Cap Product and Poincaré Duality

Orientability

Definition

A manifold

Lemma

Let be an -dimensional oriented connected compact manifold. Then

• If is -orientable then and is an isomorphism for all .
• If is not -orientable then .
• for all .
  

Fundamental Class

Let be an -dimensional oriented connected compact manifold. The fundamental class of , denoted , is the unique generator of the top-dimensional homology group that corresponds to the given orientation of .
More concretely, the orientation of is a choice of an isomorphism , and .

e.g.

• For a positively oriented point , the fundamental class is . If we instead give it the opposite orientation, the fundamental class would be .

Cap Product

Cap Product

For , we have a bilinear map called the cap product: which sends to .
This induces a map on homology and cohomology:

Compatibility with Cup Product

.

Poincaré Duality

If is an -dimensional oriented compact manifold, then the th cohomology group of is isomorphic to the th homology group of :
Specifically, this isomorphism is given by the cap product with the fundamental class : where and .