# 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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### Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...

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### An alternative description of normalized cochains in terms of tensor powers of the augmented ideal

I want to know if the following alternative of the normalized non-homogeneous cochains is already know.
Let $G$ be a group and let ${\mathcal I}={\mathcal I}_G$ be its augmentation ideal, ${\mathcal ...

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156 views

### Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?

I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...

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### Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...

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### Schur multiplier of 2-Sylow subgroups of symmetric group

Is the Schur multiplier of 2-Sylow subgroups of symmetric groups on $n\geq 4$ symbols known?
I couldn’t find much except that multiplier of $S_n$ is contained in the multiplier of 2-Sylow subgroup.

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### Abelianization of general linear group of a polynomial ring

For $K$ a field, is it known what the abelianization of $GL_2(K[X])$ is?

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57 views

### Relation between topological and differentiable Lie group cohomology for unitary modules

It is well known, that if $V$ is a $C^\infty$ $G$-module, then the differentiable cohomology groups $H^*_{d}(G, V)$ are isomorphic to $H^*(\mathfrak{g}, K, V)$. I am interested if the result can be ...

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### Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?

When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is ...

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### Examples of groups admitting a proper $1$-cocyle for a bounded representation

A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...

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### Group law in universal central extension of Thompson's group T

I'm having some troubles with universal central extensions and associated cocycles; in particular, I want to understand the group law of the universal central extension of the Thompson's group $T$.
...

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305 views

### How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...

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### Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$

I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...

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119 views

### Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...

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415 views

### Realizing inner automorphisms on Eilenberg-MacLane spaces

Let $G$ be a discrete group and let $(X,x_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi_1(X,x_0) = G$ and the universal cover $\widetilde{X}$ is contractible. ...

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### Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...

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### $G$ uncountable implies $K(G,1)$ is not a finite CW complex

I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ ...

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301 views

### Differential version of $G\mapsto H^3(G,\mathbb Z)$?

Let $\mathit{cLieGrp}^{\mathrm{inj}}$ be the category of compact connected Lie groups, and injective continuous group homomorphisms.
Is there a reasonable functor (some kind of degree $3$ differential ...

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### Cohomology with group ring coefficients and compact support

Let $X$ be a contractible space and let $G$ be a group acting freely, properly discontinuously, and cocompactly on $X$. We get an induced action on the singular chain complex $C_{\bullet}(X)$.
...

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### Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$

$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...

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### Regular growth of ranks in Iwasawa tower

$\newcommand{\rank}{\operatorname{rank}}$Let $G=H \times K$ be a torsion free pro-$p$, $p$-adic Lie group. Let $H =\mathbb{Z}_p$, the ring of $p$-adic integers and $K$ is a non-commutative torsion ...

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### Trivial homology with local system

Let $X$ be the classifying space of the Higman group $G$. It is well known that $G$ is an acyclic group
$$H_{\ast}(X;\mathbb{Z})=H_{\ast}(pt;\mathbb{Z}).$$
Now, suppose that $\mathcal{M}$ is a ...

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### To understand the description of relative group homology $H_{*}(G,H;\mathbb{Z})$ in terms of free $G$-resolution

Let $G$ be a group and $H$ its subgroup ($H$ need not to be normal). Consider a chain complex $(C_{*}(G), \partial)$ where $C_n(G)$ is the free abelian group generated over the set $G^{n+1}$ and $\...

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### Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?

Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2,
|S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ ...

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144 views

### When does group cohomology $H^1(G,M)$ depend only on the image of $G$ in Aut($M$)?

To motivate the question (and narrow it down if the one I asked is too broad), I'm doing readings from Manin's cubic forms book. A while back I was asked to compute the Galois cohomology $H^1(G, Pic(...

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### Abelian lie algebra homology

Let $\mathfrak g$ be an abelian Lie algebra over $\mathbb Z.$ We can consider its Lie-algebra homology, say as $\mathrm{Tor}^{U(\mathfrak g)}_*(\mathbb Z,\mathbb Z)$ and its group homology as $\mathrm{...

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### Extended double 2-cocycle conditions: Mathematical structure behind?

Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly.
The ordinary group 2-cocycle condition:
Let us remind the usual so-called homogeneous group 2-cocycle $...

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### Vanishing of first co-homology with coefficients modular representations of small dimension

Is the following true:
For any $n$ there exists $p_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p_0$ and any (irreducible) $\mathbb F_p$ representation $V$ ...

