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        Questions tagged [group-cohomology]

        In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.

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        Applications of one of Serre's Theorems

        This theorem is due to Serre: Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no element of order $p$ and let $H \leq G$ be an open subgroup. Then $cd_p(G) = cd_p(H)$. Where $cd_p(...
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        When is the restriction map in cohomology an isomorphism?

        When is the restriction map in cohomology an isomorphism? Let $G$ be a group, $H$ one of its normal subgroups and $M$ a $G$-module. If $H^1(G,M)\simeq H^1(H,M)^{G/H}$, can we conclude that $H=G$?
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        Actions of locally compact groups on the hyperfinite $II_1$ factor

        Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group. (1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$? (2) If so, how does one ...
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        107 views

        Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

        Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups: $$ 1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1 $$ There exists a ...
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        Extensions of compact Lie groups

        Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups $$ 0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0, $$ $$ 0\rightarrow G\...
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        Decomposition of the group of Bogoliubov transformations

        Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...
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        Singular homology: Lifting simplices gives map in homology

        Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$. Then the ...
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        215 views

        Quasimorphisms and Bounded Cohomology: Quantitative Version?

        Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...
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        1answer
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        Group cohomology with coefficients in a chain complex

        Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each $g \in G$ I have a self ...
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        Second bounded cohomology and normal subgroups

        It may be a naive question, but: If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups? But "many", ...
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        223 views

        Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$

        In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...
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        1answer
        205 views

        Classifying space of semidirect product of groups

        Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
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        Second Bounded Cohomology of a Group: Interpretations

        Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...
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        The computation of $d_2$ in the Hochschild-Serre spectral sequence

        I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups: $1\to N\to G\to G/N\to 1$ where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$. ...
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        Second homology of finitely presented group with free abelianisation

        It is known that for a presented group $G=F/N$ we have $$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$ In general, the right side seems to be difficult to calculate. I am in the special ...

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