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.

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.

Cohomology with compact support

Singular cohomology with compact support

Functorial definition

de Rham cohomology with compact support for smooth manifolds

See also