Lefschetz 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