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.
Singular cohomology with compact support.
Let formula_1 be a topological space. Then
This is also naturally isomorphic to the cohomology of the sub–chain complex formula_3 consisting of all singular cochains formula_4 that have compact support in the sense that there exists some compact formula_5 such that formula_6 vanishes on all chains in formula_7.
Functorial definition.
Let formula_1 be a topological space and formula_9 the map to the point. Using the direct image and direct image with compact support functors formula_10, one can define cohomology and cohomology with compact support of a sheaf of abelian groups formula_11 on formula_1 as 
Taking for formula_11 the constant sheaf with coefficients in a ring formula_16 recovers the previous definition.
de Rham cohomology with compact support for smooth manifolds.
Given a manifold "X", let formula_17 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 formula_18 are the homology of the chain complex formula_19:
"i.e.", formula_18 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 formula_22 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.