Graphics Reference
In-Depth Information
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