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

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

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 $\phi$ vanishes on all chains in ${\displaystyle X\setminus K}$.

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.

References

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 842190
• Raoul Bott and Loring W. Tu (1982), Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag