J
the maximal ity o f
is invariant under change of orientation
and under homeomorphism.
t hat
K .
copi~s
k
T U e.
=B
A tree is a co=ected regulae l -dimensional com­
L _
n l
Then the union of the
L
n
is
a maximal tree.
plex I{ith no non- ~oero l-cycles.
A maximal
tree in a regular' complex
Given an oriented co=ected regular complex
K
a basis for
of'
K
we construct
is a sub complex 'o'hich is a tree and cont a ins all of the verti ces
Zl (K)
as follows.
Let
K.
44.
),
"
T be a maximal tree for
K,
rea' a
and let
e
be the edges of
A}
connects two vertices in
a
B
€
in
T.
Since
T
T such that
in
a cycle.
a,
~ f a , so the
for
Zl (K)
span
L.aaaea + d ,where
the
c
If we set
d
Thus
Z
=
is a chain on
But
T
T, so
T •
n- cells" joining them; that is, a finite sequence of n-cells
Za ' not in any
z~
ao(T)
za's
Z can be written
= ao(K)
~(T)
Thus the right side
Z1 (K).
Since
T
= 1. But every vertex of K is
za's
0o, al ,·· ., as
is any regular complex
If
and
Z if
is isomorphic to a sum of
K is finite.
~ (K) - ao(K) + 1
'
copies of
We have shown, incidentally, that every l-cycle of
K is a
A simple l-cycle is a cycle carried by a
simple closed curve -- a l-cycle whose coefficients are all
+1
such
that no more than two of the l-cells of the cycle have any vertex in
We shall consider the analog of a simple l-cycle, the fUnda­
46.
as' and each pair (a.,
a l+
. 1)
l
An example of an n-circuit
IKI
such that
is an n-manifold.
Order the n-cells of
so that each
a
k
for
choose such an (n-l)-cell, call it
o~ k
for
~
Tk •
S , by induction as follows:
i < k , define
°
on
~
~
is always
ILl
+1.
cients of
wbere
4.4.
~
d'Y
Set
The chaip
multiple
~f
'Y
=
d'Y
k > 1
for each
;
~,
a
O
= 1 , and, given a i
Tk
aO' al , ... , 0k_l'
~=O~ak
has
is
Note
is crucial in
K.
K not among the
IKI
cons'isting of
By
Tk'S
IKn­ 2 1
Prop . 1.4. 2 .,
i ,s an (n-l)-chain on
L, and since
K is a face of exactly two n-ce11s, the coeffi­
are members of the set
is an (n-l)-chain on
~~.
i
Define integers
to be the subset of
L is a subcoroplex of K.
each (n-l)- cell of
a
This can always be done since
t he computation of the homology of
Define
in a se quence
K
d(aOoO + ~ol + •.. + ~ak)
so that
Tk •
Hn(K)
has at least one (n-l)­
k > 1
dimensional face in common with a preceding
and all (n-1)-cells of
mental n-cycle of an orientable n-circuit.
K
there is a "rath of
,
always a face of precisely one n-cell among
algebraic methods.
common.
0'
a'
a 0'
o
as follows.
00,Ol, • •• ,a s
This result can also be obtained using purely
sum of simple l-cycles.
and
K is an oriented finite n-circuit, we compute
Hn_l(K)
that
Zl (K)
such that
0
bas an (n-l)-dimensional face in common.
coefficient
is
~ (K) - ~ (T) = ~ (K) - ao(K) + 1
Thus
for any two n-cells
LaaC a + d
Z - L.aaZa
form a basis for
and the number of
2)
e ach (n - l )- cell is a face of precisely two n-cells
is
K is finite in the above computation.
aO(T)
1)
such that
An n- c ircuit is an n-dimensional regular complex
za
is a tree, and has no non-zero cycles.
za's
K
DEFINITION.
and
Then
Then
4. 3.
ca
We show that the
K.
T
Laifa' as desired, and the
is a tree, we have
in
A and
for some
In this equation, the left side is a 1-cycle on
T.
Suppose
B-A
For each a,
Za = ea-ca ,then
are independent.
K , while the right side is a l-chain on
is a cycle on
=
a
involves an edge,
Z be a l-cycle on
as above.
a
Za
za's
Let
de
Thus
T.
is a tree, there is precisely one l-chain
d C = B-A
a
For each
T.
not in
K
{-2,0, +2}.
Thus
L having coefficients
If an n-chain has its boundary on
'Y.
47.
d'Y
= 2~ ,
+1.
L, then it is a
Proo:f:
S
let
c = L:j =ObjO j , with
and
S=lbjO j
b~
J
o:f the
only, so
d (c -bor)
d(C-bor )
b~
J
and o:f no other cells
o:f
in
Tk
d(c-bor)
OJ
j
>
COROLIARY.
Ii'
c - bOr =
+b '
on
- 1
lies on
k •
°0
and
°1
L , so
Tk
b'
is zero.
1
Thus
r
~=m ck Tk + d I
on
is a :face o:f
Ok
,
k
to a cycle
j
Also,
i < m
dam
This is vacuously true :for
j
j = m , where
T
is zero.
\
Thus
bor •
C
+1
K has any non-zero n-cycles , they are all :for suitable
zm-
[0
K has some non-zero c
80
:T ]
m m
z -
If' the chain
K is said to be orientable.
mental cycle o:f
K.
non- orientable .
I:f
r
r
is a cycle, then the n-circuit In this case
r
is called the funda­
is not a cycle, then. K is called (Note the di:f:ference between "oriented" and °m
d"
etc., and
c~1,c~2
z
which has coefficient zero on
m
Thus
zm+l '" z
to a cycle on
on
orientable,
H (K)
n
If'
K is orientable,
is zero.
Hn (K) "" Z •
If
K
that
z = dc
c = br
4 .9 .
is non­
2~.
T
(We assume here that homology is inde­
with
°i
Tk' s ,
+C'T
S 8 +d" '
We set
zm+l
For, if
for some integer
THEOREH.
If'
Tit s.
a.nd all preceding
m
is eventUally
z
for some n-chain
to be homologous
sho.~
is an (n-l)-cycle on
z
c
b, so
K
> , where <
2~ .
(Note that
K, then it
L bounds in
on
z
dc
is non-orientable, then
2~
L
K, then, by lemma
such
4.4,
dbr = 2b~ •
K is an o~ientable n-circuit, then
If
2~
<
L •
d'
L.
i s a multiple o:f
Zn_l (L) .
COROLIARY.
Cm+l T m+l···
'
m m
"orientable. " ) 4.7.
z '" zm
and of some
c do
€
l!l
Note that if an (n-l)-cycle of
DE FU!ITION.
Then
and some Cn-l)-chain
k
is a :face of
m
1.
C dO
and note that we have produced a cycle homologous to
m m
€
O:f course
is a cycle i:f and only i:f
l<m<S.
j
Tm + t .erms involving no preceding
.
€
Consequently the coe:f:ficient
€
which has coefficients
Zj
for some coefficients
~bk ' the Sign depending on orientation.
b'
<
k
We know that
L •
n-cycle. 4.6.
:for
Tk
Suppose it is true for
Tl , the Sign depending
r
multiples o:f
is homologous for every
..heTe
is
In any case this must be zero, so
4.5.
is a :face of
< k , we have that
:for
Then
j'[e show by induction that all
Tl
d(c-bor )
j
L
zero on
know that
But
:for
= 0
lying on
L •
has coe:f:ficient
only on orientation.
Given that
lies on
~-le
are zero.
dc
Hn_I(K) ~
Hn_I(K) "" Zn ~ l(L)/
> is the subgroup of zn_l(L)
generated by
Zn_l(L) '" Hn_I(L) . )
pendent of orientation.)
Proof:
4.8.
LEMMA.
cycle on
Each (n-l ) - cy cle of K is homologous to an (n-l)­
Thus
let
~
where
let
d
be orientable.
is zero, since
z
be an (n-l)-cycle o:f
is an (n-l)-chain on
L.
K •
Then
z-"s
- ~=lb k Tk + d ,
We prove by induction that
z
--7
h omologous to a cycle
n-
as :follows.
z,
illg this theorem shows that
both are on
48.
Zl1_1 (L)
r
C I(K)
L
f : H _ (K)
n l
Proo:f:
K
on
is :free.
We define a homomorphism
z
An (n-l)-cycle
L by lemma
z,
dr = 2~ = 0 •
i s a cycle, so
is unique.
4.8.
K
is
The remark preced­
For i:f
L , then their difference bounds in
on
z, "-' z"
and
K, and must be a
multiple of
A.
But
A is zero.
in the homology class of
to
But then
zl
may set
f({z})
z,
z'
a monomorphism, for if
Moreover, starting with any
z, we get a cycle
~
zi
zl
on
1
and so they are equal.
Now
f
homologous
Therefore we
is obviously a homomorphism.
o.
f({z})
morphism because any cycle on
1
zl
But
so
2A bounds in
A itself is a cycle.
ator of
z
~
0
It is an epi­
We triangulate
K.
Thus
f
20A = d2A = 0,
lP
2
]p2, a non-orientable
as in the diagram and label the n­
is a member of some honology class
i3Jld (n-l)-cells.
on
Thus
A can be taken to be a gener-
The projective plane
Example.
2-circuit.
Therefore
1.
The corollary now follows.
Zn_l (1)
It is
4.11.
then
K and so is a cycle in
H l(K) ~ Z 1(1), as
is an isomorphism, and
n-
d"t
n­
Note that the chain "I
f
2AB + 2BC + 2CA
=
O.
Thus
]p2
00 + 01 + ••• + 05 ' so
=
is non-orientable, and
desired.
H2 (]p 2)
If
O.
=
Zl (1)
is a free group on
< 2A > , the subgroup
K is non- orientable, we consider
one generator, A , so
of multiples of
as follows.
find
z,
2A.
He define a map
Given an (n-l)-cycle
on
1
homologous to
z
z.
f: H l(K)
--7
n-
on
Z 1(1)/ < 2A >
n­
K, we use Lemma 4.8 to
Then
z·
Zl(1)/ < 2A > ~ Z2'
~ (JP2) ~
Z2'
the coset of < 2A >
morphism.
2A, so we may set
in which
z·
lies.
f( {z) )
As before,
It is a monomorphism because if
f({z})
f
is homologous to a multiple of
2A
is a homo­
and thus bounds.
Since
Exercises.
f
Hn_l(K) ~ Zn_l(1)/ < 2A >
COROLLARY.
If
1)
K is a non-orientable n-circuit,
HO
~
Z , and
H2
=
0,
2)
We may take a basis for
C _ (1) •
n l
By
H2 ~ Z,
~ ~ Z EEl Z
the Klein bottle is
~ ~ Z EEl Z2'
Z2
Hn_l(K)
HO ~ Z , by tri­
L:GA
the methods of this section. and a finite number of copies
5.
all of its coefficients are
fA
Z •
Show that the homology
the torus is
Z.
Proof:
J
angulating them as 2-circuits and using is isomorphic to a direct sum of
of
~
is zero, then
is clearly an epimorphism, it provides the desired isomorphism
4.10 .
HO(]p2)
~: A t r.~ Xft
equal to
of
z
Since
is determined, up to
cQ=ected,
an addition of a multiple of
Thus
2
lP
is
C 1(1)
n-
+1
which contains
Z 1(1)
n-
a fundamental theorem on finitely generated abelian
groups we ·can find a basis for the free group
Z 1(1)
n-
whose elements is a multiple of a basis element of
50 .
each of
C 1(1) .
n­
For this section we will need the notions of chain mapping
A since
is a subgroup of
Change of Orientation in a Complex
and chain isomorphism.
". 1 .
C'
DEFINITION.
Given two chain complexes
(fC' },o·) , a chain map
q
hOlDOmorphisms
cp q : Cq
--7
C'q
is a collection
<P: C --7 C'
such that
51.
C = ({C },o)
q
cP
0
q-l q
O·cp:C
q q q
--7
and
(CPq)
of
C'
Q-l
for
each
q.
~
is an isomorphism.
q IS
A chain map is called a cha:m isomorphism if each of the
function
The notion of chain map is important because a chain map is
easily seen to carry cycles into cycles and boundaries into bound­
aries.
~
Thus a chain map
defined by
(~*)q(Z)
If we are given a regular complex
induces a chain map
(~qZ)
~*: H*(C) ~H*(CI)
As a matter of convention we shall
dence function
CO:(K)
for
K.
be the chain complex for
a on K.
a
The
(we assume that
o
dim
0
~
~-incidence
K
except for
numbers involving
numbers involv­
induces the chain
~
is obtained from
He define a chain map
an orientation reversal.
C¥
~incidence
1). Then
C~(K) • We say that C~(K)
complex
of
Vie defi_ne a new inci­
is defined to equal
~
are defined to be the negative of the
lng
as the defining relation of the chain map ~.
~
incidence numbers involving
of a chain map or to the boundary operators of a chain complex.
~ = dl~
Let
given by the incidence function
o
Thus we write
we may reverse the orientation on a given cello·
Gil,
K in the following way.
usually omit the subscripts whenever referring to the homomorphisms
K with an incidence
CO:(K)
by
1/r: CC¥(K) ~ CP(K)
Note that isomorphic chain complexes have isomorphic homology
groups and that a chain map
~:
C ~ C'
and only if there exists a chain map
verse of
and
~
such that
C' respectively.
qxp-l
and
~
as follovrs.
1jr~
Thi s defines
1jr
~
for
e\~ry
cell
~
different from
1jro
0
=-0
is a chain isomorphism if
cp
-1
-1 cp
: C'
~
C
on all the generators of the
Cq(K) , and we extend
called the in­
1jr
are the identity on
by linearity.
It is easy to see that
is a chain map and has
1jr
C
an inverse map which reverses the sign on
once again.
0
Thus
1jr
Here we use the fact that the composition
is a chain isomorphism and induces isomorphisms of the homology
of two chain maps is a chain map.
The basic theorem on induced
groups of
chain mappings of homology groups is:
5 .2.
THEOREl'l.
If
cp: C ~ C'
and
1jr:
C' ~ Cn
and
C~ (K ) . And so H~(K) "" H~(K)
for every q.
For our general theorem on change of orientation we will
are chain maps then
need to assume the following lemma:
(1jr~)*: H*(C) ~ H*(C") •
1jr*~*
CO:(K)
5. 3.
The proof is easy and will be omitted.
I.,E}oJl.1A.
(n ~ 1)
Any regular complex on the n-sphere
is an
This theorem, to­
orient able n-circuit.
gether with the fact that the identity chain map induces the
The proof of this lemma is given in Chapter VIII, section 4.
identity homomorphisms of homology groups, shows that the assign­
ment to
C
of the chain complex
H*(C)
and the assignment to a
At the present time we may regard the proposition as an additional
redundant restriction imposed on a regular complex
K
chain complexes and abelian groups into itself.* In the future ve
q > 1
is an orientable
shall usually omit reference to the homology chain complex and talk
(q-l)- circuit.
of the homology groups or regard the collection of homology groups
o
as a graded group.
implies that
map
*See
~: C ~ C'
the map
~*
is a functor from the category of
Chapter IV.
')?
and each q-cell
0
of
K, the boundary
0
for each
Notice that R.R.2 asserts that each (q-2)-cell of
is a face of exactly two (q-l)-cells of
0
contains the ( q-l)-cycle
53·
0,
00
and that R.R.3
and hence is orientable.
the same properties:
Thus we are imposing additionally
only condition 2 of definition 4. 3 on the existence of a path of
va
(q-l)-cells connecting any two (q-l)-cells.
in the following way.
In case
K is simpli­
a
sub complex
THEOREM.
fUnctions
Given a regular complex
~
and
a
on K, then the chain complexes
C~(K) are isomorphic.
chosen so that
~a
~:
The isomorphism
for each cell
+0
a
~ C~(K)
Ca(K)
of
Ca(K)
by starting
at the chain group of dimension zero and working upwards.
V: COCK) --) COCK)
C (K) , let
l
oaa
a
El(A-B)
va
Set
be the identity isomorphism.
be a generator with vertices
for
a.
I 2
E E
o~a =
~
and
E
2
B.
on
Now
(A-B)
=
= ~l.
Note that
by linearity.
a
EVoaa
for some
E
=
Eo~a = o~Ea = o~ta
where
q
is an integer
isomorphism on each
oa
and
C. (K)
1
> 2
a a generator of Cq (K).
Ca(K)
i < q
1
for
and
~
va
= :,+:.0
Then we extend
V as defined is an
54.
for each generator of each
V over
Cq(K)
So
We then set
Z l(cr) ~
q-
z
a
We do
=
Ea.
C (K) , we have
q
Cq (K) , and
Note that
We extend
va
= +0
­
for each
V is obtained by , a collection
V in this way over all the
K, and then the result is the desired chain iso­
~ C~(K) •
If
K is a ,regular complex with two orientations
, then
H~(K) ~ H~(K) for each
q.
Homology groups are
i < q,
Proof: Isomorphic chain complexes have isomorphic homology groups.
From now on we will write Hq(K)
C (K) , commutes with the boundary operators
i
o~ , and satisfies
for
Suppose that
C.(K)
-
is a
V commutes with the boundary operators.
independent of orientation. V has been defined on
> 2.
a
This relation extends by linearity
a
Suppose now that
Now by the pre-
is a generator of
V is an isomorphism on
5.5. COROLLARY.
= -1 .
Moreover, both
a
Then, if
a
EIE2
, and of
the whole group.
orientation reversals: namely, orientation is reversed on each cell
such that
+1 •
=0
voaa
Cq (K) , and extend by linearity over
-morphism
V is induced by a set of independent
=
q
with
are cycles on the
is an orientable (q_l)_circuit, since
chain groups of
I t is clearly an
Voaoaa
=
a
on the (q-l)-cells of
of orientation reversals.
C (K)
l
o~a
thi,s for each generator of
Clearly
for _E2
+1
over the whole group, and so
aa = El(A-B) = E E E (A-B) = 0~va •
I 2 2
Extend V over all of
o~a
;jrOaa
Then
~o
isomorphism.
Similarly,
= ~:.l
El
We let
To specifY
A
ceding lemma,
and
K
o~voaa
First note that
regular complex on the (q-l)-sphere,
may be
We define a chain isomorphism V: Ca(K) --) C~(K)
Proof:
and
we define
a with the orientation induced by ~
have coefficients
K and two incidence
Cq (K)
a generator of
o~~a = 0 • Thus both o~a and voaa
course
5.4.
a
where the sign is determined by comparing
= :,+:.0 ,
cial, the condition holds trivially because any two (q-l)-faces of
have a common (q-2)-face.
For
ex , etc.
so as to have
55·
to mean
Ha(K)
q
for some
ex
5. 4 has another important
The lemma state d be fore Theorem
consequence which relates to the possibility of defining, an incidence
function on a r e gular complex
In fact we have
Given a regular complex
THEOREM.
defined on a sub complex
of
L
K
So the sum
ex
[a:Tl][Tl : P] +
1
d q- r on p, must be
[a:T ][T :P] , which is the coefficient of
2
2
Thus (iii) is verified and ex q is an incidence function for
zero.
L U K
5.6.
ex
K.
Cl q- 1 1 = 0
Now
as a common face.
K, any incidence function
ex to all of K
We continue this process and extend
q
By
taking
to be the empty sub complex we derive the third
L
can be extended to an incidence
redundant restriction R.R.3·
K .
Proof:
extend
\ole
ex
If
vertices
A
a
~ over L U Kl to be the following extension
and
ex
K not in
a, since
lying in
B
'vie define
1,
[a : A]
is homeomorphic to a
[a:B]
-1
is defined on all pairs of cells involving I-ce lls of
incidence function
K not in
ex q
L U K
q
a
Since
a.
exq- 1
He extend
as follows.
We generalize the definition of
from the set of q-cells of
Cq(K) by considering functions
K to an arbitrary R-module
G.
First If
T
is a (q-l)-cell of
is (iii) of Def. 2.1.8.
r
has on
exq
Let
(q-l)-cells of
T
•
\ole
+1
a , we set
K
choose a
on the (q-l)­
[a:T]
'[1,T
2
a •
which are faces of
By
a
ring
R with unit, the qth dimensional R-module of chains of
Given a regular complex
with coefficients in
K and a commutative
K
R.R.2
E is the set of functions defined on the
collection of q-cells of
K mapping into
all but a finite number of q-cells.
plication are defined as follows.
functions, r
E
R , and
a
R
so as to be zero on
The addition and scalar multi­
If
f
and
g
are two such
a q-cell, then
(f+g)a
equal
Then the only pI'operty
be a (q-2)-cell of
DEFINITION.
o
so def.ined which we need to verify
P
6.1.
to an
For any q-cell
is a sub complex of
a
This generator has coefficients
of the incidence function
2
2 •
ex q _ .
l
Zq- 1 (a) ~ Z
exq- l ' by lemma 5· 3,
r.
>
q
q l
These will be zero unless the second cell of the
to the coefficient which
there are
L U K ,
q
on
pair is a (q-l)-face of
cells of
K into
ex _
we must define incidence numbers for- pairs of cells
L
a.
oriented by
All element of
we make the following definition.
defined on the sub complex
generator
In
L
including
K, as the
the integers which is zero on all but a finite number of q-cells. arbitrarily.
Suppose no,," that we are given an incidence function
of
K. is then a function mapping the set of q-cells of
Cq(K) a
Cq(K) , for a regular complex
free abelian group generated by the q-cells of
L, then there are two
K not in
Cohomology.
Homology with General Coefficients.
6.
ex can be regarded as an incidence function f'or
is a I-cell of
closed I-ball.
this way
L U KO '
We have 'd efined
We define
L U KO
inductively over the subcomplexes
Clearly
L U K , •••
l
of
ex
=
fa + go
and
(rf)a = r(fa) ,
wbere the addition and scalar multiplication on the right sides are
in
R
The qth dimensional R-module of chains of
cients in
R
is denoted by
Cq(K;R).
and have p
5'7·
K with coeffi­
An incidence function
ex
on
K induces a boundary operator
the unit in
R .
a and
oaa
q
{-I, ' 0, +l}
TE
Ca(KjR)
q
a
C (Kj Z)
Note that
E
0
q
a
~ Cq- l(KjR)
elsewhere (1 and 0 are the unit and zero
R
[a:T]
lies in the set
chain complexes
groups
a
C
(KjR) .
For example, if
G is the group of rationals, there is no torsion.
G is the group
Z2' every n_ circuit is orientable, because
r
the boundary of the chain
6.2.
DEFlliITION.
Given a regular complex
G over a commutative ring
R-module
of chains of
orientation a
that for the case where
R, the qth dimensional module
K with coefficients in
This R-module is denoted by
K and a unitary
G is the R-module
R
2~
is
is
Z, the homology groups with coef-
coefficients ill a group
Cq(KjG) .
As in Definition 6.1, an
the complex
Cq(K jG)
C l(KjG) for each Q. The qth homomorphism is .the mapping
q­
® 1 , where 1 is the identity mapping on G • The collection
q
If
C = ({C q },o) and C'
chain complexes then the direct sum
(Oa
q
6.3.
®
I}
form a chain complex which we will write as
DEFlliITION.
a
C
(KjG) •
tively, of the chain c01llplex
B~(KjG) , respectively.
H~(KjG)
of
({ C Ell C'}, d Ell 0')
a
C (KjG) , and are denoted by
a
Z (KjG)
q
The qth dimensional homology module
K with coefficients in
a j G) / Bq(KjG)
a
Zq(K
•
G is the factor module
of
C
and
are two
C'
is the
whose qth boundary operator is
q
(d Ell d ' ) ( c, c' )
(dC, 0' c') •
This definition can be extended to any finite or infinite
number of chain complexes.
We remark that the homology of
is the direct sum of the homology of
each q,
C
and of
C'
C Ell C'
that is, for
Hq (C Ell C') "" Hq (C) Ell Hq (C') •
6. 5· DEFINITION.
over
58.
q
C Ell C'
({C'},d')
q
the mapping defined by K with coefficients in the R-module
G are the qth dimensional modules of cycles and boundaries, respec­
and
chain complex
The qth dimensional modules of cycles and bound­
aries of the regular complex
lie need
the following definition.
oa
together with the boundary homomorphisms
Thus homology with
This is the main result of this section.
K.
6.4. DEFINITION.
{Cq(KjG)}
Z .
G does not give any new information about
into
of modules
G can be calculated from
the homology groups .rith coefficients in
G
It turns out, however,
=0 .
f icients in an arbitrary abelian group
Cq(KjR) ~
induces boundary homomorphisms mapping
The proof of these facts is
are all isomorphic.
Homology with general coefficients is useful in some cases.
If
as defined previously.
a
Hq(KjG)
are isomoD'hic, and that the homology
Ca(K jG)
easy and will be omitted.
With this boundary operator the
form a chain complex denoted by
= Ca (K)
As in the previous section, one can show that all of the
because of
is the chain which takes
K[a:T]T, where
of elements of
chain modules
q
More precisely, if
the value 1 on
of R) we set
(/1: C (KjR)
An elementary chain complex is a chain complex
Z of one of the following three types: 59·
i)
(Free)
All chain groupS are zero except ~or an in~inite
•••
cyclic group in a single dimension.
are
o~
ii)
(Acyclic)
0
-7
(All boundary operators
d
D
q
-7
Z
q-
1
~
0
-7 •••
complexes we obtain are as follows :
~o~Z~o
~or in~inite
All chain groups are zero except
cyclic groups in adjacent dimensions,
n
n-l, ~or some
and
n , and all boundary operators are zero except
d
, which
n
q
and
(Torsion)
d
q-l
For every
p.
l
f
than
+1
or
0
6.6.
rr c
THEOREM.
Elk
~
Z~()~ •••
Z such that each
is a chain _complex over
chain group is :free abe1ian and ~initely generated, then
is the
C
direct sum o~ elementary chain complexes.
Proo~:
only the zero
stricted to
C
q
o~
D
q
Dq
~or
eac.h
q •
We set
is a monomorphism.
(dl'···'~}
respectively, such that for
such that
dd.
l
= p.z.
l
l
C
q
= Zq
Zq
is a
EEl Dq
Since
is a cycle, the boundary operator when re­
The image
o~ the ~initely generated :free abelian group
choose bases
over
and
and
(Zl,···,zm}
1 < i < k
g
60.
q
Thus we may
g-l
for
D
is a subgroup
q
and
q
o~
we have an elementary chain com­
z.
with
l
i >k
there are integers
Pi> 0
the chain complexes
C
q,
which multiplies
Pi
we have free elementary chain
of all these chain complexes is
El
The direct sum
C.
K with coefficients in an arbitrary abelian
a be an orientation on K.
group
G.
CU(K)
satisfies the hypotheses of Theorem 6.6 as long as
f i nitely
Let
mar~
cells in each dimension.
(LA ) 0 G is isomorphic to
1
E(A 0 G)
i
splits into a direct sum
chain complexes.
The chain complex
We will use the result that
Now by Theorem 6.6,
L~N. , where the
N.
l
Then it is clear that
K ha~
l
are elementary
= Ca (K)0G""E(N.0G)
,
l
Ca(KjG)
where the tensor product notation denotes tensoring in each diJllension.
Ca(K) 0 G is the cbain complex
For instance
where
(d 01)q
(] 01
q
UCq(K) 0 G},d 0 1)
Therefore, by the remark following
Definition 6.4,
H*(KjG) "" E[H*(N i 0 G)] •
Z
g-l
B,y our ~irst assertion
Pi1pi+l
is split into a direct sum over
Z
dD
we have an
We will use the theorem above to compute the homology groups
Ca(K)
From the proo~ o~ Theorem II.2. 2, we know that
direct summand in
For every
Pi •
of a regular complex
••• ~O~Z
1
q-l , and the boundary operator the mapping
is multiplication by an integer other
n
Pi = 1
complex with the non- zero group in dimension q.
The complex is the same as in the acyclic case
except that
For every
plex of the torsion type with the non-zero groups in dimension
by
... ~O~z~z~O~
The elementary chain
acyclic elementary chain complex with non-zero groups in dimensions
is an isomorphism onto.
iii)
We now split e a ch of these into a
•
direct sum of elementary chain complexes.
course zero.)
.••
-7
Computation
o~
the homology of
K is then reduced to the compu­
tat ion of the homology of the chain complexes
N. 0 G
l
If
Hi
is free , then
.•• -)0 -7G-70-7 •••
and so
Ni €I G i s t he chain complex
Proof:
H (N. €I G) = G
setting
q
',rhere
1
dimension of the single non-trivi al chain group
G of
q
is the
~(g)
contains
N. €I G
= g 0 1 for all g
v:
whose groups are all zero except for two adjacent dimensions q,
g E G,
n
q-l.
verses of each other.
is' acyclic, then
Hi
N. (19 G
1
The non-zero groups are copies of
operator between them is an isomorphism.
to see that
multiplication by
of elements of
Also,
k
Then
Hq (N.1 €I G) "" Ker G ' the subgroup
k
G whose order divides
is
q- leN.1 €I G)
H
Thus in all three cases
direct summand.
TEEOREM.
G (= G/kG)
k
H*(N i 0 G)
For each q,
k
LEMMA .
If
G
contains
,where
T
CPO : G/kG -7 G 0 ~
ng , for
coset containing
~O
V
and
~
are in-
By the fundamental theorem for finitely generated abelian
k(Gl EEl G2 ) ~ Gl EEl k G2 ' the
second assertion reduces to the case where G is a cyclic group.
G is infinite cyclic then
is of order
G is of order
(n,k).
If
Thus
Gk
=
and
T0 ~
n, generated by
are both zero.
kG <===> n
is generated by 1ll,kT
kG ~ G0 ~
6.9.
G0 ~
Then
gO '
be an integer.
n
kG
divides
kr
r
.
(n, k) • So
go and so has order
and the proof is complete.
Let
COROLIARY.
Tnen, for each
G be a finitely generated abelian group .
q,
a
Ha(K;G) ~ (Ha(K) €I G) EEl (T l(K) €I T)
qq
q
where
G/kG
T~_l(K)
is the torsion subgroup of
t ors ion subgroup of
G.
6.10.
Let
a
H l(K)
q-
and
T
is the
kG,
k, is isomorphic to
PROPOSITION.
G be a torsion free abelian group.
Ha(K;G) ~ Ha(K) 0 G
G.
q
Proof:
62.
E
o
I'
kG
n
<===> "("i1,'k) divides
in the direct sum decom­
G is finitely generated, then
is the torsion subgroup of
Let
~ 0 G
Ha(K;
G) "'" (If(K)
EEl 2: kG where the
q
q
G
k
If
rg
for which there is a chain complex of the
~.
then
~ "'" G €I ~ •
Thus
H*(Ni ) €I G as a
is an arbitrary abelian group then
G €I
V(g €I n)
It is easily verified that
•
G •
k
is isomorphic to
the subgroup of elements whose orders divide
® ~
E ~
Ca(K) •
is isomorphic t o
T
This group is written
G/kG , which we write as
torsion type with qth boundary operator
6.8.
k.
We have therefore proved the following theorem.
last sum is over all
~
The kernel of
induces a homomorphism
G 0 ~ -7 G/kG by
Suppose that
The only non-trivial complementary summand occurs
in the torsion case.
po s ition of
In this case it is easy
is of the torsion type, then N . 0 G is
1
G
k
-7 0 -) G -7 G -7 0 -7 ••• , where G is
k.
G.
groups , together with the relation
We show below that
6.7.
G, and the boundary
Ni
the chain complex
kG
is the ch ain complex
H*(N; 0 G) = 0
Finally, if
~
kG, and so
E
by
1
Define
If
~: G -7 G €I ~
To prove the first assertion, we define
For then
kG = 0
q
for all
63·
k.
Then
Exercise.
plane
JP2
Compute the homology groups of the projective
with coefficients in
groups with coefficients in
Z2.
Z and in
2
HO(JP ) "" Z
Z2
"" 0
The cochain complex
associated with an oriented regular complex
K is the cochain com­
plex
gives:
= ({Cq(K(G)},o}),
C·(K)
The
the modules of co cycles and coboundaries, respectively, of
coefficients in
G.
These modules are written
The qth homology module of
The Oth homology group on the left, Z, gives rise to the Oth group
Z2
oq = (-l)qHom(d q ,1) •
where
modules of cocycles and coboundaries of this cochain complex form
2
HO( JP : z) "'" Z2
2
HI (JP ; Z2) "'" Z2
2
~(JP ;~) "" Z2
Hl(:p2) "" Z2
~(JP2)
A tabulation of the homology
Cq(K;G) , is the R-module Hom(C (K;R),G) •
q
on the right; and the 1st homology group on the left,
factor module
zg(K; G),
K with coefficients in
Zq(K;G)/Bq(K;G), and is written
K
with
Bq(K; G) •
G is the
Hq(K;G) •
Z2'
If
u
is a cochain on
K of dimension q, and c is a
gives rise to both the 1st and 2nd groups on the right.
q-chain on K, then for the value of u on c we write
Given a regular complex
K we define the cohomology groups
of K by first associating with
Note that
(ou,c) = (-l)q(U,dC) •
K a cochain complex.
Exercise.
6.11. DEFIlHTION.
(u,.c).
A co chain complex
C over a ground ring
R
1.
state and prove a theorem about decomposing
cochain complexes over
Z whose cochain groups are free and
q
(commutative with unit) is a sequence (C } of R-modules together
q
q
with R-homomorphismP o q : C ~ C +l such that for each q,
o q 0q- 1
of
=
O.
The R-module
C, and the homomorphism
operator of
denoted by
C.
zq ,
q
C
is called the module of q-cochains
0q
is called the qth coboundary
finitely generated into a direct sum of elementary co chain. com­
Deduce from the theorem that the cohomology of a finite
plexes.
i'egular complex with coefficients in an arbitrary abelian group G
is determined by the cohomology with coefficients in
The modules of q-cocycles and q-coboundaries,
q
B
2.
are the R-modules
Ker 0q
and
q­ 1 res­
C is the factor R-module
complexes.
Define a cochain map
homomorphisms
Shmfl that
D
=
((Dq},o')
~
q
~ : C ~Dq
q
~:
such that
C ~D
C and denoted
H* (C)
m
0
regular complex
* *
*
~ : H (C) ~ H (D) •
The qth d:i1nensional module of cochains of a
K with coefficients in the R-lIJOdule
64.
-
"'q+l q -
induces a cohomology homomorphism
Prove an analog of Theorem 5.2.
DEFINITION.
be two co chain
to be a collection of
C together with the zero boundary operators forms a cochain
complex called the cohomology chain complex of
6.12
and
Im 0
pectively. The qth cohomology module of
q
Zq/B
and is denoted Hq(c). The collection of cohomology modules
of
((Cq},O)
C
Let
Z.
G, written
65.
o'm
q"'q for each
q
properties (1) - (3) for
CHAPTER III
clude that i f
REGUlAR COMPLEXES WITH IDENTIFICATIONS
f:
Accordingly, if
1.
That is, if
of equal dimension.
a
A homeomorphism
called an identification on
flo
as
1.1.
a0
carries
o
From properties (3) and (4)
con­
we
a ~:r and g::r ~ a are in F then g= f- l
f: a ~:r and h: a ~:r are in F then h = f.
The identifications
Let K be a regular complex, and let
of K
F
f
and
of
a
a < a
o
K if whenever
~
onto a (closed) face of
~
be cells
onto
a onto
is
~
a "-'
~.
~
then there exists one and only one map in
This fact will be used to construct an incidence
2.4.
fUnction in the proof of
the restriction
The family
of the same dimension
points of
F of
F
also determines an equivalence relation for
K
a
o
if a map in F carries x to y .
x"-'y
DEFINITION.
A collection
fawdly of identifications on
1. For each
a
lies in
F
2. For each
f
3. If
f:
ill
in
p~a
p ~:r
gf:
4. If f:
F
is a
K
K the identity homeomo!1Jhism
F,
f-
and
g:
l
The natural fUnction
a~ a
defines a topology for
a ~:r
~
KIF
in the famil iar way: a subset
l
is in
F
K.
Throughout
s implify notation.
We define the q-skeleton
F
in
X of
F, then
F
is in
KIF '( where sx is the class of x)
this chapter we will omit absolute value bars whenever we can to
are in
(K/F)q of KIF to be the sub-
is the identity homeo­
f
sp ace
a ~:r
IKI
F.
s(K ) .
q
morphism.
5. If f:
s:
KIF is closed if and only if s-~ is closed
is in
ao < a , then
and
complex.
fla0
The space
KIF and the skeletons (K/F)q form
point out only a few of the more important considerations.
open cell of
c'o mplex and
F
(K,F)
a family of identifications on
If (K,F)
with
K a regular
K.
Each
is a complex with identifications, the relation
a"-'~
if
f:
a ~:r
is in
F
KIF is the homeomorphic image, under s, of at
least one open cell of
that
s
maps onto
u
K.
If
sa
u, then the cells of
are precisely the cells equivalent
K.
a
Property
(4) listed for F shows that no two points of the same
cell of
K are ever identified by a map of
This follows from
boundaries.
66.
K
t~
F.
Property (5)
guarantees that if two cells are identified then so are their
is an equivalence relation on the cells of
a
The proof of this fact is routine, and we taket1me here to
is
in. F
A complex with identifications is a pair
IK I .
KIF denote the set of equivalence classes of points of
We let
K if each of the following holds:
is in
a~ a
of identifications on
67.
KIF
The complex
need not be regular, but relative homeo­
KIF
morphisms may be obtained for the cells of
homeomorphisms given for
by composition with
K
h
(En,sn-l)
s
Let
K
The torus from a disk by identification.
be a regular complex on the unit s quare with four
vertices, four I-cells and one 2-cell:
I
y
(li,D.)
PI
'i" (s la)h
(sla)h
EXAMPLE.
(a, a)
;:.
homeomorphism The map
2.2.
from any set of
is a relative homeomorphism because
B
sla
I
The homology of
x
Define
KIF
f: i\ -..;;:. P2
by g(O,y ) = (l,y).
KIF
(K,F)
be a complex with identifications.
KIF
to the homology of the space
KIF
D
f(x"O)
=
Then take
F
(x,l) , and define
KIF.
We mean that
well defined (see page 33 of Chapter II), and
1. the identity map on
1) the space
2) these homology
groups are isomorphic to the homology groups of the chain complex
that we are about to define.
l
-..;;:.
a2
to consist of
-1
and
g
l' -1
2.
f, g, f
3.
the restrictions of these mappings to bounding cells.
An incidence fUnction invariant under
F
is given by the diagram
bel ow (following the arrow-notation introduced on page 35 of
Chapter II):
Proofs of (1) and (2)
c
B
appear in section 3 of Chapter IX.
g: a
whose homology is isomorphic
does carry a regular complex, so that its homology groups are
ca(K/F)
by
°1
--"
Although
is not necessarily regular it is possible to define a chain
complex based on the cells of
P2
is a
relative homeomorphism.
Let
l'
A
2.
9
°2
The justification for presenting
the chain complex at this stage is the ease with which its homology
KIF)
(and hence that of
2.1.
DEFINITION.
under
F
[p:-rJa
=
A
may someti:a!es be computed.
An incidence fUnction a on K is invariant
if whenever
f:
p
~
a
is in
F
and
-r < p,
then
2 · 3·
EXAMPLE.
>
,D
Real projective n-space
pn
from
Sn
by identi­
f icat ions.
[fp: f-rJ a .
Let
We t ake
F
K be the regular complex on
Sn
described in I.2.1.
to be the collection of maps obtained by restricting
68.
fA.
the involution
KIF
pn
complex
T
(see 1.2.7) to the closed cells of
has one cell in each dimension.
K.
a face of
The
The incidence
:t'unction given on pages 36 and 37 of Chapter I I is invariant under F'.
LEMMA.
2.4.
complex
K.
Let
F
Then
th~re e~ists
is invariant under
be a family of identifications for a regular
an incidence fUnction on
K that
K.
n
a
F.
by induction on successive
gl
The argument resembles the proof of 11.5.6.
As
0
al
1, choose a l-cell from each equivalence class
Call the vertices of
a
Ao
and
Bo
For each
T
a
is invariant on cells of
and
1
--;..
be q-cells of
O
2
a2
T
a
to
and
Kl
a, let
f
be the unique map in
(Note that we allow
Define
identity. )
[T: B ]
T
=
1
A
f-1A
=
T
we set equal to zero.
T
that carries
a , in which case
=
and
a
F
BT
f
-1
f
Suppose now that
plex
a
a
Kl
Let
a
a
as defined
which is invariant under
equivalent to
If
P
We devote the next few paragraphs
and the preferred cells is a subcom­
Kq­ 1
a
K, and so by 11.5.6
Let
K-L
a, and let
is a face of
is invariant under
by the inductive hypoth­
f:
let
be a face of
P
Thus
To show that
1
is
by construction
since
[02: gp ]a
by construction.
a
fl
=
gf
2
F.
satisfies propETty (iii) of 11.1.8, let a
be a q-cell of
K-L, and let
preferred cell
T.
Suppose
f:
Pl
a--;.. T
identify
and
are {q-l)-d:iloensional
P2
faces of
a
and
are (q-l)-dimensional faces of
2
0
Then
1
h:f gp]a
a is invariant under
fP
T equivalent to
a
with a
with a common (q-2)-d:iloensional face
(q-2)-d:iloensional face
fP .
3
P . Then f P
l
3
fa = T with the common
Thus
[0:P l ]a[P l :P ]a + [0:P 2 ]a[P2 :P ]a
3
3
L
=
[fO:fPl]a[fPl:fP3]a + [fO:fP2]a[fP2:fP3]a
be the unique identification.
[o:p]a = [T:fP]a'
70.
