11Manifolds and Poincare duality
III Algebraic Topology
11 Manifolds and Poincar´e duality
We are going to prove Poincar´e duality, and then use it to prove a lot of things
about manifolds. Poincar´e duality tells us that for a compact oriented manifold
M of dimension n, we have
H
d
(M)
∼
=
H
n−d
(M).
To prove this, we will want to induct over covers of
M
. However, given a compact
manifold, the open covers are in general not compact. We get compactness only
when we join all of them up. So we need to come up with a version of Poincar´e
duality that works for non-compact manifolds, which is less pretty and requires
the notion of compactly supported cohomology.