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Homology and cohomology
Spring 2009
Prof. Kathryn Hess
Series 10
The purpose of the Frst part of this series is to show that the integral
homology groups of a space
X
determine the homology and cohomology
groups of
X
with arbitary coe±cients. The statements are known as the
universal coe±cient theorems; one is for homology and involves the Tor
functor, the other is for cohomology and involves the Ext functor.
Recall from lecture the following theorem.
Theorem 1
(Universal Coe±cient Theorem)
.
Let
C
be a chain complex of
free abelian groups
C
n
and let
G
be any abelian group. Then for each
n
there
is an exact sequence
0
→
H
n
(
C
)
⊗
G
α
→
H
n
(
C
⊗
G
)
β
→
Tor(
H
n
−
1
(
C
)
,G
)
→
0
with homomorphisms
α,β
natural in
C,G
. This sequence splits, by a homo
morphism which is natural in
G
but not in
C
. Here,
α
([
c
]
⊗
g
) = [
c
⊗
g
]
.
The singular homology version of this theorem follows immediately.
Exercise 1.
Observe that Theorem 1 implies Theorem 2.
Theorem 2
(Universal Coe±cient Theorem)
.
Let
X
be a topological space
and let
G
be any abelian group. Then for each
n
there is an exact sequence
0
→
H
n
(
X
)
⊗
G
α
→
H
n
(
X
;
G
)
β
→
Tor(
H
n
−
1
(
X
)
,G
)
→
0
with homomorphisms
α,β
natural in
X,G
. This sequence splits, by a homo
morphism which is natural in
G
but not in
X
, and hence
H
n
(
X
;
G
)
∼
= (
H
n
(
X
)
⊗
G
)
⊕
Tor(
H
n
−
1
(
X
)
,G
)
.
Here,
α
([
c
]
⊗
g
) = [
c
⊗
g
]
.
Let
f
:
X
→
Y
be a map of topological spaces. The purpose of the following
exercise is to prove that if
f
induces an isomorphism on integral homology,
then
f
induces an isomorphism on homology with coe±cients in any abelian
group
G
.
Exercise 2.
Use Theorem 2 together with the ²ive Lemma to prove Propo
sition 3.
Proposition 3.
Let
f
:
X
→
Y
be a map of topological spaces and let
G
be any abelian group.
If the induced map
f
∗
:
H
n
(
X
)
∼
=
→
H
n
(
Y
)
is an
isomorphism for all
n
, then the induced map
f
∗
:
H
n
(
X
;
G
)
∼
=
→
H
n
(
Y
;
G
)
is an isomorphism for all
n
.
1
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The following characterization [2, V.6.2] of Fat abelian groups will be useful
in Exercise 3.
Proposition 4.
Let
G
be an abelian group. Then
G
is fat iF and only iF
G
is torsionFree.
Exercise 3.
Let
X
be a topological space and let
G
be a torsionfree abelian
group. Use Theorem 2 to prove the following.
(a)
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).
 Winter '11
 PhanThuongCang

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