Alexander duality theorem

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This article is about a 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)

This map is an isomorphism.

Note that the specific isomorphism depends on the choice of orientation on the pair (M, M \setminus K).