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Results tagged with cohomology
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user 36688
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, ...
2
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127
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Dividing a n- cochain by a 1-cochain
Let $\alpha\in C^{1}(X,\mathbb{R})$ and $\beta\in C^{n}(X,\mathbb{R})$ be cochains in real singular cohomology. Asume that $\alpha \smile \beta=0$. … This is motivated by the similar situation, with affirmative answer, in De Rham cohomology and differential forms. …
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3
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1
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Cohomologically minimal spaces
inclusion map $i:A \to X$, $i^*$ is NOT a ring isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(A,\mathbb{Z})$"
In the other word, $A$ is the only subset of $A$ whose inclusion gives a ring isomorphism cohomology … Examples; All closed manifolds (see A closed manifold with a subset with the same ring cohomology )
Non Examples: Closed disc, figure 8,...etc. …
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209
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A closed manifold with a subset with the same ring cohomology
Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question $ …
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2
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1
answer
173
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A cohomology associated with a codimension one foliation
The total cohomology is denoted by $H^{*}(\alpha)$
Are these cohomologies finite dimensional vector space?
Are there some dynamical information in this cohomology? … Is this cohomology independent of choosing the one form $\alpha$ which kernel is tangent to the foliation? …
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2
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0
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89
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A cohomology associated with a codimension one foliation(2)
What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$? … Moreover what is the description of this cohomology for the Reeb foliation of $S^{3}$? …
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4
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0
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268
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A cohomology associated to a symplectic manifold
so we obtain a cohomology $H_{\omega}^k$.
Does this cohomology depend on choosing the symplectic structure $\omega$? … Does this cohomology contains some information about the symplectic manifold? Is there a trivial description for this cohomology? …
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1
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2
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366
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A cohomology associated to a 1- form
So we naturally obtain a cohomology.
Is each cohomology, a finite dimensional vector space? …
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3
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1
answer
380
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A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Then we have a complex of fundamental groups $$\ldots \to \pi_{1}(G_{n})\to \pi_{1}(G_{n+1})\to \ldots$$
Are there some standard theorems or results about a comparison betwee the cohomology of $\pi …
- 356
12
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3
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851
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A nontrivial principal bundle which satisfies Leray-Hirsch theorem
What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\ …
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4
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146
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A relation between Hochschild cohomology of a $C^*$ algebra and its bidual
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$? …
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8
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508
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A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem...
So we get a cohomology $H^*(M,D)$, the cohomology of $M$ along distribution $D$.
Now assume that the flow of a vector field $\tilde{X}$ on $M$ preserves $D$. … The above cohomology we introduce is a candidate for such appropriate cohomology. …
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3
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1
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356
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A second cohomology class associated to a Riemannian manifold
The first Chern class of the complexification of this line bundle determines a cohomology class in $H^2(S^*M)\simeq H^2 (M)$. … So we obtain a cohomology class $\lambda(M) \in H^2 (M)$.
Does this cohomology class have an alternative formulation?Is there a name for this cohomology class? …
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1
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270
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A cohomology associated to a Riemannian manifold
What can be said about a riemannian manifold whose all cohomology groups $H^n(G,M)$ vanish? … Namely is it true to say that two nonisometric metrics on $N$ give two different cohomology sequence? …
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7
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277
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A cohomology associated to a vector field on a Riemannian manifold
For zero vector field we get the standard de Rham cohomology.
Is this cohomology always a finite dimensional space? Does it depend on the Riemannian structure? … Is there an appropriate analogy of this cohomology in algebraic topology or other type of cohomologis? …
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11
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841
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Is "Determinant" a Hochschild coboundary?
Assume that $n>2$.
Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$,
would be a Hochschild coboundary? …
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