0
2
T be the unique preferred cell
a--;.. T
a, define
can be extended over
K with an identification
[Ol:P]a ~ [T:flPja;
F.
can be extended to an invariant function on Kq
be a q-cell of
a
By the remark on page 67 , we have
T
has been defined as an invariant
K 1
q-
The union of
L of
i = 1,2.
[T:A ] =-1
Choose a preferred cell from each equivalence class of
q-cells.
= f 2g •
is the
Bo' and set
It is easy to check that
is an incidence function on
to showing that
F
O , and so we have identification>
2
All other incidence numbers involving cells of
incidence function on
K 1
q­
The unique preferred cell
'0.~ --;.. T,
~
fl
equivalent to
invariant under
Before verifYing property (iii) we show that
f .:
of l-cells.
Kq
a
Properties (i) and (ii) 'o f 11.1.8 are clearly satisfied by a
also equivalent to
For
'de assert that the function
O.
thus defined is an incidence :t'unction on
Let
skeletons of
[a: p]a
esis, we need only check incidence numbers involving q-cells of K.
F.
construct the function
"\ole
a , set
If
P
is not
because
a
is invariant under
F.
71.
The second expression is zero
because
°
L.
proo~ o~
The
Let
we
is a cell
o~
de~ine
a chain complex, which we sha~l denote by
s#: CO(K) ----;;.
°
K that is invariant under
F .
For
cO(K/F)
to be the free abelian group
q
KIF.
on the q-cells o~
For each
~
be a co~lex with identi~ications, and let
(K,F)
q
L, and
2.3 is complete.
be an incidence function on
each integer
~or
is an incidence function
s:\K\ ----;;.
The q-th homology group o~
veri~ications
in Chapter IX and
de~ined
C~(K/F)
to
HO(K/F)
q
as the q-th
KIF.
The homology
We begin where Example 2.1
by
o~
a torus
with the
le~ o~~,
identi~ica-
t ions and orientation exhibited in the diagram below:
S~(6 aio~)
"'"
A boundary operator
0 ':
q
0'
q
a.s(oq) •
= 6
l
i
l
cO(K/F)
----;;. COq­ l(K/F)
q
go
is de~ined by
C
B
sflq_l 0°0
q
o'so
q
To show that
re~er
KIF induces a hollO­
3.
----;;.
c°.(K/F) is denoted by
In the examples which ~ollow, we will anticipate the
HO(K/F).
q
morphism
Sllq: C~(K)
Note that
cO(K/F) is a chain map.
homology group o~ the space
q, the map
cO(K/F).
P
is well-de~ined, let
cr ----;;. ~
~:
be in
~P
~
A
F.
D
0
Then
°
~,
0qSO
= s#q-l 0q 0
S#q_l
6
[o:p]op
p < ()"
sflq_l
6
p <
°
The last equality holds because
because
sIP
o ,
varies over the
~P
equals
= s#~p
°
s#q-l 0 q ~ =
Oneeasily
P <
~or
The complex
0
~aces
o~
these cells is a
~
P
0'0'
varies over the
~aces
72.
Thus
sA, so, sp
in the chain complex
and
s~
Each
cO(K/F) :
0i(so)
= s#oo = s#(D-A) =
0i(sp)
= sloP = s#(B-A) = 0
0
o~
Thus
, so the last expression
o .
cells:
F, and
0'
is trivial, so that
H (oI) ~ Z ,
2
that
~our
O2(s~) = s#o~ = s#( 0 + ~p - go - p) = 0
[~O: ~p]( ~p)
o's~ .
veri~ies
cy~le
0
is invariant under
As
•
KIF has
(C~(K/F), O~}
H(K/F) ~ CO(K/F)
~ (.;) ~
Z EEl Z ,
is
73·
and
and
Ho(oI) "" Z .
o~
4.
We start with Example 2.2.
cell
seq
0'
~
The complex
KI F
Cq (pn) ~ Z for
­
in each dimension, so that
The boundary operator
The co chain complex obt a ined from CO'(P[\:Z)
The homology and cohomology of pn
has one
0 < q < n
is determined by
is
dimension
0
1
2
3
n
Cq(JP n) Z
Z
Z
Z
Z
(0',1)
ro
e
Here,
2se
d'(se 2i ) = s#oe 2i = s#(e _l + Te 2i _l )
2i
(_l)q· Hom
5'
_
q
2i l
q
1
I
=
2
0' (se 2 i+l)
s#oe 2 i+l = s#(Te 2i - e 2i ) = 0 .
Co
CQ'(pn)
Cl
e
Z ~Z
can be written
C2
H'C!"')
C
3
odd
q
e ven
Thus,
HO(pn) ~
Schematically the chain complex
q
Z
~r
Z2
e
~ Z ~Z ~
for odd
q < n
for even q < n
for even n
Hn(pn) "'" l:2
where
e2
t
for odd
Accordingly
is multiplication by 2.
Ho(pn) ~ Z
H (pn) "'"
q
Hq(pn)
0< q = 2i + 1
n
even
n
odd.
4
Z2
The homology of the Klein bottle
and orientation shown in the following diagram:
n
n
Z n
(0
Z2
a
--,
even
odd
b
pn
~
rJ
~b
is
"a
dimension
0
1
2
3
4
n
Hq(pn:Z2)
Z2
Z2
Z2
Z2
Z2
Z2
The non-trivial chain groups are
Co
'74. with
We construct the Klein bottle from a disk, with the identi­
It follows from 11.6.'7 that the sequence of homology groups of
with coefficients in
~
Z2 .
5.
5
0
Z2
Compute the cohomology groups of
<n
ficat i~ns
3
0
Z2
coefficients in
0< q = 2i < n
is
2
1
Z
Z2 for
pn
The homology sequence for
0
for
~rZ
H (pn)
n
dimension
Exercise.
n
~
Z ,
C "'" Z EB Z
l
and
C "" Z •
2
p roduces
The boundary operator is given by
handle.
8.
Each p a ir of consecutive edges oriented and
ident i fie d as in the following di agram
=0
°l(sa)
sia
°l(sb)
s#Ob = 0
°2(sa)
s#oa = s#(a-a-b-b)
#"
-
-
-2sb
Therefore,
,
,
HI "" Z2 EEl Z
HO "" Z ,
and
H2 "" 0 •
produces a crosscap.
h
6.
Compact 2-manifolds without boundary
that e ach compact
2~manifold
as a 2-sphere with
M 1 S2 , then
0
< h
M without
handles and
bo~dary
(for some
0
<
k
< 2
can be represented
crosscaps.
can, in fact, be obtained from a closed 2-cell
M
by subdividing
'2
a
into an even number of edges and by identi­
tying these edges in pairs in an appropriate fashion.
Such a sub­
division determines a regular complex on
;2 ,
tions are carried out in such a way that
M is a complex with
identifications.
If
and the identifica­
A good description of the identification process
may be found in Chapter
h , and some
Ii i th the letters
cross cap-producing sequences
k
< 2).
k
ai' b i , ai, bi ' and we label the edges of the i-th crosscap-produ cing sequence with the letters
s:
;2 ~
2h+k
M denote the projection.
I -cells, and one vertex.
indicatrix to
Then
c ' ci
Let
i
M bas one 2-cell,
Note that by attaching a circular
2
a , and using the arrows already given, we obtain
--2
an incidence functiOn on
a
which is invariant under the
i dent i fications.
Computation of
H*(I>f)
Case 1 ) :
Topologie.
o'sa
k
2
Als o,
We have
0
si a2
In the process described there, each sequence of four
diagram,
into
We label the edges of the i-th handle-producing sequence
6 of Seifert and Threlfall 's Lehrbuch der
consecutive edges, oriented and identified as in the following
M is obtained by subdividing a'2
Each
handle-producing sequences and
We will assume that the re ader is familiar with the fact
-2
a
...
"
=
o'sa . = eSb. = 0 .
1
1
morphic to a direct sum of
Case 2 ):
k
1.
s#(Ea1.
Thus
2h
1:1e
-L:a~1
+Th.1
-Th~)
1
H (1,1) "" Z , and
2
copies of
0
Hl (M)
is iso­
Z
have
0.'
I
ofsa
2
s# oa
2
s# ( L:ai -Lai+Thi-Thi+cl+ci)
.. ,
7().
77·
2sc
l
•
H (1-1) ~
2
Thus
o.
As in Case 1, every I-cell of
The boundaries are generated by
morphic to the direct sum of
of
f'!
2S(C ) , and so
l
2h
copies of
Z
X are G-equivalent if there exists a
is a cycle .
Hl(H)
It is easy to
is iso­
and a single copy
~ne
v~rify
2.
k
\ole have K
Let
osa
Again
Thus
H (I-l) ~ O.
2
Zl (M)
2
s#oa
2
2sc
+ 2sc
l
2
.
2h+2
every cell of'
is a cycle.
xl G
Another system of generators for
Zl(M)
Hl(H)
is generated by
2h+2
K onto a cell of
A group
cellular if each
is
sCI + sc
to the direct sum of
2h+l
2
(* )
elements and has a siugle
is zero.
So
HI (l-1)
a
If
of
if
K
f
K of the same dimension.
G of homeomorphisms of
IKI
if
Z
and a single copy of
Lens spaces
F
a family
In this section we define, for each pair
tively prime positive integers, a 3-manifold
(p,q)
L(p,q)
of rela­
a < T ,
is called
IK,I
G on
satisfies the
a onto
maps
g
K, and
g
in
a, then
G is such
g
is the
g
in
G, and each
a
in
K
gFj'
is an identi­
K, and the collection of these identifications forms
of identific ations as is easily verified.
KIF
We specialize.
in the complex plane
is the space
Let
2
C
S3
The
IKI/G.
be represented as the unit sphere
That is,
family of identifications on the 3-sphere.
X be a topological space and let
X onto itself.
lence relation on the points of
It is
G is an isomorphism.
is a cell of
identification space
called a
l-le compute the homology of lens spaces, using a
homeomorphisms of
maps
Z2'
fic ation on
Let
IKII
is isomorphic
Then for e a ch
lens space.
in
g
G,
identity.
copies of
7.
x' •
fo llowing property:
that
relation, that twice
A homeo=rphi sm f
Suppose that a cellular group
sal' sbl""'s~, sbh , sCI' sCI + sc 2 .
Thus
be a regular complex.
fa < fT
the n
sal' sbl " "'s~, sbh , sCI' sC 2 .
=
.
clear that an isomorphism preserves the f a ce relation :
generators
g(l{)
X by collapsing under the action of
onto itself is called a (cellular) isomorphism of
As before, everyone-cell of l-1
is free abelian on the
G so that
identification space whose points are G-equivalence classes is
and is denoted by
Case 3:
in
that G-equ ivalence is an equivalence relation.
said to be obtained from
Z2
g
Then
G determines an equiva­
X, as follows:
78.
S3
G be a group of
x
and
x,
in
Let
f.. = e
p
and
27ri/p
q
=
((zO'Zl) E c21z0Z0 + zlzl
I} .
be relatively prime positive integers.
, and define
T: S3 ~ S3 by
79.
Let
T(zO,Zl) = (AZO,AqZ ) .
l
Then
G of order
p.
The collapsed space
~
1(p,q) • If
1(p,q)
= 1(p,q')
and
~,T3, •.. ,TP = 1
and its iterates
T
1(1,1)
q
= q'
mod p
then
If
is the antipodal map.
P
2
Thus
and
q
1(2,1)
is called the lens
Aq ' so T
T' and
of the whole 3-space
P
p
q
A= 1
similar argument holds for
T
The subset
1 , we have
A
-1, and
n S3
is the half-2-space
regular Complex on
S3
= ((zO,zl)
of
o
o
o
o
and
q
be fixed.
[(O,zl)
in
Set 2~/p)
3
8 10 < arg zl <
E
831zo
f
0
and
arg Zo
((ZO'Zl)
E
s31zo
f
0
and
0 < arg Zo <
for
k
0,1,2,3
are the cells of a regular complex on
8
and
3
= OJ
n
with
2~/p)
dim ok
S3
Zo
k
1
f
E
is a regular 2-cell.
Also, the
TnOk
Tnok
with
=0
03 defined as follows. I f
_Zo = pe ie with o<e<~. Define
E3
for
n
ie/p
cr2 is the intersection of S3
with the
E
c21zo
o
or
arg Zo
0) .
Z )
onto
03
which carries the
(-3
o ,0·3) , and similarly each
l, .•. ,p-l, is a regular 3-cell.
in a union of cells of dimension less than
I.l., Exercise 5, then, the
n·k
T 0
Tnok
k
lies
By the conclusion of
are the cells of a complex on
s3.
The complex is regular because the cells are regular.
Since
~(Tnok)
~Ok
if and only if
tion
*
set
((ZO,Zl)
lies
(zo' zl)
' 1
~nok, it follows that
of isomorphisms of the cellular structure.
p
A
There is an
Finally, it is an easy matter to show that each
clearly form a regular complex on the circle
The closed cell
Thus
= 0 or 0 ~ arg Zo ~ ~)
s31zo
is a homeomorphism of
-11
(-3
~
0 , 0·3)
•
= k , we
are disjOint and that their union is
and
is a 2-sphere.
f: E3 ~
then
= O,l, ... ,p_l , It is clear that the
0
The intersection
T~ with n = l, ••• ,p-l .
boundary sphere onto o·3 • Thus
shall appeal to ExerCise 5 of 1.1.2.
k
E3
Then
((zO,zl)
TnOk
O.
ei, ~2)
1
E
yo
S3
f ( zo'z ) = (pe
2
To show that the
with
0,
is a hemisphere and thus a closed disk.
(0,1) 1
3
0
S3
G is cellular and satisfies
obvious map
p
~
as Euclidean 4-space,
by constructing a condition (-K-).
Let
=0
p3
1(p,q)
for which xo
is a closed 2-disk, and
E3
lie COmpute the homology of
yo
2
C
and regard
Aq
1, then
is
xo + iyo
then
~
P
Zo
form a cyclic group
s3/ G
In the special case
= S3
If we set
m
= 0 mod p
•
Hence
l-1oreover
G
is a group
~(~ok)
G satisfies condi­
above.
An incidence function a that is invariant under
gi ven by:
80 .
81.
G is
[ra3 : Tn+lo2]
[Tn a 3 : Tna 2 ]1
1
['l'n a 2: ~al]
1
for all
1
[Tna l : Tn+la O] [ra l
: Tnoo)
and
and
,
for
-1
n
O,l, ... ,p-l,
m,n , If
s3jG
for
-1
n = O,l, •.. ,p_l .
has one cell in each dimension.
s
denotes the projection of S3 onto s3jG
0:
3
0:
aries in C (S jG) = C (L(p, q)) are given by
2
b' s03 = s,i'Ja 3
"
bsa 2
si
bsa 1
" 1
s#oo
Thus the homology of
L(p,q)
then the bound­
sio
0
- To ) =
X
Note that
1 "" Zp ,
5.1.
THEORill4.
X be a space that carries a complex
Let
the property that for each
q
Sn
H*( L(p, q))
is independent of
X
K with
is iso­
K
K is the irregular complex on
Then 5.1 gives the homology of
given in 1.2.2.
°
and
They
the topological boundary of each
For example, suppose that
Sn immediately.
The proof of 5.1 appears in Chapter IX.
Let
H2 "" 0
X.
may in certain cases be computed from an irregular complex on
is given by
H
But the ho=logy groups
need not be computed from a regular complex on
n
C
H3 "" Z .
2~1
denote complex n-space, and let
sented as the unit sphere in
HO "" Z ,
X agree with the
Then for each q
2
Hq(X)
q­
morphic to the free abelian group on the q-cells of' K
( p-l mi 1)
1
s# ~i=O ~ a
= pso
o
of
q-cell is contained in
°
2
sia - Ta ) =
a2
defined for
q
cellular homology groups of t he complex.
(All other incidence numbers are set equal to zero.)
The collapsed sp a ce
H (X)
complex the homology groups
an automorphism of
s2n+l
Cn + l
Each
)...
S
in
Sl
be repre­
CC
defines
given by scalar multiplication
q .
)...(z , ... , z)
o
n
()...z , •.. ,)...z ) .
o
n
For an alternative description of lens spaces, and for =re
information, the reader should consult Hilton and 1'l ylie's Ho=logy
The space
denoted by
Theory, page 223.
cpl
8.
S2n+ljsl
Cpn.
is called complex pro,jective n-space, and is
The space
is a 2-sphere.
CpO
consists of a single point;
The projection map is denoted by
s : S2n+l ~ Cpn
Complex projective n-space
For
k < n , we identify
In this section we anticipate later chapters in order to
{(zo, .•. ,zk' O, •.. ,O)}
of
state a result which allows us to compute the ho=logy groups of
and the inclusion
certain spaces very quickly.
of
81 .
S2k+l
n+l
C
.
C s2n+l
Ck + l
Then
82.
X
2k+l
k+l
S
= C
2n+l
nS
is consistent with the action
Thus we have a diagram of inclusions and projections:
In Chapter VII, we define ho=logy groups f'or arbitrary
spaces in such a way that whenever a space
with the subspace
carries a regular
83 .
Sl
C
S3
C
s1
s1
cpO
S5
C ... C
s1
point
S2n+l
Cpn
lies in precisely one
s IE2k : (2k
E , S2k-l )
s1
t he cells
C Cpl C Cp2 C ... C Cpn
Ok
~
a
k
Furthermore, since
(_k
.
CY-, CPk .. l ) .1S a rela t i ve homeomorph1sm
satisfy the conditions of Exercise 5 of 1 .1. 2 .
They are , therefore, the cells of an irregular complex
with skeletons
For
k>O
E2k
the subspace
'Cpq/2
2k+ll
{ ( zo"",zk ) E S
,zk
is real > 0 )
S2k-l .
. that
He clalID
is a closed 2k- cell with boundary
. a relat1ve
.
.
1S
homeomorph1sm
of
(2k
E , S2k-l )
It is sufficient to show t hat for each point
the open cell
to
x
Let
Zk
~
For
Since
x
lies in
q
Hq(Cr) "" [ :
S2k+ l _ S2k-l
= IZk l /zk we have
E2k _ S2k- l .
Further, if
I~! I
1 , and if
~!~
is real and non-negative in
~ ! x = (~!zo, ... ,~!~,o, ... ,O) ,
~ ! /~ = ~ ! zk/ I zk I
then
~
= ~!
equivalent to
x .
to 1.
So
lTow let
Ok
C~ _ cpl~-l
0
,
0
is real and non- negative and hence equal
and
x
°,
= CP
For each
is the unique point in
and for each
k>
°
a
k
k>O
S2k+l _ S2k-l
let
is a 2k- cell, and each
85 .
84 .
even
q
odd .
2k < 2n
otherwise .
s2k+ l _ S2k-l
M = (~zo""'~~_l' Izkl,o, ... ,O) ,
which lies in
q
ltTe may now apply 5 .1 to obtain
Sl, and this we show as follows:
= (zo" ", zk'O, ... ,O) .
°
in
= { cp (q-l )/2
contains exactly one point equivalent
under the action of
x
f-
E2k _ S2k-l
s IE2k
(CP
k , CPk-l)
onto
k
IKq I
K
on
Cpn
composition of two morphisms is the usual composition of functions.
:r
The category
CHAPI'ER IV
of all topological spaces (objects) and
continuous maps (morphisms) between them.
COMPACTLY GENERATED SPACES AND PRODUCT COHPLEXES
The composition of
morphisms is the usual composition of maps.
In this chapter we introduce the concepts of compactly
generated spaces and of the product of complexes.
Sections 1-6, 8 are
in Professor Steenrod's course in the fall of 1963.
DEFINITION.
1.1.
together with a set
~
.
M(A,B)
For each triple
from the cartesian product
If
f
f E H(A,B)
g
A and
f o (goh)
2.
For each object
H(A,B)
in
in
=
H(A,B) X H(B,C)
to the set
H(A,C)
(X,xo ) ~ (Y,y0 ) .
classes of maps
2.
of
2.1.
DEFINITION. Let
f
from
C to i)
an object
FA
~ and
Functors
JJ
be categories.
A flIDctor
is a function that assigns to each object
inJ)
and to each morphism
F
in
A
~ f: A ~ B a morphism Ff , so as to satisfY one of the following two sets of conditions: A in
~ , there exists an element
such that
folA
=
l!3"f
=
f
for each
H(A,B) •
If
is called the set of morphisms from
f: A ~ B to indicate that
f
A to
B. is a morphism in
gf
M(A,B). for
1. F(lA) IFA'
2.
~A'
F
F(lA)
satisfies (2) then
The
and
F(fog)
FfoFg Ff: FA ~ FB
and
F(fog)
FgoFf
F
F
is a covariant fUnctor. F
is a contravariant functor. The functor
morphisms
Hom(A, G)
~:
A
Hom.
Let
A and
G be abelian groups, denote the abelian group consisting of all homo-
~
G , with addition
objects are the sets, the morphisms are the functions, and the
87.
86.
If
or
2.2. Examples of functors and let
~ of sets and functions between them.
Ff: FA ~ FE
satisfies (1) then
gof. Examples of Categories
The category
are the
topological spaces with base point, and the morphisms are homotopy
M(A,C) .
(fog)oh .
M(A,A)
(x,x o )
of objects there exists a function
When no conf'usion is likely to result, we shall write
1.2.
B in
go f , aod is called the composition of
1.
f
We write
for every two objects
Two conditions are imposed:
lA
The set
The category~ , in which the objects
g E H(B,C) , then the image in
is denoted by
X g
and
and
of groups and homomorphisms, with the
usual composition of homomorphisms.
~ is a non-empty class of objects,
A,B,C
.1J
The category
Categories
A category
of abelian groups (objects) and homomor­
phisms. The composition of homomorphisms is the usual one.
based upon notes prepared by Hartin Arkowitz for lectures he gave
1.
(Q
The category
~1+~2
defined by
(ll l +1l2 )a
If a: A'
~
A and
correspondence
Il
~
then guarantee
= Illa + 1l2a • H
The cohomology functor
'1: G
~
'Ylla
is a homomorphism G'
rf(X)
If
'1': G'
~
is the identity map of
Gil
f*: rf(y) ~ rf(X).
a': A"
~A',
be a contravariant functor iTom
Hom(A,G) •
'1: G...-;:. G'
::J
to
)t
define
For each homotopy class
(f}
of maps from
Hom(a'a,'Y'Y') = Hom(a,'Y)Hom(a' ,'1') define
G the correspondence is a contravariant functor iTom
fore
CQ
CQ . For
to
TIl
A, the
G ~ Hom(A,G)
1.
group
is a covariant functor iTom
The homology functor.
o
(Y,yo) ,
f#
depends Qnly on the class
(f g )# ~ f#g#
is the identity and that
There­
G is a group, let
[G,G]
denote the commutator
Show that the assignment to each group
2.
G of the abelian
induces a functor iTom}j to () •
t-1ake up a functor us ing
@.
a to Q .
For each space
be the nth singular homology group of
homomorphism
If
G/ [G, G)
~ (Hom(lA''Y): Hom(A,G) ~ Hom(A,G'»
For each map
to
x
is a covariant functor iTom ){ to.Jj •
subgroup.
G.
o
in
o
Exercises.
fixed
corresp.ondence
group
It is known that
i~
f , that
(X,x)
Hom(A,G)
(a: A' ~ A) ~ (Hom(a,lG): Hom(A,G) ~ Hom(A',G» ('1: G ~ G')
f
(X,x )
to
rf
f#: TIl(X,xo ) ~ TIl(Y,yo)
to be the homomorphism
induced by
of
~
A
TIlf
For each pair
to be the fundamental group of X at
TIl(X,x )
o
be
The properties to
a.
The fundamental group functor.
and
are all homomorphisms, then Therefore, for a fixed
Hn( f)
be established later about co~omology groups then guarantee
~A,
a: A'
a.
X with respect to
f: X ~ Y , let
For each map
the induced homomorphism
It is easy to see that 2.
G.
to
X in ~ let
For each space
be the nth singular cohomology group of
some fixed group
Hom(lA,lG)
Hn.
:3
are homomorphisms, then the
Hom(a,'Y): Hom(A,G) ~ Hom(A',G') 1.
to be a covariant functor iTom
n
X
f: X ~ Y let
f*: Hn (X) ~ Hn (Y).
X
in
'J"
3.
let
H (X)
n
3.1.
DEFINITION.
with respect to some fixed
H (f)
n
Products in a category
be the induced
The properties of homology groups
(established later in these notes, and independently of this example)
category
the
Aa
~.
Let
An object
{Aa}
P
be a collection of objects from a
of
~
if t .h ere exists a collection
(Pa: P
~
(called projections) with the following property:
89. 88.
is said to be the product of
Aa}
of morphisms
Gi ven any object
X
in
'c
and any collection of morphisms
f: X ~ P
exists a unique morphism
each
r:t
3.2.
DE.FINITION.
(fa: X ~ Aa J
such that
Pa f
4.1.
A category
~
(Aa)
is a category with products if
t'
of objects of
exists in
If the product of any finite family of objects exists in
equivalence if there exists
and
f' f = lA •
If
equivalent.
In
f': B --;;:. A
f: A ~ B
'1,
~
in
'e.,
t:.
is closed in
ITAa
X.
Exercises.
B
'e
1.
is called an
such that
is an equivalence We call
ff' = IB
generated.
2.
A and
for example, "equivalent" means "homeo ­
Show that the product (if it exists) of a collec­
of objects in a category is unique up to equivalence.
for
Exercise.
defined in 1. 2 .
P
H of
Cle arly, "closed" may be replaced by "open" in 4.1.
Show that if
K is a complex then
is compactly
K
(Hint: use 1.4.4.)
Show that a function from one compactly generated space
into another is continuous if and only if its restriction to each
4.2.
PROPOSITION.
P
Proof.
closed and
Give examples of products, using categories
Show that
'J
Let
He X
X be a topological space.
is compact, then
F
Let
Be cause
H of
X
x.
x
be a limit point of
H
Fe X
is
H.
To complete the proof, we show that if
t hen there exists a compact subset
If
n H is closed in H by
definition of the relative topology of
is a category with products, with
the cartesian product with the product topology.
Each locally compact topological space is com­
pactly generated.
The uniqueness of products up to equivalence allows us to
write
H for each compact subset
X if and only if A n H
compact set is continuous.
Exercise.
(Aa)
A is closed in
following property: a set
then
morphic."
tion
Conpactly generated spaces
A space X is compactly generated if it has the
DEFINITION .
is called a category with finite products. In a category (; a morphism f: A
B
4.
fa' for
•
the product of any family
t;
, there
of
F
Fe X
X with
H
is not closed,
n F not closed.
that does not lie in
F.
is locally compact there exists a compact neighborhood
The set
F
n H is not closed in H because it does
not contain its limit point
Remarks.
x
For Hausdorff spaces, 4.1 can be stated in a
slightly sillIpler form because
If
closed in
90.
91.
H If can be replaced by
n
closed" (meaning" closed in X") .
With cOIIITlactly generated space
defined as it is in 4.1, Proposition 4.2 holds for any
X
in
~
To preserve this gener ality we shall assume in sections 5 and 6
which follow that
X
is any sp ace in
~.
The results of these
sections are of course applicable to Hausdorff spaces.
In later
The topology of
k(X)
compact subsets of
is called the weak topology with respect to
X
k(X) ~ X
The identity map
X is closed in k(X)
tinuous because e a ch set that is closed in
If
X is a Hausdorff space, then so is
5. 1.
X and
LEl-lio1A..
k(X)
is con­
k(X)
have the same compact sets.
sections of this chapter, as well as in the other chapters of these
Proof.
notes, we shall revert to our assumption that spaces are HausdQrff
Let
PROPOSITION.
If
X is any topological space satisfying the
first axiom of countability, (for example, if X
then
H is compact in
X because the topology of k(X)
spaces.
4.3.
If
is a metric space)
X is compactly generated.
Proof.
Let
A be a subset of X
k(X)
then
is finer than that of
H be a set that is compact in
a covering of
H is compact in
H by sets open in
k(X).
X, and let
k(X) , e a ch
many of the
Da n H cover
H, so that
{Da}
be
By definition of the
Da n H is open in H.
topology of
X •
H
Thus finitely
is compact in
k(X) •
such that for any com­
The following two corollaries are immediate.
pact subset
H of
limit point of
X,
A.
A
Since
n H is closed in H
sequence
{x }
n
Therefore
sequence
in
The set
x
{x }
n
is closed in
x
A
of points of
A - {x}
is a compact subset of
But
Y
{x } , whose closure in
n
A and
be a
which
Y consisting of the points of the
together with
Any
x
X satisfies the first axiom of counta­
bility, there exists a sequence
converges to
Let
A
X .
5· 2.
x.
Thus
x
is
is closed.
5· 3.
COROLLARY.
k(X )
are homeomorphic.
to
f:
For each space
is the underlying set
A
The functor k
X in
X
and
Y then
f
is also a function from
X ~Y
, let
k(X)
denote the space that
is closed (open) if and only if
of
H
Y be spaces.
is continuous, then
Since
f
A n H is
r estriction of
f
X
f
k(X)
to
to each compact subset of
and by Exercise 2 of the preceding section
for each comp act subspace
93·
k(Y).
X
If
is continuous,
X, the restriction of
is continuous .
continuous.
X and
is a function from
f: k(X) ~ key)
X.
Q2.
If
is continuous on
to each compact subset of
X with the topology defined by
closed (open) in
H
~
is compactly
k(X)
X is compactly generated if and only if
Let
as follows:
5.
j,
X in
generated.
already contains the
Y includes
For e a ch
COROLIilRY.
f
Thus by 5.1 the
k(X)
is continuous,
f: k(X) ~ ke y )
is
'0 k
!tIe shall denote by
the category whose objects are
compactly gener ated spaces and whose morphisms are continuous maps.
5.4.
DEFINITION.
The funct or
functor that assigns to each
1': X ---...;. Y
the map
~'j
k: ' )
is the
k
X the space
k(X)
If
and
X
Y are compactly generated spaces then their
cartesian product
'3" i s not necessarily compactly
X X Y in
generated, as an example of C. H. Dowker [1] shows (see Section 8
and to each map
f: k(X) ---...;. k(Y) •
of t ·h is chapter).
in
j
k
In contrast, their product
X
X
k
Y = k(X
is (by definition) always compactly generated.
X
Y)
I t will
be important to distinguish between the two kinds of product when
6.
6.1.
PROPOSITION.
we consider products of complexes.
Products in ) k
The category Jk
is a category with products.
7.
Proof.
If
[X.}.
l lE I
lection of objects of
~k'
(I
let
. III(k) Xi ' the product of the
'r
lE
Xi in ~k' is defined to be the space k(.II Xi) ,where .II Xi
lEI
lEI
is the product of the X. in
We show that . III(k) Xi is a
l
lE
. I f f.; :z. ---...;. x.
product in
k
are morphisms in
k ' then l
l
the
fi
Z
to
X.l
in
~.
Since
for which
are the projections
Pif = fi
unique morphism
i
in
The functor
l
l
kf: Z ---...;. k(II x . )
l
such that
kpiokf
kp.
kZ~k(IIX l. ) ~kX.
l
1
f
Z ~
k( II Xi)
IKI
to be the space
L
IIX.
then
and with a
= kfi
Xi
> X.l
in
.J'k
~ILI
K X L
together with the sub-
n
(K X L)
n
= U (Ki
~
L _
n i
i=o
),
IKI
s ian product
of two complexes
X
IL!;
K
In Section 8
L which insure that the carte­
be a complex.
and
0,1,2, ...
n
K X L is a complex.
In this section we show that
We also given an example
L whose cartesian product is not com­
pactly generated, and thus not a complex.
7.1.
The space
LEMMA.
s atisfy conditions
Proof.
Pi
l
is the product of the
k
We define the product
spaces
I , where the
[kp.: k(IIX.) ---...;. k(X.)}
l
Thus,
and
K
we discuss conditions on K and
for each
II Xi ......;;. Xi.
furnishes us with morphisms
be complexes.
L
j'
is a category with products, there exists a unique morphism
Pi
of
J
are also morphisms !'rom
f: Z ~ IIXi
and
K
then
j.
J
The product of two complexes
an arbitrary index set) is a col­
1)
~ILI
IKI
and the skeletons
(K x L)n
through 5) of Definition 1.1.1.
The spaces
(K X L)
sequence of closed subsets of
IKI
n
clearly form an ascending
~ I LI
whose union is
IKI
~ILI.
we have
(K
n
X
L) n - (K XL) n- 1=. U [(K i
l=O
94.
9'5.
-
K.l-1)
X
k
(Ln-l. - Ln-l. 1)]
•
Thus the Components of .(K X L)
form
a
i
n-i
XT
,where
(n-i)-cell of
L.
a
i
- (K X L)
n
n-
1
are products of the
is an i-cell of K
0i
To show that each cell
n-i
T
and
X Tn-i
is an
is open in
(K X L) n ,we observe that
(K X L)
- a
n
i
C
j#
XL.)) U ((K _ oi) X L . )
i
11: n-J
11: n-~
j
:r
COROLlARY.
~
(K X L)
The right hand side is the union of finitely many closed sets and
so is closed.
f: ( E i ,S 1-1) ~ (a,a.)
If
and
K X L are easily
g: (n_i
E . ,S n-i-l) ~ (_.)
T,T
are given, then
H be a compact subset of· (K X L)
H.~ = H n (K.~ X
L.)
k
n-~
plexes.
Then
K.~ X
L
.k
n-i
a X
T
is
a X
T - a
X
T =
aX
~
U
~ x
:r,
f
xg
space
IQI
7.4.
Exercise.
Shov that the product of two relative homeomor­
K and
L
LEMMA.
Q.,
~
Suppose that
are com­
.
n-~
H = U H.
IQI
is
~
(K XL)
n
Q.
~
satisfying the first
IQI
together with
satisfies conditions 1) through
is compactly generated and that,
in addition, each compact subset of
IQI
is contained in a finite
Then
Q is a quasi complex.
lies in a union of finitely many closed cells of
Proof.
closed cell of
Let
H be a compact subset of
be the projections of
Then
i = 0,1, •.. ,
union of closed cells.
IKI '1t/ L I
pectively.
L
are complexes, then each compact sub­
KXL.
q
Thus
Suppose that the Hausdorff space
the subspaces
5) of 1.1.1.
ph1sms is a relative homeomorphism. and
and
Q consists of a Hausdorff
and a sequence of subspaces
This completes the proof of 7.1.
Proof.
Ki
Ki '1t L _i .
n
Recall that a quasi complex
six conditions of 1.1.1.
set of 1.4.3,
By
H and so is a
7.2 implies that Hi is contained in a union of
Thus
is the required relative homeomorphism (see the exercise below).
If
is closed in
~
Since
n
Set
contained in a union of finitely many closed cells of
Since the boundary
LEMMA.
H is closed.
H.
finitely many closed cells of
n-i i
n-i-l
i-l
n-i)
(_ _ _ .
._)
i
fXg: ( E XE
,E XS
Us
XE
~ aXT,aXT U a X T •
7·2.
lies in a union
n
(K X L)n .
is a Hausdorff space,
n
compact subset of
Relative homeomorphisms for the cells of
obtained.
Let
(K X L)
. _ Tn-i))
n-~
k
Each compact subset of
of finitely many closed cells of
Proof.
U (K. X (L
ai
:r
7.3.
n i
X T - = ( U (K
m
union of their closures: pH
Similarly, qH lies in a
U
i=l
n
j
U
finite union
of closed cells of L . Thus H lies ill the
j=l
j
U (j'i X
finite union
of closed cells of KXL
i, j
pH
IKI '1tILI
is compact in
onto
IKI
tained in a union of finitely many cells of
By
IrKI '1tILI
IK/
1.4.4,
and
pH
Let P
ILl
res­
closed.
show that
Let
A be a subset of
Q in a closed set.
That is, since
IQI
which meets each
We have to show that
A is
is compactly generated we have to
A meets each compact set in a closed set.
is con­
K, and hence in the
97· 96.
IQI
Let
H be
a compact subset of
cell of
Q.
IQI .
Then
C
H
n
U
.
where each
(?-
i=o
i
is a
n
H
A
nHn
n
.
U
c?)
H
n
n
is closed for each
i,
n ;;l))
A n H is closed.
7.5.
A closed subspace of a compactly generated space is
8.
closed subset of
that
B
~
show that
Then
X be compactly generated, and let A be a
C is closed in
a
subset of
H n A is closed in
X
Let
sp ace.
We
H be a compact subset of
H, and so is a compact subset of
H n B = (H n A) n B is closed in
H n A
Thus
has the weak
n
K XL
satisfies
X.
8.1.
THEOREM.
plexes, then
8.2.
H.
Thus
meets every compact subset of
X in a relatively closed set .
IKI X ILl
7)
(Milnor [2])
IKI X ILl
THEOREM.
(J. H. C.
If
K and
L
is a compactly generated
We now give two results which show that
Since
H n B is closed in
of two complexes
pactly generated for a large class of pairs
A
A is closed, it follows that
IKI X ILl
is a complex if and only if
A such
C CA
C for every compact set
B is closed in
By hypothesis,
B is
Suppose that
(K X L)
The cartesian product IKI X ILl
t h at the cartesian product
X
That is,
It follows from Exercise 1 of Section 4 and Theorem 7.6
compactly generated.
Let
is a quasi complex.
n
Finally, by 7.3 and 7.4
is compactly generated.
A
Thus
IQI, which completes the proof.
Proof.
n
and the proof is complete.
is closed in
LEMMA.
(K X L)
topology with respect to closed cells.
.
U (A
i=o
i=o
A n;;i
that
(K X L)
Then
A
Since
a
K and
IKI
ILl
X
is com­
(K,L).
L
are countable com­
is compactly generated.
~~itehead [4])
If
K and
L
are
B
complexes and
L
is locally finite, then
IKI X ILl
is compactly
generated.
Since
X
is compactly generated,
B is closed in
A and so
B is closed in
X.
A
is compactly generated.
L
are complexes, then
Therefore
For a definition of " locally finite" see 1. 3·
If
7.6.
THEOREM.
If
K and
KXL
X
is a topological space, a collection
sets of
subset of
By 7.1 we need only verify conditions 6) and 7).
8 .3.
By 7.2 and 7.4,
K XL
is a quasi complex.
That is,
K XL
6).
LEMMA.
Since
If
K is a complex, then
CB .
K is countable if and K has a countable base for compact sets. Note that condition 7) merely states that each
skeleton of a complex has the weak topology with respect to closed
cells.
X is contained in some member of
satis­
only if
fies condition
of compact
X is called a base for compact subsets if each compact
complex .
Proof.
LB
is a
(K X L)n
is closed in
IKI ~ILI , we know by 7.5
Proof.
Let
for compact sets of
of closed cells of
K be countable.
By 1.4.4, a countable base
K is given by the collection of finite unions
K.
98.
99.
Conversely, suppose
compact sets.
&:>.
By
Every point of
Thus
for
' '0
set
K is contained in some member of
1.4.4, each member of
of cells of K .
~
K has a countable base
S
open in
Assume, by induction, that
vn
The following theorem thus implies 8.1.
(Weingr8lll [3])
If
X and
Y are Hausdorff
compactly generated spaces each having a countable base for compact
subsets, then
X X Y is compactly generated.
Proof.
V
n
is a countable union of finite unions
K
y, such that
Vo xwOCu n (Ao X BO) .
Bn ' contain
THEOREM.
and containing
is cQnt ained in a finite union
of cells, and is therefore countable.
8.4.
BO
X
x
and
The set
Vn
as in A
n
and is therefore compact in
compact in
Bn+l .
There
and
\{n ' open in An
y, respectively, and th at
Wn C U n (An X Bn )
exist* ,
and
Also,
An+l
U n (An+ 1 X Bn+ 1)
therefore, sets
Wn ' and open in An+l
is closed
. in
Vn+l
and Wn+l
An+ 1
as well
Similarly,
is open in
W
is
n
An+l X Bn+l
containing
Vn
and
Bn+l ' such that
and
The proof is a restatement of Milnor's proof of
Vn+l X Wn+l C U n (An+l X Bn+l )
8.1:
1': X
We know that the identity function
is continuous.
To show that
l'
Xk
Let
Y~ X XY
V
U V
n
and
W
U Wn
Then
is a homeomorphism, and hence that
X X YEV X WCU
X X Y is compactly generated, we need only show that if U is
X
open in
Xk
Y , then
Suppose that
neighborhood
V of
U is open in
f(U)
x Xy
is a point of
U
Furthermore,
XXY .
We will find a
x, and a neighborhood W of y, such that
~le
may also assume that
x E AO
x
... CAn C ...
X
y.
and
X
and
{Ai}
= U Ai
and that
Y = U Bi ' with
BO C Bl C ... C Bn C ...
and
{B }
i
all compact.
of
U
n (AO
AO
and
BO
X Bo)
is open in
Vo
open in AO
K is called locally countable if each point
8. 5.
COROLLARY.
t hen
IKI X ILl
If
K and
L
are locally countable complexes,
are bases, and that
is compactly generated.
..
AO X BO ' afid contains
are compact Hausdorff spaces and hence regular.
Thus there exists a set
This
K has a neighborhood contained in a countable subcomplex of K.
y E BO .
Now,
Y, since they
concludes the proof of 8.4.
A complex
We may assume that
X and W is open in
meet e ach set of a base for compact sets in an open set.
VXW(U.
AO ( Al (
V is open in
and containing
x, and a
*See
the theorem of Wallace on p. 142 of J. L. Kelley's General
Topology.
100.
101.
Proof.
Let
compact subset of
(x,y)
in
y
IKI X ILl
be a point of A.
IKI
£ V
A be a subset of
and
K'
Then
L.
open in
V
Now
Therefore,
n (IK' I
A
In particular,
A n (U X V)
of
IKI X ILl .
(x,y)
in
U
£
IL I
IK' I X IL' I
meets every compact subset of
set.
x
c
IK'I ,where
and
K.
