Poincare duality theorem

This article is about a duality theorem

Statement

Let $M$ be a compact connected orientable manifold. Choose $[M]$ to be fundamental class in $M$. Then the cap product with $[M]$ defines a map:

$H^{i}(M;R)\to H_{n-i}(M;R)$

The Poincare duality theorem states that this map is an isomorphism.

Related facts

Similar facts

• Alexander duality theorem
• Lefschetz duality theorem

Applications

• Euler characteristic of odd-dimensional compact connected orientable manifold is zero