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.

## Contents

* 1 Singular cohomology with compact support
* 2 Functorial definition
* 3 de Rham cohomology with compact support for smooth manifolds
* 4 See also
* 5 References

## Singular cohomology with compact support

Let X be a topological space. Then

Hc∗(X;R):=lim->K⊆XcompactH∗(X,X∖K;R)

By definition, this is the cohomology of the sub–chain complex Cc∗(X;R) consisting of all singular cochains ϕ:Ci(X;R)->R that have compact support in the sense that there exists some compact K⊆X such that ϕ vanishes on all chains in X∖K.

## Functorial definition

Let X be a topological space and p:X->⋆ the map to the point. Using the direct image and direct image with compact support functors p*,p!:Sh(X)->Sh(⋆)=Ab, one can define cohomology and cohomology with compact support of a sheaf of abelian groups ℱ on X as

Hi(X,ℱ) = Rip*ℱ,
Hci(X,ℱ) = Rip!ℱ.

Taking for ℱ the constant sheaf with coefficients in a ring R recovers the previous definition.

## de Rham cohomology with compact support for smooth manifolds

Given a manifold X, let Ωck(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 Hcq(X) are the homology of the chain complex (Ωc∙(X),d):

0->Ωc0(X)->Ωc1(X)->Ωc2(X)->⋯

i.e., Hcq(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*:Ωc∙(U)->Ωc∙(X) inducing a map

j*:Hcq(U)->Hcq(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*:Ωcq(X)->Ωcq(Y)∑IgIdxi1∧…∧dxiq↦∑I(gI∘f)d(xi1∘f)∧…∧d(xiq∘f)

induces a map

Hcq(X)->Hcq(Y).

If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence

⋯->Hcq(U)⟶j*Hcq(X)⟶i*Hcq(Z)⟶δHcq+1(U)->⋯

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

⋯->Hcq(U∩V)->Hcq(U)⊕Hcq(V)->Hcq(X)⟶δHcq+1(U∩V)->⋯

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

## See also

* Borel–Moore homology
* Poincaré duality
* Constructible sheaf
* Derived category

## 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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