L'
X IL' I)
hood of each of its points, is open in
W2
Z,
belongs to
Let
Yl
Hn z .
be a point of
X Yl CU.
The set
U inter­
V X H in an open set, by hypothesis.
V and
A open in
By
B open in
H
with
X Yl C A X B
W2
A
But this implies ilIImediately that
IK'I X IL'I •
A is a neighborhood
Xo
~
Z is open
Thus
A, being a neighbor­
It follows that
Yl
Wallace's theorem there exists
Similarly,
in a relatively open
is open, so that
H C Y be compact.
sects the compact set
U is open
a countable sub­
is open in
Let
Then, since
Let
is compactly generated, and
IK'I X IL' I
open set.
meeting each
in a (relatively) open set.
is a countable sub complex of
C I L' I ,with
complex of
IKI X ILl
lies in W
2
(xo'yo)
BC
yo
Z,
lies in
and so is open.
X H)
z / and so H n
By construction,
Y.
and
C U n (V
Z
W X Z CU.
2
is open.
Since
U is a neighborhood of
This completes the proof.
IKI X ILl .
We conclude this section by describing an example due to
8.6.
(I-leingram [3])
THEORTh1.
If
Let
X
U is open.
Suppose that
is locally compact,
is compact.
(V xyo)
The set
nU
neighborhood
Xo
V X yO
W
l
in
of
Xo
neighborhood v1
is open in
to show that
Y
'V
Since
whose closure
X
V
with
The complex
l - cells, where
The
Ai
such that
W X YO C U
l
and
W C W .
Xo
y
in
Y such that
2
l
Let
W
X yC
2
U
z
W C V.
l
be the collection
We assert that Z
Y is compactly generated, it will suffice
Z meets each compact subset of
erable collection
a common vertex,
val
o < Xi
each
Yj
B.
(A.li £ I)
IMI X INI
at
O
'
The complex
(B.lj = 1,2, ... )
N
is a denum­
of closed l-cells, all having
J
vO'
,with
Xi
A.l
=0
is parametrized as a unit interat
u
o
Likewise, we suppose
0 < y. < 1 , with
-
J
vo
We now identify
(il~ i2""')
of closed
l
to be parametrized as a unit interval
J
0
S1
u
I
with the set of all sequences
of positive integers.
Y in a (relatively)
103.
102.
N for which
is an index set with the power of the continuum.
We suppose that each
is regular, so there exists a
of
I
M is a collection
have a common vertex,
Therefore, there exists a
2
Since
We must show
is a point of U.
M and
is not compactly generated, and thus not a complex.
is a compact subset of X X Y • so
is open in
The compact Hausdorff space
of all points
(xo'YO)
lies in an open set
V X Yo
C. H. Dowker [lJ of two complexes
X X Y which meets each
U be a subset of
compact subset of X X Y in a (relatively) open set.
that
Y
X X Y is compactly generated.
is compactly generated, then
Proof.
X is locally compact, and
Thus, if
(i,j)
is a pair of
indices, with
an integer and with
j
of positive integers, we may define
dinates
(l/L, l/i j
J
in
)
P n (A. X B. )
Then
p..
lJ
X INI
Let
Thus
P
is not closed in
IMI
X
INI ,and
IHI
X
INI
is
not compactly generated.
p = (Pij} .
Exercise.
If
K 8cnd
1
are regular complexes, then
is closed.
H is a co~act subset of
If
to be the point with coor­
C li.ll
A. X B.
l
J
Pij
J
l
i = (i ,i , ... ) a sequence
l 2
then
IMI X INI
n H is
P
IKI X 111
is compactly generated if and only if one of the follow­
ing conditions holds:
finite because
(7. 2) •
M X N
H lies in a union of finitely many closed cells of
In particular,
in a finite, and hence closed, set.
Therefore,
P
is closed in
I{e show now that
For each
a
and
i
borhood of
i
b.
is not closed in
P
o in
IHI
Vo
in
Ii'll
of
P
in
IMI
will follow that
l/i~
J
Tj >
larger than
l/I-j
o
Prr
j
and
l/~
l
i
is locally finite.
K and
1
are locally countable.
J
"t:J.
We then choose
1-lith these chOices,
lJ
X
V
PrJ
fails to be locally finite at a point
Dowker's example
j =
l
as a closed subset of
(x, y).
Suppose
1141
X
INil
and this we know to be false.
X
11 ,1
X
111,
compactly generated.
would be compactly generated by 7·5,
The contradiction establishes
BIB1IOGRAPHY
1,2, ...
[1)
Dowker, C. H.,
and
Topology of metric spaces,
Am. J. 14ath.,
vol. 74 (1952).
Construction of universa'i bundles I,
Ann. Math.,
vol. 63 (1956).
[3] We ingr am, S.,
Bundles, realizations and k-spaces,
doctoral
dissertation, Princeton University, 1962.
[4] Whitehead, J. H. C.,
Combinatorial Homotopy I,
Math. Soc., vol. 55 (1949).
104.
1KI
IKI
Embed
necessity. )
[2] folilnor, J.,
a-:-
,INI
X
That
to be an integer
1/T...,.J < J/1 <
11011
y .
i, j
To accomplish this
1
1
P, it
(UXV).
so that for each
and that
Then
K fails to be locally countable at a point x
The closed subset
The pair
n
To show necessity, suppose neither condition obtains.
J
is a limit point
(A. XB.)
l
J
2
lib. .
J
IMI X INI
IMI X INI
aT or
< b-:- , so that P7.,.. lies in U
J
Both
based on the point
y. < b.
is not a point of
(iI' i ,· .. , i )
Tj >
in
Uo X Vo
X YO
is a point of
is smaller than either
we pick a sequence
both
U
V
l
of indices for which
i,j
is not closed in
P
is chosen so that
is,
Since
lEI
X
l
Uo X Vo
This will imply that
U X V •
U be the neigh-
Let
that is given by
We shall now choose a pair
lies in
b)
K,1
we may assume that
j,
x. < a. , and let
that is given by
is a neighborhood of
U X V
8.6.
IINI X IINI •
I , and for each positive integer
be positive real numbers.
J
u
in
be the neighborhood of
Then
One of
{HINT: The sufficiency of these conditions is given by 8.5 and
IMI ~INI .
let
a)
meets each closed cell of !-l X N
P
105·
Bul. Am.
~
where the is=orphism
5
Chapter
\)rm
is defined by the correspondence
THE HON:OLOGY OF PRODUCTS AND JOINS aX'''''a®'.
RELATIVE HOJ.'lOLOGY The homology of
1.
Let
K and
L be regular complexes.
the product complex
K X L.
is the computation of
regular complexes
H*(K
K .and
If
K x L
In section IV. 7 we defined .
The main result of this section (see 1.5)
x L) in terms of H*(K)
and
H*(L)
for
L that are finite in each dimension.
For
lL'
a X T
K and
L
are oriented by incidence functions
KX L
we may orient
of
KX L
is a cell
in the following way.
a' X,,
with
K and
L to be arbitrary regular c =­
[a
plexes.
Since
m-cells of
K and
KX L
L
are regular,
KX L
is regular.
X
':a'
i
U
(K . - K. 1) X (L. - L. 1)'
i+j= 1
1J
J-
(ii i) of 11.1.8, let
ated spaces.
T,
A face
+ dim T = m.
f orms:
a
If
is a cell of
K,
T
is a cell of
A is an arbitrary set, let
group generated by the elements of
any sets
A and
m-cell has the form
and
dim
~
X
a' X,,
of
a
X T be an
aX T
m-cellof
of dimension
a)
a X "
dim "
It is easy to show that for
b)
a' X "
dim a'
i-l,
c)
a' X ,
dim a'
i-2.
B
,'<
if
a
a'.
We call the
m-2
K X L with
dim a = i.
is of one of three
=m-i-2 dim "
=m-i-l
m case a), we have
F(A X B) "" F(A) ® F(B).
L
[aX ':aX p][ax p:a' X,,]
T'<p<, From this it follows that
L
\)rm:
C (K X L) ""
m
L C.(K)®C.(L)
i+j= 1
J
T.
T
numbers are defined to be zero.
denote the free abelian
F(A)
A.
L
a
and
To verify that the product incidence function satisfies property
in this chapter, are carried out in the category of compactly gener­
Each
A face of a cell
incidence function so defined the product incjdence function.
The products on the right, as well as all other operations considered
(See Chapter IV.)
if,'
(_l)dim a [,:T'lL
All other in€ i dence
where
T'l
X
The
are the components of
(K X L)m - (K X L)m_l
and
Accordingly, we set
ra: a' lK
the moment, however, we allow
a' < a
lK
,'<p<,
(_l)dim a [~:pl
L [,:plL[P:,'lL = O.
-106­
-107­
L
. (_l)dim a [p:,'l
L
Proof:
In case b) we have
Vm an i somorphism
,'J
[ax ':aX ,'][aX ,':0" X
,
= (_l)dLID
+ [aX ':0"
a [,:,'lL[a:a'lK +
d'
(-1) LID
'][0"
X
,
X
':0"
X
"l
e(K) @ eeL).
of the
mth chain groups of
defines @ '
e(K X L)
and
Thus we need only show that these isomorphisms commute with the boundary operators.
[a:a'lK[':"]L
a
aX, ~ a
We know that the corre-spondence
Let
a X,
be a generator of
em(K xL).
Then
0,
since
dim a
dim a' + 1 .
dKxL ( a X ,)
E [a X
K X L.
1.1.
D
=
e(K x L)
Given two chain complexes
R,
«(Dq},OI) over a ground ring
the chain complex whose
K
Under the is omorphism
~ X
L'
and we
E [a:a'JKa'
e
=
«(e
q
},o) and
the tensor product
e
~
E
e,
1
D is
00'
@
d)
=
It is easy to see that
1.2.
THEOREM.
Oc
@
00
isomorphism of chain complexes:
T)
maps t o E (-1) d'LID a[,:T'JLa
,'<T dim
@ T' a a@(
E [T:,'JLT')
,'<,
@ ').
I'hus
d + (_l)dim c c
@
o'd.
0(*m(a
= O.
The correspondence
X
d'
E (-1) LID ar,:T'lL(ax T'). T'<, @ T + (_l) dim a a @ O'T
o( a
j
and whose boundary operator is defined by
o(c
X,,)
0"<0'
60_ D
n
@T +
0KxL (a
(E [a:a'lKa') @ T + (-1)
mth chain group is
i+j=
'lrm_l'
0"< 0'
«(em (K x L)},~ X L)'
t o be the chain complex
DEFINITION.
0"<0'
We denote the boundary operator
associated with the product incidence functi on by
define
,,](0"
E [a:a'JK( a' )( T) +
Thus we have associated with each pair of oriented complexes
L the oriented complex
X
T'<, Properties (i) and (ii) are easily verifi ed .
and
':0"
0"<0'
Case c) is similar to case a).
X T»
1Vm- l (0KxL (a X
,»
and the proof is complete.
aX,
~
e(K X L) "" e(K)
-108­
a @,
@
We now turn t o the problem of computing
induces an
eeL).
H*(K)
•
and
H*(L).
in terms of
Because of 1.2 we may consider the problem in an
algebraic sett ing:
given chain complexes
in terms of
and
H*( e )
H*(K X L)
H* (D).
We wish t o define a homomorphism
-109­
e
and
D,
compute
H*(e @ D)
H*(C) (8) H*(D) ~ H*(C Q9 D).
a:
Let
zl
and
z2
be cycles in
C and
We now assume that
D respectively.
C
and D
are chain comple xes over
that are free and finitely generated in each dimension.
Then in
Theorem II.6.6 and write
C ® D we set
C
and
Z
We apply
D each as a sum of elementary
chain complexes of types i (free), ii (acyclic) and iii (torsion):
a( {zl}
Q9
{z2}) = {zl
(8)
z2}·
~l:
First we verify that this definition is independent of the choice of
cycles :
C ~ D-ii
~2:
and
D ~ D-l
j
.
Using the fact that the tensor product of direct sums of chain com­
a( (zl + OC l }
{Zl
(8)
(z2 +
(8)
OlC 2
OIC 2 })
+ OC
l
(8)
- a( (zl}
z2 + OC
l
(8)
Q9
(z2})
plexes is the direct sum of the tensor product of chain complexes,
O'C 2 }·
we have
But this latter cycle is the boundary of
(-1)
~l
dim zl
zl
Q9
c
2
+ c
l
(8)
z2 + c l @
OIC 2 ,
Q9
~2:
C
@
D
L:
~
.
H.
.
@
l
l,J
N .•
J
By the remark follOWing rr.6.4,
so that
H*(C @ D) ~
a((zl + OC l } @ (z2 +
OIC 2 }) = a((zl}
In order to complete the verification that
a
j
is well-defined, we
H*(C) @ H*(D) ~ H*(l1\) @H*(rn )
j
and similarly that
~ (LH*(H )) @ (LH*(N ))
i
j
a
is linear in the second component.
Thus
a:
H (c)
r
@
(i, j)
For each pair
H (D)
s
H*(C) @ H*(D)
""
a
is a well-defined homomorphism
to
@ N ).
Also,
a( {zl} @ (Z2}) + a( {Z3} @ (Z2}) = a((zl +Z3} @ (z2})
a
i
@ (Z2})·
need only note that
We extend
H*(H
L:
i,j
~
H (C
r+s
@
D).
by linearity, and obtain a degree-
a ij :
L:
i,j
H*(H.) @ H*(N.).
J
l
we have the homomorphism
H*(M. ) @ H*(N.) ~ H*(K @ N. ),
l
J
l
J
"hiGh extends by linearity to give the homomorphism
preserving homomorphism
ID
ij
a:
H*(C) @ H*(D) ~ H*(C @ D).
:
L:
..
l,J
H*(N.) @H*(N.) ~ L: H*(l,L @N.).
l
J
..
l
J
l,J
-111­
-110­
1.3.
THEORE],'!.
The following diagram is co=utative:
~
H*(C 0 D)
)
..
~)
,f,
'fl*
J
l
func tors.
IIDil
',J
"1
We remark in pas s ing that
E R* (N.. 0 N.)
(¢l 0 ¢2)*
r1*(C)0H*(D)
and the proof is complete.
f:
That is, given chain complexes
~
C'
and
g:
:> D',
D
This theorem will allow us to replace
Let
zl
and
z2
f
a
be cycle s of
by the homomorphism
where each
x.
l
lies in
Z(M . )
l
l
then the diagram below is
C and
and
D respectively.
¢2z2 = ~ Yj
* ® g*. . .
H*(C') ® H*(D')
lJ
1"
H*(C ® D)
x.
and maps
~ ...
We have
¢1 Z 1 = Ei
C, C', D, D'
..
H*(C) ® H*(D)
Proof of 1.3.
is a natural transformation of
commutative
EH-l(.(M.)0H*(N.)
l
J
f2* l,J
0'\'
C
a
(f ® g)* :>
1"
H*(C' 0 D')
The proof i s easy and is left to the reader.
J
and each
y.
J
lies in
Z(N.).
J
We now investigate the homomorphism
Then
~ij
This homomorphism
is completely determined by the individual
(1)
(¢l ® ~2)(Zl ® Z2)
Ex.0y ..
.
.
l
J
l,J
H*(Mi ) 0 H*(N j ) ~ H*(Mi 0 Nj ),
a ij :
Thus
and each
(¢l ® ¢2 )*a((zl) ® (z2))
shall see that
(¢10 ¢2)*[zl ® z2)
(by def. of
E' [x. ® y.)
i,j l
J
(by (1) above)
lJ i, j
a)
and
N.
J
l
(by def. of
J
(~ .. )( (E (x.)) ® (E [y.)))
lJ
i
l
j
J
(~ij )(¢l*[zl) ® ¢2*[z2))
(~ij)(¢l* ® ¢2*)([zl) ® (z2) )'
-112­
in turn depends only on the type of
a
lJ
~hain
and
N..
is an isomorphism except in the case that
ij
In this case we shall prove that
not relatively prime.
a .. )
Mi
We
J
Mi
are both of the torsion type with torsion numbers which are
monomorphism.
=(Ea .. ) E [x.) ® [y.)
i,j
a ij
a ij
is a
To reduce nine cases to four, we say that an elementary
complex is of type (ii)' if it is of type (ii) or (iii).
Case (1) ® (i).
That is, for some
Let
M and
p, q
N be free elementary chain complexes.
we have
-113­
Ci (M) "'"
Cj(N) ""
Then
J-1 181 N
r
r
generated by
a,
i
if'
=
Case (i) 181 (ii)'.
P
one generator
0
if
i
f
b,
if
i
=
c
r
~
in dimension
for some
H/N) ""
O.
and
d
in dimension
q,
We show that
is an is=orphism.
We have
if'
i
f
q.
a 181 b,
if'
r
= p+q
k
[,
[0
r
r
f
P-Kl.
generated by
a
i
=p
if
j
if'
M, N and
M181 N are all zero, and so
generated by
f
{a}
if
Thus
i =P
H*(M) 181 H*(N)
p + q,
e rated by
{b}
if
j
{d}
=q
has a single generator,
and of order
a 181 c
q
{a} 181 {d},
of dim-
n.
M 181 N is elementary and of type (ii)', gen­
in dimension
P-Kl.+l
and
a 181 d
in dimension
P-Kl.,
=q
vith
0
f
f
j
The chain complex
p
generated by
j
generated by
0
ens ion
i
f p
i
H/M) "'" n 0
t
N be a chain complex with
q + 1,
a: H*(M ) 181 H*(N) ----:;.. H*(M 181 1'1)
generated by
Z
~(M 181 N) ""
f
n
Let
q
'
The boundary operators in
Hi (M) ""
p.
Hi(M) "" 0
k
o(a 181 c)
=
a 181 nd
generator,
{a 181 d},
vhich maps
{a} 181 {d}
=
n(a 181 d).
of dimension
Thus
P-Kl.
H*(M I8IN)
has a single
and of order
n.
So
a,
q
generated by
{a 181 b}
if'
to
{a 181 d},
k = p+q
Case (ii)' 181 (i)
is like case
is an isomorphism.
(i) 181 (ii)'.
0'
k
f p+j
Case (ii)' 181 (ii)'.
a
{a I8Ib} ,
nd
M be a free elementary chain c=plex with
in dimension
is also a free elementary chain c=plex, with
c,(M ®N)
a:
de
with
generated by
0
z
Thus
a
two generators,
p
Let
H* (M) 181 H* (N) ----7 H*(M 181 l'!),
which maps
{a} 181 {b}
in dimension
p+l
Let
and
M be a chain complex with two generators,
b
in dimension
p,
Oa
with
=
mb,
m
f
O.
to
Let
N be a chain c=plex with two generators,
and
d
is an isomorphism.
in dimension
q,
with
de
=
-114­
-115­
nd,
n
f
O.
c
in dimension
Then
q+l
~zm'
(b) ,
generated by
if
i = P
In order that Next we compute Hp+q+l (M 181 N).
H. (M) ~
1
o
Hj(N)
~
[,
d(x(a 181 d ) + y(b 181 c)) = 0, f.
i
P
it is necessary and sufficient that
generated by
(d),
if·j='1
xm + (-l)Pyn = O.
on
j
f.
'1.
Dividing bye,
The tensor product
Z 181 Z
m
n
is cyclic and of order
(m,n), the
(b} 181 (d),
n.
Thus
of dimension
H*(!<t) 181 H*(N)
p+q
has a
e.
and of order
(_l)P+l Y ~
m
x­
greatest common divisor of m and
single generator
e
mle
Since
and
nle
e
e
are relatively prime, there is an integer
k
such that
The chain complex
ension
p+q+2,
of dimension
M® N has four generators,
a 181 d
p+q.
and
b 181 c
of dimension
a 181 c
p+q+l,
and
of dim­
b 181 d
x
~
and
e
y
~
= k
e
The boundary operator is defined by
Thus the
d(a 181 c)
= da
d(a 181 d)
= m(b 181 d) deb 0 c)
=
181 c + (_ll+l a 181 de
= m(b
(p+q+l)-cycles are generated by
181 c) + (_l)P+l n(a 181 d) (_l)P+l(~)(a
u '"
The
(-l)P neb 181 d) deb 181 d) = 0. 181 d) +
(~)(b
181 c).
(p+q+l)-boundaries are of course generated by
eu.
d(a 181 c)
Thus
We first compute H (M ®N).
p+q
are generated by
( -1 )p+l k
m(b 0 d)
and
The boundaries in dimension
p+q
As
vary
(-l)P neb 181 d).
x
and
y
Hp+q+l (M 181 N) ~ Ze' generated by
(u) .
The homomorphism over the integers,
e
(m,n).
Since
Thus the
b 181 d
xm + (-l)Pyn
describes the subgroup generated by
(p+q)-boundaries are generated by
e(b 181 d).
0::
H*(M) 0 H*(N) ---?- H*(M 181 N)
generates the cycles, we have
(b) 181 (d)
IIl&p S
Hp+q (M 181 N) ~ Ze'
generated by
(b 181 d).
(b 181 d)
is an isomorphism in dimension
0: :
-116­
to
°
~
Ze
in
dimension
p+q.
is a monomorphism.
p+q.
Thus
In dimension
In dimension
isomorphism because both the domain and range of
-117­
Ze
0::
---?-
p+q+l,
p+q+2,
0:
are
0:
O.
is an
Ze
H (K X L) ""
~ H (K) @ H (L) $
~
T (K) @ T (L),
m
p+q~ P
~
p+q=w.. l p
~
We are now ready to prove
1.4.
THEO~l.
If
C and
D are chain complexes of abelian groups
that are free and finitely generated in each dimension, then
a monomorphism of
H*(C) @H*(D )
onto a direct summand of
The complementary sUIllliland, in dimension
~
T (C) @ T (D),
~+r~-l ~
r
the
homology group of
~th
T (C)
where
m,
a
H*(C @D ).
is isomorphic to
D are elementary chain complexes.
phism.
1. 6.
Z
This follows immediately from 1.1 and 1. 4.
COROLLARY.
for a prime
11'
p
K and
L
are as above I and
we have shown that
a
is an isomor­
in each case at least one of the chain complexes has torsion-free
In case (ii)' @ (ii)' we know that
a is a mono­
morphism onto a direct sUIllliland whose complementary summand is
Proof.
Apply 1.5, 11.6.9, and 11.6.10.
Exercises.
in
H*(N)
is
H
Zn
in dimension
Zm
in dimension
~.
p.
2.
K~eth relations for cohomology.
to
n*(c')
and
H*(D·).
The only torsion
E H (M) @ H (N).
1 (M @ N) ~ Ze ~ Z @ Z ""
.
m
n
r+s=p+q r
s
The proof of the theorem is complete.
complexes with
finitel~
If
K and
L
Using the results
used in the proofs of 1.3 and 1.4 obtain a formula relating
Thus
COROLLARY (the KUnneth relations).
P X P .
of the exercise at the end of chapter II, and ideas similar to those
p~+
1.5.
2
Compute the homology of
U*(C · @ D')
is
2
1.
l(M @ N) ~ Ze'
p~+
H*(M)
is either
or the group of rationals, then
H
The only torsion in
G
In cases (i) @ (i),
The statement about the complementary summand is true because
homology groups.
p
H*(K X L;G) ~ H*(K;G) @ H*(L;G).
C and
(ii)' @ (i)
H (K ).
denotes the torsion subgroup of
C.
By 1.3 it is sufficient to show that the theorem is t rue if
and
Proof.
p
Proof .
(i) @ (ii)'
p
is
denotes the torsion subgroup of
~
T (K)
where
are regular
many cells in each dimension, then
-118­
-119­
2.
Let
of
and
X
Y be compactly generated spaces *
and
X
Y
Joins of Complexes.
is the
identifications:
for all
x,x' E X and
(x,O,y) - (x,O,y')
The projection map
tion
X
a subspace ,of
X
0
0
Y.
Y
i(x)
is denoted by
= p(x,O,y)
p.
embeds
The func­
ex.
and is written
Y as embedded in
X
E
X and y
E
Y then the line segment from
x
to
y
in
0
Y with t
[(x,t ,y )
t
f:
Y that includes all line segments
Y.
The join
X
°< t
I XX
~
XC
ex
embeds
X
as the
A space is contractible if there exists a map
FO
X with
the identity map and
The
constant.
Fl
0
X
of embedding
X
in a contractible space. 2 .1.
The j oin of a closed
< 1).
lies on a uni~ue
0, 1
TIie mapping cylinder of a map
with the pOints of
F:,
The natural inclusioJ;!
X,
cone on any space X is contractible, and provides the simplest way
Y is the subset
X
X is called the cone on
The join of a point and a space
X as
base of the cone.
Each point of
0
0
Similarly, we can regard
[x,y]
X
y,y' E Y,
Y.
If
X
0
Y defined by
0
y
Y
(x,l, y ) ~ (x ' , l,y).
and
X X I X Y ----?> X
X ~ X
i:
0
X X I X Y "lillder the following
space of
~uotient
X
The join
~
a closed
is the subspace of
Y
[x,fx],
[x ,y].
x
E
X,
LEMMA.
together
(p+q+l)-cell.
~ -c e ll is a closed
p-cell and a closed
The join of a
(p+q)-c e ll.
q-cell is
(p-l)-sphere and a closed
The join of a
(p-l)-sphere and a
(q- l)-sphere is a (p+q-l)-sphere.
Y is homeomorphic to the space
obtained by identifying the two copies of
Proof:
X X Y in the disjoint
union of the mapping cylinders of the projections
XXY~X
and
X X Y ---7 Y:
Let
(A ' ••• ,~)
o
(Jp (Jq = (Bo ' .•. ,B~)
and
be simplexes.
(Ao ' .•• ,~,Bo' ... ,Bq).
(Jp+q+l be the simple x on the vertices
We define a map ~: (Jp X I X (Jq, -----;.. (Jp+q+1 by setting
Let
[¢(f,t, g )]A1. = (l-t) f(A.)
1
*Recall t hat throughout this chapter we work in the categorJ of com­
pactly generated spaces.
[~(f,t,g)]Bj
for
-120­
f
EO,
~ (f ,O,g)
p
g
E
(J~, = ¢(f,O,g')
a c ont inuous map
one and onto.
¢:
Since
and
t
eo 1-
¢(f,l, g )
and
= tg(B j ),
If
f'
0
(Jq -----'? (Jp+q+l·
(J
0
cr q
and
(Jp
= ¢(f',l, g ).
(Jp
p
E
E
¢
Furthermore,
is c ompact and
-121­
Thus
g'
(Jp+q+l
(Jq ,
then
induces
¢ is one -
Hausdorff,
¢
is a homeomorphism.
P
E
let
(x
=
€
If I IIxll
~
Sp-l X I X Eq
11':
define a map
This proves the first statement of the ~emma.
I)
RP. We
denote the unit ball in
~ If+<l by setting
We define the augmented chain complex
Cq (K) = the
K by setting
~ (0)
and by defining
C-1 (K) = Z,
1I'(x,t,y) =
for
x
€
1I'(x,0,y)
SP-l,
y
q
E ,
€
= W(x,O,y')
11':
a continuous map
culation shows that
onto
EP+<l.
onto
SP+q-l.
t
€
I.
((l-t)x,ty)
((I_t)2 + t 2 )1/2
If
and
1I'(x,1,y)
Sp-l
0
q
E
x'
SP-l,
€
= 1I'(x',I,y).
~
If+<l.
Eq ,
€
Thus
~ induces A straightforward cal­
-W is actually a homeomorphism of
The restriction of
11' to Sp-l
Sq-l 0
Sp-l
0
Eq
defined in II.I.ll.
for any
H*(K)
I-chain
chain map
is a homeomorphism
for
This completes the proof of the lemma.
q
op
0
Exercise .
0
q
~ 0
P+<l+1
carries
aP
0
0
q
U 0
aq
0
p
IO~J
onto
1.
P+<l+
0
Show that the join of two spaces is arcwise connected.
e •
K
= C(K)
C(K)
de = (In c)e ,
K
c.
cp*
C(K)
in dimensions ~ 0,
where
In c
Iio (K)
and if
C(K)
complex.
We adjoin to the collection of cells of
eK
of dimension
of
K.
-1.
Since every
We stipulate that
e
K
K be a regular
K
an ideal cell
be a face of every cell
I-cell has precisely two vertices the redundant
restrictions are all satisfied.
Given an incidence function on
we define additional incidence numbers involving
~
by
K,
[0:"1<:]
if dim
~if
0>
°
class of
K.
The obvious
Iiq (K)
~ H (K)
q
isomorphically onto a direct :summand of
v,
where
v
is any vertex in
K.
H (K) ~ Z EB
o
Thus
0
0.
H0 (K).
For a categorical definition of reduced homology, see the exercise at
the end of
VI.4.
K and
e
1
1
j:
i(x)
of dimension
111
t:;
=x
0
IKI
0
e
=
1
an inclusion
be regular complexes, with ideal cells
of
K
and
-1.
111.
x
K
m
1
e
1_1
0
0
We have inclusions
1
i:
IKI
C
n-
j(y) = e
and
IKI
0
111.
to be the space
K
= K_l
• y
IKI
111
0
0
y.
and
and
Thus we have
We define the j oin complex
IKI
0
111
K
0
1
together with the subspaces
m
(K
0
1)
=
m.
U (K.
1
-1
0
1
. ).
m-l-1
We leave to the reader the job of verifying that
cells of K
t:;
K
For notational convenience we write
(1)
is a complex.
dim
°
are denoted by
induces an isomorphism
1=
fO =
In de
c,
A generator for the complementary sunnnand is given by the homo­
let
let
c
is the index of
II.l.11 that
The homology groups of
maps
is a chain
We now introduce a gimmick that will simplify the computation
of the homology of the join of two complexes.
K
Thus
O.
To show that
We remarked following
C(K) ~ C(K)
cp:
> 0.
o
The reader should note that the homeomorphism q-cell
and are called the reduced homology groups of
H (K).
~:
and is generated by
is a zero-chain, then
then
for any
q-cells of
"!"€K
complex, we observe that
y'
of the oriented complex
free abelian group on the
= ~ [O:T]T
q
C(K)
0
We write
1
~.
e
1
K and
1,
0
1
for the ideal cell of
are then given as follows.
ibly ideal) cells of
K
Let
0
and
thus de fined
K
0
T
respectively, of dimensions
-122 ­
-123­
1.
The
be (poss­
m and
n
respectively.
of dimension
m+n+l,
K
open cells of
a
write
a
Then by 2.1,
T
<
a
0
Int(~
T to denote the (open) cell
0
ar
is a face of
T
or
T'
T - (a U T).
0
are not joins of open cells.
following 2.1, the boundary of
so
a
with interior
L
0
is a closed cell of K
T
0
< a'
a
and
a
a
0
< a'
and
K and
on
K
By the remark
a
is the union
a
or
on
~
0
We will, however,
'j"
~
a'
<
on
T Ua
0
a
only when either
T'
Given incidence functions
an incidence function
T
0
Tt
0
T
a
for the chain
0
T,
H*(K
of
if
a
a, a'
for all cells
a
and
e ,
K
[e
e
P
K
0
of
K
T:e
D = ((D.},O'),
K
define a new chain complex
d
ator
0
0',
and
a'
p
T, T'
= (-1)
a
0
and
(C
complexes
and
K
of
L
K
L,
we define
(0
of
if
P
a
(1)
of
l+dim e
then
a
is a generator of
dao~~~a
0
T)
= oa
0
CoD,
and
with boundary oper­
D)
0
C ®D
E
q
j
i
i+j=q-l
0
0'
Hc
Oc ® d + (:'ll+dim c c ® Od.
® d)
CoD
in terms of the homology of
C and
If
C = ((C.}, 0)
is a chain complex, define the suspension of l
written
sC,
with boundary operator
cl,
by
(sC)i
=
eSc
Oc
0
IEMMA.
2. 2.
[T:T' ]~.
a
and
=
The function
q
H
that
C (K
0
q
L) ""
E
i= -1
C(K
­
c. (K)
l
0
L)
a
l+i
a
0
CT.
cp:
C
E
CoD ~
C _ •
i l
d¢
® D)
defined by is a chain isomorphism. is
~
0
Proof;
Let
It is clearly sufficient to show that
c E Ci '
cp
is a chain map.
d E Dj
® C
q-
l ' (L).
-l
(s(o ® O'))cp(c ® d)
is given by (2).
.
Here we wrlte
Cj(L),
a
0
(s(o
(81
o'))[( _l)dim d(c ®
(_l)dim d(o ® o' )(c
(81
a)]
d)
(_l)dim d ( Oc ® d + (_l)dim c c ®
T
-125­
-124­
(sC).l
H
C.(K)
and T is a generator of
l
T + (-1)
for
on the sub-
~
The verification that
L.
Ci _l
Note that if we set
L.
K [T:T']~
The boundary operator in the chain complex
If
= ((C.},o)
l
D. T'
0
indeed an incidence function is routine and is left to the reader.
It follows from
C
'j"'.
T
0
extends the incidence functions
~
L).
by
ep( c ® d) = (_l)dim d(c ® d)
Thus
0
J
then
T']
0
Cl+J+
.. l(K
otherwise
K
T',
0
of
is reduced to the following alge-
Given two chain complexes
C,
o
® C .( L)
J
L by setting
(-1) l+dim a [T:T']~
[aoT:P]ao~
(2)
L)
0
b raic problem:
Describe the homology of
[a: a'}.
C.l (K)
in the sUlll!llaJld
a ® T
Thus the computation
Thus the
T).
0
L
0
oa).
cp(o
2 . 3.
0' )( c
0
Q9
= cp(2lc
d)
COROLLARY.
Q9
Thus
d + (_ll+dim c c Q9 Ocl)
od
= 0 and 2lc
( _l)dim d(2lc Q9 d) + ( _l)l+dim C(_l)dim od c Q9 6:i
boundary of the
(_l)dim d(2lc ® d + (_l)dim c c Q9 Ocl).
acyclic .
Let
K and
( _l)q+ld.
q
E
i+j=q-l
H.(K) Q9 H.(L) +
J
1
2:
i+j=q-2
T.(K) ®T.(L)
The join of a complex
J
1
suspension of
Proof :
DEFDUTION.
H*(K) = O.
K,
THEOREM.
0
L
is
The homology of
K with a
SK
O-sphere is called the and is denoted by
SK.
In this section we prove K is a regular complex then
If
A regular complex K is called acyclic if
Equivalently, any cycle in
C(K)
bounds in
H*(SK) "'" sH*(K).
K.
If
2.5.
K
This follows from 1.4 and 2 . 2.
3 .1.
2 . 4.
and so
L,
0
is the
Then
3.
L)
0
(-1 )q+lc
(q+l ) -chain
z
L be regular complexes with finitely
many cells in each dimension .
R (K
It then follows that
PROPOSITION .
Let
K be a regular complex, and Jet
K is finite in each dimension this result follows from L be a
Corollary 2 .3. point.
Then
K
L,
0
the cone on
K,
is acyclic.
To give a proof for general
Proof:
If
arbitrary .
K is finite, 2 . 5 is a corollary of 2 . 3 .
Let
be a cycle on
z
linear combination of cells of
complex
K'
of
must bound in
K such that
K'
L,
0
K
K
0
L.
L,
0
Since
z =c
o
=
0
0';
e
z lies on
K'
and therefore in
K
=
L
+ doL,
o(c
e
0
2lc
0
e
2lc
0
e
L
L
L
where
€
C
K be
is a finite
L.
0
Applying 2 .3,
+ Od
0
C(K) ~C(SK)
Cq (K),
that raises dimension by
an isomorphism of homology groups .
d €
z
Zq(K
€
Cq - l(K).
FQr each chain
0
We have
L).
where
=
(-1)
dim
c (A
0
deL + od
L + (-l)qd
-126 ­
0
e
L
L + (-l)qd
0
.
0
ot
c - B
A and B are the vertices of the
is joined.
Thus if
c
= e K we have
o~(eK)
~(c) = (_l)dim c(c _ A
0
and that induces
c
on
K we
define
~(c)
Suppose
1,
z
L.
0
+ doL)
+ (_l)l+qc
~:
there exists a finite sub­
Alternate proof of 2 . 5 which does not use 2 .3:
Tnen
z
Let
K we define a chain map
(_lfl+dim c(A
= ~(2lc).
-127­
0
0
0
c)
O-sphere with which
If
= O.
2lc - c +B
2lc - B
0
2lc)
0
dim c > 0
2lc)
K
then
¢
Thus
of
induces a homomorphism
K with a vertex of
L.
In general
Kv L
of vert'ices, but it is easy to show that if
~* : i m(K)
Hm+l(SK), m>
--?
~* is a monomorphism, let
To show that
c
K and
d = A
0
+ B
~
Each chain
d
0
2
+ e
d
d ,
o
e
od +
1
e
ad +
e
-A
0
2
C
~+
with
3
element of t he ~sphere, and
dd
in
regular then
~-chain on
be a
K
odd - Aaed - Badd
3
1
~
C (K),
in
d2
C~+l (K).
in
~
~,
~
e
Let
(-l)~(Aoc - Bac' ).
¢(c)
X
To show that
is a
~-cycle
Also
Od l = 0,
Od
(-l)qA
l
so that
SK.
Then
so that
d
Cd.
d = Aoa
2-cell
:ry of a
A
(-l)~( dl) - d
-B
B
d
0
-B
- B
l
0
d
0
(d
0
= e(B
¢( (-l)~dl) ... d,
- B
l
1
d~
a
+
2 + dd 3
+ e ad
O.
H.(X ) ""
l
2
.)
a
2
d
aeL,
- e
d ) - e
a
S~
If
d
'.ledge
L",
and
L
denotes the
P
S~
iv-ely by
P
p
0
is an integer greater
p
Then the homology groups of
{:p
p
p
l
.)
¢.,.
is onto.
Thus
~*
X
p
is fre e abelian of rank
>
are complexes, we shall use
groups, prov ided
otherwise.
xp
p ) ""
t:
p
(defined induct-
then
i = k+l
i '" 0
otherwise.
G be a finitely generated abelian group.
G "" F EB L: Z , where
1 Pi k+l
S
V'
F
is free abelian.
Sk+l v . .. '" Sk+l v Sk.
SkX
-X , v
Pl
P2
v
Let
Then
Y denote the space k
v S X ,
where the Pn 1.
number of spheres equals the rank of
K v L,
read
"K
to denote a complex obtained from identifying a vertex
-,128­
= 0
is
Let
homol o ~J
i
p
o
so that
1
Sk+~= S(S~ )),
SX,
H.(S~
3
i
k-th suspension of
3
d.,).
complex with prescribed finitely generated
K
where
o
As an application we show that it is possible to construct a
If
2
3
l
d
0
- e
2
n
Ho
I X 1 = p2.
For example,
Therefore
d
0
+ Bod
an isomorphism, as desired.
that the given
s1,
ti:mes around
p
is a boundary.
1
~ + d
so that
0,
is a cycle.
l
(-l)q~(dl)
c
>
are given as follows:
¢* is onto, suppose that
on
This s.ays that
c,
a
~
be a regular complex obtained from wrapping the bound­
p
than one.
This implies that
'~q
H0
(Kv L) 0
IDZ "" H (K)
EBH (L).
0
the ideal
Accordingly
2
are both
is regular and
K '" L
is a sum
1 (SK)
L
O.
H (K '" L) "" H (K) EB H (L),
¢(c) = Od.
with
depends on the choice
-129­
F.
It follows that
fli (Y)
~
if
i = k+l
i f i =
{:
Cq (K)/C q (L ).
0
set of
otherwi se .
Given a sequence
set W
= Y
G ,G , ...
l 2
of finitely generated abelian groups ,
l V Y 2 v... v Y j v ... ,
where for each
j,
Y.
J
has
q
(K).
C l (L).
~
" i(Y j )
i
r
:J
if
i
=
q
H. (I·n "" G,
l
i > 0,
if
l
abe lian group of rank
l ogy
r-l
Hi "" G ,
i
i > 0
and
r > 1
°
"
oo[c]
K.
[c]
the co ­
c.
denote the boundary
°
Let
\ole define a homomor phism
"o[c]
by setting
[Oc]
is well defined because
= ~[Oc] =
[oOc]
O.
=
for any
6 maps
C (L)
q
to
Thus the collection
forms a chain complex, written
C(K,L).
= 0
otherwise.
Then
and in
The map
"
( (C (K,L)},o)
j
L
Also,
q­
if
which contains
C (K,L) ~ C l(K,L)
q­
q
C € C
been constructed so that
we denot e by
K,
be an oriented pair.
(K,L)
operator both in
"0':
is a chain on
C (K)
q
in
Cq (L)
Let
c
If
H (W) "" Z.
0
If
G
is a free
0
we can construct a complex with homo­
by f orming the disjoint union of
1'1
4.1.
DEFINITION.
pair
(K, L),
chai n complex
the pair
The
written
C(K,L).
qth (relative) homology group of the ori ented
is the
H (K,L),
q
h omo lo~J
group of the
If we wi sh to stress the orientation
we shall write
(K,L),
qth
C'Y (K,L),
of
'Y
H'Y(K,L).
q
with
4. 2 .
points.
LEMMA.
The inc lusion map
The projection map
Proof:
j:
C(L) ~ C(K)
i:
C(K) ~ C(K,L)
is a chain map.
is a chain map.
The first a ssertion is precisely the statement that
L
is
a subcomplex oriented by r estrict i on of the incidence function on
4.
A pair
(K,L)
and a subcomplex
To prove the second assertion, l et
of regular complexes is a regular complex
L of
K.
by an incidence function,
tion of
Relative Homol ogy .
