Poincare 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.

Similar facts

 * Alexander duality theorem
 * Lefschetz duality theorem

Applications

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