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.
Singular cohomology with compact support
Let [math]\displaystyle{ X }[/math] be a topological space. Then
- [math]\displaystyle{ \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]\displaystyle{ C_c^\ast(X;R) }[/math] consisting of all singular cochains [math]\displaystyle{ \phi: C_i(X;R)\to R }[/math] that have compact support in the sense that there exists some compact [math]\displaystyle{ K\subseteq X }[/math] such that [math]\displaystyle{ \phi }[/math] vanishes on all chains in [math]\displaystyle{ X\setminus K }[/math].
Functorial definition
Let [math]\displaystyle{ X }[/math] be a topological space and [math]\displaystyle{ p:X\to \star }[/math] the map to the point. Using the direct image and direct image with compact support functors [math]\displaystyle{ 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]\displaystyle{ \mathcal{F} }[/math] on [math]\displaystyle{ X }[/math] as
- [math]\displaystyle{ \displaystyle H^i(X,\mathcal{F})\ = \ R^ip_*\mathcal{F}, }[/math]
- [math]\displaystyle{ \displaystyle H^i_c(X,\mathcal{F})\ = \ R^ip_!\mathcal{F}. }[/math]
Taking for [math]\displaystyle{ \mathcal{F} }[/math] the constant sheaf with coefficients in a ring [math]\displaystyle{ R }[/math] recovers the previous definition.
de Rham cohomology with compact support for smooth manifolds
Given a manifold X, let [math]\displaystyle{ \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]\displaystyle{ H^q_{\mathrm c}(X) }[/math] are the homology of the chain complex [math]\displaystyle{ (\Omega^\bullet_{\mathrm c}(X),d) }[/math]:
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ j_*: \Omega^\bullet_{\mathrm c}(U) \to \Omega^\bullet_{\mathrm c}(X) }[/math] inducing a map
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \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
- Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3
- Raoul Bott and Loring W. Tu (1982), Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag
- "Cohomology with support and Poincare duality". https://math.stackexchange.com/q/2732445.
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