Questions tagged [cohomology]
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, ...
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Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
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Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$
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Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?
Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
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Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
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Cohomology class of fiber bundle
Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.
I ...
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Relation beween Chern-Simons and WZW levels, and transgression
3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
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English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
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Vanishing result on cohomology with support
Let $X$ be a locally contractible topological space of dimension at least $2$ and $x \in X$ a point. Is it true that $H^i_x(X)=0$ for all $i>1$? If necessary, assume $X$ is a projective variety and ...
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Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
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Renormalization from cohomology point of view
In order to construct a Euclidean quantum filed theory one usually needs to take care of the renormalization problem. Let us consider a simple model like $\phi^4$ in dimension two. In this case just ...
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Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
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Possible relation between causal-net condensation and algebraic K theory
Causal-net condensation is a natural construction which takes a symmetric monoidal category or permutative category $\mathcal{S}$ as input date and produces a functor $\mathcal{L}_\mathcal{S}: \mathbf{...
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Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
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Confusion regarding the vanishing of certain relative cohomology groups
Let $X$ be a projective variety of dimension $n$ and $D \subset X$ is a proper subvariety. Embed $X$ into a projective space $\mathbb{P}^{3n}$. The following argument implies that $H^i(X,X\backslash D)...
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Cohomology of a differentiable stack: evaluation at a point
I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
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