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7.3.1. Theorem. Let K be a simplicial complex. If T q (K) is the torsion subgroup of
H q (K) and if r q is the rank of H q (K), then
q
() ª
() (
)
H
K
T
K
free
abelian group of rank r q .
q
-1
Proof. See [Cair68].
Another point made earlier was that one important difference between co-
homology and homology is that cohomology admits a natural product. Here is how
one gets this product. First, order the vertices of the simplicial complex K and let “<”
denote this ordering.
If f Œ C p (K) and g Œ C q (K), then define f ◊ g Œ C p+q (K) by
Definition.
(
)
(
[
]
) =
(
[
]
)
(
[
]
)
fg
vv
◊◊◊
v
f
vv
◊◊◊
v
g
vv
◊◊◊
v
01
pq
+
01
p
0
p
+
1
pq
+
for all oriented (p + q)-simplices [ v 0 v 1 ... v p+q ] of K with v 0 < v 1 < ... v p+q . This product
of cochains induces a product
p
() ¥
q
() Æ
p q
+
()
»
:H K
H K
H
K
called the cup product .
Two distinct orderings of the vertices of K will induce isomorphic product struc-
tures on the cohomology groups. The cup product makes the cohomology groups into
a “graded ring.” As an example of how the cohomology ring gives more information,
consider the space X in Figure 7.15 that consists of the wedge of a sphere and two
circles. One can show that X has the same homology groups as the torus
S 1 ¥ S 1 (Exercise 7.3.1), so that homology cannot tell those two spaces apart. On the
other hand, the cohomology ring structure of X and the torus are different (even
though both have the same cohomology groups ). By the way, X does not have the
same homotopy type as the torus.
This concludes our brief overview of cohomology groups, but we shall get more
glimpses of them in the future.
Figure 7.15.
A space with the same homology groups as the
torus.
X = S 2 vS 1 vS 1
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