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https://mathoverflow.net/questions/59017/group-cohomology-with-compact-support
This response is a little late, but I have thought about the same question recently. I don't think there is a way to define cohomology with compact support in a purely group theoretic way. The problem is that compact cohomology will distinguish between multiple cusps, but cocycles can only capture one cusp.
https://en.wikipedia.org/wiki/%C3%89tale_cohomology
The étale cohomology groups with compact support of a variety X are defined to be H c q ( X , F ) = H q ( Y , j ! F ) {\displaystyle H_{c}^{q}(X,F)=H^{q}(Y,j_{!}F)}
https://math.stackexchange.com/questions/39214/cohomology-with-compact-support
For a closed manifold, cohomology and cohomology with compact supports coincide, but for open manifolds they do not. The top dimensional cohomology with compact support is always one-dimensional for a connected orientable manifold, regardless of whether or not the manifold is closed.
http://tungsteno.io/post/def-de_rham_cohomology_groups_compact_support/
But we have already seen that for the integration to be defined rigorously we have to stick to compact support differential forms. Surprisingly enough, this restriction leads to another type of cohomology. The restriction of the de Rham complex to compact support differential forms leads to a new complex
http://www.math.wisc.edu/~maxim/Topnotes9.pdf
(X)) is called the cohomology of Xwith compact support. De nition of Direct Limit of Groups (G ) 2I Let G be abelian groups indexed by some directed set I, namely, Iis partially ordered and 8 ; 2I, 9 2Isuch that and . Suppose also that for each pair there is a homomorphism f : G !G such that f = id G and f = f f . Now consider q 0G 0=˘where x ...
https://ncatlab.org/nlab/show/compactly+supported+cohomology
In compactly supported cohomology cocycles and coboundaries on some space are required to have compact support: to be non-trivial only over a compact subspace/compact subobject of the base. References General. James Milne, section 18 of Lectures on Étale Cohomology; Compactly supported de Rham cohomology
http://tungsteno.io/post/exa-de_rham_cohomology_r_2_compact_support/
de Rham Cohomology of R² with Compact Support Tungsteno is a project whose goal is to make mathematics accessible to everybody, completely free, based on open collaboration and the best pedagogical tools Free, open-source online mathematics for students, teachers and workers
https://mathoverflow.net/questions/16265/homology-with-compact-supports
Note that the usual singular homology are with compact support: the cycles have compact images. By contrast, the usual singular cohomology do not have compact support as a cocycle may take nonzero value on a sequence of cycles that run off to infinity. There is a book by Massey, "Homology and cohomology theory.
https://math.stackexchange.com/questions/770201/show-that-the-compactly-supported-de-rham-cohomology-groups-hp-dr-mathbb
Compactly supported cohomology is the cohomology of the complex of compactly supported differential forms (i.e., smooth forms that are zero outside of some compact set).
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