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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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 $H^1(G,Ad(V))$ (...
user135743's user avatar
6 votes
1 answer
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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 ...
Denis T's user avatar
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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) ...
Kevin Walker's user avatar
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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 ...
user449595's user avatar
5 votes
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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 ...
ThorbenK's user avatar
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2 answers
263 views

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. ...
Jun Yang's user avatar
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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}\...
Christopher-Lloyd Simon's user avatar
2 votes
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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....
Antoine's user avatar
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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, ...
Aidan's user avatar
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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 ...
Andi Bauer's user avatar
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1 answer
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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\...
Mikhail Borovoi's user avatar
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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 $...
semisimpleton's user avatar
12 votes
1 answer
510 views

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 ...
Patrick Perras's user avatar
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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 ...
მამუკა ჯიბლაძე's user avatar
5 votes
2 answers
391 views

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 ...
Antoine Labelle's user avatar
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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 ...
Andi Bauer's user avatar
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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 $...
Antoine's user avatar
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1 answer
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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 ...
Chase's user avatar
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7 votes
1 answer
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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 ...
David Loeffler's user avatar
3 votes
1 answer
263 views

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 ...
Andrew Dudzik's user avatar
8 votes
2 answers
678 views

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 ...
Tommaso Rossi's user avatar
5 votes
2 answers
383 views

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 ...
David Loeffler's user avatar
6 votes
2 answers
240 views

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 ...
Mikhail Borovoi's user avatar
7 votes
1 answer
349 views

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 ...
Andy Putman's user avatar
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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 ...
Lauren's user avatar
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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 $\gamma_{k+1}(F_n)$ ...
Arthur's user avatar
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1 vote
0 answers
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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 ...
RKS's user avatar
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1 vote
1 answer
265 views

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 ...
RKS's user avatar
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6 votes
0 answers
231 views

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 ...
Aditya De Saha's user avatar
9 votes
1 answer
360 views

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\...
Mikhail Borovoi's user avatar
2 votes
0 answers
132 views

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(...
MaoWao's user avatar
  • 1,027
3 votes
1 answer
192 views

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 ...
Stephan Mescher's user avatar
0 votes
1 answer
176 views

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 ...
Ronald J. Zallman's user avatar
5 votes
1 answer
272 views

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) ...
testaccount's user avatar
8 votes
1 answer
440 views

Trivial group cohomology induces trivial cohomology of subgroups

From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
Andrea Antinucci's user avatar
4 votes
1 answer
222 views

Projective representations of a finite abelian group

Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups $$ G\cong ...
Andrea Antinucci's user avatar
2 votes
0 answers
100 views

Extensions of groups with a $G$-action

Let $1\longrightarrow A\longrightarrow \mathcal{G}\longrightarrow R\longrightarrow 1$ be an exact sequence of algebraic group schemes, with $\mathcal{G}$ being an extension of $R$, an affine reductive ...
kindasorta's user avatar
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24 votes
0 answers
770 views

Revising the proof of CFSG

This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the ...
semisimpleton's user avatar
2 votes
1 answer
157 views

Rational group homology of an infinite product of finite groups

Let $G_{1}, G_{2}, \cdots$ be a countably infinite sequence of finite groups. It is well-known that the group homology $H_{n}(BG_{i};\mathbb{Q})=0$ for any $n\geq 1$. Let $X=\prod^{\infty}_{i=1}BG_{i}$...
user44651's user avatar
  • 1,039
6 votes
1 answer
384 views

Is there a clear pattern for the degree $2n$ cohomology group of the $n$'th Eilenberg-MacLane space?

Let $G$ be a finite abelian group, and its higher classifying space is $B^nG=K(G,n)$. For $n=1$ it is well known that $H^2(B G, \mathbb{R}/\mathbb{Z}) \cong H^2(G,\mathbb{R}/\mathbb{Z})$ is isomorphic ...
Andrea Antinucci's user avatar
7 votes
1 answer
315 views

Pontryagin dual of a group-cohomology class

Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence $$ 1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1 $$ This determines a class $[\...
Andrea Antinucci's user avatar
8 votes
0 answers
185 views

Universal bundles for monoids versus groups

Dold and Lashof compare their construction for a monoid M to Milnor's when M is a group G. They give an explicit comparison for the first stage of the constructions. Somewhere I've seen the general n-...
jim stasheff's user avatar
4 votes
0 answers
112 views

Homotopy type / Homology of the free loop space of aspherical manifolds

Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
Yeah's user avatar
  • 347
6 votes
1 answer
223 views

Canonical reference for dictionary between $G$-spaces and fiber bundles over $BG$?

I'm looking for a comprehensive reference (for citation purposes) laying out the basic facts of the equivalence between $G$-spaces and bundles over $BG$ for a discrete group $G$. I'd like it to also ...
xir's user avatar
  • 1,964
3 votes
1 answer
223 views

The second Tate-Shafarevich group of a permutation module is trivial

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $...
Tuvasbien's user avatar
  • 156
3 votes
0 answers
113 views

Is there a simple explicit expression of the Pontryagin square in terms of the cup product on a spin 4-manifold?

$A$ a finite abelian group, and denote $\Gamma(A)$ its universal quadratic group. The Pontryagin square $\mathfrak{P}\in H^4(B^2A,\Gamma(A))\cong \text{Hom}(\Gamma(A),\Gamma(A))$ is the element ...
Andrea Antinucci's user avatar
7 votes
2 answers
597 views

Cohomology of $\operatorname{GL}_3(\mathbb{F}_2)$

I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in ...
Noah B's user avatar
  • 403
9 votes
0 answers
185 views

Group cohomology of ${\rm SL}_{n}(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$

Let $p>5$ be a prime. For any integer $n\geq 2$, let $M_{n}^{0}(\mathbb{F}_p)$ denote the $n\times n$ matrices with trace $0$ over a finite field $\mathbb{F}_p$ of order $p$. Then we have an action ...
stupid boy's user avatar
6 votes
1 answer
336 views

Ker of corestriction of Galois cohomology

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module. Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$. On the other hand, ...
Duality's user avatar
  • 1,407
2 votes
1 answer
207 views

Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products

The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as $$ \rho_1(a,b)=(b,a) \ . ...
Andrea Antinucci's user avatar

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