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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 X {\displaystyle X} be a topological space. Then

H c ∗ ( X ; R ) := lim → K ⊆ X compact ⁡ H ∗ ( X , X ∖ K ; R ) {\displaystyle \displaystyle H_{c}^{\ast }(X;R):=\varinjlim _{K\subseteq X\,{\text{compact}}}H^{\ast }(X,X\setminus K;R)}

By definition, this is the cohomology of the sub–chain complex C c ∗ ( X ; R ) {\displaystyle C_{c}^{\ast }(X;R)} consisting of all singular cochains ϕ : C i ( X ; R ) → R {\displaystyle \phi :C_{i}(X;R)\to R} that have compact support in the sense that there exists some compact K ⊆ X {\displaystyle K\subseteq X} such that ϕ {\displaystyle \phi } vanishes on all chains in X ∖ K {\displaystyle X\setminus K} .

Functorial definition

Let X {\displaystyle X} be a topological space and p : X → ⋆ {\displaystyle p:X\to \star } the map to the point. Using the direct image and direct image with compact support functors p ∗ , p ! : Sh ( X ) → Sh ( ⋆ ) = Ab {\displaystyle p_{*},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 F {\displaystyle {\mathcal {F}}} on X {\displaystyle X} as

H i ( X , F )   =   R i p ∗ F , {\displaystyle \displaystyle H^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{*}{\mathcal {F}},}
H c i ( X , F )   =   R i p ! F . {\displaystyle \displaystyle H_{c}^{i}(X,{\mathcal {F}})\ =\ R^{i}p_{!}{\mathcal {F}}.}

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

de Rham cohomology with compact support for smooth manifolds

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

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

i.e., H c q ( X ) {\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 j ∗ : Ω c ∙ ( U ) → Ω c ∙ ( X ) {\displaystyle j_{*}:\Omega _{\mathrm {c} }^{\bullet }(U)\to \Omega _{\mathrm {c} }^{\bullet }(X)} inducing a map

j ∗ : H c q ( U ) → H c q ( X ) {\displaystyle j_{*}: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

f ∗ : Ω c q ( X ) → Ω c q ( Y ) ∑ I g I d x i 1 ∧ … ∧ d x i q ↦ ∑ I ( g I ∘ f ) d ( x i 1 ∘ f ) ∧ … ∧ d ( x i q ∘ f ) {\displaystyle f^{*}:\Omega _{\mathrm {c} }^{q}(X)\to \Omega _{\mathrm {c} }^{q}(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 c q ( X ) → H c q ( Y ) {\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

⋯ → H c q ( U ) ⟶ j ∗ H c q ( X ) ⟶ i ∗ H c q ( Z ) ⟶ δ H c q + 1 ( U ) → ⋯ {\displaystyle \cdots \to H_{\mathrm {c} }^{q}(U){\overset {j_{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(X){\overset {i^{*}}{\longrightarrow }}H_{\mathrm {c} }^{q}(Z){\overset {\delta }{\longrightarrow }}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

⋯ → H c q ( U ∩ V ) → H c q ( U ) ⊕ H c q ( V ) → H c q ( X ) ⟶ δ H c q + 1 ( U ∩ V ) → ⋯ {\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){\overset {\delta }{\longrightarrow }}H_{\mathrm {c} }^{q+1}(U\cap V)\to \cdots }

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

See also

References

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