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Results tagged with cohomology
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user 13767
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
31
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2
answers
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A manifold is a homotopy type and _what_ extra structure?
The takeaway is this: I've come to believe that surfaces are essentially homotopy types with extra structure on cohomology and homology that comes from the manifold structure. … What is really relevant here is the homotopy type of the boundary inclusion $\partial M \hookrightarrow M$, and the corresponding relative cohomology. …
- 35.4k
23
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A manifold is a homotopy type and _what_ extra structure?
Igor is giving a good reference to the topic. For completeness, my education, and satisfaction of other reader's laziness, I'm going to give a rough outline here.
A Poincaré complex is, very similar …
- 35.4k
5
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0
answers
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Pull back group cohomology onto handle decomposition
in the Dijkgraaf-Witten invariant uses the following ingredients:
An oriented, (assumed here to be smooth) manifold $M^n$
A finite group $G$ (and a field, chosen to be $\mathbb{C}$ here)
An $n$-th cohomology … We can then abstractly construct the cohomology class $c^* \phi^* ([\omega]) \in H^n(M, U(1))$.
I'm interested describing this cohomology class very concretely. …
- 5,565
4
votes
0
answers
273
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Cap product for (co)homology from handle decompositions/Kirby diagrams
It is known how to compute the cup product $\smile\,\colon H^i(M) \otimes H^j(M) \to H^{i+j}(M)$ for the Morse cohomology $H^*$ of a smooth manifold $M$. …
- 5,565
13
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1
answer
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A cohomology theory for fusion categories
Ideally, the cohomology theory could just be formulated given the based fusion ring. … I.e. is such a cohomology related to the cohomology of the Brauer-Picard groupoid? In which way is its "tangent cohomology" the Davydov-Yetter cohomology (as discussed here? …
- 28.1k