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

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Let   be a topological space. Then

 

By definition, this is the cohomology of the sub–chain complex   consisting of all singular cochains   that have compact support in the sense that there exists some compact   such that   vanishes on all chains in  .

Functorial definition

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Let   be a topological space and   the map to the point. Using the direct image and direct image with compact support functors  , one can define cohomology and cohomology with compact support of a sheaf of abelian groups   on   as

 
 

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

de Rham cohomology with compact support for smooth manifolds

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Given a manifold X, let   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   are the homology of the chain complex  :

 

i.e.,   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 XU) is a map   inducing a map

 .

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: YX be such a map; then the pullback

 

induces a map

 .

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

 

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

 

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

See also

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References

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  • Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190
  • Raoul Bott and Loring W. Tu (1982), Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag
  • "Cohomology with support and Poincare duality". Stack Exchange.