Alexander duality theorem: Difference between revisions

Latest revision as of 19:31, 11 May 2008

This article is about a duality theorem

Statement

Let $M$ be an orientable manifold and $K$ a compact subset of $M$ . Denote by ${\bar{H}}^{i} (K)$ the direct limit of cohomology groups for all open sets containing $K$ . Suppose $(M, M ∖ K)$ is $R$ -orientable. Choose a generator for $H_{n} (M, M ∖ K)$ (this group is a free module of rank one over the coefficient ring). Then cap product with this generator yields a map:

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

This map is an isomorphism.

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

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 ==Statement==
-Let <math>M</math> be a [[manifold]] and <math>K</math> a [[compact space|compact]] subset of <math>M</math>. Denote by <math>\overline{H}^i(K)</math> the direct limit of cohomology groups for all open sets containing <math>K</math>. Suppose <math>(M,M \setminus K)</math> is <math>R</math>-orientable. Choose a generator for <math>H_n(M, M \setminus K)</math> (this group is a free module of rank one over the coefficient ring). Then [[cap product]] with this generator yields a map:
+Let <math>M</math> be an [[orientable manifold]] and <math>K</math> a [[compact space|compact]] subset of <math>M</math>. Denote by <math>\overline{H}^i(K)</math> the direct limit of cohomology groups for all open sets containing <math>K</math>. Suppose <math>(M,M \setminus K)</math> is <math>R</math>-orientable. Choose a generator for <math>H_n(M, M \setminus K)</math> (this group is a free module of rank one over the coefficient ring). Then [[cap product]] with this generator yields a map:
 <math>\overline{H}^i(K; R) \to H_{n-i}(M, M \setminus K ; R)</math>