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:

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

Poincare duality theorem states that this map is an isomorphism.

Related results

• Alexander duality theorem
• Lefschetz duality theorem