The pair is oriented if
'Y
say, and
L
K
"OjC
=
"o[c]
=
[Oc]
= j(Oc).
is oriented by the r estric­
We thus have the f ollowing commutative diagram:
C (L)
q
i:
C (L) ~ C
q
the pair
q
(K, L),
(K,L),
(K).
~
C (K,L),
q
-130 ­
q-chains of
to be the quotient group
C (K)
+
q
j
.
-1jr
we have an inclusion homomorphism
We define the group of relative
written
Then
C (K).
q
€
K is oriented
'Y'
For each pair
C
1° ,,-- -
~
~
C (K,L)
q
l~
Cq- l(L) ~i Cq- l(K) ~
Cq­ l(K,L)
J
-131­
K.
By
4.2,
i
and
j
induce homomorphisms
j*:
H (K) ~ H (K,L).
d*:
Hq (K,L) ~ Hq- l(L).
q
some
vle
q
C E C (K).
in fact a cycle, on
chain on
L,
The map
d*
L.
we vary
Let
z
Then
z E Zq (K,L).
We set
ex:
chain
and
so
o*(z} = (ex:}.
d*
c
L.
Thus
ker d*
Suppose that
is well-defined.
~:
~:
and
THEOREM.
(K,L) .
,
7
7, y'
C7 (K) ~ C (K)
are orientations of the pair
L to chains on
C7 (K,L) ~ C7 '(K,L).
i*
~ H (L) ~
(1) ...
q
Hq (K)
j*
d*
~ H (K,L) ~ H
q- l(L) ~
q
Then
¢
~[c]
then
= a[¢c]
A
Thus
is a chain isomorphism, and
¢
Hq7 (K,L) ~ Hq7' (K,L).
i nduces an isomorphism ¢*:
i*
Let
L and so induces
c E C7(K),
q
If
A
d*
(K,L).
be the chain isomorphism of 11. 5.4.
[d¢c] = [¢ex:] = ¢[ex:] = ¢d[c}.
The sequence of groups and homomorphisms
He have
The proof of 4.3 is complete.
A
4.3.
(z} = j*(c-d).
(1) is called the relative homology seguence of
carries chains on
(K,L).
d(c-d) = 0
ill j*.
the oriented pair
by a
is called the boundary or connecting homomorphism for
the oriented pair
on
The sequence
is a chain,
If we vary
by a boundary, and so
for
[c]
ex:
d
shown that
wish to define a homomorphism
j ex: = djc = ~ = 0,
He have
q
Hq (L) ~ Hq (K)
i*:
groups are independent of orientation.
Relative homology
Furthermore, the triple
A
( ¢ , ¢ IL, ~ )
is exact.
induces an is omorphism of the exact homology sequences
a ssociated with the orientations
Proof:
(i)
Exactness at
c E Cq+l(K).
Then
ker i*:?ill d*.
H (L).
Let
q
z E
i*d*(Z} = i*(ex:}
Suppose
Z
l(K,L),
q+
z = [cl,
(iex:) = (dic} =
i*(z} = 0
for
z E Zq(L).
o.
c E C l(K).
q+
Then
It is then obvious that
Exactness at
(ii)
So
ker j* :? ill i*.
Let
~[c] = [ex:] = liz] = 0,
Also,
d*([C]} = (z}.
Let
Hq (K).
Thus
i*
such that
z E Zq (K)
j*(z}
q
,
q­
(¢,LI·l ·
j*i*(z} = (j i z} =
Then
= O.
¢* ~
¢* 11 ~
A
1
o.
(¢,LI·l"
~ Hr ' (L) ~ H'" (K ) ~ H7 ' (K,L) ~ Hr'l (L) ~
q
1*
q
J*
q
*
q-
A
jz = d[C],
z
d*
q
[c] E Z l(K,L).
q+
A
so
j*
~ Hr(L) ~ Hr(K) ~ Hr(K.L) ~ Hr l(L) ~
z = ex:
so
Then
That is to say, the
di agr am below is commutative:
ker i* = ill d*.
z E Z (L).
q
7'.
Thus
q
for some
rand
CEC q+l(K).
He have
is homologous to a cycle on
ker j*
j(z-ex:) =jz - jex: =jz - d[C] =0,
L.
Thus
Z
E im i*
and
The proof that this diagram is commutative is routine and is left to
the reader.
ill i*.
Remark:
(iii)
Exactness at
H (K,L).
q
d*j*(Z} = d*(jZ} = (m} = O.
be such that
d* (z) = o.
If
Thus
Let
z E Z (K).
q
Then
dill
Let
z
then
ex:
ker d*
z = [c],
j*.
C E C (K),
q
€
Zq (K,L)
The pxact sequence
o~
C (L) ~ C (K) ~ C (K,L) ~ 0
q
q
bOlJUds a
-133 ­
-132­
q
is of course split exact .
h
Cq (K,L) ~ Cq (K )
k:
and
Thus we have maps
restricts chains on
K
be de fined as follows.
where
d
of
K
not in
k,
we could
If
is a chain on
L.
Then
L
such that
L.
is a chain on
and
k([c])
define a map
= 1,
hi
to the cells of
c
~:
c'
cq (K)
h:
jk
~
= 1.
then
k
c
of chain complexes and chain maps such that for each
d+c ' ,
Using the splittings
C (K,L) ~ C
q
q-
l(L)
h
~
by
O~L~K~]'!~O
(1)
to
is a chain involving only cells
= c'.
short exact sequence of chain complexes to be a sequence
q
The map
We may take
K,
C (L)
Thus
~r
O~Lq ~Kq ~Mq ~ O
(2)
= hdk.
is exact.
We may proceed as above to define a connecting homomor­
d*.
1jrd
-dt.
phism d*:
Hq (M) ~ Hq- l(L)
carries cycles to cycles and boundaries to boundaries, and
H (K,L} ~ H l(L).
q
q­
~r*:
induces a map
1jr*
A tedious calculation shows that
Note that the splitting maps
It is easy to prove that
h
and
are not chain maps.
k
j
is onto, we have
for some
Let
vertex of
L.
K
and
L
The complex
be complexes,
K \/ L,
v
a vertex of
obtained by
K,
identi~Jing
K U L,
v'
a
the
vertices
v
and
v'
of the disjoint union
is called the
wedge of
K
and
L.
The wedge can be regarded as the subcomplex
Z
d
=
is a cycle.
de + did'
c
of
K X L.
l ogy of a pr oduct, compute
Using the KUnneth relations for the homoH*(K X L,K v L).
sH*(K X L,K
v
L) "" R*(K
using the res ults in section 2 concerning
Show that
R*(K
0
We set
= i(d+dd l
).
id'
Since
= Cd}.
for some
d
since
q
j
is a
(2) ,
and since
d*CZ}
Thus
Z (M),
de = id
i
is a
To shuw that
d ' E L.
q
d*
Then
varies by a boundary.
We
receive a sequence
j*
d*
(3) .•• ~H q (L) ~H q (K) ~Hq (M) ~Hq- l(L) ~ ...
which is calle-d the homology sequence of the exact sequence (1) .
One
proves just as in 4.3 that the homology seuqence (3) is exact .
L),
0
c E K .
q
Z E
By the exactness of
by
i*
K X v' U v X L
If
q.
ida = did = dde = 0,
is well-defined, we vary
d(c+id')
= o.
dZ
dEL l' Then
q­
for each
for some
jc
= djc =
jde
chain map,
monomorphism,
Exercise:
the sequence
and
A
(See diagram p. 13 ~)
q
L).
If
0
~
L'
~
K'
~
M'
~
0
is another short exact
sequence of chain complexes, then a homomorphism of short exact seIn defining relative homology groups we had occasion to refer
to the exact sequence
o
A
quences of chain complexes is a triple
(~,~,~)
of chain maps such
that the following d iagram is commutative :
~ C (L) ~ C (K) ~ C (K,L) ~ 0
q
associated with any pair
q
(K,L).
q
More generally, we may define a
O~L ~K ~M ~ O
1~
"' l~ .,
l~ O~L ' ~K' ~M' ~O
-134 ­
-135 ­
4.4.
PROPOSITION.
A homomorphism of short exact
of chain
se~uences
A = cCt(L),
B
= Ca(K),
C
=
cf(K,L).
complexes induces a homomorphism of the associated homology sequences.
the relative homology group with coefficients in
In other words, the following diagram is commutative.
H (C;G).
<\
i*
----?>- H (K )
----?>- H (L)
~l·*
Cli
j*
----?>- H (tIl)
"1·*
i~
q-lq;*
~ H (L') ~ H (K') ~ H (W) ~ H
q
q
i*
Exercise:
~
q
q-
Cjl*i*
(Cjli )*
(~j)*
=
=
(j~)*
The third square is more complicated.
C
€
C (K),
by definition.
Thus
be
=
id
=
is exact.
G
A ----7 B ----7 C ----7 0
be
Prove that the associated
Hom(A,G) ~ Hom (B,G) ~ Hom(C,G) ~ 0
d
Cl'cpc
Z
Z (1<1).
€
Z
€
q-
l(L).
Then
jc
Cl*(z} = (d) ,
Cli(~} = (cpd).
and so
i'cpd,
z
Then
q
(cpd) = Cjl*(d} = Cjl*Cl*(z}.
Verify that the following constructions are valid:
G be an abelian group, and let
0 ----?>- A
~
be an exact sequence of free chain complexes.
Define coboundaries in these cochain complexes as in
II and verify that
(Relative homology with coefficients in an arbitrary abelian
G.)
Let
s equence
l(U) ----7 j*~*
Let
for some
~ = j 'cpc,
He have
Cli~*(z} = Cl~(~}
Exercise:
group
q
G.)
(iCjl)* = i*Cjl*.
~*j*
for some
0 ~
an exact sequence of free chain complexes.
i~ o~
Proof:
to be the group
(Relative cohomology with coefficients in
be an abelian group, and let
Cl~
G,
q
~
'\
----?>- H 1 (L )
"1 "*
ji
H (K,L;G),
Then we may define
B
~
C
0
5*:
~
are chain maps.
Finally, define a
Hq(A;G) ----7 Hq+l(C;G).
Obtain a se­
quence
* Hq(C;G) ~
* Ifl(B;G) ~
* Hq(A;G) ~
* Hq+l(C;G) ~
*
~
Prove that this sequence is exact.
If
(K,L)
C =£f(K,L).
group of
Then
and
connecting homomorphism
Let
~
Cjl
Cha~r
is a pair of complexes, set
The group
Hq(C;G)
A =c?(L),
is called the
(K,L) with coefficients in
B =cf(K),
qth relative cohomology
G and is written
~(K,L;G).
It f ollows from the exactness of the sequence above that we get an
O ~A®G~B®G----7C®G----""'O
is exact, and, as in the case
~
Z,
e xact sequence:
we have a relative homology
----7 Hq(K,L;G) ----?>- Hq(K;G) ----7 Hq(L;G) -----7 Hq+l(K,L;G) ----7
sequence
Tnis sequence is called the exact cohomology sequence with coefficients
~ H (A;G) ----?>- H (B;G) ----7 H (C;G) ----?>- H
q
which is exact.
q
q
q- l(A;G)----?>­
For an arbitrary oriented complex pair
-136­
(K,L)
in
G of the pair
(K,L).
set
-137­
Chapter VI.
1.
Let
THE DWARIANCE THEOREM
a complex
Remarks on the Proof of Topological Invariance.
and
K
be regular complexes.
K'
Suppose that
is a continuous mapping with the following property:
of
K,
there is a cell
fa ; T.
(We say that
T
f
(as long as
f:
K ----;;.. K I
for each cell
a
of the same dimension such that
K'
maps cells onto cells . )
K'.
dence function for
of
~
Let
Define an incidence function
0
K
be an inci­
K by
for
[o:T10 = [fo:fTl~
KIF
a
and
ify that
cp:
and
d~
~
as follows;
= Ea.foi •
Then
~
cp
if
Then it is easy to ver­
K.
is an incidence function for
cp(Dl..o.)
~
K.
are arbitrary cells of
cf(K) ~ C~(KI)
then
dO
0
T
We define a chain map
L.a i °i
is a
is a homomorphism.
Moreover, if
are the boundary operators associated with
respectively, and if
C (K),
is a generator of
a
~
0
~,
and
then
First, in section 2, we extend it to mappings
TEl{
is
acyclic *
f-image of any cell
Such mappings we call proper mappings.
Then, in sec­
3 and 4, using the notion of subdivision, we show that an arbi­
trary
mappi~g
K ----;;.. KI
f:
by
f
of finite complexes
can be factored
He define the homology homomorphisms induced
to be the composition of the homomorphisms induced by the fac­
tors of
f.
Returning to our first topiC, we note that if
the identity mapping then
H (K) ----;.. H (K)
f*:
~
f:
~
K ----;.. KI
gf,
and
in homology theory.
E [fo:fTl~fT
which have the prop­
tions
so does
0
f
erty that the minimal subcomplex containing the
if
cpd0: a = cp E [o:Tl T
by identification
K
In this chapter we extend this definition of induced homomorphism.
K,
of
~-chain
obtained from
is regular).
into proper mappings.
where
KIF
to the complex
and
(gf)*
f
f:
maps cells onto cells, and
is the identity isomorphism for each
g:
is
K~K
KI
----;..
= g*f*.
K"
Also,
~.
map cells onto cells, then
These two properties are fundamental
In section 4 we show that they hold for the induced
homomorphisms of arbitrary continuous mappings of finite regular com­
TEl{
(4.9) is a corollary of this result.
d~fo
plexes.
The invariance theorem
d~ cpo.
We emphasize that in this chapter we prove the invariance theorem only
for finite regular complexes.
By linearity, this relation holds over all of
chain. map.
cp*:
Consequently,
H (K) ~ H (K')
~
phisms
~
cp*
cp
C (K).
~
Thus
cp
is a
2.
induces homomorphisms
for every
~.
are induced by the mapping
Suppose we are given two topological spaces
..,Ie then say that the homomor­
f
and write them as
f~.
Examples of mappings satisfying the condition given above are
Chain Homotopy, Carriers, and Proper Mappirlgs.
continuous mappings
*See
fo
and
fl
from
v.2.4. for a definition.
a) the inclusion of a subcomplex in a complex and b) the projection of
-139­
-138­
X to
Y.
X
and
Y,
We say that
and two
f
o
is
homot9pjc to
f ,
l
written
I X X~ Y
F:
=f l ,
fo
such that
F(O,x)
PI
from
into
X
Then two mappings
I X X
fo
and
by
fl
o
fo
We define mappings
Po
1):
q
F(l,x)
and
= (O,x)
X
I X X
F:
We
and
X to
from
only if there exists a mapping
and
P
= fIX.
fox
can reformulate this condition as follows.
and
2.2.
if there is a continuous mapping
PIX
Y
such that
Fpo
that we are given two chain complexes
fo
and
fl
cr.
C to
from
Let
I
closed unit interval consisting of a
I.
and
Let
I
I 0 C
0
defined b y
fl
o
fo
Proof:
I-cell
I
19c
q
Suppose
and two vertices
I -
I
0'
=fl
fo
F
and
for
c
€
o'~c
=I
d'f)q
C.
q
to
C.
(8)
= Feplc
with the
d'
,$q c
19.q-l Oc
~)
q
a chain map
0 Oc,
+
CPo
and
Cj)l
from
0
and
for each
C'
q,
respect­
f
o
and
f ,
l
set
F:
c -
00
c - I 0 Oc)
j}q- lOc
- FCj)oc -
= flc - f c.
0
is given satisfying the conditions of the Lemma, define
10 C
~
fOC'
by
C'
= flc,
F(I0 c)
and
F(I 0 c) =
c
a
C.
q-chain of
Then
F
clearly maps
J)-;,
q
q-chains of
I 0 C
C
into
q-chains of
C' ,
and
c)
d'£)C
1 0 C by
O'F(I 0
=0
DEFINITION.
0 c
and
Two chain maps
CPl(c)
=1
C
fo,f :
l
homotopic if there exists a chain map
F:
0 c.
~
1 0 C
fo
and
and we write
FCPl
fO
= fl·
=fl
F
C'
=
are chain
~
C'
is called a chain homotopy of
to indicate that
fo
and
= f 1c
- f c
0
-,e Oc
F(I0 c) - F(O 0 c) - F(I 0 Oc)
fl
F( o( I
(8)
c)).
such that
Thus
Fepo
C
10 c - 00 c - I 0 Oc,
We define chain maps
Cj)o(c)
2.1.
= fl - f
Then
= F(1
for
a chain of
c
Then
d' ,F(I 0 c)
0
F(O 0 c)
for
19q- 1 0
+
is a chain homotopy of
and
0(1 0 c)
cr.
Fd(I (8) c)
If
d(1 0 c)
C to
be chain maps from
are the boundary operators of
F(I (8) c)
The n in the chain com­
0.
fl
such that
C' 1
q+
be the cell complex for the
01
00 Oc,
and
If
we have
0(00 c)
and
are chain homotop ic if and only if there are homomorphisms
~
C
q
fo
and two chain maps
C' ,
also represent the chain complex for
boundary operator
plex
and
C
Let
ively.
Fpl = fl·
We define in a similar way the notion of chain homotopy.
and
where
(l,x).
Yare homotopic if and
~
~~ .
fO
and
f
l
F
is a chain map and provides the desired chain homotopy.
,
The collection
are chain homotopic.
deformation for
f
o
L)
and
of homomorphisms
f ,
l
E
q
is called a chain
and is said to exhibit a chain homotopy
-140­
-141­
of
f
2.3.
f
o
and
CD
fl·
Let f
THEOREM.
and
fl
chain homotopy of
o and
fl
be chain maps from
C to
C'.
I:f
f l ® 1.
and
(f O )*
H*(C) ~ H*(C ').
(f l )*:
=
fo 0 1
and
o
f
l
,
J)
then
o
is a chain de formati on
® 1
{(-l)q+~omvC1q , l)}
Furthermore,
Hom(f ,1)
cochain deformation for
are chain homotopic, then (f O )*
for
f
(f l )*:
H*(K;G)
*
(f l ):
H* (K';G)
It follows that
Hom(f , 1) .
l
and
-----'?
is a
H*(K' ;G)
and
Proof:
topy of
Let
f
o
1) = (j)}
q
and
(f o )*
be a chain deformation exhibiting a chain homo­
Let
fl·
z be a
q-cycle of
C.
=
(flz - foz)
+ [}q- lClz)
{o,nz}
since
q
z
sub c omplex
is a cycle
Analogous.ly, let
fo,f :
l
C·
~
C·
and
(K,.L)
D· be cochain maps.
homomorphisms ,[) :
q
Cq ~ Dq - l
A collection
f) = r1}q)
1
- f
5fJ+ f) 5
a cochain deformation for the cochain maps
(fo) *
*
= (f l ):
Now let
fo,f l :
that
group.
Then
fl
o
fo,fi:
~
C·
D·,
is
then
H*.
(D ) ~ H* (C . ).
K and
K'
fo
and
fl
If
~
Suppose that
cr < T then X(cr)
X from a pair of complexe s
is a carrier from
K to
K'
which 1{hen
L is a carrier from
L to
f:
K~K'
is a carrie r
is a continuous functi on then a carrier f or
X such that for each cell
cr of K,
fa
C X( a) .
Carriers for a map always exist -- set
X( cr j = K'
a.
The minimal carrier for
f or each
assigning to each
f
is the carrier
a in K the smallest subcomplex in K'
contain­
f' a.
A carrier
the subcomplex
X from
X(cr)
of
K to
K'
K'
is acyclic if for e ach
is acyclic.
A map
f
f:
cr
K ~ K'
is acyclic.
K
in
is
We may
r e lativi ze the s e notions in the obvious way, noting that if
f
0
and
is a chain deformation exhibiting a
-142­
A carrie r
c alled prope r if the minimal carrier for
and cochain maps
C· (K'jG) ~ C·(KjG).
are chain homotopic.
G be an abelian
induce chain maps
C(KjG) ~ C(K'jG)
,1), Hom(fl,l):
Let
in such a way that if
f.
ing
are chain maps.
(K', L')
cr of K a non-empty
X is said to carry
be oriented regular complexes, and suppose
C(K) ~ C(K')
fo ® 1, fl ® 1:
Hom(f
£)
One proves as in 2.3 that if
is called a cochain deformation.
to a pair
I:f
satisfying
0
K'
K'
L'
of
f
f
of
r egarde d as a function on the cells of
be cochain complexes, and let
D·
X( cr)
is a sub complex of X(T).
O.
K to a regular complex
X which assigns to each cell
i s a function
{o',Qq z
H* (K;G).
Then A carrier from a regular complex
(fl)*{Z} - (fo)*{z) ~
f:
CK,L) ~ (K' ,L')
is given, then the minimal carrier for flK is
automatically a carri er f rom
(K,L)
-143­
to
(K',L') .
The c oncept of a
proper mapping is fundamental in cellular homology theory because a
since
proper mapping from one regular complex to another can be shown to
o''P0o
induce homomorphisms of homology groups.
n-chain
Finally, we call a chain map
for each
O-chain
c
on
K
and
C(K) -----;.. C(K' )
rtl:
In q;c
K,
proper if
oa
a.
is a chain on
rv00a
c
= O.
on
X(o)
such that
C (K)
n
.
erators and is carried by
o'c
'Pda.
'P
and then
xC a)
We set
rtla
Let
KI
be oriented r e gular complexes,
If
to
K
then there exists a proper chain
K'
acyclic carri er
map
C(K) ---""" C(K' )
rtl:
such that for each c ell
a
of
K
commutes with the boundary op-
rtll
X( a).
(We say
------
X
carries
rtl.)
Fu...-rthermore , i f
rtla
are any two proper chain maps carried by
there e x ists a chain deformation
X,
then
cP
C(K)
C(K'),
into
0
rtlo
and
CPl
K,
'PoA - rtllA
f) A
equal to a
is a vertex of
rtlo -:: rtll
and
A
of
We construct.
K,
of
KI.
We set
X.
by induction on dimension.
We then set
l-cell_
In
rvOa = In Oa
c
on
X( a)
I-chain on
a
c
of
= O.
a
rtlOa
is a
X(a)
Since
and
O'rtla
C (K)
o
In q;c
O-cell
K, X(a) ,
O-chain on
0' c
0 and 0'.
We set
'PO a
rtla
= c.
cP
so that
We extend
Then
cP
by
I-chain on
X( o).
has been defined on
C (K)
q
a
be an
n-cell of
K.
Then
-144­
cpO a
is an
f)
exl1ibiting
If
A
be an
- ,8Oc
X(A) having
Co (K) .
'PoA - 'PIA Then if
is a c
since the last term is zero.
has been defined for all dimensions less than
n-cell of
K.
X
and satisfies
Then
0',9
= CPo - CPl -
rtloo - 'PIa - j} oo
Eo.
is a chain on
\.Je have
o'rtloa - o'CPla - or/do a
0' (cp a - cP a - jj o cr)
o
1
'Pooo - rtll Oa - CCPo - CPl
is a
-/Jo)oa
= O.
commutes with
over
C (K)
l
Then since
by
for all
q < n-l
so that it commutes with the boundary operators and is carried by
Let
a
I-chain
X( a)
rtl
a
and
'Pa
be
has index zero, and thus bounds in
'Poc - 'PIC
so that it is carried by
Let
linearity.
Suppose that
o,f) c
Suppose that
n
In c.
is acyclic, there exists a
o'c = rtl0a.
the boundary operators
K we have
O-chain of
K,
such that
X(a)
and extend over all of
'P(A) = B,
Then for
For each
B of the non-empty subcomplex X(A)
we select a verte x
linearity.
Given a
rtl
'PI
both carried by the
by induction as follows.
exhibiting this chain homotopy
O-chain on
Proof:
and
n.
and
as boundary and extend by linearity over
which is also carried by
for all
is
X{ A).
£J
rtl
CPo
;;( . We construct a chain deformation
a chain homotopy of
a chain on
We extend
X
two proper chain maps of
is an acyclic carrier from
c.
In this manner we define
X.
because
we may choose an
To prove the second assertion of the Lemma, let
LEMMA."
2.4.
X(a)
is a cycle on
Thus by the acyclicity of
by linearity over
In c.
rvOa
ALso,
(n-l )-chain on
X( cr)
is acyclic we can choose an
whose boundary is
tend by linearity over
X.
X(o)
rtlocr = 'PIa - E ocr.
C (K).
(n+l)-chain
We set Eo = c,
c
on
and ex­
We continue in this manner for all
n.
n
~o defined is carried by X and exhibits the desired chain homo­
topy of
'Po
and
'Pl'
The proof of the Lemma is complete.
-145­
Now suppose that
X
is an acyclic carrier from
(K,L)
to
~ H (K I.G )
Hq (K·G)
,
q
'
*
cp:
Hq(K';G) ~ H~(K; G)
cP*: Hq(L; O) ~ Hq(L';G)
-*
cP :
Hq(L 'j G) ~ Hq(L;G)
"cP*:
"*
~ :
Hq(K',L';G) ~ Hq(K,L;G)
cP*:
By L2mrna 2.4,
(K' ,L' ).
x,
~:
carries a proper chain map
cP
to
C(L)
as an acyclic carrier from
C(K) ~ C(K ').
~:
is a proper chain map
[qc],
C
map.
= [ Cq:c ] = [cpOc] =
~o ' CPl
Furthermore, l et
E
~o
Thus if'
Cq (L),
c
E
ij [c] = ESc]
and
f or
CPl'
,G
c
c
be proper chain maps carried by
€
Cq+l(L').
C (K ),
q
then
for any coefficient group
j}
i s carried by
X.
C (K, L)
q
~ C'1.+ l(K',L
')
~homomorphism_of'
defines
(K,L )
The triple of homomorphisms
~* '
cP*, and
(K' ,LI ).
the r elat ive homology sequences of t he In other words, the diagram below is
commutative:
'0*
i*
j*
'0*
i*
••• ~ H (L;G) ~ H (K;G) ~ H (K,L;G) ~ H l(L;O) ~ •..
q
'1.
q
q-
l~
exhib iting a chain
B
G.
,.. X.
It follows that if we set
S:
Hq (K , L·G)
~ Hq
(K'' L'·G)
,
,
pairs
C (K,L) ~ C (K ' ,L ' ) .
q
'1.
~[Oc] = ~[c], ~ is a chain
and we may suppose
€
Thus if we set
Cq (K),
Then by 2.4, there exists a chain deformation
homot opy of
L'.
-
which is
~:
1{e obtain a well-defined homomorphism
~[c] = ~[qc]
to
K',
CfJ*
~[c]
Since
L
to
The restriction of
C(L) ~ C(L')
X as an acyc l ic carrier from
carri ed by
K
'o!
l~
i~
iI~
j!
­1
Cf'*
'0'
*
i'
*
••• ~ H (LI; G) ~ H (K';G) ~ H (K',L';G) ~ H
(L'· G) ~
q-l
'
'1.
'1.
q
A similar result holds for cohomology.
is well - defined. PI·oof :
""
(/3 '0
,,/'-..
+ 'O ,,.e)( [c])
L$'Oc + 'o',8c]
and that follows from v.4.4.
[~oc - ~lcl
"
"
2 . 6.
= cpo[c] - CPl [cJ.
Thus
CPo
and
CPl
are chain homotopic .
the triple of chain maps
( ~,~, ~) ,
Everything but the commutativity of the diagram has been proved,
and
In othe r words,
X
ind:uces
defined uniquely up to a chain
COROLLARY.
~
If
(K,L) ~ (K',L ' )
f:
are any two acyclic carriers for
l ogy and cohomology homomorphisms for
that
f
Xo
f,
and
is proper, and if
then the associated homoXl
homotopy.
2. 5.
COROLLARY .
An acyclic carrier X from
(K,L)
to
(K',L ')
induces a unique set of homology and cohomology homomorphisms
coincide.
induces these homomorphisms and write them as
f*:
H (K,L;G) ~ H (K' ,L';G)
q
q
(f!K )*:
Hq (K;G) ~ Hq (K'
(f!L)*:
Hq (L·G)
~ Hq
(L''·G)
,
and s.imilarly for cohomology.
-146­
-1 47­
Xo
jG)
We say
Proof:
Let
X(o)
C
then
<p
'S.
X be the minimal carrier for
Xo(a) n ":t(a).
is also carried by
as carriers from
'S.
and
2·7·
Thus, if
are
<1'*'
COROLLARY.
*,
X
o
K to K'.
f.
Then for each
0
of
b)
K,
~ is a proper chain map carried by X,
'S.;
and
here we are regarding
X, Xo ' a
The unique homomorphisms induced by x
Automorphism of a complex:
Exercise.
for
X
E
Let
K,
~*' etc.
If
X is an ac~clic carrier from
the dimension of
X(a)
K to
T.
such
~at
11' is also carried by X.
a collection of homomorphisms
=11'. I f
So .l) q 0,
is a q-cell of
0
~q carried by
K, then
which is a (q+l)-chain on
is zero and
<p
x( o)
Show that
q:>(c 0 d)
'fCc 0 d)
=d
Let
® c
(-l)pq(d ® c)
=
defined b y
a,
is at most that of
~
in­
c
E
Cp(K),
C(K) 0 C(L) ~ C(L ) ® C(K)
t:
is not a chain map.
n l
n l
R + -?>- R +
T:
for
be given by a diagonal matrix,
X.
By the Lemma, there is at least one proper chain map
<p
(y, x )
is proper, and the minimal carri er
T
Note that the mapping
with
SUPFose that
Then
L.
dEC (L ).
q
K'
Exercise.
X.
E
T( X,y )
preserves dimension, so, by 2.7, there is a unique chain map
duced by
~
then there is exactly one proper chain map carried by
by
Y
K X L ---? L X K be defi ned by
o
for each cello£! K,
Proof:
T:
<p
carried
plus ones.
c)
Then by the Lemma there is
X which gives a chain homl
is of dimension at most
X( a), mus t be zero.
Thus each
q. tlJ q
minus ones starting from the upper left, and then
k
f:
Compute
in
T*
H (Sn) .
n
Projection of a product onto its factors:
K ],i, L -?>- K be the projection mapping.
X for
f
X( a
satisfies
a unique chain map
X T)
= a.
= 11'.
0(0"<)"
Thus
f
Let
The minimal carrier
is proper and induces
Note that
by 2.7.
~,
n-k+l
G
if
dim T> 0
if
T
Let
K and
is a vertex.
Examples of proper mappings:
We assume throughout that a regular compl~ on a closed cell is
acyclic.
This will be proved in section
4 of this chapter.
d)
Then
f:
Simplicial mappings:
K -?>- L
of each simplex of
(a) Inclusion of a subcomplex:
Let
(K,L)
is a simplicial mapping if
i:
and so
i
i
L -----> K denote the inclusion map.
assigns to each cell
is proper.
a
of
L the sub complex
The induced homomorphisms
the homology and cohomology sequences of the pair
L
and is linear
i*
and
i
*
A mapping of the vertices of
K
The minimal
0
of
L
can be extended to a simpliCial mapping if
K,
and only if it maps vertices spanning a simplex of
K onto vertices
are those of
s panning a simplex of
L.
opposite a vertex
then
If
a
is a simplex, and
T
is the face
(K,L).
by V.2.5.
map
~:
A,
a
is the cone on
T.
Thus
a
is acyclic
It follows that a simplicial mapping induces a unique chain
C(K ) -?>- C(L ).
-149­
148
carries the vertices
K into vertices bf a simplex of
into the vertices of
carrier for
f
be a pair of regular
in terms of barycentric coordinates.
complexes, and let
L be simplicial complexes.
3.
Subdivision of a Regular Complex
Proof of Theorem 3.2:
3.1.
DEFINITION.
Let
K be a regular complex.
We define a homeomorphism
step wise extension over the sub complexes
subdivision of
K, written
Sd K,
A
K
K
form the vertices of a simplex of h:
o
order so that each is a proper face of the next. 0
Sd(K
'1-
Sd K
is a simplicial complex.
is a s implicial subcomplex of
to
ooSda.
L
is a subcomplex of
Sd K.
K,
then
Hote also that in general
Sd L
Sd(K ) '1 (Sd K) . '1
If
TREOREI1.
K
IKI
is homeomorphic
ISd KI.
Let
'1-1
o
to a homeomorphism
o
be a '1-cell of
We choose a homeomorphism
a ----->
Sd
'1-1
Sd
onto
0
Sd
so that
K.
E'1
to be
0
h
a~
K.
f: E'1 ~>
S
We extend
f
-1
h
'1­
1
which sends the vertex
'1-1 •
a.
fg.
extends
We note t hat on
h
'1­
E'1
is
f-lh
g
into the origin.
Sd IT
of
'1-1
By hypothesis,
to a homeomorphism
0
h
Then, by Le=a 3.3,
is a homeomorphism and so the mapping
IT
is a homeomorphism.
on
is a regular complex then
K
h
homeomorphic to the join of the origin -with
h
Note that if
0
onto
1)
ISd K I b:y giving it the weak topology with respect to
is not equal to
Sd K,
Suppose that we have extended
Sdo
It is obvious from the definition that
closed simplices.
is also a vertex of
Sa(K) - - > K •
Sd K if and only if the cells of the collection can be arranged in 3.2.
K
and whose simplexes are defined as follows :
finite collection of cells of
We topologize
Sd(K ). In dimension zero, every
'1
so we have the obvious homeomorphism
is the simplicial comple x whose
vertex of
vertices are the cells of
- - > [K[ by
h: [Sd K[
Tne (first i t e rated) we have
h
: Sd
'1-1 mapping
We define
1
ff
-1
h
'1
1 = h 1
'1'1­
We proceed in this manner for every '1-cell of
g
fg
a ->
h
extends h
and is a homeomorphism since a
'1-1
'1
function on a regular comples is continuous if and only if it is continuous
The resulting mapping
h
In the proof of this theorem >Ie ..!ill use the following le=a.
on each closed cell of the complex.
m'lMA.
3.3.
If
is a ce l -l of a regular complex, then
0
a complex is the join of the vertex
Sd
with the subcomplex
0
as
0,
Sd
a.
a function
h:
[Sd K[-----> [K[.
The collection of mappings
The inverse of
it is continuous on each skeleton of
h
A cell
,
of
Sd
is a collection of faces of
0
be arranged in an increasing order:
Ok =
0
or
,
of the vertex
k
0
then
,
is a cell of
a
0
0
Sd o.
with the cell
is the vertex
If
~
0
1
< .•• < ok'
= a,
< ••• <ok_l'
00
o.
<
o
wlion of subcomplexes
Either
L,
0;
K. The function
then the topology on
",ith respect to the closed subsets
,
then
which can
[Lo;[.
Thus
h
L
[L[
h
is also continuous.
is expressed as a
is the weak topology
is a homeomorphism and
is the join
the proof is complete.
unless
Thus the cells of
Sd
k
0
O.
If
Note that the homeomorphism
h
of
i[ Sd K[
onto
[KI
given in
are those
the proof of 3.2 preserves the cellular structure, in the sense that
of the join of the vertex
0
with the subcomplex
-150­
Sd o.
defines
is continuous because
This follows from the fact that if a regular complex
Proof:
(h}
'1
sq·
h
carries
Sd cr
onto
of the homeomorphism
cr
h
for each cell
K.
of
cr
The construction
involves choosing, f or each cell
omorphism with the Euc lidean ball of the sazne dimension.
call a function
homeomorphi sms if
a)
cell
h
ISdKI. ~ IKI
h:
h
of
K. b)
If
K
a home ­
He shall h- 1
and its inverse
subdivision
satisfies the following two properties:
is a homeomorphism which carries
cr
cr,
Sd ~
onto
cr
for each R2q+3
R2q +l
which c ontains
2q l
R + •
of
K,
h
carries the vertex
of
cr
,
of
Sd K
onto the barycenter Sd cr,
terms of bar yc entric coordinates, into
h
maps
,
One easily embeds each
ISd KI ~ IKI
n
COROLLARY.
such that
IK I
cr
IKI
can be embedded in
K
,
there exists
DEFDUTION.
The
Oth iterated subdivision of a r egular complex
K itself.
The
(n+l)si iterated subdivision of
is
for the
Sd%
nth iterated subdivision of
nth iterated subdivision of
K
is the
is homeomorphic to
Sd K.
If
n
ISd K I by 3.2.
Sd K
Thus
K
is a finite
all of its iterated subdivisions.
Let
Choose a subdivisi on homeomorphism
n
Sd K ~ R
g :
o
He write
K be a finite regular complex.
Sd K,
metric on
then
can be embedded in Euclidean
Sd K.
(It suffices to embe d
n+l
and
K.)
ISd K I,
The embeddings
vertices, where
-1
hlg o
us ~ the metric on
morphisms
~:
Sdk+~ ~ Sd~.
t e rms of the metric on
We metrize
and
~,
hl
n+l
For each
R2n+l.
[Hint:
( Cl+l) -cell
cr
of
-152­
Suppose that
K,
K
~
k > 2,
q
choose a hyperplane
go:
p
In the future we shall always regard a finite complex
cr
Sd K ~ K and of the embedding
in
-153­
h :
l
in
be an iso­
K
and
\{e note that the metriza­
n
is embedded in
is
define a
tion depends upon the choices of the subdivision homeomorphism
can be embedded in
as a
in the obvious
I.Sd~ I,
Sd K by demanding that each
its subdivisions as metrized in this marmer.
Prove that a c olmtab le regular complex of dimension
Sd K
By the remarks preceding 3.4, we have uniquely defined homeo­
metry.
K.
IK I
and an embedding
SdK~K
which is linear, in t e rws of barycentric coordinates,
subcomplex of the standard simplex with
way.
on all of the
cr
is the number of vertices of
Rn ­ l
K.
K.
Rn.
space of dimensi on one less than the number of cells of
R2q+l
cr,
3.5.
h :
l
Given a finite regular complex
can be embedded in
Exerc ise :
.
He nm. describe a way of metrizing a finite regular complex and
satisfying a) and b ) above. simplicial complex, a subcomplex of the simplex
vertices of
P
2q 3
R + .]
the number of cells of
Proof:
in
in
on each closed simplex of
3.4.
cr
Kq+l
cr. Simplicial complex, there is one and only one homeomorphism 8.
except
linearly, in He leave a s an exercise t o the reader the verification that if K
is
P,'s,
is itself a Simplicial complex, then for each simplex and f'or each simplex
cr,
contains
cr
2q
so that cr n R +l
cr
and then one must verify that the result is indeed an embedding of
in
first subdivision of the
of
P
a limit point of the set consisting of all the other
K
cr
in such a way that no
Sd K ~~.
3.6.
Given a metrized finite regular complex
DEFDIITION.
mesh of
K
K,
is the maximum of the diameters of the cells of
vertices
the
K.
°1
and
°1
and
°2 are mapped int o the barycenters of the simplices
°2
To compute the l ength of the edge
r es pectively.
compute the distance b etween the barycenters of
It is obvious that the diameter of a triangle linearly embedded
in Euclidean space is the length of its longest edge .
f ollows
°
that if
space, and
x
is an
and
y
From this it
i
are two points of
°
then ne ither
0,
x
nor
y
0·
Thus
and
°
is the
length of its longest edge.
and so the diameter of
0,
j
E Bi
vector s in
x
k
E Bi
~
and
j+l i=O
lies
in the interior of a straight line segment contained in
must be vertices of
~
such that the distance
Rn ).
3.7.
~
j
j+l i=O
Lim
j+l
K be a fin i te regular c amplex.
k+l i =O
k .
Then
j
E B
-J
Mesh(SdrK) = O.
j
It is clearly sufficient t o show that if
plicial complex of dimension
L
k-j)(~ E B.
(k+l j+l i=O ~
is a finite sim­
linearly embedded in
'1,
n
R ,
then
Mesh(Sd L) < ~1 Mesh L.
- '1+ '
Sd L
homeomorphism
h:
inverse
of
e
e
ISd LI ~ ILl
L
be an isometry.
b e the longest edge of
k
~
q.
are those of
and
°1
assume that the vertices of
°2
as
1
k
E
Bi
- -
1
k
E
B
k+l..
l=J+1 i
1
k
k-j . E B.).
~=j +l ~
1
g-image of
i s the barycenter of
E Bi
01;
the
j+l i =O
.
k
E
Bi is the barycenter of the simple x of
k-J i=j+l is defined by requiring that the subdivision
are the simplices
tice s of
!..-
g -image of
spanned by
c l osed simplex
Let
B.~
j
The inverse
The metric on
int o the pOints
°2
k+l i=j+l
(j+l )(k+l) ~=v
. -.f> i
r-4OO
Proof:
into a
k
E B.
i=O l
1
j
_ _ ) E Bi
( 1
Let
~
The
This vector i s :
1
k+l
EBi
l-simplex.
THEOREM.
A.
(Here we add the
complex linearly embedded in same Euclidean space is the length of its
longest
°2·
we
·The di stance between these tw o points is the length
of the vector joining them.
Therefore the mesh of a finite simplicial
and
°1
respectively.
k+l i=O
and
carries each vertex
and the barycenters of
B ,
n-s1mplex embedded in some Euclidean
b etween them is the diameter of
y
point
n
L ---7 R
g:
linear embedding
°1
e
°1
01
Sd L.
°2
are
of
Suppose that the vertices
L,
with
Ao,~,···,Aj'
t ogether with
Under the subdivision homeomorphism
-154­
h:
01 < 02·
We may
and that the ver-
Aj+l,Aj+2' ••. '~'
[Sd L[~[L!.,
where
the
Aj+l,Aj+2'···'~·
°2
and so the distance between them i s at most
Thus the length of the edge
times the mesh of
These barycenters b oth belong t o the
L.
e
i s at most
Me sh L.
(k-j)/(k+l) ~ k/(k+l) ~ '1/('1+1 )
Thus , by the comments after Definition 3.6, the
proof of 3.7 is complete.
