Cohomology with compact support

Let ${\displaystyle X}$ be a topological space. Then

${\displaystyle \displaystyle H_{c}^{\ast }(X;R):=\varinjlim _{K\subseteq X\,{\text{compact}}}H^{\ast }(X,X\setminus K;R)}$

This is also naturally isomorphic to the cohomology of the sub–chain complex ${\displaystyle C_{c}^{\ast }(X;R)}$ consisting of all singular cochains ${\displaystyle \phi :C_{i}(X;R)\to R}$ that have compact support in the sense that there exists some compact ${\displaystyle K\subseteq X}$ such that ${\displaystyle \phi }$ vanishes on all chains in ${\displaystyle X\setminus K}$.

Let ${\displaystyle X}$ be a topological space and ${\displaystyle p:X\to \star }$ the map to the point. Using the direct image and direct image with compact support functors ${\displaystyle p_{*},p_{!}:{\text{Sh}}(X)\to {\text{Sh}}(\star )={\text{Ab}}}$, one can define cohomology and cohomology with compact support of a sheaf of abelian groups ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ as

${\displaystyle \displaystyle H^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{*}{\mathcal {F}},}$

${\displaystyle \displaystyle H_{c}^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{!}{\mathcal {F}}.}$

Taking for ${\displaystyle {\mathcal {F}}}$ the constant sheaf with coefficients in a ring ${\displaystyle R}$ recovers the previous definition.

Given a manifold X, let ${\displaystyle \Omega _{\mathrm {c} }^{k}(X)}$ 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 ${\displaystyle H_{\mathrm {c} }^{q}(X)}$ are the homology of the chain complex ${\displaystyle (\Omega _{\mathrm {c} }^{\bullet }(X),d)}$:

${\displaystyle 0\to \Omega _{\mathrm {c} }^{0}(X)\to \Omega _{\mathrm {c} }^{1}(X)\to \Omega _{\mathrm {c} }^{2}(X)\to \cdots }$

i.e., ${\displaystyle H_{\mathrm {c} }^{q}(X)}$ 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 ${\displaystyle j_{*}:\Omega _{\mathrm {c} }^{\bullet }(U)\to \Omega _{\mathrm {c} }^{\bullet }(X)}$ inducing a map

${\displaystyle j_{*}:H_{\mathrm {c} }^{q}(U)\to H_{\mathrm {c} }^{q}(X)}$.

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

${\displaystyle f^{*}:\Omega _{\mathrm {c} }^{q}(X)\to \Omega _{\mathrm {c} }^{q}(Y)\sum _{I}g_{I}\,dx_{i_{1}}\wedge \ldots \wedge dx_{i_{q}}\mapsto \sum _{I}(g_{I}\circ f)\,d(x_{i_{1}}\circ f)\wedge \ldots \wedge d(x_{i_{q}}\circ f)}$

induces a map

${\displaystyle H_{\mathrm {c} }^{q}(X)\to H_{\mathrm {c} }^{q}(Y)}$.

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

${\displaystyle \cdots \to H_{\mathrm {c} }^{q}(U){\overset {j_{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(X){\overset {i^{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(Z){\overset {\delta }{\longrightarrow }}H_{\mathrm {c} }^{q+1}(U)\to \cdots }$

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

${\displaystyle \cdots \to H_{\mathrm {c} }^{q}(U\cap V)\to H_{\mathrm {c} }^{q}(U)\oplus H_{\mathrm {c} }^{q}(V)\to H_{\mathrm {c} }^{q}(X){\overset {\delta }{\longrightarrow }}H_{\mathrm {c} }^{q+1}(U\cap V)\to \cdots }$

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

• Borel–Moore homology
• Poincaré duality
• Constructible sheaf
• Derived category

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190
• Raoul Bott and Loring W. Tu (1982), Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag
• "Cohomology with support and Poincare duality". Stack Exchange.