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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Geometric interpretation of shuffle product

Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a_0,\dots,a_k] \in B^k(A)=A^{\...
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Is there a cohomological interpretation of the bilinear form arising from Clifford theory?

For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
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2 votes
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Cohomology of compact open subgroups of semisimple groups over local fields

Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
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Non-abelian group cohomology, additional information

Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module. We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...
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3-cocycles on outer automorphism groups

Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is ...
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2 votes
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Different definitions of p-fusion and Mislin's theorem

Currently, I am trying to understand and compute homology of finite groups with coefficients in a field of positive characteristic. So, I was searching for some results that could reduce this problem (...
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5 votes
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Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\to H^2(G,V)$ is injective

Let $G$ be a group and $F$ a field. I am particularly interested in the case where $G$ is a uniform lattice in a Lie group and $F=\mathbb{F}_2$, or in finite groups $G$ where $\operatorname{char} F$ ...
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How can I detect the homology image of a unipotent group in the general linear group?

Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements. Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
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8 votes
1 answer
357 views

Abelianization of $\mathrm{GL}_2(R)$

$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
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Augmentation ideal of a free group

If $F$ is a free group then it has cohomological dimension one, which implies that the augmentation ideal $IF=\operatorname{ker}(\epsilon:\mathbb{Z}G\to \mathbb{Z})$ of its group ring is a projective $...
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Injectivity of a module over tensor product of group algebras

Let $R$ be a commutative ring and $G$ be a finite group that is the direct product $G=H_1*H_2$ of two subgroups $H_1$ and $H_2$ with $H_1$ Abelian. Let $R[G]$ be the group ring and $M$ be a left $RG$-...
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On the linearizability of the action of a finite group on a formal polydisc

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
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Tensor product of group rings [closed]

Let $R$ be a commutative ring with unity and $\mathbb{Z}_n$ be the cyclic group of integers modulo $n$. If $R\mathbb{Z}_n$ is the group ring of $\mathbb{Z}_n$ formed over $R$, then I want to compute ...
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Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules. See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$ and of the cup product pairings \...
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Direct limit of groups rings of finite quotients of a profinite group

Background: Let $G$ be a profinite group, for $M \leqslant N$ open normal subgroups we have the projection map $p_{M,N} \colon G/M \to G/N$ which induces a transfer map on rational group rings $$ p_{M,...
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8 votes
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132 views

Non-finitely presented FP groups with cohomological dimension $2$

In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
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If a module over a group ring has no self-extensions, can it have any self-extension over subgroups?

I use the terminology of K.S. Brown's "Cohomology of groups". Let $RG$ be the group ring formed by an arbitrary group $G$ over a commutative ring $R$, and let $X$ be a finitely generated ...
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Does Kudo's transgression theorem still hold when the coefficient is ℤ/pⁿℤ?

I am trying to compute a Lyndon–Hochschild–Serre spectral sequence with coefficient $\mathbb{Z}/p^2\mathbb{Z}$, where $p$ is an odd prime, and Kudo's transgression theorem looks like a good tool for ...
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Kan–Thurston theorem and R-completion

A corollary of the Kan and Thurston theorem states that the space $X$ (path connected) can be recovered, up to homotopy, by applying the fiber-wise $\mathbb{Z}$-completion functor of Bousfield–Kan (...
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Salvetti complexes and cohomology of affine completion of Artin groups $E_6$ and $E_7$

After the solution of the Brieskorn-Arnold Pham conjecture on the asphericity of a space for affine Artin groups by Paolini and Salvetti MR4243019 (arXiv), I would like to know if there are ...
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2 votes
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Chain-level representability of simplicial cohomology

There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...
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2 votes
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Are tamely ramified representations $\widehat{\mathbb{Q}_p^\text{tr}}$-admissible?

Let $K$ be a finite field extension of $\mathbb{Q}_p$. Let $G_K$ denote the absolute Galois group of $K$, $I_K$ the inertia subgroup and $I_K^{(p)}$ the $p$-Sylow subgroup of $I_K$, i.e. the wild ...
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8 votes
1 answer
441 views

Importance of third homology of $\operatorname{SL}_{2}$ over a field

$\DeclareMathOperator\SL{SL}$I am reading some papers about the third homology of linear groups. In particular for the $\SL_{2}$ over a field. Why is it important to study these homologies? I have ...
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1 answer
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Coboundary operators, 1-cocycles and computing cohomology

My question is about the compatibility and consistency between two definitions of cohomology in two books. I asked this question about 10 days ago on MathSE and I set a bounty on it, but I didn't ...
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1 vote
1 answer
242 views

Exact sequence, de Rham representation

Let $k$ be a $p$-adic field and $G_k$ its absolute Galois group. Let $B_\text{dR}$ be the de Rham period ring with the usual filtration given by powers of $t$. For $i < j$ integers we have an exact ...
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16 votes
1 answer
650 views

Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL_2(\mathbb{Z})$ via $$ \chi(\SL_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \...
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2 votes
1 answer
173 views

Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]

$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are ...
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1 vote
0 answers
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Number of generators of a group

Given a (finite) group $G$. Is there any bounds on the minimum number of generators $d(G)$? For example, it is clear that $d(G) \geqslant d(G^{ab})$. Where the right hand side can be easily computed. ...
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2 votes
0 answers
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Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative ...
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0 answers
236 views

The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field, and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$. By local class field theory, there is a canonical isomorphism $$...
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2 votes
1 answer
113 views

Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit. We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...
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1 vote
0 answers
104 views

Inflation in degrees $0$, $-1$, and $-2$ for Tate cohomology of finite groups

Let $\pi\colon G'\to G$ be a surjective homomorphism of finite groups, and let $A$ be a $G$-module. I need explicit formulas for the inflation maps $${\rm Inf}^{r}\colon H^{r}(G,A)\to H^{r}(G',A)$$ ...
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5 votes
0 answers
204 views

Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$

Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional. There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
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2 votes
0 answers
126 views

Cup product in modified (Tate) group hypercohomology

Let $G$ be a finite group. Let $$ M^\bullet=\,(\dots\to M^{-1}\to M^0\to M^1\to\dots) $$ be a bounded complex of $G$-modules. One can define Tate hypercohomology groups $H^n(G,M^\bullet)$ for all $n\...
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0 votes
0 answers
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Obstruction to lifting homomorphism of groups

Is there a "cohomology" group that encodes obstructions to constructing a lift in a diagram of groups below? If $X\to Y$ is an extension and the bottom row is the identity map it's just $H^1(...
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3 votes
0 answers
181 views

Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$

$$ \newcommand{\Z}{\mathbb{Z}} $$ Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...
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3 votes
1 answer
311 views

Relation between the cohomology group of a curve and the cohomology group of its jacobian

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...
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  • 407
4 votes
0 answers
158 views

What is known about the cohomology of the U-duality group?

$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
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3 votes
0 answers
145 views

Cohomology of the ring of integers of a local or global field

Let $\mathcal{O}$ denote the ring of integers in the separable closure $K^{sep}$ of a local or global field $K$ with absolute Galois group $G_K$. What is known about the cohomology groups $H^i(G_K,\...
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8 votes
0 answers
207 views

Representing the fundamental class of an aspherical manifold in the bar complex

Suppose $M$ is a compact orientable aspherical manifold and $G$ its fundamental group. Is there a nice description of representatives of the fundamental class of $M$ and its dual in the (homogenous) ...
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  • 663
8 votes
2 answers
608 views

Groups all of whose extensions are split

Is there a sensible characterization of groups $G$ with the following property? Every extension of groups $1\to G\to H\to K\to 1$ is split. A complete group $G$ has that property and in fact such a ...
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4 votes
0 answers
66 views

Which cocycles (function classes) compute $H^*(BG, A)$ for $G$ a compact Lie group?

Let $G$ be a compact Lie group. Given a class $F$ of functions (continuous, measurable, piecewise smooth, $L^2$, bounded ($L^\infty$), polynomial, …) one can define group cochains as in the finite ...
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10 votes
1 answer
266 views

Induced map on $H_4$ of Eilenberg–MacLane spaces

$\DeclareMathOperator\Hom{Hom}$It is well-known (see Breen, Mikhailov, Touzé - Derived functors of the divided power functors for example) that for $A$ a free abelian group we have $$ H_i(K(A,1); \...
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5 votes
1 answer
147 views

Rational cohomological dimension of a locally finite group

$\DeclareMathOperator\cd{cd}$Recall that the rational cohomological dimension of a group $G$ is the supremum of the set of integers $k$ such that there exists a $\mathbb{Q}[G]$-module $M$ with $H^k(G;...
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  • 53
4 votes
1 answer
159 views

Central extensions of torsion groups by $\mathbb{R}^n$

Let $\Gamma$ be a torsion group (i.e. every element has finite order). I am interested in understanding central extensions of the form: $\require{AMScd}$ \begin{CD} 0 @>>> \mathbb{R}^n @>\...
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4 votes
0 answers
129 views

Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field. Let $A$ be a local integral domain with maximal ideal $M$, residue ...
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  • 229
8 votes
1 answer
348 views

Finite group with squarefree order has periodic cohomology?

Is it true that a finite group with squarefree order has periodic group cohomology (with trivial coefficients)? I cannot see why this would be the case, but I'm looking at a paper which seems to ...
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8 votes
0 answers
114 views

What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?

Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
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8 votes
0 answers
203 views

Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?

Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...
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3 votes
2 answers
254 views

Quasi-isomorphism preserves group hypercohomology

I am looking for a reference for the assertion in the title. In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts). ...
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