Information AboutAlexander-spanier Cohomology |
| CATEGORIES ABOUT ALEXANDER-SPANIER COHOMOLOGY | |
| homology theory | |
| duality theories | |
| differential forms | |
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Given a manifold ''X'', let 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'' are the Homology of the Chain Complex : :; i.e., 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 :. They also demonstrate
\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 :. A Mayer-Vietoris Sequence holds for Alexander-Spanier cohomology. |
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