-155­
We are about to prove the main theorem of this section.
a mapping
each
r
f:
K
and
~
s,
into a c ompact subset of the positive real numbers.
Given
of finite regular complexes, we have, for
K'
bounded a:<tIay from zero by some
o<
the following diagram:
<
5
Let
E.
W be a set of
W is contained in t he open
K
f
Thus for some
jg'
division homeomorphisms.
and
g
~
K
and
g'f'g.
is the mapping
f'
We shaJ.l sometimes write
f
We say that
as a mapping of
for
f
f
induces
SdsK
into
-s pace and let
is proper as a map of
The complex
Proof :
f
l
a>
Then there exists a number
covering
- l( St A.)'*1
i
For each
x
X
where
be a compact metric
X.
X of diameter less than
a
l
E
X and every
i,
let
d(x,i)
contained in
x
Ui ·
be the radius of
Aj
for the covering
t
than
s > N,
f:
Given
5.
c omplex of
a
Then
d(x)
is finite for each
d:
~
R
x
since
X is bounded.
We sbow that
a
is continuous.
dey ) I <
is positive for e ach
d( X).
:::
and
K' ,
f
a
a positive integer.
r
s > N,
such that for each
the map
into
SdrK '.
of open sets
SdrK '. Since
varies over the vertices of
"\,e
t
may by 3.8 choose a Lebesgue number
N be such that the mesh of
SdNK
K
°
is less
we show that the minimal carrier for
is acyclic .
We note first that the smallest sub-
X is the union of
containing a given point set
fa
Thus
E
x,
Letting
whenever
p be the metric on
p(x, y) <
since the
-156­
U.
l
E.
cover
X,
d(x~
X.
X
Thus
d
maps
'r A
of
c;: StA.
BdrK'
'*
1:
which have a non-trivial inter­
be a cell (open simplex ) of
~,
1)
If
so that
fa
aC f
-1
We can thus write1:
opposite
contai ning
StA
A:
fa is
~
SdsK.
Then the diameter of
for some vertex
~
meets a cell
we
Next we note that
SdrK '
X.
is less than
vertex of
!d(X)
SdrK'
section with
Let
i
have
N
SdsK
Let
SdSK ~ SdrK'
Define
= l.u.b . d(x,i).
X
€
be finite regular complexes,
K into
the closures of those cells of
the largest open ball around
d(x)
<
"I.
U..
is contained in some
Proof:
8
€
is a Lebesgue number of the
5
is covered by the collection K
J
call ed a Lebesgue number of the
0,
such that every set of
(U ),
Let
be an arbitrarily
indexed open covering of
­
(U.)
But
SdrK '.
is a compact metric space,
(Lebesgue Covering Lemma ).
LEMMA.
x.
x
f'
prove the following lemma from general topology.
3.8.
about
V
Choose
f'
Before stating our theorem we
f'.
So
K'
continuous function mapping
3.4. ) The mapping
f
or that
0,
such that
i
Then there exists an integer
f
<
are the iterations of sub­
(See comments before
is defined by the relation
a
of diameter
o-ball
wCvCU..
l
i,
3.9. THEOREM.
g'
X
Choose any
is
(U ) .
covering
f ' ) SdrK'
In this diagram the mappings
> O.
d (X)
K'
)
gl
Sdl K
€
So
of
SarK',
as the join of
A of
then
A
r
Sd K' .
is a
A with the face
A· ~A' Thus the smallest subcomplex of
U1:
fan.--/4J U A
~
0
~A
A
0
Ij
~
1:A ,
Since the
I f A is a vertex of a regular complex K, the ( open) star of A,
written st A, is the union of all (open) cells of K having A as a
vertex ,
join of a vertex and a complex is acyclic , by V.2.5, the smallest subcomplex containing
fa
minimal carrier for
f
is acycli c for each
i s acyclic, and
f
a
of
SdsK.
4.
The Indu.c ed Homomorphisms and the Invariance Theorem.
Thus the
In this section
i s proper.
describe the process of associating with each
we
continuous mapping of one finite regular complex into another a
sequence of homomorphisms of the homology groups of the complexes,
and similarly for cohomology.
We have already shown that a proper
Thus Theorem 3.9, which asso­
mapping induces such homomorphisms.
ciates with an arbitrary mapping of a finite regular complex a proper
mapping, will play an essential role in all that follows.
4.1.
Each continuous mapping
THEOREM.
(K,L) ~ (K' ,L')
f:
pairs of oriented finite regular complexes induces
f*:
Hq (K , L·, G)
for each
a)
each
Hq
(K'' L'·, G)
and
*:
f
homomo~hisms
Hq(K' ,L'; G) ~ Hq(K,L; G)
q, such that the following conditions hold:
If
f:
Hq(K,L; G)
(K,L) ~ (K,L)
is the identity mapping, then, for
Hq(K,L; G) ~ Hq(K,Lj G)
q, f*:
b)
~
of
q
f*:
and
H ( K, L; G) ----;.
are the identity isomorphisms.
If
f:
(K,L) ----;. (K',L')
and
g:
(K',L') ~ (K",L")
are arbitrary continuous _mappings, then, for each
(gf)*
g*f*:
q,
Hq(K,L; G) ~ Hq(KI,L"; G)
and
(gf)*
Pr oof:
Let
A
n
*
f g
*=
q
q
H ( K" ,L"; G) ----;. H ( K, Lj G)
be the statement that the theorem holds for all
finite complexes of dimension at most
s t atement:
-158­
dimension of
If
n.
Let
Bn
be the following
a is a cell of an oriented regular complex and the
a is at most
n, then
159
a
is an acyclic complex.
Then
EO' AO'
and
Bl
An _l ~ Bn"
Let
0 be a cell of a regular complex and let
dim 0
m < n
Then the dimension of 0 is < n-l. He have a
m-l
.
f: o
~ S
,where Sm-l is the regular complex
homeomorphism
on the
are all obvious.
em- l) sphere given in Section 1.2.
Then by
from
B
n
4.2.
LEMMA.
~s
at most
n
are induced homomorphisms f *: H (0) ~ H (Sm- l)
q
q
l
(f) -)E- : Hq\ISm-l) ~ Hq (0) for each q. From b ) and a ) ~~ derive
\
a)
g
and
b)
g*
induced by
-1
flC_(f
Similarly,
(f-
l
f* ~
)-)(_
(ft'
-1)
c)
*
Thus
{z
f*
if
0
q
g:
g
and
g
are proper maps.
*
, the homology and cohomology homomorphisms
given by 2.5, are independent of the choice of
g
g*
-1
g*
and
is
By 3.3 and V.2.5, if
or
0
m-l
otherwise.
then
Sda
cell
o of
proper.
is acyclic.
Tbe minimal carrier
K into the subcomplex
Next we show that
SdK ,then
C (a) ~ C lea) ---;;. ... ----;. C (a)
m
m0
a
1eqUalit y
is mapped by
m- l( 0) ---;;. ... ---;;. C0 (cr )
C
~_1 ( 0) ""Z
cell of
o.
is generated by
The cycle
0
Z l ea) =
m-
dO f
0
J
so trmt
dO actually generates Zm_l(a)
Hq (a)
zm­ l( 0)
=
0
Bn
that
is zero for
o
1
<
q _<
m-2
J
~(a)
=0
so that
by the above diagram.
160
q
Thus
If
0
is acyclic.
g
B
~
J
and
All of
~
B and by
g.
This proves b).
Now let
and
'P2
Let
0
of
q
0
is
K containing
go
o
q
n , we have from
X_
- ~
for
g
is
for
g
-1
and
B
are
it follows that the homoloBY and
of the choice of
respectively.
is
We have proved a).
cohomology homomorphisms induced by
'Pl
-1
The vertex
Thus the minimal carrier
is proper.
g
0 is a cell of
is a cell of dimension at most
q
maps a
K which can be arranged
0 < 0 < ... < 0q
1
0
i ndependent of the choice of
X
2
SdK.
of
, so the smallest subcomplex of
Since
0q
g
into an interior point of
g
-1
~ for
is also proper.
g
Since the minimal carriers
is acyclic.
rle first derive some preliminary results, 'wh ich follow
q
o
acyclic and so
dO has coefficient + 1 on each (m-l)
He have therefore shown that
Bn ~An
He have
This implies f irst that
and secondly that
11
-m- lea)
o.
is
Sda
is a collection of cells of
a
mapped into
jequality
C (a)
m
is a cell of a regular complex,
a
in an increasing sequence:
Here
g
are isomorphisms.
is an isomorphism so that
Consider the following diagram:
z ""
(K, L) is a subdivision
(SdK, SaL) ~
1 .
1 _)E-
Proof:
1 .
H (0) ""
q
)-)(
and if
K
homeomorphism, then
A I ' there
n-
and that
If the dimension of the oriented regular complex
g
-1
are independent
be proper chain maps carried by
0 be a cell of
161
K.
Then
cp]O
~
and
is a chain
L: aTT
~(a) = SdG
on
For each cell
a
closed cell contained in
The carrier
X
acyclic by
En
1.
X
Since
X
-1
)*
=1
~2~1
is a
(gr, s)
a
is a chain on
(g r,s
a is
to
and also the identity chain map
If
>
r
*
>* = (gr,r+l) * ( gr+l,r+2) *
This of course implies that
~2~1 = 1 (g-l)*g*
and
a
be a cell of
ao <O·l < •.. <aq
~2(a)
(gr,s)*
1 .
SdK.
a
a
If
of cells of
is a chain on
K
For each cell
q
x2 (a)
then we have
T
of
a
Gq
*
*
*
(gr,r-l) (gr-l,r-2)
..• (gs+l,s)
*
Proof :
q
is then carried by the acyclic carrier which sends each cell
is a chain on
minima l carrier
X
for
acyclic by 4.3.
Thus
IT
Qr,s
is proper .
map carried by the minimal carrier
< aq
SdK
~lcp2 ~ 1
map, and so
g *(g -1)*
of
1
Thus
into
by 2 .4.
g*
and
SdG
But so is the identity chain
q
It follows that
g*
(g
-1
then
)*g*
=1
and
K
If
K
of dimension at most
a
n, then
Bn .
Sdra
For
-) "'" H (Sdr+l-)
H ( Sd r a
a , by Lemma 4 .2.
q
r
>
q
for
g.
a, and so is
is a proper chain
. l' r
l,l+
Sds -ra .
r
~
is acyclic for every
0 ,and
Thus
Sdra
q
>
r
arbitrary,
Also, by Corollary 2.5 and the exercise at the end of Chapter II,
is acyclic for
For each pair of no=egative integers
g
r,s
we may derive analogously that
rand
for all
r, s
If
--
K
Now suppose that
(SdrK, SdrL) ~ (SdsK, SdsL)
If
is of dimension at most
r < s , then
n, then
r
gr,s
g
r,s
is proper
Each
gi,i -l ' for s+l
>
s.
)*
We have
gs+l,s gs+2,s +1 ... gr,r-l .
< i < r , maps cells into (open) cells since
163 162
(gs-l,s
s
obtained by composinG subdivision homeomorphisms.
rn,·u-1A •
By Corollary 2 .5,
(gS-l,S)* ... (gr,r+l)*
0
( gr , s) * = (gr r+l) *
NOTATION.
< i < s -l ,
-­
(~s-l)* ..• (CJl)*
0
we have a homeomorphism
4 .5.
cpi
s-r-
(cpS-l~s-2 ••. CJlr )*
,
4 .4.
If
The
and by Theorem 11.5.2,
a is a cell of
is a regular complex and
is acyclic by
l
is a chain on
(gr,s)*
COROLIARY.
Proof:
~r(a)
X.
Sd
SdrK
are isomorphisms and the lemma
is proved.
4 .3.
CJl _ CJl _
s I s 2
a to
sends
gr,s
SdG
q
a be a cell of
r < s , and let
Suppose that
,we have
~1~2
cpl~?(a)
Consequently,
( gr,r-l )*
(gS+1,s)*(gs+2,s+1)*
Thus
= Sd-:r C SdGq
O
)*
is the collection
Xl (T)
a <
(gs-l,s
s , then
( g r,s)
Let
(gr,r+l)*
(gS-1,s)*(G S -2,S-1) *
does not increase dimension, we have from Corollary
2.7 the fact that
g* ( g
saG, X (T)
2
sendinG each cell a
K
carries
of
~2~1(a )
Thus
K to
from
T
it is a subdivision homeomorphism.
and so is proper by
SdrK
Thus
More explicitly, let
B
n
There are uniquely determined cells
with
0"
0"
r
The minimal carrier
1
for
X.
1
sends each face of
then
qli(c)
that
qls+lqls+2 .•. qlr 0"
g
r,s
relatiou ~
go,s
i
Sd K , s < i < r
in
sends
gi,i-l
is a chain on
Xi' and
(g
is a chain on
r,s
)*
to
O"i_l.
Thus if
O"i_l
c
Also ,
is a chain on
CPi
O"i
From this it follows immediately
0i_l
0"
carries the proper chain map
.'.nvolving
O"i
In this diagram
Thus the minimal carrier
S
ql
s+
(g
)*
r,s
and
ql
1
•
r
The desired
morphisms and
g o ,..s
and
,
f'
f
gr,o
f'
> (SdrK', SdrL')
COROLLARY.
g'
r, o
Go,s
are iterated subdivision homeo'
go,s' fT , g r,o
.
The mapp1ngs
ar e
all proper, and vie make the following definition.
4 . 7.
DEFnUTION.
If
f:
(K,L) ----?> (K' ,L')
regular complexes- of dimension at most
cohomolo~
is a mappinG of finite
n, then the homology and
f , wTitten
homomorphisms induced by
f*
and
f
*
respectively, are the compositions
follow immediately.
f*
4.6.
i g~, o
1
t SdSK, sasL)
s+l < i < r .
for
to a subcomplex of
0".
1
is a proper chain map carried by
for
0".
1
be a cell of
0"
> (K',L' )
f
(K,L)
and
(g 1,1. . 1)0"·1 C
- 0".1­ 1
X.
maps cells into cells
gr, s
Given nonnegative integers
r, s ,
and
= (G~,o )*(f ·)*( go ,s)*
t,
and
(gr,t)*
(gs,t)*(gr,s)*
f
and
*
( go , s) * (f') * (g~,o) * .
In order to justify this definition, we prove:
(g
Proof:
r,t
)*
(gr,s) * (gs,t) *
This corollary follows immediately frcrm Lemma 4.5 and the
fact that
[(gi+l,i)*]-l , proved in Lemma 4.2.
(gi,i+l)*
i·le proceed with the proof that
(K',L')
B
n
==9A
n
Let
(K,L) ----?>
f:
< n
Let
r
be a positive integer.
3.9, there exists an integer N such that for
as a mapping of
(SdsK, SdsL)
into
the following diagram:
164
Then by Theorem
s > N
(SdrK', SdrL') .
f
of
rand
Proof:
be an arbitrary mapping of pairs of finite regular complexes
of dimension
LEMMA . The induced homomorphisms
4 .8 .
Let
such that
into
and
f*
are independent
s
r
be given and suppose
f , as a mapping of both
(SdrK', SdrL') , is proper.
are induced by
saud
t , with
(SdsK, SdsL)
and
In the diagram below,
f :
is proper
165
Thus we have
f*
s < t , are
(SdtK, SdtL)
fO
and
fl
(K,L)
go,s
1
Thus
=
(g 0, t)*' and similarly for cohomology.
(fO) *
=
(gs,t) * (f l ) * .
Let
cr be a cell of
for
fO' fl
Then
~la
fOa , so
XO' ~ be the minimal carriers for
and let
~
~, ~O' ~l
If
Sdt-s
If
T €
on
Xo ( cr)
Sd s K , then
Sdt-s Ci , and
is a chain on
a , then
and so
gs,t' fo ' and
be proper chain maps carried by
i s a cell of
a
fl -r:
~l cP
,0
i g~"r
(Sdr' K', Sdr'L')
(f O)*
= (g~, ,r) * (f l )*
( fO ) *
=
CPl cpa
~ fo
X()
a
a~ Xo( cr)
Thus
1"
Sd sK .
~(a)
is a chain on
Sd
X ' ~
O
be the minimal carriers
CPO' CPl proper chain maps carried by XO'
and
rr_r
Let
Xo(a)
. xO (cr)
~
.
is a subcomplex containing
is a subcomplex of
Sd
r'
K'
containing
flcr
Thus
X, Xo ' and
Sd t-sa
i s a chain on
fl '
€
U
Sdt-s cr
CP1CPcr
Xl ( cr) ~ Sd
cpa
So
~ (1")
is a chain
Thus
CPCPl cr
if
cP
r '-r
Xo ( cr) ,and
g , r ' ,r (~ ( cr» ~ Xo ( cr) .
is a chain map carried by the minimal car:rier for
is a chain on
equalities follow .
The desired equalities follow.
:::: CPO .
(f l ) * (g~, ,r) *
(fl)*(gs,t)*
and that
x,
gr'
and
(f O)*
Let
~
By 4.6, we need only prove that
need prove only that
we
(SdrK ', SdrL'»)
~
(SdrK', SdrL')
/r1
(Sd t K, Sd,t L )
By 4.6, (g s, t)*(g o,s )*
f
(SdsK, SdsL)
~
gs ,t
\
r,o
go,s
f
~
(SdsK, Sd s L )
go,t
1
i g~,o
1
jg.
> (K',L')
I
(K,L )
> (K' ,L')
f
Xo ( cr)
Consequently
CPo ::::
~l
" ' ,r
Dr'
' and the des ired
The proof of the l emma is complete.
We show now that the induced homomorphism satisfies properties
Let
rand
r'
be given,
~~th
r < r' , and suppose
s
is
a) and b ) of the theorem.
chosen so that
f
is proper as a mapping of
r
r
(Sd K', Sd L')
and into
(Sd
r'
K', Sd
r'
(SdsK, SdsL)
To prove a) we note that the identity map
into
is proper by
L').
En
If
f:
(K,L)
~
(K',L')
is an arbitrary pr oper
In the diagram below,
mapp ing of finite regular complexes of dimensi on
f O and
fl
a re induced by
< n , then by an
f :
argument similar to the proof of Lemma 4.8, the induced homomorphisms
166
167
,
f*
f*
and
given by Definition 4.7 coincide with those previously
defined (Corollary 2.6) for
f
as a proper mapping.
of
Property a)
(K",L")
< n.
B and so
X (-r) Cst B.
2
gl(~) , can be written in the form
(K,L) ~ (K' ,L ' )
f:
and
(K' ,L')
g:
~
plex of
SdrK"
Let
r
be a positive integer.
SdrK"
A
is a vertex of
such that
We subdivide
(K',L')
depending on
B 0 M~
~ E ~ (0)
SdsK ' , then there is a vertex
g(St A) C St B.
For, if' 5
B
B 0 M
~
B
Thus
-r .
a U M.r
~ E ~ (0)
S
Sdr ](" ) ,
If
s
t>1esh(Sd K') <
so large that
order to ensure that stars of verti_c es 01'
is a subcom­
~(~)
U
is an acyclic sub complex of
st -B
U
X2(~).
~ E ~ (0)
4
then it suffices to take
~
is a Lebesgue
X ( 0)
a vertex of
1>1
We define
number of the covering
{g -1 (st B) IB
-where
T E ~ (0)
U
so
As in the proof
, which is the minimal Subcolllplex containing
be mappings of pairs of finite regular complexes of dimension
finely that if
of
C st
~ ( ~)
of Theorem 3.9 ,_
follows as a special case.
To prove b), let
~(o), g'(~)
SdsK'
~. In
a ' < a then
~(a')
C >S.(a)
is an acyclic carrier.
Since
is a chain on
So
are acyclic , we
X (a).
4
~la
X4
x4 (a') C x4 ( a)
so
is a chain on
.
~
Thus
~ (a), ~2~1(a)
is an acyclic carrier for
~2~1 .
But
s > 1.
also require that
is a cell of
Then we take
SdtK , there is a vertex
t
and
g'
are the mapping;'! induced by
f
1
g's,o
glf'(a)
such that
and
respectively:
g
(K" ,L")
11
and
g'f'
are all proper.
~,X2'
and
X ,
carries a proper chain map
~i
a cello
Xl (0)
C st
of
Sdt K, f' a
A , and
for some vertex
B
~l 0
of
Xi
C st
A for some vertex
is a chain on
r
Sd K"
U gl(~)
~ (a)
C
~ ( 0)
X
A
But
of
~
E
~ (a) X2( ~)
X4
and so we have
CP2~1::: ~3
CP2~1
and
X ( a)
4
CP3 are both carried by
Consequently,
and
(g'f'
)*
(f') * (gl) * •
1 .
Thus
Given
Sd s K' .
g'(St A)
4 carries g'f' . Thus
(g~,s)*(g~,o)*
3
Thus
C st
This implies that for each cell
168
E
Fr om Lemma 4.5 and the proof of Lemma 4.2,
f', g'
Then each
~
(g'f ')* = (g')-j'(f ' )*
Arguing as in Theorem 3.9, 'we show easily that
Let their minimal carriers be
so
1g~,o
g'
'a,s
(Sdt k , Sd t L ) ~ (SdSK', SdSL') ~ (SdrK", SdrL")
respectively.
C gl~(O)
In the diagram below,
(K' ,L')
(K,L)
go,t
SdsK'
A of
faC St A , as in the proof of Theorem 3.9.
f'
a
so large that if
~
B
g*f*
(g~,O)*(g')*(g~,S)*(g~,O)*(fl)*(go,t)*
(g~,o)*(g' )*(f' )*(go,t)*
(gr" , o)*(g'f')*(go ,t)*
(gf )* •
169
.,.. .,..
A similar argument shol{s that
(gf )'"
He have t hu s shol{l1 that
A
n
morphisms lead to the same result.
(K,L) --.;;. (K' ,L')
then for
nand
n'
Thus
is true for all
n
the processes used f or different
f':
f g
> An
B
n
n
We note that
in defining the induced homo­
is a mapping of finite regular complexe s,
both greater than the dimension s of the com­
A
n
and
An'
coincide.
Also, the properties a} and b) follm{ since in anyone instance vre
may take
n
large enough and apply
A
Theorem 4.1 is proved.
n
It is important for later considerations to note that in our
proof ve .d o not assume Lemma I1.5.3.
that
K
and
K'
K
and
K'
Stated another
our proof yields the fact
(see belo\{) t hat their homology groups are isomorphic.
This is one
,ray to prove, incidentally, that homology groups are independent of
orientation!
regular complexes, t h en
q
(The Invariance Theorem).
If
and
homomorphism.
4 .10 .
H (K
f *:
Hq(K' ,L'; G)
L'
G)
~
f:
"
H (K'
q"
L"
Thus
f*
=
(ff
-1
)*
=
is an isomorphism.
1*
=
K'
induces
into those of
K'
Similarly, ,ore have hOmOmOIllhisms
f*, (fIK)*, (fIL).:,
,re
which
(f IL)*
defines a
(K,L)
have
The exact homolog y and cohomology sequences of' a
~
Hq(K,L; G) •
1 .
Similarly,
By similar reasQning,
(f
(Reduced homology groups.)
uni qUe map
f:
~ (K) ~
g:
H* ( Kl )
L
f*
= ker
H.*(K)
is a continuous map, then
--.;;.
H* (IS: ) .
K, the
is induced by
Hq (K) _1*
f?:.
o·
to
Shovr that if
i ndu ces a map
Shov that the sequence beloy; is exact,
K
~ Hq (L) ~ Iiq ( K) ~
i:
L
c: K,
Hq (K , L)
3'*
> Hq (K,L) ,,-,here
j:
g*
> Hq-l (L)---->
is giVen by the compos it ion
K r'::., (K,L)
is the inclusion,
Cl* is i n duced by the boundary homomorphism of t he ordinary
h omology sequence.
-1)
Set
is a subcomplex of
i*
For each comple x
K --.;;. point induces homomorphisms
H*(point) .
~ y~
Kl
Hq (K) :,.r
G)
Thus t he h omology and coh omology groups are topological invariants.
-1
f*(f )*
Thus
COROLLARY.
Ex e rcise.
and
Proof:
into
pair are topological i nvari ants .
Here
q
( f IK)*
K
K
(K' ,L') , using Corollary 2.5 and the dsfinition of- the induced
(K,L) ~ (K',L')
f:
i-l~have
*
as a mapping of
of oriented f inite
homomorphism of the exact homology sequences associated Mith
is a homeomorphism of oriented finite regular complexes, then for
each
f
I t is easy to see that the triple
g* :
(See Chapter VIII, Section 4 . )
COROLLARY.
(K,L) ~ ( K ' ,.L ' )
f:
homomorphisms of the homology groups of
if
4.9.
a re g iven a map
He only assume, in effect,
satisfy the redundant restrictions .
'.fay, no matter hoy vre orient
Vie
we shall writ e
),jore precisely , if
plexes, the induced homomorphisms given by
If
Hq (K, L)
~
One mu st sho'.-I' that the composition
II q- l(L) ----> Hq- 1- (point)
More generally, let
t:
is zero.
be a category vrith a m2.xilrlal object
*f* =' 1 .
sati s f'yirlg
is also.
(i )
For each object
170
morphism
f'A
E
A
€
r:: ,
M(A, p) .
171
there is one and only one
P ,
(ii )
then
fB
0
A and
f
fA
?; -.;>
F:
Let
If
a
B are objects of
and
f
€
5.
11(A,B) ,
be a covariant functor from
<::
to the category
In this section we state the seven basic properties of cellular homology
peA)
f
€
peA)
M(A,B) , since
into
a sequence of homology groups and t o each continuous mapping of pairs
of regular complexes induced homomor phisms of the homology groups. *
Define the reduced functor
by setting
If
The Properties of Cellular Homology Theory.
Cellular homology theor'J assigns to each pair of regular complexes
of abelian groups and homomorphisms.
F
r:: ,
fB
Thus
F(B)
ker F(fA)
we
A
€
theory .
<::.
fA' it follo~~ that
f
0
for
It turns out that these properties charac.terize cellular
homology theor'J.
F(f)
maps
A description of the precise sense in 'iThich this
is true and a proof of the characterization may be f ound in Foundations
of Algebrai c Topology. by Eilenberg and Steenrod.
may define
He keep the coeffi cient group
F(f)
for
to
f
~
functor
€
l1(A,B)
Verify that
=
fixed throughout this section.
G
F(f)lker F(f )
A
F
is a covariant functor from
~
I.
If
f:
( K, L)
-.;>
(K,L)
is the identity, then
£'
*
is the
ident ity isomorphism in each dimension.
Carry out an analogous definition for a contravariant
F:
t:
-.;>
~,
II.
using cokernels.
f:
If
(K,L)
-.;>
(K' ,L')
and
arbitra.r y continuous mappings, then
Ill.
If
Let
~*
f:
and
(K,L)
O~
-.;>
=
(fIL)*O*
Corollary
IV .
172
-.;>
(KI! ,LI!)
are
g*£'*.
be an arbitrary continuous mapping.
(K,L)
and
(K',L'), respectively, then
Property III follows from the remarks after
4.9.
The homology sequence of a pair is exact.
proved in Chapter V.
*
(gf)*
(K' ,L')
are the boundary homomorphisms of the exact homology
s equences associated with
O_~f*
(K' ,L')
g:
Property IV
~~s
We note that properties II and III imply that
We restrict ourselves in t h is section to finite regular complexes.
We will use s ingular homology theory in the next chapter t o extend
our results t o infinite complexes.
173
It is easy to show that homotopy equivalence is an equivalence relation.
the induced homomorphism is actually a homomorphism of the exact
property which, "hen possessed by an arbitrary space
homology sequences.
sessed by all spaces homotopically equivalent to
V.
If
then
f
and
are homotopic mappings of
g
(K,L)
(K '/ L') ,
into
f* = g* .
Let
PO and
Pl
mapping
K, into
I XK
be the embeddings defined in the first paragraph of Section 2.
and
Thus
Pl
Po
Then
are proper and their minimal carriers preserve dimension.
and.
Pl
CPo
and
Note that
Section 2.
Properties I, II, and V imply that homology
induce unique chain maps
and
CPo
CPl
Now, since
f
and
then
f *:
If
(K, L) ~ (K' ,L')
f:
Hq (K,L; G) "'" Hq (K' ,L ' j G)
= F*(PO)*
(Pl)*.
(po )-*
J!)(c)
for
d'
Consequently,
of
CPo
and
f)
f) :
and
If
e(I X K)
J9 (c)
~
Cq+l (I
d and
Let
g
be a homotopy inverse for
I
=
CPlc - CPOc -
c - (5
d'
X
K)
Simil arly
by
are the boundary
(po)*
(Pl)*
(8)
c - I
(8)
~ (dc)
f*g*
1.
f*
invariance under excision.
By p r operties II,
f
1 .
1_)(,
is called an excision if
•
VI.
If
i*:
Hq (K,Lj G) "'" Hq (K',L'; G)
i:
(X,A)
and
K
L
~
(K,L) ~ (K',L')
Proof of Property VI:
and our proof is complete.
is an isomorphism.
An inclusion mapping
dC
Two pairs of topological spaces
DEFINITION.
(gf )-;:_
Thus
i,
(K,L) ~ (K',L')
K' - L' .
is an excision, then
for each q .
It is only necessary to observe that if an
orientation is chosen for
5.1.
Homotopically
The sixth property of cellular homology theory is called
is a chain deformation exhibiting a chain homotopy
CPl • Thus
q.
V, and I, in that order, we have
g*f*
respectively, we have
=
(8)
for each
' and we must show that
eq(K)
eq (K)
C E
e(K)
= F*(Pl)*
g*
He define
I (8) c
operators for
and
is a homotopy equivalence
e quivalent complexes hav e isomorphic homology groups.
Proof :
are homotopic, there is a mapping
g
THEORElvI.
respect.ively.
are the chain maps defined at the start of
CPl
5.2.
Fpl
f*
X, i s called a
groups are homotopy invariants.
Proof of Property V:
Po
homotopy invariant.
X, is pos­
K, inducing orientations on
L, K' ,
(Y,B)
and
L' , then
C(K,L)
e(K ' ,L') •
are said to be homotopically equivalent if there exist mappings
f:
(X,A) ~ (Y,B)
(X,A)
F:
for
and
fg ~ 1
and
on
g:
The map
homotopy inverse
f
such that
gf ~ 1
on
VII.
If
K is a complex wh.ose only cell is a single vertex, then
This means we can find a homot opy
(Y,B)
(I X X, I X A) ~ (X,A)
(Y,B) .
(Y,B) ~ (X,A)
of
fg
and the identity, and similarly
Hq (K)
""
[G0
is called a homotopr equivalenc e with
175
g .
174
if
q=O
otherwise.
A
Thus, a point is an ac yclic complex.
Chapter VII.
Exercise :
SDWULAR HQIvlOWGY THEORY
1.
state and prove the corresponding seven basic properties
Direct Spectra.
Up until now, we have defined h=ology groups only for regular
of cellula r cohomology theory.
complexe s and induced hom=orphisms only for mappings of £inite
regular complexes.
The process of direct limit, introduced in this
section, will enable us to extend our domain of definitions over
the entire category of topological spaces and continuous mappings.
"He will define the singular h=ology groups of a pair of spaces
(X, A)
in terms of mappings of pairs of oriented finite regular com­
plexes
(K,L)
into
with each mapping
of
(K,L),
(X, A).
(K,L) ~ (X, A)
f:
the
qth homology group
and the collection of these groups will form a direct
spectrum of groups.
The limit group of this direct spectrum will
be by definition the
DEFD~ITION.
1.1.
l-lore specifically, we will associate
qth singular homology group
of the pair
A directed set (class) is a set (class)
gether with a binary relation
< on A.
The relation
<
(X,A).
A to­
is re­
quired to be reflexive and transitive, and to satisfy the following
property:
For any
such that
m< p
1.2.
and
DEFINITION.
~ = ({G }, A,< ),
a
m and
S p.
in
A there is an element
The ordering
{G}
a
a
<
~
(ii)
(a,~)
For each pair
(A,S),
G
a
into
Whenever
A
is said to direct
A.
G~ ;
satisfying the following
of elements of
there is given a nonempty collection
mapping
176
(i)
€
is a collection of abelian groups in-
dexed by the directed set ( or class)
conditions:
<
p
A direct spectrum of abelian groups is a triple
where
-
n
n
{
a~~}
A such that
of hom=orphisms
these h=omorphisms are called admiSSible;
a~~: Ga--;;"G~
and
177
a~,.: G~--;;"G,.
are
i
If
r
°a!3
J
and
are admissible, and
°a!3
k
and an admissible homomorphism
i
j
admissible, then so is t he c omposition
G.
(iii)
G
g EGa'
then there exists a
a
k
such that
°!3r
~
°!3rOa !3 :
r
i
'
k
j
g
°!3roa !3 •
°!3roa!3g
Then
g2Rg3 '
and
and
81
have a common successor
g3
the common ancestor
g2'
for any pair
°a!3 Note ;
./:J =
of some
o~!3g
8
is called a direct s ystem of abelian groups .
He do not re'luire in this case that
Let
g
(a, !3 )
a
G,
a
­
°a,a
be a direct spectrum.
Given an e lement
and an admissible homomorphism
~-successor
is a
is a
([ G }, A, < )
be t he i dentity.
~-ancestor
° i A'
we say t hat
a~
( or successor, for short) of
g,
and that
( or ancestor, f or short) of
g
°a!3 •
that two e l ements of the same group which have a common ancestor
common ancestor
g~
E
j
G •
r
and
G 's
for s ome
both elements of
gl
and
5.
GE>'
The elements
By conditions
Ga
g2
84'
R
g2
have t he
G!3
E
J g
for s ome
°a!3 3
=
k
i
°aE-° ,erg 3
(ii'
k' j
and
The element
g6
is a common successor
we den ote the
R-equivalence class to which
gl
{gl) •
Given a direct spectrum of abelian groups
DEFn~ITION .
({G } ,A,< ),
a
­
the limit group of
~
is the abelian group whose
R defined
elements are the e'luivalence classes of the relation
Gi ven two equivalence
Addition is defined as follows.
classes
{gl}
and
{g2 }
represented by
there exist admissible homomorphisms
{gl} + {g2}
i
°a?,
{O~rgl + O~?,g2} '
gl E Ga
L~
o~r
and
8 2 E G!3 '
for some
r·
where the addition in
the brackets on the right is the addition in
is written
and
G ,
r
The limit group
Ga ,
k
To justify this definition, one must show that addition as
are
g
and (iii), these elements
g4 is a common successor
def ined is independent of the various choic es made and gives a
group structure on the
R-e'luivalence classes.
an exercise to the reader.
This we leave as
'de note that the zero of
Lim G
a
a
is the
G'S.
a
on the set theoretic disjoint uni on
Given
and
Given an
i .s an equivalence relation.
R
°a~
°!350r!3 3
gl E G
a
and
g2
€
G ,
13
gl and 8 2 have a common success or.
gl' g2'
g5 have
Now, since the indexing set
But then
clearly reflexive and symmetric.
suppose elements
1; =
g6'
R-e'luivalence class containing all the zeroes of the
as follows.
if and only if'
belongs by
1. 3.
Thus
g3'
gl EGa'
We define
g2'
He define a relation
of the
j
ora' °r!3'
have a common successor
of
and
is directed, there are admissible homomorphisms
a
k'
°!35
gl E Ga
- i '"
gl = Gra°3'
Thus
i
admissible homomorphisms
of the
Let
element
above.
He note first that condition (iii) of 1. 2 states
must have a common successor.
and
gl
and
g4
g2
i
i-re show that any tiW elements having a common ancestor have a
c ommon successor.
of
But then
g5'
and
g4'
so by what we proved above, t.hey must
have a common successor
A direct spectrum which has at most one admissible homomorphism
have a common successor
g2
To show that
g3
R
then
Then
8
R
1
Rg2
is
is transitive,
are given such that
g Rg
l
2
We prove now a theorem which gives an interesting application
of the direct limit process.
r e gular complex.
Let
~
Let
K be an arbitrary oriented
be the collection of finite subcomplexes
and
179
178
of
ordered by inclusion.
K,
be a fixed integer.
q
Then
t
is a directed set.
lies on some finite subcomplex
Let
We define a direct spectrum of groups by
L'
containing
L.
iole then have
=0
i(L,L')u
setting
G(L)
iolhenever
H (L)
L \: L I ,
i(L,L'
q
»)
= Hq (L)
for each
L
E
and so
~.
U
The collection
q
inclusion homomorphisms
(i(L, L' »)
together with the
(H (L»)
q
forms a direct system of abelian
groups.
1. 4.
of
For each q,
q
(K ).
O.
If
Thus
z
qJ
z
To see that
is a cycle representing
as a cycle on
DEFJNITION.
direct spectra.
Lim H (L ) "" H (K).
LE t q
q
is one-one.
L.
L,
= u.
qJ(u')
JJ
Let
Let
The collection
He define a homomorphism
whenever
qJ:
An element
(u)
Lim H (L) ~ H (K) .
LE ~
q
q
of Lim H (L)
L€ t
L C K.
some
qJ(u)
=
i*u.
is represented by
To show that
qJ
L C K by
o E Hq(Lt) is another representative of (u).
L"
It follm.'s t.hat if
such that
i ':
H (L)
€
L'
for
f:
= ((Ga),A,:s:)
A
~
Thus
qJ
is an isomorphism and ]..l
((H(:)) ,B,:s: ' )
=
i(L,L")(u)
i,
collection
be an order preserving function. B
C K and i":
LU
E
H (L").
q
C
- K, then
Thus if
i!. Uo
Suppose
qJ(
(u)
z
= O.
is well defined.
T'nen
z
bounds in
K.
Suppose
Ga "
E
then
a I(Y )
qJ
is an
.1f ----:;..']1
qJ:
to indicate that the 'JJ = ((H(:)) , B, :s:')
(B,:s:' )
(2) for all
=
((G)
e ,A,<)
x-
if (1)
aEA,
is contained
(A, :s:)
Ga=Hex'
is a subset
and (3)
- admissible homomorphism is an 'JI-admisSible homomorphism.
J;
is contained in
U €
z
If
is a homomorphism.
qJ
H (L)
such that
2x:.
The chain
q
)J
c
J;
W,
is contained in)J,
the inclusion ) J \:
J.J is a
we say that
,J;
is cofinal in
if the following two conditions are satisfied :
(a)
~-admiSSible
180
y
homomorphism of direct spectra.
It is clear that
is a cycle of a homology class
#(l,-successor
(qJ)
in the direct spectTUIIl
evexy )
qJ
a ~ Hf(a)"
G
qJa(Y).
(or subclass) of
i*u = i~ i(L,Ln)( u) = i~ i(L',L" )(u )
O
and so
has a
He say that the direct spectrum ;U
L
Then there is a
o
qJa:
is a homomorphism. It is clear that a homomorphism a
of direct spectra induces a homomorphism of their limits. we set
i(L',L")(u )
=
Ga
E
be is called a homomorphism of direct spectra if, We shall sometimes write
is well-defined, suppose that
U
finite subcomplex
u
q
Denoting the inclusion of
x
(qJa J
J.I -successor of
q
lies on z
Suppose that for each a E A we have a homcrnorphism
Proof:
is onto, let is the homology class q
then
then
u,
u' E H (L)
If
qJ
and the theorem is proved. 1.5.
THEOREM.
H
=
s ome finite subcomplex
the inclusion induces a homomorphism
H (LI ).
E
(u)
For every
(:) E B,
there is an
homomorphism mapping
181
H(:)
into
ex
E
G0;.
A and an
(b)
If
~-admissible
g EGa'
and
Ga~
aCt(3 :
homomorphism, then
and
g
a g
exi3
..);J -successor. Gi3
is an 2.
Singular Homology Groups
have a cammon
For convenience, in this and the next two sections of this
chapter, the word "complex" will mean" oriented finite regular
1. 6.
If)J
THEOREM .
1J c;: 'f.j induce s
is cofinal in
an isomorphism
'JJ
complex".
then the inclusion
Hhen we make refe renc e to a pair of complexes
will assume that the orientation of
Lim G "" Lim HQ •
a ex
i3
f-'
the orientation of
Proof:
cp:
Condition (a) of cofinality implies that the homomorphism
~
Lim G
a
a
For if
(i).
h
E
Since
Lim HQ
f-'
13 H ,
i3
induced by the inclusion
bas an
b
aar:x :
G
ex
common~-successor a~.
of itse lf.)
(h)
fin ality to show that
be such that
~-successor
~
(g
h
E
Hi3
a
Thus
h
and
Then
gl
13.
Now
for some
J
E
~
and
to the class
(g)
E
and
h
has
g2
Lim G
a ex
bave a cammon
an~ -successor
~
It follows that
cp
is one-one.
as follows.
(K',L') ~ (X,A)
g:
such that
let
f
(X, A),
and
f
U
and
g,
be given.
(K',L').
g:
see that
mapping
Let
Then
(K,L)
(K" ,LiI)
and
f
The theorem is proved.
de­
If
are given, then
f = gh.
The ordering
f < f Ug
and
and
< directs S(X,A),
To see that
(K' ,L')
< is
respectively, into
be the disjoint union of
(K,L)
induce a mapping
g
(K" ,L") ~ (X,A),
and using the obvious embeddings we
g ~ f
U g.
Then
(S(X,A),~)
is a directed
set.
For each mapping
f E S(X,A)),
nically,
let
Hf(K,L;G)
q
f:
U
in
E
Hq(K,LjG),
~ (X,A)
(K,L)
f
H (K,L;G)
q
(i.e., for each
be a labeled copy of
is a group isomorphic to
wi th a definite isomorphism
= {g3} = {g2 } .
