# Alexander duality theorem

Let $M$ be an orientable manifold and $K$ a compact subset of $M$. Denote by $\overline{H}^i(K)$ the direct limit of cohomology groups for all open sets containing $K$. Suppose $(M,M \setminus K)$ is $R$-orientable. Choose a generator for $H_n(M, M \setminus K)$ (this group is a free module of rank one over the coefficient ring). Then cap product with this generator yields a map:
$\overline{H}^i(K; R) \to H_{n-i}(M, M \setminus K ; R)$
Note that the specific isomorphism depends on the choice of orientation on the pair $(M, M \setminus K)$.