# 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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### H^2 of symmetric group

I'm a number theorist in need of some group cohomology lemmas, and I'm rather bewildered by the level of generality used in the literature. Specifically, the result I need is as follows: the ...
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### Interpretation of $H^3(\mathrm{Gal}(L/K),L^\times)$

During my work I came across the group $H^3(\mathrm{Gal}(L/K),L^\times)=H^3(L/K,L^\times)$ for certain (infinite) Galois extensions $L/K$ (for an arbitrary field $K$) and I wondered if there is an ...
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### Alternating bihomomorphism is a skew 2-cocycle

It seems to be a well-known fact that every alternating bihomomorphism $G\times G\to\mathbb{C}^\times$ for a finite abelian group $G$ is the skew of some 2-cocycle (see for instance Symmetric analogue ...
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### Extensions of $G$-modules parametrized by $H^1$

Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
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### Bounding the dimension of $H^1(G, V\otimes V^{\vee})$

Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified away from a finite set of primes, $S$. Let $V$ be a finite dimensional, $G_S$-representation over $\mathbb{F}_p$ (...
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### What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?

In Cassels' paper Arithmetic on curves of genus 1. IV introducing the Cassels–Tate pairing the following lemma is stated. Lemma 5.1: Let $q$ be a rational prime and $\Gamma$ the Galois group of the ...
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1 vote
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### Second group cohomology of a twisted fundamental group

Let $X$ be a smooth hyperbolic projective curve defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes, and let $\pi:=\pi_1^{\text{ét}}(X, \overline{b})$ denote its étale fundamental ...
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### Shapiro's lemma for group derivations [duplicate]

Let $G$ be a finite group, let $H$ be a subgroup of $G$, let $W$ be an $H$-module and let $V$ be the $G$-module induced from $W$. Shapiro's lemma says that $H^1(G,V)\cong H^1(H,W)$. I was wondering if ...
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1 vote
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### How can we construct a non-trivial central extension of a Lie group

Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
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1 vote
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### Stable homology of general linear groups

For what class of rings $R$, is the stable homology (with various choices of coefficients) of $GL_n(R)$ known? Borel computed it rationally for number rings, Quillen computed it for finite fields. Are ...
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### Bounding dimensions of Galois cohomology

Let $G$ be the absolute Galois group of the rationals, and $V$ a finite dimensional $p$-adic representation. Is there a way to provide a general bound on $H^i(G, V)$ in terms of $\dim_{\mathbb{Q}_p} V$...
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### Are isomorphic maximal tori stably conjugate?

Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
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### Interpretation of Kazhdan T property cohomologically

$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology. In general, we heuristically have $H^1(G,Ad(V))$ (...
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### Do acyclic amenable groups exist?

Is there an example of a nontrivial discrete amenable group with vanishing integral homology? To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
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### Explicit representatives for Borel cohomology classes of a compact Lie group?

I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
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1 vote
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### When is $B^G\backslash(B/A)^G$ finite?

Let $G$ be a locally compact group, let $A,B$ be (not necessarily abelian) connected reductive complex groups equipped with continuous actions of $G$ via algebraic automorphisms. Let $\phi:A\to B$ be ...
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### Group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients

What is known about the group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients and what are strategies to compute it (or at least some groups for low degrees)? Here I want to consider ...
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### Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$

As $\rm{PSL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/3\mathbb{Z})$, its cohomology groups $H^n(\rm{PSL}(2,\mathbb{Z});\mathbb{Z})$ are easy to get. Let $N$ be a product of distinct primes. ...
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