This is also naturally isomorphic to the cohomology of the sub–chain complex consisting of all singular cochains that have compact support in the sense that there exists some compact such that vanishes on all chains in .
de Rham cohomology with compact support for smooth manifolds
i.e., is the vector space of closedq-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 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.