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 $Omega^k_{mathrm c}(X)$ 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 $(Omega^ullet_{mathrm c}(X),d)$ :

: $0 o Omega^0_{mathrm c}(X) o Omega^1_{mathrm c}(X) o Omega^2_{mathrm c}(X) o ldots$ ;

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 $Omega^ullet_{mathrm c}(U) o Omega^ullet_{mathrm c}(X)$ inducing a map

: $H^k_{mathrm c}(U) o H^k_{mathrm c}(X)$ .

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

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

induces a map

: $H^k_{mathrm c}(X) o H^k_{mathrm c}(U)$ .

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

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