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https://math.stackexchange.com/questions/754338/singular-cohomology-with-compact-support
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https://math.stackexchange.com/questions/521265/understanding-cohomology-with-compact-support
I am trying to understand the definition of (singular) cohomology with compact supports. ... Understanding cohomology with compact support. Ask Question Asked 6 years, 1 month ago. ... Simplicial Cohomology With Compact Support. 11. Reference Request: Equivariant cup product in singular cohomology ...
https://en.wikipedia.org/wiki/Singular_homology
Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
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://en.wikipedia.org/wiki/%C3%89tale_cohomology
Cohomology with compact support is the special case of this with S a point. If f is a separated morphism of finite type then R q f! takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then R q f! vanishes on torsion sheaves for q > 2n.
http://people.bath.ac.uk/jlpn20/dt2013/wrap.pdf
Singular cohomology with compact supports It is possible to define singular cohomology of Mn with compact supports. It is the cohomology H∗ c (M) of the complex C∗ c M), where Ck c (M) ⊆ Ck(M) is the subgroup of cochains α such that there is a compact K ⊂ M …
https://www.physicsforums.com/threads/cohomology-with-compact-support.570117/
Jan 23, 2012 · So in singular cohomology with compact support, you consider the subcomplex of C^k(X) of those singular k-cochains f having the property (P) there exists a compact subset K of X such that f(c)= 0 for all k-chains c sitting outside of K.
https://en.wikipedia.org/wiki/Sheaf_cohomology
For arbitrary topological spaces, singular cohomology and sheaf cohomology (with constant coefficients) can be different. This happens even for H 0. The singular cohomology H 0 (X,Z) is the group of all functions from the set of path components of X to the integers Z, whereas sheaf cohomology H 0 (X,Z X) is the group of locally constant ...
https://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf
A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science
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