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.

de Rham cohomology with compact support for smooth manifolds

Given a manifold X, let \Omega^k_{\mathrm c}(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^q_{\mathrm c}(X) are the homology of the chain complex (\Omega^\bullet_{\mathrm c}(X),d):

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

i.e., H^q_{\mathrm c}(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^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X) inducing a map

j_*: H^q_{\mathrm c}(U) \to H^q_{\mathrm c}(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^q_{\mathrm c}(X) \to \Omega^q_{\mathrm c}(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^q_{\mathrm c}(X) \to H^q_{\mathrm c}(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^q_{\mathrm c}(U) \overset{j_*}{\longrightarrow} H^q_{\mathrm c}(X) \overset{i^*}{\longrightarrow} H^q_{\mathrm c}(Z) \overset{\delta}{\longrightarrow} H^{q+1}_{\mathrm c}(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^q_{\mathrm c}(U \cap V) \to H^q_{\mathrm c}(U)\oplus H^q_{\mathrm c}(V) \to H^q_{\mathrm c}(X) \overset{\delta}{\longrightarrow} H^{q+1}_{\mathrm c}(U\cap V) \to \cdots

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

References