Graphics Reference
In-Depth Information
(
[]
[
]
)
=
k
nk
ss
-
I
,
a b c
.
*
k
],[s
n-k
]) is well defined.
7.5.2.2. Lemma.
The intersection number I = I ([s
Proof
.
We have to show
(1) I does not depend on the order of the vertices
p
i
and
(2) I does not depend on the choice of n and t
*
.
A proof of these two facts can be found in [SeiT80].
As we pointed out earlier, although the dual cells in the dual complex K
*
may not
be disks, the fact that we have a homology manifold means that, algebraically, they
are linear combinations of simplices that act like disks from the point of view of
homology. If one worked through the details one would be able to show that repre-
sentatives of homology classes can be replaced by linear combinations of the dual
cells and that a homology theory that uses chain groups based on dual cells would
produce the same homology groups as before. Furthermore, there is also no problem
with defining incidence matrices for the dual cells and these would play the same role
as the incidence matrices for the simplicial theory.
7.5.2.3. Theorem.
Given K and K
*
, if one orients the cells of K and K
*
in such a
way that the intersection number of dual cells is +1, then the incidence matrix E
k-1
for K is the transpose of the incidence matrix E
n-k
for K
*
multiplied by (-1)
k
.
Proof.
See [SeiT80].
7.5.2.4. Theorem.
(The Poincaré Duality Theorem) The kth Betti number of an ori-
entable homology manifold
M
n
is the same as the (n - k)-th Betti number and the k-
dimensional torsion coefficients are the same as the (n - k - 1)-dimensional torsion
coefficients for k = 0, 1,..., n.
Proof.
See [SeiT80].
Theorem 7.5.2.4 applies only to orientable manifolds. If the manifold is not ori-
entable, then the theorem is false, but one can prove a duality theorem provided one
uses
Z
2
for coefficients. In that case things get simpler because one does not have to
worry about orientations.
7.5.2.5. Theorem.
(The Mod 2 Poincaré Duality Theorem) The kth connectivity
number of a homology manifold
M
n
(orientable or not) is the same as the (n - k)-th
connectivity number for k = 0, 1,..., n.
Proof.
See [SeiT80].
7.5.2.6. Corollary.
The Euler characteristic of an odd-dimensional homology
manifold is 0.