S(X,A)
if and only if there is an isomorphic embedding *
g
(K,L) ~ (K',L ' )
h:
G
a
{81 }
S(X,A)
might not be a successor
cp
Let
g
for some a
by (a). Thus gl and g2 have the common
3
3
fl-successor g3 E G · By (b), gl and g3 have a cammon
ex3
-successor, and the same is true of g2 and g3· Thus in Lim G
a
a
we have
g~
We define an ordering in
(K,L) ~ (X,A)
f:
f
G.
be a pair of topological spaces.
by
E Lim H . Next we use both conditions of co­
i3
i3
cp is one-one. Let gl E G
and g2 E G
~
a2
= CP(g2}.
CP{gl}
ex
is induced by restriction of
K.
obviously reflexive and transitive.
It follows that applying
yields the class
g E G
for some
a
(X,A)
we
note the class of all continuous mappings of pairs of complexes into
we bave a ;l1-admiSSible (and tbus fItiJ-admissible) homomorphism
have the
onto.
­
(X,A).
'].J -successor
a < a in A,
j; C J;!! is
Let
L
(K,L),
we write
H (K,L;G).
q
H (K,L;G)
q
Hf(K,L;G) ~ H (K,L;G).
q
[u,f]
q
Tech-
together
If
for the element corresponding to
u
Hf(K,L;G).
q
*An isomorphic embedding is an embedding which maps cells onto cells.
He shall often use "embedding" to mean "isomorphic embedding".
182
183
by the identifications
f
q
\'le \i ish to show that the triple
((H (K,L;G)l1 S(X,A),
-
­
togethe r with appropriate admissible homomorphisms, i s a dire ct
spectrum.
f,g
To define the admissib le homomorphi sms , we suppose tha t
S(X, A)
€
f ~ g.
and tha t
g:
(K" , L I) ~ (X,A),
h:
(K, L) ~ (K ' ,L")
Thus
(K',L')C: (K", L" )
Let
h:
gl:
(K" ,L")
~ (X,A)
and
an d there is at l east one embedding
satisfying
= gh.
f
homomorphisms induced by embeddings
h
(K, L) ~ (X, A)
h
g'x
(l,x) - h 2 X,
for all
be the obvious inclusion .
x
E
(K,L ) .
Define
by setting
g(h-1X)
f or
x
fy
for
x
h(K')
E
=
{
The admissible h omo ­
~q (K,L; G ) to Hg(KI,L
'i G)
q
morphism mapping
precise , if
f:
(O,x) - hlx,
<),
(t, y)
E
I X K.
are then the homology
such that
f = gh .
Then
To be
is well .de fine d because
g'
is cont inuous.
that is such an emb edding , then the mapping
Als o ,
f
gh
g'h
g
l
= gh 2 .
It is e asy t o s ee
by c onstruction, so the map
is an admissible homomorphism, [u,f] ~ [h*u,g]
is an admissible h omomorphism.
h *:
Not e that
there may be more than one admissible homomorphi sm betwe en a given
pair of groups.
2.1.
( k~~)
PROPOSITION.
For each
f
((H (K,L; G»), S(X,A ), <),
the tr iple
q,
q
.
­
with the admis sible homomorphisms de fined above, is a direc t spectrum
of
abeli~an
Proof:
groups, wri tten
.JJ (X,A; G).
\'Ie verify t he properties of a d irect spectrum, referring to
Definiti on 1. 2 .
Property (i ) is clear.
Property (ii) follows fr om
Property II of cellular h omol ogy theory.
f:
(K,L) ~ ( X,A)
that
f
hl
gh
l
and
gh ,
2
h2
and
and l e t
[u, f ]
€
(K, L)
Hf(K, LjG).
q
[(hl '*u,g ]
lt1e de fine a comple x
To prove (iii) J let
(K', L ' ) ~ ( X, A)
g:
are embeddings of
homomorphism carrying
ment.
lk
fl
q
and
(K" ,L")
on the unit. int erva l with v e rtic es a t
in
be g ive n.
(K',LI)
Suppose
satisfy ing
He find an admissible
into the same ele~
[(h )*u, g ]
2
as follows.
0, 1/ 2 ,
Let
and
the n the c omplex obtained f rom the d isjoint union
I
1.
be a comple x
(K" ,L" )
is
Sinc e
in
hhl
(K" ,L"),
and
hh2
we have
hhl
(K,L)
onto opposite ends of
=hh2'
Thus
erty V of cellular b omol ogy tbeory.
(hh )*
l
[(hhl)*u,g']
[(hh 2 )*u,g']
and the proof of the Lemma i s complete.
185
I X (K,L)
= (hh 2 )* by Prop-
Thus
I X (K,L) U (K' , L')
[h*(bl)*u,g']
184
map
[b*(b )*U,gl]
2
2. 2 •
DEFINI'I'ION.
coe ffici ents in
The
G,
qth s ingular homology group of
written
HS(X,A;G),
is
(X,A)
Lim Hf(K,L;G),
-
f
3.
,,,ith
In ce llular theory we found it relatively easy to compute h omo ­
the
q
The Properties of Singular Homology Theory
logy groups and difficult to prove that they are topol ogical invariants.
limit group of the direc t spectrum lJ.(X,Ai G ).
q
The r everse i s true of singular theory.
Notation:
If
then we write
taining
f:
(K,L) ~ (X,A)
{[u,f])
i s given and
for the e quivalenc e class of
[u,f]
E
Hf (K,L;G),
q
Given a continuous mapping
q
~ (X,A; G )
con ­
(X,A) ~ (Y,B)
f:
of pairs of
t opol ogi cal spaces, we defi ne the induced h omomorphism
[u,f].
~(X,A;G)
~ ~(y,B;G)
as follows.
q
q
f*:
mapping
g
of a pair of complexes
(K,L) ~ (Y,B),
f g:
S (X,A)
~g :
to
S(Y,B).
i nt o
( X,A)
the mapping
we obt ain an order -preserving function from
For each
g
S(X,A),
E
define
~ Hfg(K,L;G)
by setting ~ [u, g ] = [u,fg ].
q
g
Hg(K,L;G)
q
collection
(K,L)
If we associate with each
{ ~)
The
i s c learly a homomorphism of direct spectra (1. 5 ),
g
and is said t o be induced by the mapping
f:
(X,A)
~
(Y,B) .
The
r esulting homomorphism of limit groups i s called the homology homo­
morphism induced by
1.
and is written
(X,A) ~ (X,A)
If
f:
(X,A) ~ (Y,B)
f*:
~(X,A;G)
~ HS(Y,B;G).
q
q
is the identity, then
~(X,A;G)
~ ~(X,A;G)
q
q
(1 )*:
II.
1:
If
f
i s the identity f or each
and
arbit rary continuous mappings, then
g:
(Y,B) ~
g*f*
q.
(Z,C) are
( gf)*.
It f ollows that the s ingular homol ogy groups are topological
i nvariants .
If
(X,A)
is given, we define a boundary homomorphism
S
~~q­ l(A;G)
"*: ~(X,A;G)
q
assoc iate to each mapping
( fI L):
186 L ~ A,
f:
f or each
q
as follows.
(K,L) ~ (X,A)
If we
the mapping
we obtain an order-preserving functi on fr om
187
S(X,A)
to
SeA).
For each
f
S(X,A) ,
E
(fIL)
~(K, L;G ) ~ Hq _l
(L;G)
*f:
c\
where
define
.
by sett1ng
_f
~f[u,f] = [o*u,( r IL)],
is t he boundary hom=orphism of the relative homology
sequence of the pair
(K,L) .
To show that the collection
g:
is an embedding
(u, f]
E
(K',.L')
h:
If(K,L;G);
~
(K,L)
then
q
eX,A)
----?>
[h*u,g]
f = gh.
with
is a successor of
('f l
thus, to show that
CfJ'fi
Let
[u,f].
morphism for
(o*u,(fIL)]
= [ (hIL)*O*u,gIL']
(K',L ').
Thus
where
oS*:
HS (X,A;G ) ~ ~ l(A;G)
q
x!J (X,A;G)
groups of
III. If
q-
f:
spaces, then
(X,A)
~
q-
Let
(Y,B)
·1 4_1(A;G)
E
Hg (K, Lj G),
q
= [o*u, (fgIL)].
is the boundary homo­
is a successor of
Therefore
On
the other hand
(*f}
~(gIL)Yg[U, g ]
induced by
=
~(gIL)[o*u,( g IL)]
[o*u,(fIA) ( gIL »)
[o*u, ( fgIL» ).
("'f l.
is a continuous mapping of pairs of
Thus
TV.
q
,'<p
If
~
and the proof i s complete.
(X, A)
is a pair of spaces, l et
denote the inclusion mappings.
homo lo~J sequence of the pair
~1I (Y,B;G)
-
(u,gj
By
Proof of III: Consider the following diagram
q
Then i f
we have
S
S
O*f* = (fIA)*o*.
11 (X,A;G)
Similarly for
it suffices t o show that
(K,L) ~ (X, A) be given.
g:
We define
l(A;G)
~.
and by
(fIA)*d~,
t o be the homomorphism of the limit
it
and
q
o~
~_l(A;G).
is a homomorphism of direct spectrum.
d;f*
,+,'
i s the compos ition of the
*fg~g (U,g] = yrg[u,f g ]
(o~h*u,(g I L')]
in the direct spec trum
'¥'<p
the above diagram is commutative.
property III of cellular homology (VI.5),
[o~h*u,(gIL')]
is a homomorphism of direct spect r a;
the hcmmnorphism of limits induced by
be a mapping such that there
(K' ,L')
*'~
h omomorphisms of limits induced by
preserves the succ ess or relation and thus is a homomorphism of direct
spec tra, let
The composit ion
before III.
.
J*
~C;
••• ~ H- l (X, A;G)
q+
1·'
~ ~_l(B;G)
Proof of TV:
i:
AC X and
j :
xC:
(X, A)
Then the sequence below, called the
(X, A),
is exact:
.
.
t
~
o~_
S
1*
S
J*
---.S
*
~ Hq (A;G) ----:;;.. Hq (X;G) --?- lr(X,
A;G) --?­
q
S
Hq (Xi
We prove exa.ctness at
G)
and l eave the rest as
an exercise to the reader. Here
~
is the h=omorphi sm of direct spectra induced by
induc e d by
(fIA ) ,
and
V and
"
f,
q;
is
are defined as in the discussion
a)
Tm i*
~:
L
c: ker
~
j*.
( L,L)
Suppose that
f:
L
denote t he inclusion.
188
189
~
A is gi ven.
Define
g:
Let
(L,L ) ~ (X, A)
so t.hat the diagr'C\m bela'..! i .s commutB.tive:
L
h omology groups induced by a map
(L,L)
If
f
q
[u,f) E H (L;G),
an embedding,
Ker j*
([u, jif)}
c;: ira
(K' ,L')
but
i-l(.'
Suppose that
f:
K
Since
A*U = 0,
and
h*u
A
is
of the pair
O.
Then the re exists a map
h:
K
c;:
(K' ,L')
We denote the embedding
O.
hiu
(kl)*v
for some
c;: K'
K
by
h ',
(Y, B )
(f O )*
=
(x, A)
induces a homomorphism of' e xact sequences.
such that
fO
If
=f l ,
fO
and
fl
are maps of
(X,A) then ~( X, A;G ) --->- H~( y, B;G). (f l )*:
v E R (L').
q
Let
Hg (K,L;G).
q
E
co=e cting
= 0,
1
fO
by
(K,L) ~ ( X,A)
g:
Let
and
Define
PO(x) = (O,x)
I X (K,L) ~ (Y,B)
G:
GP
1
=
fIg·
Since
theory implies that
A
(Y,B;G).
q
and
by
Pl(x)
G(t,x)
(p )*
o
be a h omotopy
(K,L) ~ I X (K,L)
Pi:
= (l,x),
for
= F(t,g(x)).
x E K.
Then
Gp o
f or
Define
= fOg
property V of cellular homology
Po Z PI
have the co=on success or
He show
I X (X,A) ~ (Y,B)
F:
fl'
b e given, and suppose that
(P )*.
Thus
1
[(PO)*u,G)
(See di agram below.)
[u,fOg ]
and
[u,flg)
in the direct spectrum
It follows that
(fO)*([u, g J) = (fl)*([u,g] }. k
1
(glL'J
A
[u, g )
and
Then
i*[[v,(gIL'))} = ([u ,f)}.
L'
f
Invar'iance under homotopy :
into
i
such
By the exactne ss of the r e lative homology sequence
(K' ,L'),
--->- (Y,B) co=ute with
so
is given and that
X
kl
k2
L' ~ K' ~ (K' ,L' ).
and ,,'e have inclusions
(k 2 )*h~u = h-l(u
=
~
j*([u,f)} = O.
sat isfies
~ (X,A) and an embedding
that gh = jf
that
([A*U,g)};
([ u , jif)}.
Thus
Proof of V:
[u ,f) E Hf(K;G)
q
g:
j*i*([u ,f)}
(Y,B).
V.
> (X, A)
g
= O.
j*i*( [u, f)}
b)
then
and
~i ,1 (X,A)
the homomorphisms of the exact homology sequenc e s of the pairs
> A
f
f:
>
k2
K' ------=--~~
(d K') .
l
~K/K'l'::')
~X/
j;:'
fog
;:. I
>
( K,L)
J(
( K,L)
~=---3>-
PI
(X,
A)
fIg
Now
i*([v,( g IU))}
i(gIL')
and
(gIK' )k l
[u ,f)
spectrum
is represente d by
and
(kl)*v = hiu,
have the common successor
1;q (X;G).
[v,i( gl u)J.
Since
it follows that
[hiu,(gIK'))
[v,i(gIL' ))
3. 1.
COROLLARY.
A homotopy equivalence between tw o pairs of spaces
in the direct
induces is omorphisms of their singular homology groups.
Thus
i *([v,(gIL'))} = ([ u ,f)}
and
ker j* = im i*.
Note that II and III imply that the homomorphisms of Singular
190
191
3.2.
DEFINITION.
exc ision if
closure of
VI.
If'
morphism
Y - B
Y - X
(X, A)
An inclusi on mapping
X - A.
c: (y, B)
(X,A)
The excision is c alled pr o~er if t he
is conta ined in the i nterior of'
c: (Y,B)
A counterexample f or improper excision :
the line
is a proper eXClsion, t hen the induced homo­
HS(X,A;G
) -----...,.. 1f
q
q (Y, B,oG)
is an isomorphism for each
q.
y
= -2,
and between the lines
be the boundary of
(X,A),
Let
xy-plane that lies below the curve
x
c~:
X.
=0
X be the closed
y = sin !,
x
and
above
= 2/rr.
x
~(x) = ~(X ) = O.
h omotopically equivalent to a point, so
for
He give the proof of VI in section 4.
VII. If
4.1.
regi on of the
B.
The excision property.
!+.
is c a lled an
X is
Let
A
By the exactness of the homology sequence
~(X,A) ~ ~(A).
X is a point then
HSC Xo G)
q
,
~ [G0
if
q = 0
°
y
Sln ­
VII follows from the fact that a map
f;
.K -......,..
1
x
otherwise.
X can be
fact ored through an acyclic complex.
x
x
0
2
rr
x
y
~(A)
He show first that
=
O.
-2
\-Je claim that it is enough to show
that
(1 )
~
If
f:
L
L into
A,
then
f
A is a mapping of the finite regular complex
is homotopic to a constant.
Suppose that we have proved (1).
and suppos e that
of
192
[u,f]
E
f
Hl (L).
Let
f:
L~A
We have the inclusion
L as the base of the cone on
L.
193
Since
f
be given,
h o LC CL
=c onstant,
there
exists a map
the class
so
0.
~
CL
[h*u, g ]
h*u =
Proof of
g:
as a suc ces sor.
{[u,f] ) =
That is,
(1):
A such that
Suppose that
that part of
feLl
f:
But
°
x
f.
CL
has
Sln
homology groups we find it conveni ent to de fine a new direct spectrum
is acyclic (V. 2.5),
'jf(X,A;G)
show that
\~e
ass ociated with a pair
q
S
0.
on
S (X,A)
by s aying that i f
g:
(K',L')
~
h:
of points in
{v)
n
< a < 1.
form
(O,a),
-1
f(w )
a
=
v '
n
Since
L
the direct spect rum
Then there exist s
f ew)
c o=ected at
l~ore
(in
A)
of
(O, a )
f «g,
For each
n,
choose
{w )
n
Now
(O,a).
wn E L
so that
has a limit point
w,
'J.I,q (X,
A; G)
nected,
(O,a)
q
f-l(U)
x
=
0.
Since
L
c ontains a connected neighborhood
if there exists a continuous
such that
f
=
are then the same as be f ore : If
f, g
E
S( X,A)
(K,L) ~ (K' ,L')
h:
0.
claim that
W. (X,A;G)
s pectrum.
(S(X,A), « )
U
= ((~q (K,L;G) ) , S(X,A), « )
q
f
=
gh.
is then a direct
is a directed clas s since the relation
<.
{w )
n
the fact that
has
w as a limit point.
{w ),
n
are immediate.
W of
w.
f(l-I)
4.3 be l ow. Pr operty (iii) i s given by Corollary
Since
must lie
~q (X,
A;G).
cofinal in the new di rect spectrum
1; (X,A;G)
q
The firs t conditi on
Y
But
+1,
B
and l et
B
Y - (X-A).
second conditi on of cofinalit y is contained in the foll owing l emma:
x = 0,
2
x
We have as before that
is homotop,ically equiva l ent to a Circle , and so
Conse quently
~(X, A )
by the excis ion
(X,A)
=
0,
H~(y,B) ~ Z,
c: (Y,B)
=
rr'
y
-2,
H~(y, B ) ~ ~(B).
~(B) ~
Z.
and the homomorphism induced
is not an isomorphism.
To prove that a proper excision i nduces is omorphi sms of singular
LEMMA.
and
f
(K,L)
h:
gb,
(K,L)
fl:
If
f:
~
(K,L)
----0;..
(K',L')
(X,A),
g:
(K' ,L' )----7 (X,A)
are continuous mappings satisfying
then the re exi st isomorphic embeddings
and
(K',L'),
(K",L")
~
hl
and
respectively, in a pair (K" ,L" ),
(X,A),
such that
(h 1 )*
=
4.3 .
COROLLARY. ~q
'11 (X,A;G)
flh l
= f,
f'h 2
h2
of
and a mapping
= g,
and (h 2 )*h*.
satisfies conditi on (iii) of De finition
.
1. 2 and is thus a direct spectrum.
194
The This contradiction
4.2 .
Y be the closed regi on bounded by
is which contradicts
establishes (1).
Let
«
Properties (i) and (ii) of Definition 1. 2 of cofinality is obvious since we don't have any new groups .
w not containing any points of the sequence
We U which
W i s a ne i ghborhood of
But that implies that
as above and such that
He assert that our original direct spectrum
x
The groups of is l ocally con ­
the continuous image of a co=ected set is connected,
on the line
gh .
are the homol ogy homomorphisms induc ed
q
by continuous maps
precisely, there exists a neighb orhood
lies on the line
g
and then the admissible homomorphisms mapping contains the relation
contains
(K',L')
~ (K,L;G) ~ Rg(KI,LI;G)
and
A f ails to be lo~ ally
such that the co=ected component of
f «
(K,L) ----7 (X,A)).
q
which c onve r ges to a point of t he
is compact,
then it fo llows that
(O,a).
S
He define a new ordering
(K,L) ~ ( X, A)
f:
then
~
(K,L)
is
S
{Hf (K,LjG)lf:
a sequence
(X,A),
(X, A).
be
mapping
1
x
Suppose not:
0.
[u, f ]
A is given, and l et
.
Y
Thus
(1) implies that H~(A )
and so
~
L
which lies on
bounded away from the line
gh
195
Proof of Corollary :
Suppose we are given the commutative diagram i n the closure of the unique open simpl ex containing
belm,: Si nce
f_
~ (K' ,L')
----'=''---......,..
f2
Here
fl
and
Then by the Lemma,
~-successor
a c orumon
1f
are arbitrary continuous mappings.
~(K,L;G).
q
[u,f] E
common
f2
,f; -successor
-ancessor
page 178).
[u,f],
vI'
v2 ·
.)f-succ essor (and thus
and
Since
[u,f]
[u,f]
vI
and
and
[{fl)*u,g]
~ -successor)
v
and
v
and
Suppose that [{f )*u,g]
have a
2
2
have
[{fl)*u,g]
[(f )*u,g]
2
v3
(see
have the coramQn
3·
and results. s
> 0,
K ~ L
f:
f
is a map of regular complexes, then for each
induces a map
f':
f l
and
SdsK ~ Sd L
such that the follow­
4 .5.
THEORE.t\l.
a map of
K
integer
}
f
)
L
Let
a vertex
A
meter of
st B
of'
vertices of
verte x
3>
.t..
]\) .
k
and
k'
L.
~neor em
Sd L
x
E
SdsK.
K ~ L
Then for some
such that
0/2.
Let
of
SdsK,
B
S d.
L
f' (St B ) ~ St A,
Sd L.
of
BO,B , ... ,B •
l
g
and s e tting
from the
g (B)
SdsK,
a C n St B.,
-
=
J
j
B,
A.
a
To
it is sufficient
SdsK
a be a simplex of
Let
Then
there exists
g
maps the vertices of any simplex of
sim~lex
be any
by choosing, for e ach
extends t o a simplicial map of
g
s
since the dia ­
Define a function
5.
to those of
VI,3.9, except that we choose
f' (St B) ~. St A,
such that
is less than
SdsK
g
For each vertex
onto the
SdsK,
and so
f '( a ) C n St(gB .).
Sd L
-
Here
for each
K be a f in ite regular complex, f:
'
f'
Sd L
N so that the mesh of SdNK is l e ss t han
wit h ve rtices
k
in
to an arbitrary r egular complex
vert ices of some
1
f' ( x )
We proceed as in the proof of
t o shaw that
K
and
--+ Sd L t o :t'.
SdsK
Proof:
g ( x)
lie i n a closed simplex, there is a
> 0 there exists a simplicial appr oximation
s
show t h at
ing diagram is coramutative
SdsK
f '(x )
are homotopic.
g
integer larger than
We prepare for the proof of Lemma 4.2 with a few definitions If
Thus
g:
have the common
1; -successor
they have a common
Consequently
and
line segment j o ining
fl
(K,L)
g{x )
f'{x).
are subdivision homeomorphisms.
(See VI. 3 for
This implies that
nStg(B . )
J
J
j
is non-empty.
Let
'I'
be any simplex
j
details. )
whose interior meets
n
j
4.4
DEFINITION.
map
g:
A s implicial approximation to
SdsK ~ Sd L
such that, for each
1 of-
f
is a simpli.c ial
x E SdsK,
g{x)
lies
St g( B )
j
vertices
n
'I'
F 0,
(g{B.»)
J
st g(B. )
J
and so
in a non-empty set.
g (B )
j
is a vertex OfT.
span some face
T'
197_
of
T.
For each
j,
Thus the
Itf'ollowsthat
g
Proof of Lemma 4 .2 :
extends to a simplicial map of
SdsK
into
Sd L.
Let
x
E
a.
Let
simplicial approximation to
n St
f'(x) E
g (B. ) .
(x)
,,'
in its interior contains each
g (B,)
Therefore
g( x ),
simplicial approximation to
4.6.
DEFIIHTIOH.
K
L
~
Let
K
which of course lies in
L
f
folla.ls.
The vertices of
those of
L.
and
i f the
f
the simplex
is
g
is a
L
f
K
K
together with
A collection
a
in
K
and the
ILf I
L
J
LEMMA.
Then
i ::." jf.
Let
are embedded as subcomplexes in
j:
LCLf
and
map.
fx
x E K.
lies in the simplex
Thus the simplex
line segment from
ix
Then
x
L .
f
Note
(f~, ... ,fAs)'
since
(~, ... ,As' fAo, ..• ,fA )
s
to
jfx.
It follows that
of
f
(~, ... ,As)'
is a simplicial
L
f
i::." jf.
Let
(Sd K' , Sd L')h'
h'.
(K" ,L")
is ob tained fr om the d isjoint union
[I
(K,L)] U (Sd
X
(O,x') _
j ( k'
)
-1
(K" , L")
K',
Sd L')h' U [I X (K',L')]
x'
X
E
K, i: (SdsK,SdsL) C (Sd
x'
E
K',
(Sd
j:
K',
K',
Sd L') C (S d
Sd L' )h'
K',
Sd L' )h'
Furthermore , v e have embeddings
contains the
K',
of
(K,L),
Sd L'),
i v e ly, in
:t x.(k~l)
is a regular complex. hl , (SdsK,SdsL), and
(K" ,L").
(K',L')
r espect­ (Stl ~: S.a l!)~
- - - - --~frl
(See picture.) i-le want t o shov that (h
l
)* =
hl ::." ik,
(h )*h*.
2
It is clear that and t hat
a ddi t ion , Lemma
198
as folla.,'s.
f,
denote the inclus ions.
lies in some simplex
(K" ,L")
be the simplicial mapping cylinder of the simplicial map
i , j, h2
Suppose that
::> (X,A)
g
(K',LI)
h' _ ::> (Sd K',Sd L')
\</e define the comple x
(Sd
Proof:
~
Ik'
(SdsK,Sd.sL)
Then
KCL ,
f
h"
t
(l,x) - ikx
is not homeomorphic to the mapping cylinder of
i:
h" = k'h'k.
b y making the f oll owing identifications:
de fined i n V.2.
4.7.
in
span
B.' s
L.
and
are subdivision homeomorphisms, and
(K,L)
of vertices spans a simplex
L}
E
Consider the diagram belm"
h" ::." h.
is the simpliCial complex g iven as
K, B.
J
h.
Then the simplicial mapping
He include all faces of such simplexes together
fa.
k'
be simpliCial c omplexes , and let
are the vertices of
f
Thus
and
k
(1)
span a simplex
with all simplexes of
als o that
E
l
i
and so
are assume d to be dis joint.)
A.
A 's
Not e that
L
L
{~, ... ,AS' Bl,···,Bt :
L
and
of the mapping
(K
f'(x),
or' ,
fl.
be a simplicial mapping.
cylinder
of
as a vertex, and so has
J
as a fac e .
be a
containing
whi ch
contai ned in the closed simplex containing
f:
Sd L
J
j
f'
Thus the unique simplex of
(Sd\,sdsL) ~ ( SdK ' ,Sd. L')
h':
Then
h 2k' ::." j.
In 4.7 implie s that 199
--
-
(k')L')
IK(~'
i
Z jh' .
It f Qllows that
(h l )*
Tn the pr,?of of Theorem 4.9, we will work with the direct spec­
On I X (K, L) :
homotopy between
f
1;CJ. (X, A;G).
j*h~k*
(since
i ::: jh l )
results c oncerning the successor relation in
(h2 )*k~h~k*
(since
h kl :: j)
2
results about
(h2)*h~
(since
hit = k'h'k)
(h 2 )*h*
(since
htl :: h).
Since
and
f
f l:
gb. ::- ghlt
gklh'k
gk'hlk,
to define
f'
on
I X (K,1)
CJ.
Proof :
we may use the
so We prove first that
that
i*([u,f]}
f ' (l,x)
m:
for
(x
O.
=
fl
on
and m(jB)
(Sd K', Sd 1 1 )h l
I X (K', L' ) :
On SasK,
if
gh
is a monomorphism.
and
h*u
for
for
f
~
f'
the identifications yielding
and
4.8.
f lh
2
COROLLARY.
S
H (X,A; G)
CJ.
g.
B a vertex of
Sd K'.
gklm.
The sets
g
x
E
By construction,
is cofinal in
flh
l
f
-1
Tnt B and
Tnt B and
g
Y - Y-X
-1­
(1 - Y-X)
r
~ ( X,A;G),
covering
(K,. 1) ~ (K' ,1')
h:
1· ~ (Y,B)
form an open covering of
form an open covering of
Y.
K'.
Thus
He
so that:
(k1B -1 Tnt B, klg -l( Y
SdrK'
of
SdrK'
gk
-1
l
Y-X)},
Let
is a subdivision homeomorphism.
which are mapped by
into
lies in an open set of the
where
r
~ • Sd 1 1
Tnt B}.
U {closed simplexes
K"
Let
KI ~ SdrK"
k :
l
of all closed simplexes mapped into
be the subcomplex
X by
-1
gkl .
Set
and so
L"
is isomorphic to the limit group of the direct spectrum
= K" n 11"
complex of
J./ q(X,A;G).
such
K' .
The proof of Lemma 4.2 is complete. ~CJ. (X,A;G)
Hqf (K,1jG)
m(iA) = h'A
as defined above is consistent with (Kit, 11t ).
€
) (X, A)
g
(K ', L')
(1) Every closed simplex of
As the reader may verify,
[u,f]
and
Suppose that we
O.
(K, L)
choose an integer
g(x ')
i*
and an embedding
K).
€
to be the composition
f' (t, x ' )
Set
=B
is a proper excision, then
K)
€
(Sd K', Sd 1 1 )h : Define a simplicial map
'
(Sd KI , Sd 1')h' ~ 'Sd K', Sd 1') by setting
A a vertex of
(Y,B)
This means there exists a map
On
Define
(c )
= gklh'kx
c;:
(K,1) ~ (X, A)
f:
(K' ,1 1 ) ~ (Y,B)
such that
(x
(X, A)
i:
CJ.
are given a map
g:
f'(O,x ) =fx
If
to deduce
CJ.
~ ( X, A; G ) ~ ~(y,B;G).
i *:
(KIt,11t) ---;;.. (X,A ) .
'J.I (X,A;G )
);jCJ. (X,A;G).
THEORDf.
4.9.
that (b) How that we have proved Lemma 4. 2, we can use
hI :: ik)
He must nOl, define the mapping
(a)
trum
(since
= i*k*
Finally, since
(K ' , 1' ) .
We set
have the diagram below, where
h
is an embedding,
(K"' ,1" 1
k2' k3'
200
? nl
)
h (K, 1)
r
r
(Sd h(K }, Sd h(1) .
and
k4
is a subThen we
are the obvious
inclusions and
k5 = klh.
[(k k )*U, g ' ] e
4 5
4.2,
(K,L)
( K'.t~
(K ', L') ~ (Y,B)
k2
0
-C
-l( Y
O _klti
L"
=
t
. (Y, B)
=
Y - Int B
n ontI'ivially.
-l-C
° _ x.
Thus
gkl
~
He have shown tha t
- L".
k3
t
is an
k3
be an open simplex of
0
and s o
0(; K"
In other words,
H (K", L"i G) ~ H (SdI'K',LliG)
q
~
k3 k 4k 5 = k 2k l h:
(K,L) ~ (SdrKI, L ),
l
(k k 4k )*u = (k k h)*u = (k k )*h*U = O.
2 l
2 l
3
S
phism,
(k 1. k )*u = O.
f 5
de note the mapp ing
spect rum
Pro -
Note t hat
( gk~lI K" ):
'JIq ( X,A; G),
[u,f] E
k4k5
(K",L")
~ (Y, B)
Here
and since
k3
SdrKI - Ll (; K" - L" i
is an excision of complexes.
is an isomorph ism .
..Ie have
and s o
Since
(k )*
3
is an isomoI'­
is not an embedding .
~ (X,A).
Let
l
g
/
(K ' ,L ')
SdrK' - L ·
l
By (1 ) ,
<;: K" ,
i
2
g'
kl
is an excision of complexe s .
(k )* :
3
v
is a sub division homeomorphi sm,
Hq(K",L" i G) ---::>- Hq(SdrK ', L )
l
=
(k ): 1(k )* (k )*U € Hq(K" , L"j G) .
l
2
3
gk~\ (SdrK' ,Ll ) ---::>- (Y ,B). Then
[v ,i g ']
€
. ,
lg
H
q
(K" , L"; G)
k2
is an inclusion, and
Thus
i s an isomorphism.
Let
g"
Let
denote t he map
[u, f] e H! (K', L I; G)
and
have t he c=non succ e ssor
[(k )*v, g"] € ~" (SdI'K I,Lli G) in the di r ec t spec t rum
3
It follows from Lemma 4 . 2 that
i* ( [v, g I ])
( [u , g]) .
the proof of 4 . 9 .
In the dir ect
~(K,Li
G) has t he s ucc e ss or
q
203
")1"'1..'")
,.. (X,A)
3
I
K" - til.
Let
int ersects
Y- X) ,
g'
(SdrK', Sdr L I)
By Property VI of cellular' homo l ogy t heory,
~
Suppose th at
is a monomorphism, COBstruCt the
( SdrK" Ll) ~
"
g
the reverse inc l usion is obvious, and
(k~ ) * :
is ont o .
k
1'k
SdrK ' - Ll
~,
n
K"
i*
(K" ,L")
V
excision of c ompl exes .
-1­
i*
complexes and maps in the diagram be low:
gkl
(K' ,L')
gkl
is a mon omorphism .
q
ceeding as in the proof that
Ikl
Then
It follows that
q
Ii
( SdrK ' ,S drL')
He show that
and so by Lemma
is given , and t hat [u,gk Hg( K', r!; G).
(SdrK I ' Ll)~l
j
H ( X, A; G).
We prove in a simil a r manner that
g:
~3
(k k )*u = 0 ,
4 5
represents the zero of
H
~ (X,A)
( Kill ,L'" )
But
S
~C;
(X, AiG) ~ H-'(Y,B : G)
q q
'
i*:
~k5
h
[u,f]
H~'(K",L"iG).
)Iq (y,B;G) .
This comple tes
continuous mapping of complexes.
5 . Equivalence of Cellular and Singular Homology
Theories.
The
Invarianc~
Theorem for Infinite Complexes.
homomorphisms
f*
respectively.
Since
with
IJet
regular complexes.
direct spectrum
~
IJet
denote the collection of finite sub-
K,
ordered by inclusion.
We define a direct spectrum
~
where
(K,L)
is finite the single group
Hl(K,Lj G)
q
the identity, is cofinal in the
'J.Iq (IKI, ILl iG).
JJq ( IK ' I, lL' liG).
final in
complexes of
.
for cellular and slngular homology,
(K, L) ~ (IKI, ILl)
1:
be an oriented pair of (not necessarily finite)
(K,L)
S
f*
and
Then we have induced homology
Likewise,
Hl(K ' ,L' iG )
q
is co­
So we have the followinG diagram,
denotes the isomorphism defined by taking equivalence
of abelian eroups by setti.ns
classes under the relation of baving a common successor:
G(M)
The collection
= Hq (M, M n L)
(G(M)) ,
MEt .
for each
together with the homomorphisms induced by
inclusions, forms a direct system of abelian groups.
(G (M) )
f:
is cofinal in the direct spectrum
(K',L')
plexes
~
(K',L ')
(K,L)
into
(K,L),
then, since
IK'I
be factore d through some finite subcomplex of
is compact,
K.
f
can
5.3.
DEFDUTION.
1-1
and
M'
f)
be categories, and let
t
G o.e two covariant functors from
1.9.
to
qJ
from
F
to
G is a collection
are finite subcomplexes of
(qJclc
E
~),
n L) in the direct spectrum
morphism induced by inclusion.
H
q
(M, M n L)
']J.q (IKI, ILl)
E
M(F(C ) ,G(C )), such that if C and
f E MeC, D),
and
where each
D are objects of
M~ M',
K such that
F
A natural transform­
Condition (b) is trivial in this case,
then the only admissible homomorphism mapping
Hq (W, M'
and
""
This proves con­
qJC
because if
t
IJet
> ~(IK'I,ILII;G)
C£
H (K' L"G) = Hl(K' L"G)
q
"
q"
ation
dition (a) of cofinality.
l~
q
1:*
First, if
is a map of the pair of finite regular com­
> rt'(IKI,ILI;G)
'"
(1 )
We claim that
~q (IKI, ILl).
qJ
Hl(K
q , L'G)
,
Hq (K,L; G)
~
and
then
to
~ • F(r)
is the homo-
= G(r)
0
qJC'
The latter is of course admissible
One defines a natural transformation of contravariant functors ana­
in the direct system
(G(M)).
Thus the following theorem follows from
logously.
1.4 relativized, 1.6, and 4.2:
5.1.
5.2.
THEOREl-l.
COROLLARY.
5.4 THEOREM.
S
The di agram (1) above is commutative.
Thus
~
yields
H ( IKI, ILliG) "" H (K,Li G).
q
q
a natural transformation of functors.
Cellular homolosy groups of infinite regular com­
IJet
Proof:
[u,l]
plexes are topological invariants.
[u, f]
We can prove a little more.
80ry of finite regular complexes.
E
Hf(K, L;G ).
q
E
Hl(K,L;G).
q
But
[f*u,l]
Then
E
We restrict ourselves to the cate­
IJet
f:
(K,L)
~
(K' ,L')
be a
205
s
f*{ [u,l])
Hl(K' , L'j G)
q
is represented by
is a successor of
[u,f]
in t he di rect spectrum
f~, ( [u, III
Exe rcis e .
the diagr am
?J
q
( IK'
I
J
IL'l ;G),
and so
Chapt,,; ' VIII ( [ f*u,l] ) .
Let
(K,L)
be l~l
INTRODUCTORY HOMOTOPY THEORY AND TEE PROOFS be a pair of fini t e r egular complexes .
i s commutative f or every
q
I n t h i s chapter we will be touching upon a main prob l em in t opol.ogy ­
t he e x tens i on problem as defin.eCi, in the Introducti o n, page ii.
'> lIS ( K I, ILl; G )
q
""
f*
q:
I
:£
R (K,L, G)
OF TIlE REDUNDANT RESTRICTIONS Prove that
Our main
goals are the horllQtopy e xtension theorem and t he t heorem on i nvariance o f
t
S
0*
Sf!
Hq ­ l( L,G )
;.
""
domain .
S
H l(IL I;G )
restr ictions imposed on :cegul
<1­
This fact) t OGet he r "I>'it h Them'em 5. 3 and the result t h at
cp
is
an
i s omorphi sm of cellular and singul ar homology groups for fi nite
r egular c omplexes, cons titut e the s tatement that
qJ
(See Ei lenbe:cg and! S teenr o~,
Foundations of Al€ ebraic TopoJ.?,gy . )
comple xe s i n Chapter II, and 'by proving
t nat each r e gular qua si compl e x is a comple x. ( See 1 . 5.)
1.
Solid s-paces .
1.1.
DEFINITION.
Be'l:;rac ts .
defines an
e quival e nce of ce llular and singular homo l ogy theorie s on t he cate ­
gory of finite r egular c omplexes .
We will compl.ete the chapter' by proving the r-edundance of the
normal space
f :A
- -> Y,
spa ce
A
i s solid if wheneveI ' there i s given a Y
X, a closed subspace
then
extends over
f'
A of
X, and a continuous func t i on X.
The proper ty o f b eing s olid ie topologic al .
of a single
~o int
1s solid .
dosed unit interval is
The space c onSisting
The Ti etze exte nsion t heorem asserts t hat the
so~id .
The n ext pr oposition thus impl i es that the
closed n- ball is sol.id .
1.2 .
PROPOSITI ON.
Proof :
Le t
be nor mal,
p :
~
Y
a
,
a€J\, be an arbitrary fami.J.y of sol id spa ces.
A closed. in
--->
~ Y
a€ A a
A product of solid space s is solid.
Y
-~
X, and
f:A -->a7J.AYa
g: X
a
--> Ya , sin ce Y
a
X
a cont i nuou s mapping .
is t he projection then for each
be extended to a mapp i ng
Let
If
a, Pa f :A ---> Y
a
is solid .
A point
c an
of 't(a ) € Y ·
a'KA Ya is a function 't on 11 mapping each a t o eo point
a
Define g: X - > nY
by [ g(x ) J(a) = g x. Then p fl.
g
for each a,
a
a
0:a
and
206
so
g
i s c ontinuous and extend s
f .
~· 3·
DEFINITI ON.
a retract
A
X
01'
Let A be a subset of a space
if t here exists a map
is the identity .
The map
Note that if
Y, every map of
n-l
S
r etracts:
the origin.
A
to
Then
---> A whos e r es triction to
is called a retract ion of
r
X i f and
A
is a retract of
Y
e x t€ n d s to a map ot'
is a retract of
(But not of
N
n
nn
X onto
X
to
o~y
y.
X
r,
A.
to the algebraic extension problem .
a homomorphism
is closed
Example s of
En
In the product
i s a r etract.
sn - l
is not a retr act of
If
A
X
C
X x X is a retI'act of
> H (A) , for each n .
identity map
l:A
- ->
A
X x X.
A
extends over
X.
X x y.
Thus t he
F inall y , suppose
is a retr act of
X
X,
A space is c a lled
1' ,
is normal ,
fo:!' the
gi
f.
(It is not in G<2neral a
and
and
r:X
I t fol lowS that
Hn ( X)
=
for each
1m i* IS: Ker r *
a solid torus .
Let
n.
i * :H ( A)
1
A
i*
then
1,
is a monomorphism
A be a s imple closed cuX've in
-
HI ( X) ""
Although
z.
X.
is no t a. retract of
8.
eClu ivalent to a retraction problem . f :A ---> Y
1.4. DEFll'lITI ON . each
H (X)
n, is there a homomorphism ¢ such that ¢ i.*
g: X
---> Y extending f
x eA
then "e can set
¢
f.*
1'* ?
If there is
and obtain a solutio
space
-
X Uf Y
PROPOSITION .
extensio n of
-
and suppose that
The adjunction space of the mapping
vii th its image
• Hn (Y)
1.5 ·
a map
'De a subspace of X
is the space obtainecl- from the disjoint union
Note that
*
For each
A
f
to
Let.
X
xuY
f :A -
contains
f :A
- >
y
Y
by i dentify-ing
if and only if'
X U y.
l'
2 0Q
as a subspa c e .
be given.
Y
> Y
f, vritten
f( X) € Y .
