Alexander-Spanier cohomology

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In mathematics, particularly in algebraic topology Alexander-Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. It is similar to and in some sense dual to de Rham cohomology. It is named for J. W. Alexander and Edwin Henry Spanier (1921-1996).

Given a manifold X, let \Omega^k_{\mathrm c}(X) be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative.
Then the Alexander-Spanier cohomology groups H^k_{\mathrm c}(X) are the homology of the chain complex (\Omega^\bullet_{\mathrm c}(X),d):

0 \to \Omega^0_{\mathrm c}(X) \to \Omega^1_{\mathrm c}(X) \to \Omega^2_{\mathrm c}(X) \to \ldots;

i.e., H^k_{\mathrm c}(X) is the vector space of closed k-forms modulo that of exact k-forms.

Despite their definition as the homology of an ascending complex, the Alexander-Spanier groups demonstrate covariant behavior; for example, given the inclusion mapping for an open set U of X, extension of forms on U to X (by defining them to be 0 on X-U) is a map \Omega^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X) inducing a map

H^k_{\mathrm c}(U) \to H^k_{\mathrm c}(X).

They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let f: U → X be such a map; then the pullback

f^*: \Omega^k_{\mathrm c}(X) \to \Omega^k_{\mathrm c}(U): \sum_I g_I \, dx_{i_1} \wedge \ldots \wedge dx_{i_k} \mapsto (g \circ f) \, d(x_{i_1} \circ f) \wedge \ldots \wedge d(x_{i_k} \circ f)

induces a map

H^k_{\mathrm c}(X) \to H^k_{\mathrm c}(U).

A Mayer-Vietoris sequence holds for Alexander-Spanier cohomology.

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