Alexander-Spanier cohomology

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 <math>\Omega^k_{\mathrm c}(X)</math> 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 <math>H^k_{\mathrm c}(X)</math> are the homology of the chain complex <math>(\Omega^\bullet_{\mathrm c}(X),d)</math>:

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

i.e., <math>H^k_{\mathrm c}(X)</math> 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 <math>\Omega^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X)</math> inducing a map

<math>H^k_{\mathrm c}(U) \to H^k_{\mathrm c}(X)</math>.

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

<math>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)</math>

induces a map

<math>H^k_{\mathrm c}(X) \to H^k_{\mathrm c}(U)</math>.

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