Английская Википедия:Cohomology with compact support
Шаблон: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 <math>X</math> be a topological space. Then
- <math>\displaystyle H_c^\ast(X;R) := \varinjlim_{K\subseteq X \,\text{compact}} H^\ast(X,X\setminus K;R)</math>
This is also naturally isomorphic to the cohomology of the sub–chain complex <math>C_c^\ast(X;R)</math> consisting of all singular cochains <math>\phi: C_i(X;R)\to R</math> that have compact support in the sense that there exists some compact <math>K\subseteq X</math> such that <math>\phi</math> vanishes on all chains in <math>X\setminus K</math>.
Functorial definition
Let <math>X</math> be a topological space and <math>p:X\to \star</math> the map to the point. Using the direct image and direct image with compact support functors <math>p_*,p_!:\text{Sh}(X)\to \text{Sh}(\star)=\text{Ab}</math>, one can define cohomology and cohomology with compact support of a sheaf of abelian groups <math>\mathcal{F}</math> on <math>X</math> as
- <math>\displaystyle H^i(X,\mathcal{F})\ = \ R^ip_*\mathcal{F},</math>
- <math>\displaystyle H^i_c(X,\mathcal{F})\ = \ R^ip_!\mathcal{F}.</math>
Taking for <math>\mathcal{F}</math> the constant sheaf with coefficients in a ring <math>R</math> recovers the previous definition.
de Rham cohomology with compact support for smooth manifolds
Given a manifold X, let <math>\Omega^k_{\mathrm c}(X)</math> 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 <math>H^q_{\mathrm c}(X)</math> are the homology of the chain complex <math>(\Omega^\bullet_{\mathrm c}(X),d)</math>:
- <math>0 \to \Omega^0_{\mathrm c}(X) \to \Omega^1_{\mathrm c}(X) \to \Omega^2_{\mathrm c}(X) \to \cdots</math>
i.e., <math>H^q_{\mathrm c}(X)</math> 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 <math>j_*: \Omega^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X)</math> inducing a map
- <math>j_*: H^q_{\mathrm c}(U) \to H^q_{\mathrm c}(X)</math>.
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
- <math>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)</math>
induces a map
- <math>H^q_{\mathrm c}(X) \to H^q_{\mathrm c}(Y)</math>.
If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence
- <math>\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 </math>
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
- <math>\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 </math>
where all maps are induced by extension by zero is also exact.
See also
References
