Cohomology with compact support

Singular cohomology with compact support

Let X be a topological space. Then

This is also naturally isomorphic to the cohomology of the sub–chain complex C c ∗ ( X ; R ) consisting of all singular cochains ϕ : C i ( X ; R ) → R that have compact support in the sense that there exists some compact K ⊆ X such that ϕ vanishes on all chains in X ∖ K .

de Rham cohomology with compact support for smooth manifolds

Given a manifold X, let Ω 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 ) are the homology of the chain complex ( Ω c ∙ ( X ) , d ) :

i.e., H 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 ) 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

induces a map

If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence

called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X.

De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then

where all maps are induced by extension by zero is also exact.