# 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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### Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$

Let $G \leq \mathrm{PGL}(2,\mathbb{C})$ be the subgroup of upper-triangular matrices. I am interested in the natural morphism on the Schur multiplier (i.e. group homology as discrete groups)
$H_{2}(G,...

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### Crossed homomorphism as morphism in the ambient category

Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...

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### Difficulties in the proof of finiteness of n-Selmer group using cohomology

I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.
1st ...

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### Finiteness of Selmer group

I was reading the proof of finiteness of $S^n(E/\mathbb{Q})$ but I am unable to understand from the following lemma how it follows that $S^n(E/L)$ finite.
LEMMA 3.13 For any finte subset $T$ of $\...

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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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### 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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### 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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### 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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### 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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### Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...

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### Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms

Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....

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### Reference for isomorphism between group cohomology and singular cohomology

Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...

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### Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions
$$N\rightarrow G\rightarrow Q$$
are classified by a group homomorphism
$$\rho: Q\rightarrow \operatorname{Out}(N)$$
subject to a constraint ...

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### Transfer for the group of coinvariants: a reference request

Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...

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### Can modular representation theory be used to prove Sylow's existence theorem?

Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...

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### Generators for the first cohomology of free groups

Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...

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### Can K$_3$ of finite fields be related to Teichmüller cocycles?

This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...

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### How is the classification of groups extensions by $H^2$ related to Yoneda Ext?

It is well-known that group extensions
$$1\to A \to H \to G \to 1$$
where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...

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### Explicit examples of 4-cocycles over finite 2-groups

By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...

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### G-modules vs. $\Delta(NG)$-modules

Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...

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### Pontryagin product on the homology of cyclic groups

Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...

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### Lifting SL2(k) to a subgroup of Witt vectors

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$).
Does there ...

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### Examples of Lie groups where $G\to G/H$ splits topologically but not as groups

Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.
My motivation here is to ...

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### Pullbacks of classifying spaces

In what follows all the groups will be discrete, not necessarly finite.
Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...

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### Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$

Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...

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### Group homology for a metacyclic group

Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...

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### Classifying abelian (but non-central) group extensions using homotopy theory

Let $G$ be a group and let $A$ be an abelian group equipped with an action of $G$. Group extensions
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
inducing the ...

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### Spectral sequence construction of Euler class of group extension

Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...

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### Euler class of extension of free nilpotent groups

Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...

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### Cohomological dimension of the kernel of a homomorphism induced by a singular fibration

I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be ...

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### Cohomological dimension of kernel

Let $M$ and $N$ be two connected, smooth (possibly non-compact), aspherical manifolds (that is, $\pi_k(-)=1$ for $k\geq 2$) of dimensions $n$ and $n-r$ respectively. Let $f:M\to N$ be a smooth map ...

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### Generalization of $H^*(\Gamma; \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$?

Let $\Gamma$ be a group acting freely and cocompactly on an acyclic space $X$. I know that there is a isomorphism $H^*(\Gamma, \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$ of $\Gamma$-modules. The ...

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### For which subgroups the transfer map kills a given element of a group?

$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...

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### Strong converse of Kazhdan's property (T)

In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...

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### Subgroups of top cohomological dimension

Let $G$ be a geometrically finite group, i.e. there exists a finite CW complex of type $K(G,1)$.
By Serre's Theorem, every finite-index subgroup $H$ of $G$ satisfies $cd(H)=cd(G)$, but what about the ...

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### Modular forms and the cocycle condition in group cohomology

I am interested in $H¹$ right now and the cocycle condition $φ_{jk} • φ_{ij} = φ_{ik}$ because of how it is said to relate to automorphic forms. I can't quite see the relationship between factors of ...

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### Extension of base field for modules of groups and cohomology [duplicate]

Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...