# 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 $X$ be a topological space. Then
    $\displaystyle H_{c}^{\ast }(X;R):=\varinjlim _{K\subseteq X\,{\text{compact}}}H^{\ast }(X,X\setminus K;R)$
By definition, this is the cohomology of the sub–chain complex $C_{c}^{\ast }(X;R)$ consisting of all singular cochains $\phi :C_{i}(X;R)\to R$ that have compact support in the sense that there exists some compact $K\subseteq X$ such that $\phi$ vanishes on all chains in $X\setminus K$.

## Functorial definition

Let $X$ be a topological space and $p:X\to \star$ the map to the point. Using the direct image and direct image with compact support functors $p_{*},p_{!}:{\text{Sh}}(X)\to {\text{Sh}}(\star )={\text{Ab}}$, one can define cohomology and cohomology with compact support of a sheaf of abelian groups ${\mathcal {F}}$ on $X$ as
    $\displaystyle H^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{*}{\mathcal {F}},$
    $\displaystyle H_{c}^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{!}{\mathcal {F}}.$
Taking for ${\mathcal {F}}$ the constant sheaf with coefficients in a ring $R$ recovers the previous definition.

## de Rham cohomology with compact support for smooth manifolds

Given a manifold X, let $\Omega _{\mathrm {c} }^{k}(X)$ 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 $H_{\mathrm {c} }^{q}(X)$ are the homology of the chain complex $(\Omega _{\mathrm {c} }^{\bullet }(X),d)$:
    $0\to \Omega _{\mathrm {c} }^{0}(X)\to \Omega _{\mathrm {c} }^{1}(X)\to \Omega _{\mathrm {c} }^{2}(X)\to \cdots$
i.e., $H_{\mathrm {c} }^{q}(X)$ 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 $j_{*}:\Omega _{\mathrm {c} }^{\bullet }(U)\to \Omega _{\mathrm {c} }^{\bullet }(X)$ inducing a map
    $j_{*}:H_{\mathrm {c} }^{q}(U)\to H_{\mathrm {c} }^{q}(X)$.
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
    $f^{*}:\Omega _{\mathrm {c} }^{q}(X)\to \Omega _{\mathrm {c} }^{q}(Y)\sum _{I}g_{I}\,dx_{i_{1}}\wedge \ldots \wedge dx_{i_{q}}\mapsto \sum _{I}(g_{I}\circ f)\,d(x_{i_{1}}\circ f)\wedge \ldots \wedge d(x_{i_{q}}\circ f)$
induces a map
    $H_{\mathrm {c} }^{q}(X)\to H_{\mathrm {c} }^{q}(Y)$.
If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence
    $\cdots \to H_{\mathrm {c} }^{q}(U){\overset {j_{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(X){\overset {i^{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(Z){\overset {\delta }{\longrightarrow }}H_{\mathrm {c} }^{q+1}(U)\to \cdots$
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
    $\cdots \to H_{\mathrm {c} }^{q}(U\cap V)\to H_{\mathrm {c} }^{q}(U)\oplus H_{\mathrm {c} }^{q}(V)\to H_{\mathrm {c} }^{q}(X){\overset {\delta }{\longrightarrow }}H_{\mathrm {c} }^{q+1}(U\cap V)\to \cdots$
where all maps are induced by extension by zero is also exact.