Cohomology with compact support

Template:No footnotes 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

$H_c^{\ast }(X;R):=\underset{K\subseteq X\text{compact}}{\underrightarrow{\lim }}{H}^{\ast }(X,X\setminus K;R)$

By definition, this is the cohomology of the sub–chain complex $C_c^{\ast }(X;R)$ consisting of all singular cochains $\varphi :C_i(X;R)\to R$ that have compact support in the sense that there exists some compact $K\subseteq X$ such that $\varphi$ 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 $ℱ$ on $X$ as

$H^i(X,ℱ)=R^i{p}_{*}ℱ,$

$H_c^i(X,ℱ)=R^i{p}_{!}ℱ.$

Taking for $ℱ$ 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 ${\mathrm{\Omega }}_{\mathrm{c}}^k(X)$ 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 $({\mathrm{\Omega }}_{\mathrm{c}}^{\bullet }(X),d)$:

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

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 $j_{*}:{\mathrm{\Omega }}_{\mathrm{c}}^{\bullet }(U)\to {\mathrm{\Omega }}_{\mathrm{c}}^{\bullet }(X)$ 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

$f^{*}:{\mathrm{\Omega }}_{\mathrm{c}}^q(X)\to {\mathrm{\Omega }}_{\mathrm{c}}^q(Y)\sum _I{g}_{I}d{x}_{i_1}\wedge \dots \wedge d{x}_{i_q}\mapsto \sum _I(g_I\circ f)d(x_{i_1}\circ f)\wedge \dots \wedge d(x_{i_q}\circ f)$

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

$\cdots \to H_{\mathrm{c}}^q(U)\stackrel{j_{*}}{⟶}{H}_{\mathrm{c}}^q(X)\stackrel{i^{*}}{⟶}{H}_{\mathrm{c}}^q(Z)\stackrel{\delta }{⟶}{H}_{\mathrm{c}}^{q+1}(U)\to \cdots$

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

$\cdots \to H_{\mathrm{c}}^q(U\cap V)\to H_{\mathrm{c}}^q(U)\oplus H_{\mathrm{c}}^q(V)\to H_{\mathrm{c}}^q(X)\stackrel{\delta }{⟶}{H}_{\mathrm{c}}^{q+1}(U\cap V)\to \cdots$

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

See also

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

References

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