Cohomology with compact support

Functorial definition

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 ${\displaystyle p_{\ast },p_{!}:\text{Sh}(X)\to \text{Sh}(\star )=\text{Ab}}$, one can define cohomology and cohomology with compact support of a sheaf of abelian groups ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$ as

${\displaystyle {\displaystyle H^i(X,\mathcal{F})=R^i{p}_{\ast }\mathcal{F},}}$

${\displaystyle {\displaystyle H_c^i(X,\mathcal{F})=R^i{p}_{!}\mathcal{F}.}}$

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

de Rham cohomology with compact support for smooth manifolds

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

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

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

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

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

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

induces a map

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

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

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

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

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

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