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https://mathoverflow.net/questions/317640/invariants-in-relative-cohomology-and-compact-support-cohomology-of-the-quotient
Invariants in relative cohomology and compact support cohomology of the quotient. Ask Question Asked 12 months ago. Active 12 months ago. Viewed 203 times 7 $\begingroup$ Let $\cal H$ be ...
https://en.wikipedia.org/wiki/Sheaf_cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central figure of this study is Alexander Grothendieck and his 1957 Tohoku paper.
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
https://math.stackexchange.com/questions/137489/an-isomorphism-in-relative-de-rham-cohomology
I don't think that what you're asking is true exactly as stated, but what you're getting at is the Thom isomorphism. The proof of the isomorphism really only uses basic facts about vector bundles and axioms of homology, so pick any proof that you like and you should be …
https://math.stackexchange.com/questions/1523309/what-is-dual-to-relative-homology
What is dual to relative homology? Ask Question Asked 3 years, 11 months ago. ... We can also define relative de Rham cohomology of the pair $(X,A)$. I believe that for this you consider the double complex $\Omega^{\bullet}(X) \to \Omega^{\bullet}(A)$ (where we are pulling back forms by the inclusion map), take its total complex and then take ...
https://arxiv.org/pdf/1511.00324.pdf
To introduce differential characters with compact support, we follow the well-known con-struction of compactly supported cohomology as the colimit of the relative cohomology functor H♯(M,M\ −;G), with Gan Abelian group, over the directed set KM of compact subsets of M.
https://en.wikipedia.org/wiki/Borel%E2%80%93Moore_homology
Definition via sheaf cohomology. For any locally compact space X, Borel–Moore homology with integral coefficients is defined as the cohomology of the dual of the chain complex which computes sheaf cohomology with compact support. As a result, there is a short exact sequence analogous to the universal coefficient theorem:
https://arxiv.org/pdf/1907.13336.pdf
Moreover, the relative de Rham cohomology is isomorphic to the de Rham cohomology with compact support, while it does not hold true for the Morse–Novikov case anymore (see [11, 17]). So we are not able to apply the relative cohomological method via the compactly supported cohomology directly as [11, 12]. Acknowledgement.
http://export.arxiv.org/pdf/1511.00324
To introduce differential characters with compact support, we follow the well-known con-struction of compactly supported cohomology as the colimit of the relative cohomology functor H♯(M,M\ −;G), with Gan Abelian group, over the directed set KM of compact subsets of M.Cited by: 2
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
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