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 :

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

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 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.