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as follows: Let g ΠC i ( X ), a ΠC n ( X ). Then, using the product of cochains, there is a
unique element in C n-i ( X ), denoted by g « a, with the property that
(
) =◊
(
)( ) ,
fg
«
a
f g a
(7.14)
for all f ΠC n-i ( X ). In the case of simplicial complexes and an oriented n-simplex [s],
in i
« [] = ()
(
-
)
(
[
]
)
[
(
)
]
g
s
1
g
back i - face of
s
front
n
-
i
-face of
s
,
(7.15)
where the front k-face of s consists of those points with barycentric coordinates (t 0 ,t 1 ,
...,t k ,0,...,0) and the back k-face of s consists of those points with barycentric coor-
dinates (0, ...0,t n-k ,...,t n ).
7.5.2.8. Proposition.
The map «
defined by equation (7.13) is well defined,
bilinear, and satisfies
(1) (g ◊ h) « a = g « (h « a)
(2) 1 « a = a
(3) ∂ (g « a) = (dg) « a + (-1) dim g g «∂a
Proof.
See [MilS74].
Property (3) in Proposition 7.5.2.8 implies that « induces a well-defined bilinear
map
i
() ¥
() Æ
()
«
:H
XX
H
H
X
(7.16)
n
n i
-
Definition.
The map « defined in equation (7.16) is called the cap product for X .
7.5.2.9. Theorem. (The Poincaré Duality Theorem) Let M n be an orientable homol-
ogy manifold and let m be a generator of H n ( M ) ª Z . The homomorphism
i
() Æ
()
H
M
H
aa
M
ni
-
m
Æ«
is an isomorphism for all i.
Proof.
See [MilS74].
Theorem 7.5.2.9 and Theorem 7.3.1 imply Theorem 7.5.2.4.
7.6 Where to Next: What We Left Out
We have covered a lot of algebraic topology, but there is much more and we have only
scratched the surface. Of course, we have left out many details and proofs and these
should be filled in and understood before moving on, but we would like to mention
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