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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54 views

Profinite projective and free modules

I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
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Group extensions with non-abelian kernel

If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are ...
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Cohomology of commutative monoid acting on module

I have a some naive questions about how to define the cohomology of a commutative monoid. One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i_{\...
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Top cohomology of profinite Poincaré duality group

The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...
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Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
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Galois cohomology with coefficients in the unit group of a cyclotomic field

While understanding Fermat's last theorem's proof for regular primes, I bumped into a proposition (http://www2.biglobe.ne.jp/~optimist/algebra/fermat2_proof.html#proof10, written in Japanese) stating: ...
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Group cohomology with coefficients in a graded module

I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows: Let $G = C_4 = \langle \sigma \rangle$ be the ...
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What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?

The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
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Neutral cohomology classes and restriction maps for $H^2$ in group cohomology

$\DeclareMathOperator\res{res}$ Let $G$ be a profinite group. Let $A$ be a finite $G$-group (a finite group on which $G$ acts continuously), not necessarily abelian, with center $Z=Z(A)$. We say that ...
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A road map through group cohomology

I am a PhD student in algebraic topology, and I would like to learn something about group cohomology. The final goal would be to present one or two seminars on this topic, in order to give my mates a ...
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Finite $\Gamma$-modules with trivial $H^2$, where $\Gamma$ is a group of order 2

Let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a finite $\Gamma$-module, that is, a finite abelian group on which $\Gamma$ acts. It is a hopeless problem to classify finite $\Gamma$-...
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Constraints on the homology of amenable groups

Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one. Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything ...
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Using HoTT, why is twisted cohomology of BG group cohomology?

I've been reading Michael Shulman's blog posts defining cohomology in homotopy type theory, and I'd like to understand (using HoTT) why cohomology of BG is group cohomology. if I understand correctly, ...
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Standard terminology for “observable” subgroups of discrete groups

I've encountered in Bass, Lubotzky, Magid, and Mozes - The proalgebraic completion of rigid groups (Remark 1. p. 7) the following terminology: A normal subgroup $N$ of $G$ is observable if every $N$-...
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Cohomology of finite $p$-groups over integers in local fields

Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2). In ...
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Betti numbers and lower central series quotients of finite-index subgroups of nilpotent groups

Let $G$ be a finitely generated nilpotent group and let $A\le G$ be a finite-index subgroup. I have two questions about $A$: Is it true that the inclusion map $A \rightarrow G$ induces isomorphisms $...
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Regular conjugacy classes and irreducible representations in the infinite, projective case

Let $k$ be an algebraically closed field and $G$ a (not necessarily finite) group. Let $\alpha\colon G\times G\to k^*$ be a multiplier, meaning that $\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
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A conjecture of Adams on the mod p cohomology of classifying spaces of Lie groups

In the paper On the mod p cohomology of BPU(p), the authors say that there is a conjecture of J. F. Adams as follows: Conjecture (J. F. Adams) Let $G$ be a compact connected Lie group, and let $p$ be ...
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A finitely presented group whose rational cohomology is not nilpotent

Does there exist a finitely presented (preferably $\text{FP}_{\infty}$) group $\Gamma$ and an element $\alpha \in \text{H}^{\ast>0}(B\Gamma;\mathbf{Q})$ that is not nilpotent? If non-discrete ...
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Effaceing group cohomology of restricted representations

Suppose we have a group $G$ and a normal subgroup $N$ in $G$. Everything below in characteristic zero. My question is simple: given a finite dimensional representation $V$ of $G$, a positive integer $...
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How many cells needed to build the classifying space $BG$?

Let $G$ be a finitely presented group of cohomological dimension $n$. Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional ...
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What is the meaning of this coboundary homomorphism for group hypercohomology?

$\require{AMScd}$ Let $\Gamma=\{1,\gamma\}$ be a group of order 2. In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules (abelian groups ...
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$\ell$-adic cohomology of a quotient by group action

Suppose $Y \to Y/G$ is the Galois cover induced from a finite group $G$ acting on a scheme $Y$ and that this is indeed a Galois cover with $Y/G$ a scheme. In my case $Y$ is the Drinfeld curve $\mathrm{...
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A conceptual explanation for a simple fact about twists of objects with abelian automorphism group

Suppose $X,X'$ are two objects (say, genus 1 curves) over a field $k$ such that over the algebraic closure $X_{\overline k} \cong X'_{\overline k}$ and moreover, $Aut_{\overline k}(X)$ is abelian. ...
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Rational cohomology cohomology of $p$-adic analytic groups

It is a result of Lazard that given $G$ a compact $p$-adic analytic group then we have an isomorphism \begin{equation} H^*(G; \mathbb{Q}_p) \cong H^*(T_eG; \mathbb{Q}_p) \end{equation} where $T_eG$ is ...
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Non-trivial example of $H^2(G,M)$ where $M$ is a non-trivial G-representation

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}...
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Which groups have undetectable third U(1)-cohomology?

Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map $$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
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Central extensions of orthogonal group by $C_2$

Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...
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A description of the fundamental class of the group cohomology of the fundamental group of an orientable surface of genus $g$

I asked this question on Mathematics Stack Exchange some months ago but I got no answer. Suppose one has an orientable compact surface $S$ of genus $g\ge 2$, $x\in S$, and $G=\pi(S,x)$ the fundamental ...
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kernel and cokernel of corestriction map in cohomology of a profinite group

Let $G$ be a profinite group, $N$ a normal open subgroup and $A$ a discrete $G$-module. We have a corestriction map $cor: H^1(N, A)_{G/N} \to H^1(G, A)$. Are there any results on the kernel and ...
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Conjugation action on relative homology

Let $G$ be a group and $K$ be a subgroup. Suppose $g \in G$ commutes with every element of $K$. Is it true that conjugation by $g$ will act trivially on $H_*(G,K)$?
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Homology $H_{\ast}(T, V)$

Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$. We have that $T$ acts ...
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Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group

I am reading a paper which makes the following claim: let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$. Let $X' = X \vee S^2$ be the wedge sum of $X$ with the ...
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Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$

Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$ I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
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Are the Galois actions on automorphisms of twists isomorphic?

This might be a trivial question and I might be overlooking something: Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct ...
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What is the image of the diagonal map on the cohomology of Lie groups

Consider a simple Lie group $G$ and its mod $p$ cohomology $H^*(G, \mathbb{Z}_p)$. A good reference is the book Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by ...
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Computing cohomology using bounded hypercovers

Let $G$ be a Lie group (paracompact, not necessarily compact), and $A$ an abelian Lie group. I want to write down cocycles in $\mathrm{H^n}(\mathbf{B}G,A)$, the cohomology in the cohesive $\infty$-...
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Cohomology of the Baer-Specker group

Let $A = \prod_{i \in \mathbb{N}} \mathbb{Z}$ be the Baer-Specker group; that is, a countably infinite product of the integers. We will consider this as a discrete abelian group. Are the higher ...
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Is Lie group cohomology determined by restriction to finite subgroups?

Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
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Elementary $p$-subgroups of a compact Lie group

How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$? Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}_p$, there is an ...
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Find $a$ satisfying $x \cup_1 y = \delta a$ when $x,y \in Z^2(G,\mathbb{Z}_2)$

Let $G$ be a finite group. Let $x,y \in Z^2(G,\mathbb{Z}_2)$ be 2-cocycles. Find $a \in C^2(G,\mathbb{Z}_2)$ such that \begin{align} x \cup_1 y = \delta a. \end{align} Is there a general solution? Is ...
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loop space of a finite CW-complex

Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...
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Colimits in cohomology of profinite arithmetic groups

Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...
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Rational cohomology of p-adic general linear groups

I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows. First ...
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Which classes in $\mathrm{H}^4(B\mathrm{Exceptional}; \mathbb{Z})$ are classical characteristic classes?

Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...
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What is $H^1(\mathbb Z, GL_k(R_n))$ for a ring closely related to the cyclotomic rings of integers?

Let us consider $R_n = \mathbb Z_\ell[\theta_n]/(\theta^{\ell^n}-1)$, an auxiliary prime power $q\equiv 1 \pmod \ell$ with an action of $\mathbb Z = \langle \sigma\rangle$ by $\sigma(\theta_n) = \...
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surjective algebra homomorphism $k\widetilde{G} \to k_\alpha G$

I am having trouble understanding the following statement: Let $Z$ be an abelian group written multiplicatively and let $1 \to Z \to \widetilde{G} \xrightarrow{\pi} G \to 1$ be a central extension of ...
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What can be said about the Chow rings of classifying spaces of semi-direct products of groups?

For instance, what can we say about the Chow ring of the classifying space of a semi-direct product $CH^*(B(G\ltimes H))$, in terms of the Chow rings of $CH^*(BG)$, $CH^*(BH)$, and the singular ...
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When is the natural map of Tate cohomology an isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
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Non-abelian Ext functor and non-abelian $H^2$

Let $G$ be a group and $$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$$ a short exact sequence of groups. Now these are abelian groups, if I want to show that $\text{Hom}(G,M)\rightarrow \...

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