Cohomology with compact support

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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[edit]

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 XU) 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: YX 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 = XZ 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[edit]

  • 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