Graphics Reference
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7.5.2
Manifolds and Poincaré Duality
This section discusses a very important property satisfied by the homology groups
of manifolds. It helps greatly in their determination and is a cornerstone in their
classification.
Throughout this section we assume that M n is a closed , compact , and connected
n-dimensional homology manifold and that K is a simplicial complex that triangu-
lates it. Identify the homology groups of M with those of K.
First, consider the case where M is an oriented manifold. The orientation induces
a well-defined orientation on all the n-simplices of K. Consider an oriented k-simplex
[s
k ] and an oriented version of its dual cell [s n-k ]. Both correspond to a union of ori-
ented k-, respectively, (n - k)-simplices of sd(K). Let n be any k-simplex of sd(K) con-
tained in s (= s
k ) and assume that [n] has orientation compatible with [s]. Similarly,
let t * be any (n - k)-simplex of sd(K) in s n-k and assume that [t * ] has orientation com-
patible with [s n-k ]. By Proposition 7.5.6(3) n and t * have a single vertex in common.
Let
[] =
[
]
[
] =
[
]
n
a
pp
...
p
and
t
b
pp
...
p
,
01
k
*
k
k
+
1
n
where the p i are vertices of sd(K), p k is the vertex that n and t * have in common, and
the integers a and b are ±1. Let
[] =
[
]
c
pp
01 ...
p
,
n
where the integer c is ±1 and is chosen so that the oriented n-simplex [x] has the ori-
entation induced by the given orientation of M . The integers a, b, and c clearly depend
on how the points p i are ordered .
7.5.2.1. Example. See Figure 7.33. We have two 2-cells with a counterclockwise ori-
entation. Here n = 2, s = p 0 p 3 , n = p 0 p 1 , k = 1, s * = p 2 p 1 » p 1 p 4 , t * = p 2 p 1 , x = p 0 p 1 p 2 ,
a = 1, b = c =-1.
k ] and its dual [s n-k ], denoted by I([s
k ],
Definition.
The intersection number of [s
[s n-k ]), is defined by
P 3
s = P 0 P 3
s * = P 2 P 1 »P 1 P 4
P 1
t *
P 2
P 4
x
u
Figure 7.33.
Defining the intersection
number of dual cells.
P 0
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