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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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      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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      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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      197 views

      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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      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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      92 views

      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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      91 views

      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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      222 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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      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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      154 views

      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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      1answer
      137 views

      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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      124 views

      Survey on homological stability

      Background: homological stability is the phenomenon that for many natural families of groups $G_0 \to G_1 \to G_2 \to \dots$, the group homology $H_i(G_n)$ stabilizes for $n \gg i$. This is e.g. the ...

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