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 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 (mathematics) 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 and exact differential forms k -forms modulo that of closed and exact differential forms 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

: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.