.. ' .. ~
":.l
f
Let
is a c ontinuous mapping.
x Uf , Y,
Hn ( A)
x
that e very extens i on p:coblem is actually
i :JCX
corresponds to the e xtension problem diagrammed b elow:
;/
of a spac e
is a retract __ ,-lhic h Vie call a r etract i on problem -- is of course an extension
Given any diagram o f spaces and maps "Ie obtain a co r r esponding
I i' -we let
A
c losed
problem . The proposition below ShO'.'IS
i* /' a
X ,,'hich ,
is a monomorphism t he image is not a direct summand and
> R1 ( X)
Thus a solid space is an obsolute retract.
denote inclusion, then the extension problem for a mapping
r_*i* '" 1 : an absolute
retract i f it j_s a retra ct o f any nor mal space containing it as
i agr am of homology groups and induced homomorphisms .
X, s o that
is a r etract of
Suppos e , foX' example , that
as a cycle , repres ents t'll ice the gemorator of
so
A
n
S1 x D2 ,
x
X
Note that i f
The problem of d etermining vlhethe r a given subset
subs pace .
of
is a necessary co ndition f or the
---> A s atisfying ri
H (A) -
X, then
is sol id and closed in
En .
a s ,~e shall prove l ater in thi s
xo x y
n,
The above reasoning -was used i n the introduction to prov<2 that
i:A C
XxY of arbitrary spaces a cross sec t i on
= 1'*
s uch that
',ve have
consid ered as a set of ordered pall's , is a I'etract of
diagonal
g
minus
f: X ---> Y is continuous, then the funct ion
Also , i f
¢ i*
such t hat
Thus the exis t e nce , for each
sufficient conditic,n . )
n
cilapter. )
¢
e xistence of a mapping
i f for a ny space
minus the origin and o f'
or of
is called
A
is a Hansdorff space, each r etract of
X
It is e asy to see that
r: X
X.
The n tnere e xis-ts an
is a retr act of the adjunction
Proof' :
Let
x Uf y.
of
f.
j :X
- ->
X Un y
X E
X to its equi valence class in
-.- -> Y i s a re traction, then rj
r:X U f Y
If
map
I
On the other hand , sugpos e
--->
g :X
Y
extends
n
Then we define
1' .
f- ~ II.
A"
is an extension
Sin ce
g( z. ) if z
r(z)
V'
n
X
if z €
z
A'n
V" (
for all
X U Y
f
1. 0.
z
E
Then
X U-r y.
onto
n
C X
U"
m
V'
U
V'
n n
A"
A'
s atis:f'ying
Y
ax'e all solid , t hen
y
Y
If
Y'
and
Y' U y lI> and i f
Y"
Y', yll, and
n
V"
n A"
2.
is a metric space .
- V' •
and
PROPOSITI ON.
and
'
\ole have
For each
U' nAil
W
A".
Suppose m < n.
mZn n
n
n,
U'
n
V'
n V"
V'
U V"
are disjoint open sets s uch that
n
n
n
=X
X'
V" n A'
m
¢.
Let
X be normal,
A I' = 1'-1 y" .
X
such that
'We can extend
is solid .
A
Suppose we have found closed s ub spaces
X ' U X" , A '
f fA' n A"
C
C
X' , A"
to a map
X", and
- -> y. Se t
A'
X'
and
X
f I (X I n X" ) U A"
~
g2
agree on
extends
f.
Thus 'We only have to find
s ince
g2: X" _ _> yll .
and
X"
Then
Y' n Y"
gl: X' - -> Y'
Then since
g:X - -> Y whic h
X' n X", they defi ne a mapping
X'
of
x 'n X" n A = A' n A".
fl(x ' n X") U A'to a map
to a map
X"
f -i.,
X and
sat is fying the conditions
the mappin@from
to
X
¢.
Thus
UA ' (
A ' - A ' nAil
n n ~
The r elation of homotopy is
Suppos e
X'
each map
g :X - ->
Y
rr( XiY)
is another' spac e , and
'de
f :X' - -> X is a map .
can assign the composition
> rr(Xi Y) . Similarly,
is a map , t hen
f
y.
Thus
the collection of these equi valence
gf :X' - -> y .
as s ignment pr eser ves t he relation of homotopy and so
rr(Xj Y) -
X to
Y are partitioned into equivalence classes of'
He denote by
classe s .
*:
conta i ns V'n
- V" . Y be topological space s .
homotopi c mappi ngs .
f
set
The Homo t opy Extension Theorem .
Let
f:A
f: X' n X" - -> Y' n Y"
Similarly, we can extend
and we can extend
gl
X, and le t
WE:
Thus we may take
is soli d .
cl osed in
'
n
an equiv al ence re lation on the colle ction of mappings from
Proo f :
¢. =
I
and so
n ~
Thus
U A" (
V", and V I n A"
n n ~ and
V' (
Then
nA
such t hat
For each
¢. Similarly,
n V"m
n
U"n
and
Un'
= A"
Then, for each
U" - LJ
n
A' - A'n A" .
n
n A"
n
¢.
V"
n
n
and
n
U AI
A'
n,
ij"" n A'
n
n
n
cont a ins
V"
and
X" '" X - V '
Y
Y'n y "
_
A" - A I n A".
U ' -~ ij""
n
n
The following porpos i tion is called a Polish add i t i on theorem .
Suppose
HA~
YI
n n
f- ~'
A'n
set
"\ole
is normal, we may choose open sets
X
m ~
is well- defined and continuous , and r etracts
y.
are closed subspaces of
y ' ny"
r
U YI
A" ( U" and
A' ( U
'
n ~ n'
n ~ n '
X
€
n.
are closed for e ach
n
Since
n
Similarly, r: X U Y ---> Y by setting
f
y"
and
Y'n
Then
if'
y l
X
and
This i ndu ces a function
is another space , and
induces , by composition , a function
Thus t he a ss ignment to every two spaces
f
To f:Y - -> Y'
f* :rr( Xi Y) - > rr(Xi y l ) .
Y of the set
rr( Xi Y), with
the induced functi ons described above , is a function of t\iO variable ,
given above.
We defi ne
and
Y'n (YEY 'ld(y ' n y ll , y )
> ~n
y"
{y € y " Id(Y' n Y" , y )
>
n
210
1
n
J
contr avariant in
If
y
Y
X and covariant in
y.
is arc- wis e connected , then any tl,W cons t ant maJls from X
are homotopic.
to
Thus the constant maps are all contained in a s ingle class
21.1
DEFINITION.
2 .1.
of
A map fr om
"(XjY).
to an arc - wise connected s pa ce
X
Y
Let
(X,A)
be a PBir of spaces .
The space
A has
i s called
X with r espe ct to a spa ce
t he homotopy extens ion property i n
homo top ically tr ivial or inessential t f i t i s homotopi c to a constant map.
f:X - -> y , and a homotopy
any map
G:I X A ---> Y of
Y
if, given
flA ( i. e . a map
All other maps are called essential.
s uc h t hat
The homo t opy c las s ifi cat i on problem fo r the s paces
X and
Y
fX
G(O, x)
F: I x X - -y Y
e numerat e the homotopy classes in
for
x €
A), then there exi sts a homotopy
is to
of'
f
which extends
IT( X;Y) and to g ive an al gor i thm f or
X
homotopy extens i on -property i n
d e ciding t o which homotopy class any g iven map from
X to
Yare homotopic is an extension problem .
0 X X U 1 >.: X
on
f
and
fr om
g
A ha s the absolute
A has the homotopy extens i on pr operty
if
That is , we have a map
'With r espect to every spa ce
X
y.
X to
F
definet
PROPOSI TION.
2 .2 .
is clos ed in
in I X X by
X. I f
Let
(X,A)
be a pair of spaces , and assume that
fx
sets and f unctions is 8xact at
A
X with
A has the homotopy ex tens i on pr operty i n
re spect t o an arc, "liSe connected space
F(O,X)
The space
Y belongs .
in
The problem of de c iding whether blo maps
G.
s e que nce of
Y, then t he f'ollmli
IT( X jY)
F(l,X) = gx
*
*
Tr(XjAj Y) -p - >
and
f:: g
that if
Y
i t' and only it
is solid and
e:;.,,'tends to a map of
F'
I XX
is normal, then
I X X to
y.
class onlY, that of the inessential maps .
Les s t rivially, s uppos e
X
Here
X L->
ident i f ying
is a soltd metric space,
Y
i s ar c wise conne cted.
Xj A
A
is solid.
By
1.6, the set
Z
Since the compos ition
By the Tietze extensio n theorem,
0 X X U I X Xo U 1 X X
=
€
o
X.
It follows that
Z
is a retract of
connected, we can exte nd the map
F
I X X.
Since
defined above over
Y
Finally, s uppose
I XX
~-y
K and
Z
K'
F
are finite
s~ licial
by the simplicial approximation theorem
f :K - - y K'
over all of
Sd~
to
complexes .
K'
Sd~
to
K"
n
ex .
*
i ex = 0, f
Yo € Y '
Let
to the inessential
IA satisfi es
i *ex
O.
Let
is homotopic to a constant
f iA and a constant map .
Since
A has t he homotopy extens ion pr oper t y in
'Wi t h re spect to
G extends to a map
X
Then
Y,
and F (l, x) = yo
for all
x
€
A.
F :I X X --> Y sati s :t'ying
The map
g
yO '
Thus
defined by
F(O,x )=fx
g (x) = F (l ,x)
for s ome n. f
and maps
A to
" (KjK , ) is countable . 21 3
?l ?
A t o a point,
G:I X A --> Y be a homotopy of
there are only f i nitely many s implic i al maps f rom i t 1'ollo\IS that
maps
IT( XjA jY )
ex E IT (XjY )
Since
A to Borne point
i s homotopic to
Since for each
by I X X.
(v Ir. 4.5) every cont inuo us map
is homotopiC to a simpl i c ial map of
On the other hand , suppo se
f:X --> Y represent
Z, and t hen '.e
to extend
sends every clas s in
is arc wis6:
map of
can use the r e traction
i *P*
A ---> X ---> X/A
is s olid for any
class .
X
J{
to a point . the composition
I
is pI'o j e ction ont o the spa ce obtained f rom
I X X is
Proof':
normal, and
IT(A jY)
I t fDllows
consis ts of one
" (XjY)
"(XjY ) 2....->
g
define s a map
h:XjA --> Y
such that
hp
Let
Exe rcise.
It then follows that if
g.
v
be a point of
Sl.
h
~€
re presents
Prove that
has the absolute homotopy extension Property in
S\13l
rr( x/ A;Y) , p*~
SIx v U v x Sl
= a.
C SIx
S
Proof : Embed
I X En
( XO ' Xl'
xn )
"'J
I X En
Sl X Sl.
as the subset of
0 X En U I X Sn-l
to
Then
PROPOSITION.
Let
X,A) be a pair of spaces with
A
A has the absolute homotopy extension llroperty in
I x A U 0 x X
i s a retract of
c l osed in
X.
X if and only if
I X X.
the r etraction of
K:n
Kn ULand
each
Proof:
f:X
Suppose
0 X
U 0 X X
X
c..
A
has the absol.ute homotopy extension property in
I X A U 0 X X be the obvious embedd ing.
b e the inclusion. G
I X A U 0 X X ,- > I
A in
and
A U 0 x X.
X
I XX
onto
x
€
A
and a homotopy
are given,
then
f
and
G:I X A _> I X A together define the identity map I X A U 0 X X, t hi s map extends to give G:I X A -> Y
[!;
such that
G(O,x)
fx
for
H:I X A U 0 X X _> Y.
define a map
Extend
Hover
2.4.
THEOREM: ([,he Homotopy Ext ension Theorem for Ree;ular Complexes).
I X X by compOSing with the retraction
r
n
L.
on all of
n
n
r
wi th respect to the cells of
lli , r n
is an obvious retraction
of
Then
r~
n > 0, let
retract's
union of all of the
m < n, r
rO
Mn
onto
m~ppings
is well-defined.
Vb
r' :M
n n
M_
define the function
For
n
0
t here
M_ l •
onto
-->
uSing 2.5 .
is given by the weak topology
M
n
i s continuoup.
n
is
e
If'
I Xe
defined on
n
n, let
The n for
-> j.1n-l as follows.
:M
Since the topology on
1ft.
n
For each
I X L U 0 X K.
­
be the composition
Iii
-1
rOrl···r n ·
Define r: I X K - -> M-1 by taking the
l ·
r'n'
Since
The map
r
r'
m
and
r'
n
agre e on
is continuous because
the weak topology with r espect to closed cells.
r.
K -
These retractions together with the identity map on M _1.
n
X'
Then project
I X L U 0 X K by applyine;
M1
n
K-L, we have the retraction
For each
r : IXX -> I X A U 0 X X is a retraction.
I X K U 0 X K, with
1.
( 2,0,0, ••• ,0).
2 . 5 to the individual cells of
M
n
?
i~l x~
and
1
I X K onto
we define as retraction
n-cell of
an
I X A U 0 X X.
On the other hand, Suppose
f:X - > Y
Let
USing the homotopy extension property for X with respect to the space
a retraction of
If
f
Let
X.
n > 0
~
.
from the pOlnt
We obtain a retraction of
2·3·
consisting of all points
n
Xo
0 ~
such that
Rn+.L
1'1
m
for
I X K has
This completes the proof'
of 2.4.
If
fK,L
is a pair of regular compleses, then
extension property in
L
has the absolute homotopy
It follows from the homotopy extens i on theorem that extend ibility of
a map depends only on its homotopy class.
before 2 .1, if
Proof: He show that
I XL U0 XK
is a retract of
I X K.
P"ropos i tion
2·3 then applies to give the theorem.
2·5·
LEJ·ir.1A.
is a retract of
En be the closed unit
into
I X En.
n-ball.
Then
K and K'
are finite simplicial complexes there are only
countably many extension problems involving maps of a subcompl ex
f
Let
Thus, in the light of the remarks
K.
K'.
Also, if a map
extends to all of
L
COROLLARY.
f : L --> Sn
K
f:L --> Y is homotopic to a constant map, then
K.
I X Sn-IV 0 X En
2.6.
of
If the dimension of
is extendible over
K.
214
21 5
L
is l ess than
n, any map
Proof :
Let
h:
Sd~ --->
Sn
be a simplicial approximation to
some simplicial decomposition of' Sn .
IT
e
is an n- simplex of
Sn, and
is a point in the interior of e, then h maps Sd~ into Sn _p.
n
Sn_p is contractible in 8
and s o h is homotopic to a constant .
But
P
Exercise.
t heorem is f al se in general .
2
there is a homotopy F: I x S2 --- > 8
~.
z
integer .
of the identity and
This means
Let
Let
1
X be the subspace of
1
n
of the origin t ogether with all points of the form
R
with
consisting
n
a positive
Y = X U [ -1,0 ] .
A be the origin, and set
She,,, that in the group of linear fractional transformations of
the Ri emann Shpere, the identity is homotopic to the map
f or each
We give an example to show that the conclusion of the homotopy extension
f, using
A
- - - - +- -- -..l.~----~_ _ _ _ __
-1 o
~1
'-------------v---­
1
such that
z
X
-.~~. \
t, F ( t,z) is a linear fractional transformation . "'----- y
------­
2
Exercise . Define the degree of a map f' : 8 ---> 8 2 to be the unique integer
n
nu, ~heTe such that f*( u)
,,,ith
P
and
U
of t he polynomials
ha\~ no common roots .
P
I f the degree of
coefficient 1 .
be a polynomial in z
n- l
.
Let p ( z) = zn + ~ a.z l
i=O
mapping of
S2
to i tself.
F (O, z)
n
n-l
F ( t,z) = zn + (l - t ) ~ ai:
i=O
is a homotopy of P as a
p ( z) and F ( l:z) = zn.
i n the s econd exercise above that the degree of the map
Therefore if
n > 0, P
z ___> zn
is
i s not homotopic to a constant, by propel' ty ~
cellu~ar homology the ory, sect ion VI. 5 .
poi nt to zero .
It was shown
In other ~ords,
P
Thus
p
n.
of
y
to
to
0
-1 .
vie claim that there is no homo topy of
f
to
a map wh i ch sends 0 to - 1 .
F ( t,x)
X, and
property in
f
must satisfy
Y is a f inite Simplicial complex,
If
X and
I X X are normal, then
X with respect to
F ( t,x) =fx
m\iHA .
If
A
A
is closed
has the homotopy e xtension
y.
K is a finite Si mplicial comple x and
then there exis ts an open neighborhood
of
of
X
t .
THEOREN .
2.8.
f lA
This is so because, except at the origin,
is d iscrete, and so any homotopy
in
Then
U of
L
L
such that
is a
L
subcom~lex,
is a retract
U.
is onto and so maps some
must have a roo t .
Fe have sketched a
proof of' the fundarnental theorem of algebr a using methods of algebraic topolog
Proof of 2 . 8 ;
L, and let
s
Let
b e the set of vertices of
T be the set of ve rtices of
U =
ra E K
(
216
is homotopic in
f :X ---> Y is the inclusion.
the map sending
2 ·7·
with leading
Define
F
Suppose
for all
z •
of degl'ee
l
I f we define PCco) = co and F Ct,oo) = co then
is the maximum of t:
f
n
g(z )
p(z )
= p ( z)/Q( z)
n, eY~ibit a homotopy f ~ g
is
f
fe z)
Q, assuming that P and Q
and
in t he Space of rational :f'\mctions, where
Let
IT
pol ynomials, shmi' that the degree of
Q.
algebraic degrees
Example .
9
H2 (8-) .
generates
I
.E
VES
a ( v) >
~
J
K not in
L.
K which lie in
We define
a
If
€
L,
then
E_ a(v)
ves
1,
it meets every simplex of
U onto
L
so
LeU.
The set
in an open set,
K
U
is open because
We define a retraction of Proof of 2.9:
vertices of
h: X
by s etting
--> a.
retraction
(r(a))(v ) = (
a(v)
if'
v'~S
o
S
V €
if'
€
V
T
If
a
is a simplex of
of' t-wo of its face s,
Int a
lies on a
~ of T.
uni~ue
Remark:
t.r(a).
U.
a
-where
T
TO,.',
may be exmessed as the join a
n L.
Each point
U
€
n
line segment joining a point of T' to a point We note that the retraction
r
is homotopic to the identity
For example, -we may def'ine a homotopy
F
rCa)
lies in
F(t,x)=(l-t)a +
by
a
Since
a
is solid, -we may extend
By 2.8, there exists a neighborhood
if it is homotopic to the identity on
F:I X X - > X -with
mapping
A
F(O,x)
is called a deformation retract of
d eformation retract of
U of Y
h-~ and define
LEMNA,
normal space and
Let
A
g:V
->
is closed in I X X, G extends to a mapping H:V - -> Y
V is open and contains
the compactness of
containing
x
-which is open
W = x~ !'l(x).
t hat
X
Then
X
X ---> Y
I XA U
I, that for each
° X X.
x €
It is easy to see , using
A there exi sts a set
X and such that
I X N(X) ~ V.
is normal there exists a Urysobn function
heAl = 1
I X A U 0 X X
C
E
= O.
heX - H)
and
C
I X I~
C
Let
By 2 . 9, since
be given.
is an open set containing
I'T
--> A is a deformation retraction A
N(x)
Let
and I X A .c..- I X '.1
h:X --> [0, 1]
E = {(t,x)lo
~
t
~
hex)}.
C
V.
such
Then V, and -we define a retraction r:I X X
--> E (h(x),x) if t ~ hex)
i f t ~ hex)
( t,x)
X; that is, if there exists a x and
X.
F(l,x) = rx,
In this case In the proof of 2 .8, L
Y be a finite simplicial complex.
a closed subspac e of
If
X
X, then for each map
V of A and an extension of
Then
Hr:I X X ___> Y extends
G and the proof of 2·7 is complete.
is a f
is a
f :A
The exerc i ses in Chapter I In particular, -we note that O. Hanner proved (Archiv For i'iathematik, 1951 ) that a locally f inite --> Y
to a map
s implicial complex is an absolute neighborhood retract (abbreviated ANR). A me tr i c space
X
is an ANR if given an embedding of
ubspa ce of a metric spB,ce
218
to
Y
I X AU 0 X X
°
The hypotheses of 2.7 can be relaxed.
Y.
->
by -with
U. there is an open nieghborhood
g:V
a and a
G:I X A U
of Ru 's book, Homotopy Theory, provide references.
2,9·
in
Let
r(t,x)
r:X
to a mapping
f
Proof of 2 .7:
U.
In general, a retraction
V
Then set
r:U --> y.
on the
a
rho
Since
This -works because the line segment joini ng a point
its image
a
rea) = ~. We set
mapping on
a
K, then
y.
as a. subcomplex of the simplex
Y
be the composition
where
vie may describe the retraction as follo-ws.
Regard
Z, then
X
219
X as a c.l osed is a retract of some open neighborhood
i n Z.
Lemma 2. 9 i mpl i e s tha t.
fi ni te c ompl e x i s an ANR.
8!
r esul t i t f ollows that 2 .7 hol d s "it h
2.10. DETn U TION•
A spa ce
X
From Hanner 's
Then for
Y a l oc all y f init,e simplicial comple!
i s contra cti ble i f the ident:it y map of
mapping
g
aO
i nto
h
gf.
(X,a ) ~ (X,A)
O
wi th the i nclus i on
(Y, yO)
and s o
ha ,
The map
ha s t he quot i ent topolo~J i nduced b y
(Y, yO)
continuous since
compos i ng
X
gyO
a E A, gfa
(X,A) •
We show that
g
g
is
f.
By
we may r egard
as
g
i s a homotopy i nvers e
i s homotopi c to a constant map .
Exer c i se .
Prove that i f
HS (X )
q
f or' q > O.
"" 0
X
i s contr a ct i ble t hen
fo r
~ (X) "" Z a nd
Firs t,
(Hint : ShO"I, tha t
c ontra ctible.
X
ident i ty .
I X X is no rmal, then
i s solid and
Also , i f
ex
i s any spa ce, then t he coue
X
gfx
X
H( t , y)
is
= fG ( t , f
(See V .2 f or
we have
A
X.
i s closed i n
map
X
i s contractible .
b y ident i fyi ng
f:(X ,A)
Proof :
A
Since
- -> (y,Yo)
A
If
A to t he po i nt
(Y, yO)
- -> X wh ich extend s F and sat i sfie s G(O,x)
Define
h: X -
Not i ng that the i de ntif i cat ion map
g
mapping
and the
f G(t, f -ly )
=
f:(X,A)
i nto
A.)
i s one - one on
f
-1
f
YO ' the n
Y
YO' Thus
H
Ne xt, define
X - A, H( t, y)
i s we ll
A, b u t since G( t , A )~ A
Y
i s s i ngle valued and leaves
yo fixe d .
H i s continuous , ,Ie us e t he f act t hat t he topol ogy on
> I X y.
I X Y
(For 1:1e have
the follml ing d iagram :
(I X X, I X A) ~>
h
~l
aO A
€ .
Since
maPS A
x
f or all
t o the poi nt
--> (Y , yo)
f
H
x€X.
a '
O
i s one - one on
Since
(Y, yO )
f G i s contiuuous, the fact that
wiGh respe ct to
H(O ,y)
H
Y to X b y
t
X f
( I X Y, I X YO )~>
A
(X, A)
X, there e x i sts a homot opy
G:I X X
Then
if
F: I X A - -> A
A to a po i nt
ha s t he absolut e homotopy extens ion property in
X - A, we de f i ne
Si nce
y ).
is a homotopy equival ence of pair s .
by h( x )=G(l, x ).
gf
a proof , se e Hilton , An Introduc t ion to Homotopy Theory, p. 109 · )
i s the pair obtai ned
i s contrac t ib le t here e xists a homotopy
X
i s a homotopy of
yo' t hen t he ident i f ic at i on
of the identity to the cons tant map s e nd i ng
>
G
i s the qu oti e nt topo l ogy i nduced by the map 1 X 1':1 X X -
i s a pair of spaces such t hat
and has t he ab so lute homotopy extens i on pr ope rty i n
Suppose also that
from (X,A )
(X,A )
,- 1 ,
YEY - YO'
To sho\, t hat
Suppose that
so
I X A
G maps
(No t e t hat
d e fined fo r
a defi nition ) i s contra ctible.
2 . 11. PROPOSITION.
= G(l ,x) ,
lLx
contrac t i ble space has the
8!
homotopy t ype of a poi nt . )
Note t ha t i f
f.
~
implies that
1 Xf
ff
f'G( O, f'-ly)
is a homotopy of
-1
y: y
and I X Y has t he quot i ent topo l ogy
H is conti nuouS . Si nce
H( l ,y). = :fG (l , f1Y ) = fhi' -1 Y
fg and the ident i ty on
(Y,yO) '
f gy,
Th is compl ete s t he proof of 2.11 .
g( y )
:
hf
f
-1
a
y
O
f or'
y
E
if
y
= yo
Y - YO
Examples : 1.
Let
K
b e a fiui t e conne cted
A b e a maximal tree , 2.11 shows that
??()
K
I - dimen s i onal complex .
Le t ting
i s homotopi c all y equival e nt to a
wedge of c ircles .
2.
Let
3. Invariance of Domain
K be the 2 -torus 1-lith t'Wo disc s adjoined along t he
In this section "We prove
genera tors of the fundamental group .
Then if
A
of the n-spher e.
d i scs , p inch i ng
A
t o a pOi nt shows that
K
Our principal result i s the theorem on invarianc e of
is homotopically equival e nt
domain which states t hat if
to the 2- sphere.
then m
2 .12. PROPOSITION.
Let
(X,A)
be a pair of spaces s uch t hat
A
is
n.
1i1O)
Then the identification map U
C
m
R
and V e
Rn
are homeomorphic open sets
The proofs of the follo"Wing classical theorem and corollary
"Were given in the Introduction.
X a~d has the absolute homo topy extension property in X. closed in
some classical the orems about the topology
is the union of t hese
3 .1 .
de fi ned in 2.11 induce s
THEOREM.
If n
~ 1 , then the unit Cn - 1 ) sphere Sn-l is not
a retr act of the closed unit n-ball
En.
isomorphisms of Si ngular homology groups.
Proof:
Let
j: I X A
C
3 .2 .
CA
denote the inc I us ion of'
COROLLARY. (The Brou"Wer Fixed-Point Theorem. )
Any continuous map
I X A, as the bottom
of t he closed unit n-ball En into itself has a fixed point.
half of the cone CA.
X l\CA
Consider t he following commutative diagram , where
is the adj unc tion space of the inc l usion
the cone
k
of'
A as the base of
CA:
Note that the proof of 3 .1 utilizes the concept of induced homology
homomQrphism and thus depends upon Theorem VI 4.1.
Since \ore are going t o use
3 .1 to prove r es ults "Which will lead t o proofs of the r edundant restrictions,
(X , A)
2->
(X U I X A, I X A)
..iT....>
(x Uk
r~ CAl
it i s essential that we remark that the proof of 3.1 may be obtained using
simplic ial homology theory alone.
We verified at the start that simplic ial
complexes s atisfy the redundant r estr i ct ions.
f
(Y'YO )
Vr.4 .1 , stated f or simplicial complexes, to derive Theorem 3.1 above.
If a space
The maps are given as follows:
by
j:I X A
C
to t he point
and
J
CA, and
YO '
g
i
is the inclusion,
J
is a proper excision.
i
g is a homotopy eqUivalence.
USing 2.3, it is e asy to see that
is omorphisms of Singular homology groups.
CA has
X Uk CA, and so, by 2.11,
Thus the composition
f
=
is homeomorphic to [K [
for some r egular complex
K,
gJi induces
thus
X
is said to be triangulable and
K
is called a triangulation of
X.
3 .3.
THEOREM. (Characterization of dime nsion of a finite regular complex) CA
is a homotopy equivalence of pairs,
the absol ute homotopy ext ension property in
X
is induced
is the projection defined by identifying
The inclusion
Tbus"We may us e Theorem
Let
K be a finite regular complex .
equivale nt for each
The n the foll o"Wing t"Wo statements are n: (i ). The dimenSion of
K
( ii l .For each closed set
i s less than or equal to
A
C iKi,
every mapping
,..,,..,-,
n. f:A --> Sn is extendible Theorem 3.4 follo'..: s :;"rom the follo'\-ling theorem.
[K [ . over
Sta tement (ii) expres s es a topological proper ty. Thus the dimensions of two
triangulations of IKI are equal and dimension is a topological property of
triang ulable spaces.
=>
Proof that (i)
with
A closed i n
A and a map
K.
in
U.
Set
L
K
g[L
n-
pl
=
L
n-
1.
Using the compactness of
so finely the-I-, each simplex of
A.
Then
SdsK
Then
Thus we may define
g iL
h:K
and g'
_ >
Sn
meeting
of
A,
A lies
consisting of all closed
extends
f.
Let
By 2. 6, we may extend
Sd s Y .
M is a subcomplex of
Then
s
Sd K
LCUand ( g ' L):L ___> Sn
1 to a map g ':M ---> Sn.
nL
f.
equal to the subcomplex of
simplices meeting
M = Ln- 1 U -K- L.
~ich extends
U
agree on
L
n­
1
f:X ---> Sn
a finite subset
is given
Then by 2.9, there e xists an open neighborhood
g: U ---> Sn
we may subdivide
f ~ ___>G n
Suppose dim K ~ n, and that
since
mapping
g(x)
xe:L
g ' ( x)
x e:101
Suppose dim
K, and let
a
g :cr ---> Sn
f:En+l
--->
K
(fIS
n
~ n+l
n
h:K ---> S , then hf:E
---> S
n
).
some point
Let
a
g
be an ~ n+l)-ce l1 of
n~
O.
If'
X does not separate
Sn
onto
a.
Define
were extendible to a map
would be a retraction.
This is impossible ,
be a closed proper subset of
S
n+l
3.5:
3.3,
'KI
--->
=
and let
f:X
F'
Since
_>
F or
Sn.
p
Sn+l
Sn+l - X
Since
which
is counect
In either case 3.5 asserts
(sn+l - x )
o
and!
g'
of the mapping
I' for
is contractible, g, and hence
Sn+l
.
,then e v ery mapplng
3.3,
gives a mapping
n
l
of
F
each
xae:V
i
i
K
not in
--> a.
a
LUK
n
UK) - > Sn.
Then
n
of course extends
x e:a
a
L, choose a point
n,
is
n
g
f.
and a
The composition of these retractions with
h' : (K - {x )) _> Sn.
a
i
Each
Bn+l
n
n
x )
which lies in
a homeomorph
g': (L
h:L U K - -- -> Sn, and h
:.1.<
-
{V.
{x}.
a
p oints of
is
-
to a map
n
(n+l)-cell a of
ra: CG
Since the dimension of
f.
(g il L)
extend
Sn+l ,where
f :X ---> S
X~ L, and such that there exists an extension
of the mapping
retraction
similar to the one used in the proof of
we take (K,L) to be a pair of finite regular complexes such
define a map
For each
arg~nt
USing an
Sn
homotopically tri v ial .
224
Sn+l
xoe:sn+l - X.
we may, by
h
X
Let X be a closed proper subs et of
O
Now let
THEOREM. (Borsuk) Let
of the
f, is homotopic to a constant map.
and the proof is complete.
3·4.
Sn+l - X,
g :( s n+l - F') ---> Sn
the existence of an extension g:(Sn+l - x ) ___>.Sn
that
If
is
by
be a homeomorphism sending
to be the inverse of
and an extension
has just one point, and so
Theorem
> n.
F' of F
does not separate
F
F
If
is an arbitrary continuous mapping, then there exists
3.5 => 3.4:
Proof that
g:L
Proof that (i i ) ~ ( i):
Sn+l
be a closed proper subset of
f.
Proof of
hex) X
a set consisting of exac t ly one point from each component of
and if
(ii):
Let
THEOREvl.
3·5·
­
< kI
be the components of
Sn+l - X
x e:V. is an interior point. Let y.
a l l
V.•
l
Since
V.
-l
be the point
js open and connected, there exists
of the closed unit
and Yi' and such that
which contain
n+l
Bi
(n+l)-ball containing simultaneously
~ V •
i
'Let
r.
l
be a retraction of
n+l
Bi
- Yi
·n+l
Bi
onto the boundary
Then the r e tractions
hll: (Sn+l _ (y. ) )
l
Let
Sn+l
Y
together with the mapping
r.
l
_ ,> Sn which extends f.
• n+l
B. •
l
H(O,x)
= gXo(X)
i
(hIIY)
give a mapping
Since
Xl
h"
g
U
Int
To be precise,
is defined
by
Xl
H(l,x) = g
and
h'x
lies outside of the ball
:X ----> Sn
Thus
B
and
gxo
containing
g
are homotopic.
Xl
X, the image of
Thus
is contained in a hemisphere.
Now s uppose
for
x €
Y
hlr.x for
x €
n+l
Bi
l
may assume that
3.6. THEOREIl . Let
g
X be a compact subset of
lies in the unbounded component of
g :X -->
-Xo
Sn
Xo
Xo
the ill1it ball Bn+l.
The Theorem is now proved.
R
n+l
•
.
A pOlnt
€ R
O
X
n+l
- X
Rn+l - X if and only if the mapping
defined by
­
xo
-b,
,x,
(x)
Suppose that the map
g, , and so also g ,
Xo
Xl
X.
Thus
are normal.
"n+l
g:B
->
n
S
Since
Suppose
Xo
lies in the unbounded component
U of Rn+l -
and that
g
Xo
We
X is contained in
,which is defined by
Xo
to X U V.
Note that the boundary of
X U V is compact.
Clearly
VIe must verify that
V is
X U V and
Consequently
X is closed in X U V, and so we may
g':X U V ~> Sn
which extends
g.
VIe define
by
is homotopically trivial.
Proof:
X UV
V of Rn+l - X.
Then we shall apply the homotopy extension
of 2.7 are satisfied.
apply 2. 7 to obtain a mapping
Ix - Xo I
o
is the origin of Rn+l
is inessential.
h~~otheses
the
I X (X U V)
x - Xo
lies in a bounded component
theorem (2.7) to obtain an extension of g
contained in
~ (x)
g"x f~xTxT
x. if
x €
X U V
if
X €
Bn+l - V
X is compact there exists a solid ball B about the origin containing Choose a point
there exists a path
and
(xl.
'
is homot-opic to a constant map.
h"x
x.
Xl
h(t)€ U
Xl€U
lying outside
h: [0,1]
for every
t €
B.
Since
U
is open and connected,
_> Rn+l such that h(O) = x ' hell = Xl'
o
[0,1].
Define a homotopy
H(t,x)
by
Then
g"
g
a(t,x)
Ix- h(t)1
226
X
€
X, tdO,l]
Bn+l
V is a neighborhood of the origin xO.
onto
Sn, which is impossible .
But
Consequently
is essential and the proof is complete.
Xo
X
is continuous since
is a retraction of
3·7·
x - h(t)
g"
THEOREM. (Borsuk) If
separates
Sn+l
wh i ch is essential.
X
is a closed proper subset of
if and only if· there exists a mapping
Sn+ll
f::l(
--_>
~
Sn
Proof:
If
X separates
components of
from
If
Xo
and
xl
be pOints in different
.
..
Under stereographlc proJectlon
p:S
n+l
n+l
--> R
3.1.1 . COROLLARY.
then
m
By theorem 3.6, the map
X does not separate
3.8. COROLLARY.
n
are homeomorphic open sets,
gp(xl):p:x:- > Sn
is essential.
n+l
S
,then apply Theorem
4 . Proofs of the Redundant Restr ictions.
3.4.
n
S.
h
Let
h:B - > Sn
Sn - h (n-l)
S
Then
be an embedding of the closed unit
K, if
h ( Bn - Sn-l )
has exactly two components,
'I"
e..
P(q,r )
is a
a
a.
be the following statement :
q-cell of K and
an
T
for all regular complexes
0, then
r-cell which meets
We prove
(n
Sn -hB).
and
n
B
Proof: Since
into
is contractible, so is
Sn~l is homotopic to a constant.
Sn _ heBn)
Sn. Therefore
so is h(Bn _ Sn-l).
embedding, by 3.7.
But
h(B ).
By 3.7,
is connected.
h(Sn-l)
n
Since
separates
It follows that
Sn
Thus any map of
n
h(B )
since
is connected,
h
is an
Sn _ h(Bn) , h(Bn _ Sn-l)
p (q,r )
Proof:
The proof is most conveniently given in three steps. are the
Step 1.
P(q,r)
Step 2 .
P(q+l,q)
Proof:
.~
Let
(Invariance of Domain)(Brouwer)
no
( r S),
n
of R
U is open in
Proof: Let
such that
hen)
and
R
x
n
~
h
.
U In
embeds
If
U is an open set
no
( r Sn ), then the
R
h-image of
+,¢.
nO-
Be.. U.
By 3.8,
is a neighborhood
of
U.
There exists a closed ball B containing
h(:B- B)
hx, and so
is open in
h(U)
R
n
is open in
(or Sn).
R
n
of f
Thus
(or Sn).
3.10. COROLLARY.
for any
A none~ty open set of
k < n.
R
Can not be embedded in Rk
be a
r.
Let
q
f:S
=--> cr
'1".
and
f
K, and let
sq.
onto
0-
is closed, T
n o- +¢,
St ep 3 .
is true.
If
we have
'I"
na
n 0-
n
cr.
-1
f
~ n
Since
in
'I"
p (q,r) for all
By Step I, p(q + l,r)
Let
a
be
a
ts c losed
r, then
t he closures of the q-cells which intersect
mee ts only finitely many cells, by
closed cell:; and so
A is closed.
1.4.4.
Since
228
229
'1",
na
'I"
'I" 1'-1'1" the restriction
'I"
is open in
is connected,
'l"e.. a.
is true for all
K.
to
and
is homeomorphic
'I"
Since
p(q + 1, r)
( q+l)-cell of
f
crC-
But
'1".
in ~.
and so
= 'I"
a e.. Kq
Since
The restriction of
'I"
is an embedding of
'I"
with q:::: r. be a q- cell such that T
By the theorem on invariance of domain (3.9),
Since
'I"
-1
f-l'l"
r
be a hOIlleomorphism.
is open in
q
and
q
q. (q+l)-cell of
K , i-l'l"
to
Proof:
n
and
is true for all
is a homeomorphism of
q
to R •
(or Sn).
be a point of
q
is vacuously true for all
a
is open in
3.9. COROLLARY.
is true for all
h(B )
does not separate
Bn _ Sn-l
R.R.l.
n
Sn _ h(B n ).
components of
x
V e.. R
n.
Let
.
n-ball In
m
U C...- R
and
If
projects into a bounded component of the projection of
xo' xl
Sn+l _ X.
Sn+l, let
n+l
S
- X.
for all
r ~ q+l.
Define
a.
Thus
r.
By Step 2, P(q+l,q)
A to be the union of
Since
cr
is compact, it
A is a finite union of
P( q+l,q) is true, each of the
q- cells which meets
cr C
_ AUK q- 1 0
0
aC_ A U Kr
T,le assert that
by a descending induction on
vle
A then
,.
l'
Let
for all
~
Suppose that r
,.
r.
This is proved
q-l
and that
K.
be an r-cell of
We ne}.:t prove R.R.2.
By construction If
,.
that for each cell
crCA
UKr .
_
,.
A, and so, by P(q, r),
of
,.
cr CA.
n
a = jJ,
Then since
p
o.
a
Thus
cr
and so
=
A.
aC A U Kr _l •
Let,.
Thus
cr
be an r-cell of
K
is a union of closed q-cells, ,.
contained in
A.
By P(q,r),,.
Thus p( q+l,r) is true for all
C
A UK
r
for all
such that
r
by Step 1.
4.1. lEMMA.
Let
is contained in
union of all cells
Proof:
Let
1"
of
n cr
of
So p and so is contained in
o. If
0, choose a cell
show that
1"
p.
meet s
,.
q.
P(O.r) is 1"
)
a cell of K.
o.
Then
Let
does not contain
o.
By R.R.l,
P
does
Po < P theR Po
CP
Thus U is open as desired. and so
Suppose
1"
d.im p, and so 1"
is open in
We
is an arbitrary cell of
p
Cp.
K
one other n-cell
1"2
of
IK.I
Let
1"
p contains
o. whose closures
K
meets
p
in an open
The theorem on invariance of domain
0 be an (n-l)-cell of K.
of
0
Thus
p.
Thus
K such n. "1
that
IKI.
K whose closures
dimension.
~~imal
Using R.R.l it follows that 1"
0, dim,.
Q.E.D is an n-cell
U is open.
of
Among those cells of
is of' maximal dimension among the cells of
Now let
U be the
~
is open in IK I.
1"
set, and so
Po does not intersect o.
Since
Thus Po C IKI-u. ByR.R.l,p is the union of the proper i acfS ot p.
each proper face is contained in IKI-u, pC IKI-u. So IKI-u is a union of the
Therefore [KI-u is a 8ubcomplex. cells of a collection satisfying 1.4.2.
not intersect
contain
K.
0 be a q-cell of
implies that dim 1"
K such that
p
Let
contain
Step 3 provides the inductive step.
Then
Proof:
Since
We show that the complement of U is a sub complex and thus is closed.
P be a cell of IKI-u.
Each q-cell
K is a face of at least one n- cell, and each ( n-l )- cell is a face of
that
jJ. r.
be a regular complex, 0
K
,.
r•
meets some closed q-cell
The proof of R.R.l is completed by an induction on
true for all
be a regular complex on the n-sphere .
Let K
,.
be a homeomorphism.
f restricted to f- l ,. is an embedding of f- l ,. in ,. . But 'r is homeomorphic
r
to R , and r < q-l < q . By invariance of domain ( 3.10), f- l ,. is empty.
