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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 contravariantContravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system bein 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
induces a map
A Mayer-Vietoris sequenceIn algebraic topology and related branches of mathematics, the Mayer-Vietoris sequence is an exact sequence that often helps one to compute homology groups. It is somewhat analogous to the Seifert-van Kampen theorem for homotopy groups. Homology groups ca holds for Alexander-Spanier cohomology.
Homology theoryHomology theory Algebraic topology In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces. At th Algebraic topology