Cohomology with compact support

In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. Let X {displaystyle X} be a topological space. Then This is also naturally isomorphic to the cohomology of the sub–chain complex C c ∗ ( X ; R ) {displaystyle C_{c}^{ast }(X;R)} consisting of all singular cochains ϕ : C i ( X ; R ) → R {displaystyle phi :C_{i}(X;R) o R} that have compact support in the sense that there exists some compact K ⊆ X {displaystyle Ksubseteq X} such that ϕ {displaystyle phi } vanishes on all chains in X ∖ K {displaystyle Xsetminus K} . Given a manifold X, let Ω c k ( X ) {displaystyle Omega _{mathrm {c} }^{k}(X)} be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative. Then the de Rham cohomology groups with compact support H c q ( X ) {displaystyle H_{mathrm {c} }^{q}(X)} are the homology of the chain complex ( Ω c ∙ ( X ) , d ) {displaystyle (Omega _{mathrm {c} }^{ullet }(X),d)} : i.e., H c q ( X ) {displaystyle H_{mathrm {c} }^{q}(X)} is the vector space of closed q-forms modulo that of exact q-forms. Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping j 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 j ∗ : Ω c ∙ ( U ) → Ω c ∙ ( X ) {displaystyle j_{*}:Omega _{mathrm {c} }^{ullet }(U) o Omega _{mathrm {c} }^{ullet }(X)} inducing a map 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: Y → X be such a map; then the pullback