Thus
is a regular subcomplex. exactly two n-cells.
is open in AUK. Let
r
Then f- l ,. is open in sq. AB in Step 2,
a
a
of a regular complex ,
interesects
n A = jJ. We claim that ,. n a :: jJ. Since
K,
and A is closed,
r
0
Nm, that we have proved R.R.l, 1.4.2 implies
Thus the following theorem implies R.R.2;
4. 2. THEOREM.
Suppose that
is open in
f: sq -->
r.
AC cr.
Thus
intersects a closed q-cell contained in
,. C A Ca.
,.
cr C A U Kr­
s how that
cro
is contained in
such that
0<1"1'
K such that
is a face only of the n-cell
n
-
'i'le have already proved that there
We show that there is at least
0 < 1"2'
1"1'
Suppose, to the contrary,
By 4.1, 1"1 U 0
n-l
S
.
f: E --> 1"1
be a homeomorphism carrylllg
is open in
onto
.
1"1'
Let
n-l
U = f - 1 oC Sn-l • Regarding En as a subset of Rn , the set (n
E - S
) UU
n
is not open in R.
.
It follows by invarlance
of domain that f [( En - Sn-l )U U] ="lUo
is not open in IKI.
This is a contradiction, and so
n-cell
1"2 distinct from
0
is a face of some
1"1'
He complete the proof by showing that
only n- cells having
o
as a face.
1"1
and
1"2
ar'e the Any nei ghborhood of a face of a cell contains points in the interior of the cell, so it is clearly sufficient
230
23 1
to ShOH that
is a neighborhb.od of
1"1 U a U 'r 2
Son-l
be a homeomorphism carry i ng
We define a homeomorphism
h
such that
E+
a C heE l.
Let
open equatorial disc
with
D.
Finally, define
n
E~
E
be the upper half of E
E
into
*
Cla
f;l ( a)
'rl
~
onto
fl a.
-1
with apex at the origin
f2-1 a
and such that
f2h2 [D
flhl[D .
r.,le define
-
carrying
->
*
C a
2
D
carryir
h:E-> 'rl U a U or;:
a.
h
Thus
maps
E
if x e: E+
does not separate
'rl U a U 'r 2
Sn.
@_l
C1· a
is connected.
of
K
can be joined by a path of n-cells.
n-cell of
K.
Set
1
EB
Eon
*
C a
f2 a
To see this, let
a by a path of n-cell.s .
K-K _2 ·
n
a bean
Then
A - K _2
n
is both
A = K-K _ ·
n 2
Consequently
K is orientable we are going to use V1.4.9
and carries
D onto
[K[.
The
n.
Let
C
be the statement that Theorem 4.1 is satisfied in dimensions
Let
Dn
be the statement that all regular complexes of dimension < n
n
can be oriented (R.R.3).
that for any complex
By Theorem 11.5·6,
C => D l'
n
n+
K of dimension at most
n+l
For in proving
there exists an incidence
K, we only need to know that the boundary of each q-cell of
(q-l)-circuit for
q
Dn+l => Cn+l .
~
n+l.
groups f or
In detail, if
K, and by topological invariance,
H l(Sn+l) ~ Z.
n+
Thus
K is indeed orientable.
C
n
11.5.6, 4.1 implies R.R.3.
is true for all
n.
K is a complex on
H l(K)
n+
Since
is isomorphic to
C
l
is obviously
This proves 4.1.
As in
The proofs of the redundant restrictions on
regular complexes are complete.
K
Using the theorem of topologica:
the (n+l)-sphere, then by R.R.l, R.R.2, and D l' we can define homology
n+
true, it follows that
~ @_l ~
K 2
n-
equal to the union of the closures of all n-cells
A
invariance, we see that
a
is of dimension n-2,
From this it follows very quickly that any two n-cells
which can be connected to
is an orientable
~.
)/
E
f
K _2
n
Consequently the union of the n- and (n-l)-cells
K
function for
En
Since
K is
(Topological Invariance) and so we have to be a little bit careful.
aC h(E), and by invariance of domain, h(E) is open in
proof is complete.
K.
of
if x e: E
homeomorphically into
Then
we know from the proof of 2.6 and from Borsuk's Theorem (3·7) that
<
Then
Sn.
n-circuit.
In proving that
f2h2x
K be a re G~ar complex on
The preceding proposition states that every (n-l}-cell of
open and closed in
by
h:X: = ( fl hl x
Let
the face of precisely two n-cells of
together
with apex at the
~:E
K is an orientable
Proof:
Cla - (origin} is homeomorphic to the product
Similar l y, there exists a homeomorphism
THEOREM. (Lemma 11.5.3)
U a U 'r 2
together with the
be the lower half of E
be the cone on
4.1.
is a copy of En. )
*
a, and so there exists a homeomorphism hl:E + - > Cla
(0,1]
D onto
( Here
ala ~o be the cone on f~l (c )
C a
2
Then the set
EO'
'r 2 '
n
f2 :EO - > 'r 2
Let
sending an open n-ball
D, and let
origin of En, and let
of
onto
a.
Proof':
5. Regular Quasi Complexes
Let
~a'
r-cells,
On this section we prove that a r egular Cluasi complex is a complex.
a point
of
Le t
IEI,lMA.
be a regular Cluasi complex .
K
K such that the number of cells of
a
K contained in
is a union of finitely many cells of
Proof:
a be a cell of
Le t
is finite .
a
dimension
° or 1,
number of ce.lls of
is never used.
which meets
a
If
a
a.
contained in
T ea
Let
a = A.
By the inductive hypothesis, T
It
~.
Let
and so any cell contained in
Since the number of cells contained in
A
is the union of their closures,
of finit ely many cells.
of
IEMMA·
K.
Let
~
be
is
a.
be a metric for
Each
L be the set of limit points
Then if
whose di stance from
is closed .
Le t
L
Un
U ,n=l,2, ... , denote·s the
n
~, we have
is less t han
n
must contain all but a finite number
point .
Thus the interse ction of the Un' s contains all but countably many
of the
xa 's.
K .
is compact and so a ny
a
For some
a, xa
€
L.
But
a
a
x
So
a
IEMl-lA.
xa
infinite subset has a limit
€
~a'
and
~a
is an open set
x
ex which contains only one element
cannot be a limit point of
Let
S.
This contradiction establishes
K be a regular quasi complex .
Tben the number of cells of
Proof:
is a union of
Step 2 .
a
be a q-cell
be a cell of
a
K. cr is f inite . K contained in
The proof is by a double induction.
Q(O, r ) is clearly true for all
Let
Q(q,r)
be the statement Q( q ,o) is true for all q because the closure of a cell is compact and the zero skeleton
Step 3.
r. KO
is discrete. Suppose that Q(s, t)
is true for all pairs (~,t ) with s < q. Suppose that Q(q,t) is true for all t < r.
a
Let
t hat the number of r-cells contained i n the closure of each q- ce ll is finite. Step 1.
itself is a union
K b e a r egular quasi complex, and let
Then the number of r-cells which interSect
is thus a neighborhocd of
~
r
is finite,
a
This completes the proo f.
234 A).
€
is
xa's, because
Proof:
5.2.
= ( xa1a
S
:n:
We claim that some point
A.
€
of the
5. 3 .
A be the union of the
Since there are only finitely many (q-l)-c ells contained
a
a
the l emma .
finitely many cells.
and
for each
Thus any (Cl-l )- cell
T.
0,
L
L, since
1.1.1
the same is true for
in
d( x, y)
n Un
of S.
follows, just as in Step 3 of the proof of R.R.l, that
Then
Let
a
Recalling Step 2 of
closures of the finitely many (q-l)-cells which are contained in a.
A.
na
Choose
be a q-cell such that the
( Property 6) is not used either.)
a (q-l) -cell in
a
a.
Suppos e that the lemma
is finite.
a.
~
A, which intersect
is of
a
§4, we note that property 7) of Definition
a must be contained in
€
a
K, and suppose there are uncountably many
ranging over an index set
set of points in
in
contained in
K
the proof of R.R.l in
Let
S.
n
the lemma is obviously true.
is true for cells of dimension <q.
Then
K.
The proof is by induction on the dimension of a.
x
a
a limit point of the set
The key step in the proof, Lemma 5.3 below, is due to Dennis Sullivan.
5.1.
be a q-cell of
a
Let
a be a q-cell of K.
Since r < q, Q(r,t)
is countable .
is finite .
Le t
is true for · 11
By Lemma 5.1,
i
T
Then
Q(Cl, r ) is true. be an r-cell contained in
a. t, and so the number of cells contained in
T
is a fin ite union of cells.
(* )
The number of' r-cells contained in
a is infinite. Assume that closed subset
This will prove Q(~)r).
We will show that (* ) leads to a contradiction.
f
steps 1)2 and 3 prove the leIlllIla by double indu.c tion.
Since Q(q)t) is true for all t < r) the collection C of all cells
<r
of dimension
T Cal
i
which are contained in
CJ
is finite.
is a finite union of cells of the collection
ove:c all of
X.
The proof is given on page 83 of Dimension Theory .
;For each T-cell
C.
r-l
f :C ---> S
) there is an extension of
C and each mapping
There are only
X
with
lim sup Ti )
follows that
Sr-l
C
Sr-l ) and f
= identity :
Applying this theorem Sr-l ~> Sr-l
) it
is a :t'etract of lim sup T..
l
finitely many such unions.
many r-cells T )T ) .••
l 2
Thus (*) implies that there are infinitely
contained in
We show that this is impossible.
Xi€ T
i
for each
cells
T..
l
i.
CJ)
Let
all with a common boundary
(x . )
l
be a
use the following theorem, proved on page 82 of Dimension Theory. We define the limit su£erior of the cells T.) written lim sup T.)
l
l
K is closed) lim sup T.l r~ K.
r
r-skeleton of
K ) and any fundamental
r
se~uence
se~uences.
Since the
Since each r-ce ll of
in the cells
Ti
K
is
intersects
any (open) r-cell in at most one point) no fundamental sequence has a limit
point in any r-cell.
Ti
CKr _l
0
Thus lim sup 'r.l
By lemma 5. 2 ) K _
r 1
CJ.
closed sets of dimension less than
C Kr- l'
na
r.
Since
is closed) lim sup
CJ
is the union of countably many
th~orem
for
dimension (Hurewicz and Wallman) Dimension Theory) p. 30)) the dimension
of K 1
r-
na
is
< r-1.
This implies that dim(lim sup T.)
l
< r-1.
­
But this
is impossible) as shown by the following lemma.
5 .4.
LEMt.ffi.
f
C be a closed subset of the separable metric space
a mapping
C over which
f
of
C
to
Sr-l .
X) Then there is an open set containing can be extended. lim sup T ) X
CJ)
we may extend the
i
retraction obtained above to a retraction r of some open set U containing
Applying t his theorem with
lim sup T.·
l
to
Suppose that
C
~.l
Ti gives a retraction of
C U for
Thus for each
i
onto
T.
there exists a point
-
converges to some point
But each
y
since
lies outside of
Xi
some
contradicts the fact that
CJ
i.
r
sn-l) which is impossible by 3.1.
X.€
l
CU.
The se~uence
Ti - U.
is compact .
U) and since
lim sup Ti
The restriction of
U
(x.}
l
By definition) y
is open)
y ~ U.
€
lim sup T
This
Thus the proof of 5.4) and
.::T.:::h.:::e--=.s.:::t;::a.:::te.:;m=e.:;n.:::t~(,-*_)<-.:im=p.:::l:.:i:.:e.:::s--=t.:::h.:::a:.:t---=.t=h.:::e_d=im=e.:;n.:::s:.:l::..:
· o:.:n.:......:o:.:f:.-::l.:::im=-.:::s.:::U""P.......:.T i _is_
5 ·5.
COROLLARY TO LENHA 5·3.
A regular quas i complex satisfies R.R .1.
Eac h closed cell is a union of finitely many cells.
The common boundary
lim sup T .
i
r-l
S
of the cells
Ti
Suppose that the dimension of lim sup Ti
is a closed subset of
is
< r-l.
Theorem
3.3 of this chapter can be generalized as follows:
THEOREM.
and
Let
hence that of 5.3) is complete.
> r.
Proof:
THEOREM.
l
Thus) by the sum
He now CJ.
I~i th
of points
Such a sequence we call a fundamental sequence in the
to be the set of all limit points of fundamental
open in
se~uence
Next) an easy arGUlllent shows that lim sup Ti is closed in
Sr-l
A separable metric space X has dimension < r-l and only if for each
Proof:
5.6.
Proof:
This follows immediately from 5.3
THEOREH.
Let
together with 5.1.
A regular quasi complex i s a complex.
K be a regul~~ quasi complex.
given q) if A C Kq
We have to prove that for any
meets every closed c:e ll of dimension < q
in a closed
s e t, then
A
is closed in
K.
Thus we must prove that
A
meets any
Chapter IX
closed cell of
K
in a closed set.
Let
is a union of f initely many cells, by
5· 5·
union of finitely many closed cell s .
But
closed cells in a closed set .
Thus
A
a
n -;
be a cell of
In particular,
A
K.
Then
a n Kq
a
is a
HOMOLOGY
SKELET~L
intersects each of these
In thi s chapter we int roduoe t he concepts of skel etal
is a finite union of c l osed
decomposition of a apace and of s keletal homo logy groups.
sets, and so is c l osed .
The proof is complete .
We show that there is a natur a l isomorphi sm of ske le tal
homology groups and singul ar homology groupe of the under­
We also justify the methode and r e sults of
lying space.
Chapter III.
1.
1.1.
Ske l etal Homology.
DEFINITION.
~
fil t r ation of a topologic a l s pac e
is a n increasing sequence
Ixl
s ub spaces of
such that
X -
iX rl r-O, 1,
U
X
.. r
-
Ixl·
...J
'xl
of closed
A filtration X is
cal l ed a skeletal decomposition of the space
Ixl
if the
f ol l ow ing two conditions are satisfiedl
i) Fo r all
we set
ii)
Ixl
r ~O
X_ l - "
PROPOSITION.
+r,
S
H (X ,
q
r
xr _ l )
- 0;
for convenience.
has the weak topology with respect to the
subspacea
1. 2 .
and for all q
Xr •
If K is a (not necessarily regular)
S
S
Hq(Kr' Kr _1 ) - 0 for q t r, and Hq (K q ,K q- 1)
is isomorphic to the f ree abelian group generated by the
com plex. then
9 -ce11s of K.
239
,,~ ()
Proofs
Let r be given.
relative homeomorphism
Un -
V
, fJ txt Erl
W • Kr - U2 •
Ixl
1.3.
For each r-cell ~i o~ K ohoose a
r
r-l
_.
~il (E ,S
)--'{or-.~). Define
< 1/n1]
Let
V· Kr -
COROLLARY.
The sequence of skeletons ot a complex It
is a skeletal deco mposition of ths underlying apace
U3 and let
Exercisel
Give a proof of 1.2 using VlII.2.l2.
Then V and Ware open in Xr , and the olosure
of W is contained in V.
The i nclusion
V)
(X • Xr _l ) S; (X r '
r
condi t ion ii) of Definition 1.1.
V by
n • 0. 1, .. .
radial projection to K _ and dilates
r 1
U3
along radius
homology groups.
The inclusion
(X -'if. V-'if) C (It • V)
r
-
r
is
1.4.
(Xr-W, V-W) - (U 2 ' U2-U ) is
3
a pair to (U2 ' U2 ). Thus uS(r:, Kr 1)
q r
­
Finally. the pa ir
PROPOSITION.
8S
S - , U
. ) ' and the conolusion of the
Hq(U
2
2
proposition no. ~ollows from the tact that W
2 is a union
of disjoint closed r-cells, one for each r-ce1l ot K.
XIII
We next construct an explicit ieomorphism ~ mapping the
tree abel ian group on the q-oells of K to BS (X ,K
q q
q­1)'
eaoh q -cell tT of K, choose a relativ-e homeomorphism
fer' (E q , s q-1)--+(cr,Cr)
For every g!
Then we define
represented by
'S (cr)
[urI
to be the class in
aB(X
• Kq-1)
q q
~f..]E a~'(Eq, SQ.-I), where
is the inclusion map of the pair
(crI t!r)
It follows ~rom the proof of 1.2 that
isomorphism.
5 so
in
h:
m-
induced by includon.
T
~ a:(X n ).
n
­
aqs ( I xl) :::: Lim
as(X).
n
q n
Suppose
aSCx
)--to
q n
h:
n
u, a:(x.)
has the
I f.,e let i denote the inclusion o~
Xn , this me&os that
S
1* u
- v.
I t follows. using
as(X
)--+aqs ( Ixl ).
q n
Lim
n
We ahow that h i 8 an i 80morphism.
Y i8 a compaot set contained 1n
in Xn for 80me n.
First note that if
lxi, then Y i8 oontained
( One prOTes t hia using sequential com­
pactneas, aa in the first part of the proor of 1.4. 4.)
\r
(X,
q Kq_ 1) •
defined is an
Nezt, let
uE Hq(K)
xE
H:( Ixl)
be repr esented by
[u,
rJ
t
t]
where
and t is a map of t he f ini te complex X in t o
Sinoe rex ) i s compact, f maps K i nto Xn f or some n.
(u.
240
in
X e X . m5 n.
property II of singular homo l ogy theory, that hn( v)­
S
hn(i.u) - hm(u) . Thus the hn'e induce a homomorph1am
For
q
u,E Bq(E , sQ-1) .
and a generator
H:(IXI )
succeS80r
is isomorphic to
fB:(Xn)s q fixed,
Por each n we have a homomorphi8m
Proofs
satisfying
J be the direct eystem with admis8ib1e homo­
a proper exoision, and thus induces isomorphisms of singular
homotopy equivalent
Let
1II0rphisms induced by the inc l usions
vectors onto Kr--and so induces iso.morphisms of singular
Ixl
Let X be a filtration of the space
is a homotopy equival ence of pairs--a homotopy inverse maps
homology.
lIe!.
represen t s a clas 8 y in H~ (Xn)'
241
Clsarly
lx,.
Thus
hnY. x,
that
h
n
x • O.
XE KS( X)
Nex t, suppose that
and so h is onto.
q
n
is such
With each skeletal decomposition X we associate t he
Choose a represent ative [u, fJ fo r x, where
u is in Hq (K) and f mape the fiI.ite complex K i nto Xn .
The
For each q, Cq (X) - USCX
q q • Xq- 1)' and
class
hn x is represented by [u, jf]. where j denotes the
inc l usion of Xn inlXI. To say that h x • 0 means that
and we lIlay assume that
of Xn in
Thus
x
-
0,
L-+JxJ
a nd a map g:
X.'
m ~n.
suoh that
U:_l(X q _1 , Xq _2 )
monomorphism.
aq
incl usion, then
(l')~X - class of
Lilli
n
H:(X n )
(k* u, til
space Ixl.
-
j.~*.
S leX 1)--+
Bqq­
See the diagram below:
and so h 1s
'8.
c
Let X be a skel etal decomposition of the
h
l( X) • <-1 (X q _l , Xq-{.\'H
I ~_1
J.
Cq-2 (X) -
g. r, with 9< r,
Then for each pair of integers
Xr in Ixl.
b .j*
Since
Let k be a non-negative integer, and consider the portion
US
X )~ KS(
)
q+l (X r +k +l ' Xr +k )--+ HSe
q r+k
q X r+k+l ) --. Ji3(x
q r+k+l' X
r +k
(~
r+
k
+ l ' Xr +k) '
Since
H~_2(Xq_2' Xq _3
(X
S
q-2
,...-,;..
)
Q-l
(X
2)
q-
- 0 in the homology sequen ce of the pair
(X q _1 , Xq _2 ) , ~q-l~q - O.
~
of the homology sequence of the pa ir
q_l
t i•
H
q
(j r )*5, Ffl(X
), where
j
denotes the includon of
q r )~ HS(IXI
q
-r
Proof:
is the bound.a ry
j.:
c (X) - HqS( Xq ' Xq - 1) ~""
~-S
• O.
The proof of 1.4 is complete.
CO ROLLARY.
X 2)'
q-
i.s the homomorphism induced by the
q-
1.5 .
and
xq- l'
If i' denotee the inolusion
then i t follows that
repr esen ts the zero of
(X,
Xq- 1)
q
operator for the pair
Since L is compact, g maps L into X. for some
gk - jf.
lilt
k. u
(X,
q
That is, if" ~*: Jti(X
, Xq- l)~HSq- l(Xq­ 1)
q q
n
complex, such that
aq : Cq (X)-+C q-l(X)
is the boundary operator of the triple
k: K S L, where L is a finite
there exists an embedding
C(X) - (iCqCX)~,~), defined as follows.
chain complex
e(x) is oalle d the skeletal
complex of X; the homo l ogy groups of C(X) are called
the sk.eletal homology groups of X and a.r e denoted by B.(X).
After looking at an example, we will prove tha t the
skeletal homology of X is isomorphic to the singular homology
of the underl ying space
Ix\.
(See Theorsm
1. 7 below.)
q < r, the two r e l ative groupe a re zer o, and so i . is an
This 1eomorphism, together with t he co nclusions of 1.2 and
isomorphi sm.
1.3, give a method of computing the homology groups of th.e
~H~(Xn):
Thus the limit group
q fixed, n - 0. 1, •••
!
of the direc t system
1s i somorphic to
H~ ( Xr) '
space underl y ing an arbi t rary irr egular complex.
and we apply 1.4 to comple te the proof.
Example:
242
The homology of a torus
243
SPx sq, where p and q are
posit i ve integers.
SPx SQ.
is the underlying apaoe of an
irregular complex with four oel l s of dimensions
and
0, p, Q.,
Consider the skeletal decomposition K of
p+q.
given by skeletons of ths oomp1ex.
0(1)
boundary operator in
are retraots of
SPx sq
of the homology groups of K and those of L, must map mono­
morphically into the homo l ogy groups of K "L.
p+q,
In dimension
Let X and Y be skeletal decompositions of spaces
there 1e a commutative diagram:
c
p-t-q
(X) -
-
al
C
p-t-q-l
rrp+q (sP>' "I.SQ. '
Iyl.
SPv sq)
(K)-B S
p+q-l
'
to
X
)
p+q-2
p+q-
l( SPy sq) ~HS
p+q­
·l (SPX sq)
•
spvsq)
and thus also
,
is a monomorphism.
a:
(SP)(sq sPvSq)~HS
(SP'IIS q )
p+q'
p+q - 1
0 p+q (K) ~e..<
p...q- I\K).
of the proof that ~=o
HqS (Y q ,Y q- 1)
is 0
PROPO SIT I ON.
vertioes
vII
K)( w V
L
v)(
S:
and
as an exercise to the reader.
Kk L
Yq
for each q.
e(y).
'rhe
K "'L
induoes a monomorphi sm of homo l ogy groups.
is a sum of a cycle on K and a cycle on L .
244
H~(Xq'
Any
Xq _l )
lxl .
oycle on
But K and L
For each q,
inclusions induce homomorphisme of singular homologyl
d.
H!(Xq)---'B~(Xq' Xq _1 ) • Cq(X)
~P
H~(Xq+l)
Choose
Then t he in olusion iJ I",L.
In the proof we use ce llu l ar homology.
skeletal
This ohain map induces
1.7.
THEOREM..
The homomorphisms
oc.
an isomorphism ~X: Bq (x)-..gS(lxl)
q
~,
de1'ined above induce
for each 9.
morphiem ie natura l under skeletal mappings I
Proofl
A
for all q, and it £ollows from properties
isomorphic to the singular homology of
q
SPXS •
Let K and L be regular complexes.
w6.L.
into
epace \xt then the skeletal homology of X is naturally
•
skeletal homo l ogy of the decomposition K is then free abelian
1.6.
is called a skeletal mapping from
We now show that if X is a skeletal decomposi t ion of a
we leave the reet
on generators oorresponding to the four cel l s of
Xq
and
a homomorphism of skeletal homology, denoted by f."
By
~Srrf3
..
Ixl.-.trr
dsfine a chain map from C(X) to
the exactness of the singular homology sequence of the pair
( SPKSq
fr
\xl
I I and III of singular homo l ogy that these homomorphisms
The proposition below asserts in particular t hat
i., HS
map
mapping from X to Y induces homomorphisms from
J,j"
(Spv sq
A
X to Y if' f maps
Ifp+q-11(;PvsQ.)
-
Thus the homology groups of KVL,
whioh in dimensions greater than zero split as the di rect 1Jum
We claim that the
is zero.
KXL.
1s a
slceleta~
This iso­
i.e •• if f: X-fY
mapping then the diagram belo" is commutative for
each 9.
245
..",
S
H (x) - 4 H (
q
q
'·1
Proof;
1
,!
1'­
Hq (y)--4HqS(
fJy - 0, and so
then
IXl )
HqS (X r ) ,. 0
First note t hat
'>"x(u) - ( Jq+l).py - O.
Ixi )
u - 0.
for
<q.
r
proved by induction on r, using co ndi tion i)
q
t Xl)'
r
r-
is a monoaorphiSIII.
1). 0, and so 0(.
q-
Next, by 1.5, the inclusion of Xq+ 1 in
Ixl
(X
induces an iso­
A
I'"
I
HS(X ) -+HS{X 1 )
q q
q q+
l'x.
the skeletal cyc l e
Let
S
z 'Hq ( Xq ,X q- 1)'
Since oc. is a lIonomorphism,
y E Hq (X q ), and since
0(
z' - z - ~ x
Z
+01:
).x -
-0(
0(
a.x
(y +
If
Xq+l
u
in
y
-« c).z
- O.
1"x
is an isomor phism.
is natur al with respec t to skeletal
I I and I I I of s ingular homology and is left to the reade r.
COROLLARY.
(Theorem 5. 1 of Cha.pter III )
9
t he topo l ogi cal boundary of each 9-cel l l ie s Then for each 9
Then we set
Proof:
pOSition of X by 1.;, and we shall de note this de compos i t i on
by K.
By 1. 7, the e i ngular homology groups of X a r e isomorphic
70'
It i s clear t ha t
tx
Suppo se that 1"x(u) - 0 .
boundary ope r a t or in
+ ~.x) - ( j q+ l) --I'Y + (jq+ l ).pa.x
'1 a. - O) •
C(I) is identi ca lly ze r o. Ac cording to
~( Kq' Kq _ 1)
is iso­
cor responding to a given q-ce l l a is q
[ucr
' \rf",] £ H!-'fcr(E , s q-l ) . The class
~~(~)
.;)
in Cq- le x) -
asq - I(X q- l'
then
247
246
Cq(X) -
t he ge nerator ! (V)
as de f ined above is a homomorphi sm.
z - «y,
We show t hat the
morphi c to the fr e e a bel ian group on the q- cells of K, and is well -defined.
I f u is repr esented by
H* (K) .
­
repre s ented by
it f ollows that ~x{u)
is iso­ The seque nce of ske l e t ons of X i9 a ske l e tal de com­
to the skele tal homology groups
S
x i Hq+ 1 ( Xq+ l' Xq ),
then
Since • (j q+l).py·
HS(X)
q
morphi c to t he free abe l ian group on the 9-ce11s of K.
1.2 and the r emar ks f oll owi ng ,
(jq+l).~ ( Y
Let X be a for sOllie
is well-defined.
txl by jq+l'
and ao ')"X
in t he (g- 2)-akeleton of K.
is also repr esented by 21'. 80 t hat
f or some
a.x).
z -oc.y
Thus
is a monomorphism,
Denote t he inclusion of
1x(u) - ( j q+I ).P y.
a.1I - O.
be represented by
"'e then havs ~ z
and so z ill a boundary and
mappings is an easy exercise in the application of proper tie s
that for each
uEHq{X}
:But
topological space untie rlying a complex I with the property i s on t o.
We now construct
XEH:+I(X q +l • Xq ).
Finally, t he fact thatj3 is onto impl ies imme di ately
1'x 1s onto,
1.8.
morphisa of singular homology, and by a similar &.rg'Ullent
tor some
a. x -?> x,
The p.r oof that
of Def inition 1.1 and the exact homo l ogy sequenoes of the pairs
5
In partic ula.r , H (X
that
This is ~*x
y -
z -GeT -0(
Since (jq+l). is an isomo rphism,
Kq ­ 2)
is represented by
[a.~, j(~ISCl-l)] £ R~~i-'t>Cl-1) (SCl-l).
18 the inclusion.
By hypothesis
where
i:r c:- Kq­ 2'
jJCr~(KCl_l'
K _ )
Cl 2
Thu8 the composition
j ( 1j,.JsCl-l) sende sq-l into K _ , and ao induces a map
q 2
the pair
(sq-l, sq-l)
(sq-l, sct-l)
to
(X
q-
l' K
q­
2)'
It
danotea inclusion, t hen [a.u.. ,
baa the aucceuor
[k. a
it
1lcr'
h] f. H! -1 (sq -1.
latter group is zero, and so ~ !(cr )
• O.
h
of
j(~lsq-l)J
sq -1) •
(K q ,Kq_ 1 )
q
u_i
Hq (;P,;')
v
denotes inclusion a n d
generator chosen inductively ae followe.
cla8ses
But thia
Thus the boundar,y
U
A
is a
In dime ns ion zero
' RO(A), A a vertex of K, be homologous in K.
In dimension 1,
~u..
[crt
-
~e
How as s ume that
H~(X) - Hq (X)
Ucr s o as to s a ti sfy
c hoose
1aci.uA +
vertioes offJ' and
in e(x) is identically zero. and so
[0"1
~_j.~,
j l B~h- are the inclueions.
1: ASci' and
u.e' Hr(f, +)
where A and B are the
haa been chosen for each
r-cell "t of dimensio n less than q, whe re
Cl~2.
q-ce1l, and euppose that'l" is a (q-l) -face of
• Cq(X)
11
• free aba1ian group on the
(r, 1:-)
~ (ci',.r-t')
tr.
Ths inclueion
so induces i somorph i sms of cellular homo l ogy groups.
next consider the inc l usion
1.10.
LEJnlA.
The regul a r complex
ir...-r
er-T is
acyolic.
;--r has
Let e(K) denote the oel lular chain complex for the oriented
Proof:
regular complex X and let D(K) de note the skeletal chain
non-bound ing cycles exoept perhaps in dimension Cl-l.
oomplex associated with the akeletal decompoeition of X giTen
dimension q-l, any cyole inCr-'t
b7 the skeletons of X.
and hence a mul tiple of the fundame ntal cycle ino-.
By comparing
the fundamenta l cycle in
1 .9.
THEOREM.
The chain complexes e(l:) ~ D(K) are chain
with;" we s e e that
ir
We
fr C; CG-t c1'-T) .
For the final theorem of thia section we suppose that
«.
Let tr be a
1 8 an exoision of regular complexes and
q-ce11a of K, b7 1.2.
K ia a regular complex oriented by the i .n cidence function
~
1er: (cr, Oo)
(u cr • ier] E Hi,(i',ir ) , where
we demand only t hat in each connected component of K the
~
It. I sq-l
represe n ted by
IIlUS t
no
In
a l so be a cycl e in
ha s ooefficient
±1
ir
But
on the ce11 1" t
and so there are no (q-l ) -cycl es in (,--T.
isomorphic.
Proofl
We define a chain isomorphism
Let tT be a q-cell of K.
¢ mapping e(K)
Thus the inc l usion
Then rq
rl. 1 eq (K)~D q (x) - SS(K
q q ,K q­ 1)
mapa the generator ~ of 0q(X)
248
klCT
C; (cr,ir-1')
induoes isomorphisms
to D(K).
of cellu lar homology groups in dimensions greater than zero.
It fol l ows that we have isomorphi sms:
to the aingular homology class
249
i .
-..I;.
(1)
H l(?"~ )---+H
q-
~
..
k*
l~<r,cr-t')+-- H
q-
~
e
We deno te t he c ompo s i tion by
Ucr
Finally, we s e t
a*
q-
Since
::::
q
(cr, 't" ) : Hq- 1 (f, -to ) -+ Hq (ii, iT) .
[cr: T].. 9(<r,'t")
=
_.
l(b)'--H (6',<7' )
[G'":-r'] [T': p] =0
[G'":TJ['t'~f] -+
it is suffi c ient
to prove that
eCcr,T) a(T1
0)
=- e(~'t') &("r~ p).
p)
u t'.
In the pro of of (3) we wil l u se the f ollowi ng " he xagon al
1. 11.
LEMMA.
The de fini t i on of
Ucr given above is
l e mma", wh ich is proved in
independe n t of t he choice of "t".
Proof:
Foundations of Algebraic Topology, p . :5'8.
If 't" is another (q- l) -face o f tT the n, since
iT
is
an orien t able ( q -l)-circui t , the r e is a c hain of (q - l )-c ells
from ~ to ~'.
cel ls
T
and
1. 12.
LEMMA.
Con s i der the d iagram of S!0uEs and homomor-
Ehi s ms be lo w:
Therefore i t is s u fficient to sho w that t he
def in it ions of
T'.
~Gl~
are t he same if we start f r om adj acen t
u~
We let
f
be a (q-2)- face of
T
G6
and ~'.
j
2
We have defined isomorph i s ms
H let, i' ) -+ H (~,
qq
I
e (a-; 'l"')
: H 1 cF~ i') ~ H (i, iT )
qq
f) : Hq _2
cf.
t'1
(7
4
2
i3
Assume that
a) Each tri angl e is commu t a ti ve
-.
b) i2 a nd j 2 a re isomorphi sm s
Hq - 1
CT'' 'l' ') .
= 1m
c) Ke r f2
II
~
Ker g2
1m gl
d ) k 2 kl - 0
We wan t to sho w that
(2 )
G2
~1 k
G
5 ~2~3
j3
cr)
P)~ Hq_ /~' t)
e (,r: p) : Hq -2 (P' p)
ki
t~G.-:(T
1
~7~
G
e (cr, T)
e(T,
Eil en be rg and s tee n-rod,
(CT :'t]G(9 (CT, 't")U-r
The n it follows that
= [cr,T']9(a',T')u....,.
Substituting e x pressions in terms of
up
for
~
and
u.r, •
we replace (2) by
(cr:-r][-r:pJ&(CT/'t) e(t',p) lAp
i3 i;l i l
~ -
= [cr:'t'][-t' :p]
eCa:;"r') e(T~p) "p.
In applying the he xa gonal lemma we consider the diagram
on the next page.
The homomorphisms are boundary maps and
map s induced by inclusio n .
The composition of the vertical
251
250
j3 j;l jl"
T
.,
factors through
Hg-l(tT)
(eJ(iSio oocur.
:.:y
(. . (
~ ~':-l
,.. 1
Z~Z.Z
~
tool,t'''''l',\t'vt' -p
~ \, '/' jct«<~ \
According to 1. 2 and the r emarks before 1. 3, the map
Jlq so defined
of
is an i somorphism fo r each q.
H!;"3.
(i-,i-:> (
Hta(ot" i'-&
~~ H,.q(~,e)~) 252
Thus t he pr oof
1.9 wIl l be c ompl ete when we show t ha t the diagram be l ow ie commutative for each q:
C (K) A HS(K • Kq 1)
q
~
q
q
1a~
­
H~_l(Kq _l )
1
! k~
~,-t
H,_t.(V)
i:
Thus the hexagon al lemma
applies to yiel d (3) as de si red.
H,.i't',i')
)
The commuta tivity of
--+Z.
al l t he necessary triangles follows by nat u r al ity and the
"*f / 1 ~ ~/~
D.
(i... i" ,(i'v'i")~
t) ~ H q- lef... -r-',t ... t') ~ H q­ lCi .. :r',-r",t)
split exa ct seque n ce
~T"i"~ T.+J . ')
t o H 2(tut-')
q_
H q_2(~v+~-,), and eo two e ucce8s~ve maps
boundar y axi om of cel l ular homology.
-It,~(i~t')
Ho-2 (t) H /,- .If.'
le;... ?(i- ... i'J-..1P
is easily s e en to be exac t --in fact i t ie isomorphi c to t he
_. /~(fI~i"i~~~~ _
11,-, ('t"/t')
H l (T'",T:t-..,r)
q-
The sequence
H l(~'
q-
,'" ...... H,.,{Cr, cr-t')
~_,(f.)r~ (tvi')-f)
H\_I(f~'F, -tu~
Hq-
vi a
of the exaot homology sequence of the pai r
.. ~ 1~.
-·
H 2(O,P')
q- l
to
is zero because the map froID
"altl z
Mo., (cr,O"'-t)" ~_
R(ii,t1)
q
maps from
H,,(5',o-)
( inC l u sion)
Cq_1 CK)~ H~ _l (Kq_ l '
eCt;e)
LetfT be a q-cell of K,
f or
[u.,...
q
s
1a]'H~(ci.cr).
[~*Ucr' k~f'<:l (cr).
q~2 ;
(We assume tha t
0 o r 1 is easy.)
Then ~ (~)
Thu s
where
q
k~b ~¢q (cr)
j :Cr~
253
Kq _2 )
the proof
i s represen t e d by
is repre se nte d by Kq _ l
is t he inclusion.
Let L denote the (q-2)-ske l eton of<T.
oarriee L into tbe (q-2)-skeleton of X and so [().
u.,.,
t h e suooessor [h.l.ueJ"
(is', L)~
(K
q-
l' K. 2) and b:Cr C.
q-
­
g]Ea:_1(Cr, L), where g'
(cT,
r p" :·t'lt'.
[ut-' Lr] has
jt'l
q­
2) factors through
the Bucoeesor ~ jt') .. ~,
(f, 1:-) ~ (G-,
sented by
q-
l' K
~
L) is t he inclusion.
[t ~:'t]
(Jt') ..
u.r,
-'
cations on K.
The
(0-, L) and so
B:-1 (<rt L). where
ThuB
~q -1 ~6'
g) S:-l (0', L).
Complexes .ith Ident ifications.
Let K be a regular complex, I' a family of identifi­
X/F
[~, it'J~ fiq!'l ('t, 't').
T of CT, "q-l ('t') is represented by
(K
has
For each (q-1)-face
i
't'
,l
(f,t) C
-
k j]
L) are inclusions.
On the other hand, aCT-
inclusion
2.
The oomposition kj
is repre­
We complete
(See Chapter III for definitions.)
Let
de.note proj e ction onto the identification s paoe.
general.
K/F
under F .
Let
C(I/:-)
0(.
Ixl--+
In
(K/F) q - S(lCq ) .
is an irregula r complex with
Choose an inoidence timotion
s:
for K .hioh is invariant
denote the ch.ain oomplex defined as
in III.2 i n terms of the inoidence function
«.
Finally. let
~enote the skeletal ohain complex assooiated to the
D(K/F)
deoomposition of K/F as a oomplex.
the proof that the diagram above is commutative by showing that
2.1.
h.~. Ucr - I: @- I't"] (.1t-) .. 'lor '
(4)
Cr,
is the direot sum of the images of
1.2 implies tbat Ii lCer. L)
q-
Bq_l(f,~)
.bere"t' varies over the (q-l)-faoes offl'.
under
(J,).,
Proof:
tion onto eaoh direct summand of B leG; L).
q-
C(IC/F) and D(K/F) are chain isomorphic.
(See (1) above)
k.). ~
q
which i e mapped isomorphical17
-[cr I't"] p.(~) * ur.
:But by definition
This is true for
each (q-l ) -face ~ ofer, and so equation (4) 1s proved.
"
s.1
C (K)
The homomorphism p. maps each d i rect eumma.nd of Bq -1 (G-, L) to
(by the excision axiom) onto Ii 1(0.,0--1").
q­
Tbis
~, ,01
~(Xq'
\!
Xq _1 )
~T
~
~
~
~ cq­1(1)
'¢!
)a:_l(X q _l , Kq
_ )
2
~l
~«K/F)q' (X/F)q_l)--+H~_l « K/F)q_l'
t
Dq(K/F)
~
completes the proof of 1 . 9.
254
Consider the diagraml
Cq(K/F) -----~ Cq _ 1 (K/P)
Consider the mapsl
CT, 'bjf(ir, L)~(~Cr-'t)f-k <r.
1r),
Theorem 1 . 7 implies that i t is sufficient to show that
To show that (4)
h olds i t is sufficient to cheok tha t (4) h.o lds under projeo­
zero exoept for (jT). Eq_l(f,
The homol ogy grOUpS of C(X/F) are isomorphic
to the singular homology groups of the space K/F.
T
Sinoe L is the (q-2)-skeleton of
THEOREM.
255
)
1
Dq_l(X/P)
(K/F )q_2)
lJtrk1~'1'
The map
¢
~
is the chain isomorphism of 1.9.
freedom of choioe in the definition of
zero.
s6 occurs
The only
in dimenaion
We s t ipulate that for each pair of verticea A and
mapping by s to the same vertex of K/F
a. U A •
B
s+ u •
B
The first or uppermost square of the diagram ia
cOlllllll1tative by construction.
(See Chapter III.)
aeoond square ia oommutative because
The
~ia a chain map.
The
third square 1e cOllUDutative because a is a skeletal lII.a pping.
J
We define a obain lIlap
Let (1' be a q-cell of K/F.
Then
S
).
by induction on q that tbe definition at
!
(0') - s . ¢(T) ,
and
and so
l
Ji
It is eaay to show
J (D")
1e independent
Aleo, if't' is mapped onto (J""by s, then
80
l
is a chain map sinoe the diagram
on page 255 is commutative.
of 1.2 i mply that
follows.
'12 (CT) £0 Bq ( (K/F) q ,(K/F) q­ 1 )
.s
rUt; 8 ~ fH: ir(lf, t
of the choice of t".
8a
Suppose that"t is a q-oell of K
which is mapped by a onto 0".
i8 represented by
eeK/F) --t D(K/F)
z
The remarks following the proof
is an isomorphi sm in each dimenaion,
is a chain isomorp.h iem.
of Theorem 2.1.
2
8
This completeB t he proof