Lefschetz duality theorem

This article is about a duality theorem

Statement

Let $M$ be a compact manifold with boundary, let $\partial M$ denote the boundary. Suppose the pair $(M,\partial M)$ is $R$-orientable. Then choose a generator for $H_n(M,\partial M;R)$ and use cap product with this generator to get a map:

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

This natural map is an isomorphism.

Related results

• Alexander duality theorem: This can be used to prove Lefschetz duality theorem, by first using the fact that the boundary of a manifold has a collar
• Poincare duality theorem: This is a special case of the Lefschetz duality theorem, where the boundary is empty