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### Do there exist acyclic simple groups of arbitrarily large cardinality?

Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...

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293 views

### Imperfect Tate (cup product) pairing in Galois cohomology?

Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$
and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$.
Let $M$ be a finite $\Gamma$-module, that is, a finite abelian ...

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483 views

### An intuitive explanation for group cohomology via cochains?

I'm fairly new to topology, and so far I've understood cohomology via cochains.
First we build an object called a cochain ($C^n$), then define a differential map that takes you from $C^n$ to $C^{n+1}...

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### Where should I search for computations of group cohomology rings of not-too-complicated finite groups?

A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
...

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187 views

### An action of the symmetric group $S_n$ on group cohomology $H^n(G, A)$ of abelian groups

Let $H$, $A$ be discrete abelian groups, and for simplicity suppose $A$ is given the trivial $H$-action.
When considering the second cohomology group $H^2(H,A)$, it is natural to talk about the ...

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120 views

### Computation of extension groups in the category of pairs $(M,f)$

Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...

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### Multiplicity of indecomposable stable summands of $BG^{\wedge}_p$

I am reading the article Homotopy stable classification of $BG^{\wedge}_p$ by Martino-Priddy. Let $P_u$, $P_v$ be $p$-subgroups of a finite group $G$, such that $P_u\leq x^{-1}P_v x$ for some $x\in G$,...

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### Unifying “cohomology groups classify extensions” theorems

It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean:
1) Derived functor of hom, $\text{Ext}^1_R(M, N)$. ...

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### When are there continuous families of pull-backs of a discrete cohomology class of a compact Lie group?

Let $\mathcal{G}$ be a compact Lie group. Then define $H^n(\mathcal{G},\mathrm{U}(1))$ to be the cohomology of measurable cochains $\mathcal{G}^{\times n} \to \mathrm{U}(1)$ with the usual coboundary ...

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### Extension of $FP_{n}$ group

I am reading a paper (Finitely presented residually free groups by Bridson, Howie, Miller III, and Short, Theorem 5.2) where they write the following:
Since $S_{0}$ is a group of type $FP_{n}(\mathbb{...

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### A formula that proves that $G$ acts trivially on $H^*(G,M)$

If $G$ is a group and $M$ a $G$-module then for $n\geq 0$ we have an action of $G$ on the cochains from $C^n(G,M)$. If $s\in G$, $a\in C^n(G,M)$ then $(sa)_{s_1,\ldots,s_n}=sa_{s^{-1}s_1s,\ldots,s^{-1}...

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### On normalized 2-cocycle

Let $G$ be a group acts trivially on an abelian group $A$. Let
$\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume
that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times
...

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186 views

### Cohomology of linear algebraic groups

Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...

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167 views

### Cup product in Tate Cohomology Ring

Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^*(G, \mathbb{F}_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}_pG$-module.
There is a ...

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65 views

### Lattices with trivial coinvariants for finite groups

Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...

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### Definition of the Gauss symbol [closed]

Tahara refers to the "Gauss symbol" in the article, On the second cohomology groups of semidirect products, Math. Z. 129 (1972) 365--379. For a fixed $n$, let $S_{ij}$ be the expression
\begin{...

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### When is the cohomology of a fiber bundle a tensor product?

Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...

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### Analogue of integration for group cohomology

Consider some oriented surface $S$ with fundamental group $\pi_1(S)$. The group cohomology of $\pi_1(S)$ with coefficients in $\mathbb{R}$ is isomorphic to the de Rham cohomology of $S$. In degree 2, ...

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### Finite generation of group homology

I am reading 'Subgroups of direct products of limit groups' of Bridson, Howie, Miller and Short (http://annals.math.princeton.edu/wp-content/uploads/annals-v170-n3-p11-p.pdf) and I am finding similar ...

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150 views

### Group cohomology: Why does the trivial Z coefficient produce nontrivial cohomology [closed]

Let $G$ be a group and $M$ be a $G$-module. Then group cohomology $H^q(G,M)$ is defined as the right derived functor $\operatorname{Ext}^q_{\mathbb Z G}(\mathbb Z,M)$. Here $\mathbb Z$ is the trivial $...

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502 views

### Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...

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### Coboundary in Kummer theory

Let $K$ be a non archimedean local field whose residue field is of characteristic $p$. Denote by $G$ the absolute Galois group of $K$. Denote by $\mu_p$ the group of $p$-roots of unity and assume it ...

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431 views

### First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....