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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Finite groups inside an infinite group with the same homology
Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:
$G$ and $H$ are finite groups and $K$ is an infinite group.
there exist two monomorphisms $G \rightarrow K \leftarrow H$...
23
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1
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Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
9
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Known results in the Cohomology of finite groups
I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
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1
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Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?
A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group.
I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...
3
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1
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Transfer map in group cohomology
Let $H$ be a subgroup of a finite group $G$, and let $M$ be a $G$-module. Are there any simple conditions on $H,G$ and $M$ which would ensure that the transfer map $H^p(H,M)\to H^p(G,M)$ is the zero ...
14
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2
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Group cohomology of the cyclic group
It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...
15
votes
3
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Lower central series quotients in terms of (co)homology
Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...
3
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1
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Third (co-) homology of Cyclic groups
Is there a general simple theorem for the third cohomology of cyclic groups $H_3(\mathbb{Z}_n, U(1))= ?$. In particular, I am interested in finding $H_3(\mathbb{Z}_8, U(1))$. I know the answer can be ...
6
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0
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Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
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1
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Cohomologically trivial $G$-modules
Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle \prod_{i=1}^{2^d}...
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Does the Tate construction (defined with direct sums) have a derived interpretation?
Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
16
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1
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Cohomology of central extensions of groups
Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
5
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0
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Cocycle condition for 2-groups
I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
3
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Have locally principal crossed homomorphisms been studied?
Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...
2
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1
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Existence of class modules for finite groups
I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$:
1) $H^1(H,...
6
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2
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comparing homology of a space and homology of the classifying space of its fundamental group
Let $X$ be a (connected) closed $n$-manifold and $G=\pi_1(X)$ be the fundamental group of $X$. There is a classifying map $f: X \rightarrow K(G, 1)$ which induces an isomorphism on $\pi_1$. I would ...
18
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A cohomology class associated with a complex representation of a group
$\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex ...
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"Three great cocycles" in Complex Analysis as cohomology generators
In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...
14
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2
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Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$
This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...
6
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2
answers
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First Galois cohomology of Weil restriction of $\mathbb{G}_m$
Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...
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Continuous cohomology via model category
Is it possible to formulate notion of continuous cohomology in terms of model categories?
If yes, then is there a reference for this?
5
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2
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$2$-cohomology group of semi-direct products
Let $G=N\rtimes T$ and let $A$ be a $G$-module with a trivial $G$-action.
The action of $T$ on induce a natural action of $T$ on the second cohomology group of $N$. Denote by $H^2(N,A)^T$ the $T$-...
6
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Is central extension of a group equivalent to a bundle with gauge field?
Let $\tilde G$ be a central extension of a group $G$ by $U(1)$.
One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$
However,...
5
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0
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Generalization of a 1D unitary representation
Probably a very naive question, but I'd be grateful for any input. Consider 1D representations of finite group $G$: $\chi(g) \chi(h) = \chi(g h)$ with $\chi(g)\in \text{U}(1)$, and $\chi(1)=1$. The ...
7
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Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$
$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...
4
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2
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Behaviour of cohomology groups under extension of scalars
Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let
$\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong H^n(G,\hat{M}...
3
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0
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Integral cohomology of elementary abelian groups
Let $p$ be a prime. I am looking for a reference or a short proof for the fact that cohomology groups $H^i((\mathbb{Z}/p\mathbb{Z})^n,\, \mathbb{Z}),\, i>0,$ have exponent $p$ (i.e., that they are ...
1
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0
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When can a 2-cocycle on a subgroup can be extended?
This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?
I am asking this as a new question as I already asked that user but got no ...
2
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1
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How to claculate the $T$-stable subgroup of second cohomology group
Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \...
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1
answer
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Generalization of a lemma of Livne
Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} \...
12
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1
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Homological stability for orthogonal groups
In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal ...
20
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In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?
First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...
12
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2
answers
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Cohomological obstructions to lift $\pi_0$ of a topological group
Let $G$ be a topological group. Denote the same group with the discrete topology by $G^\delta$ and denote the group of connected components of $G$ by $\pi_0G$. I am interested in the question when we ...
4
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2
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rational cohomology of symmetric groups
Let $\Sigma_k$ be the $k$-th symmetric group and $B\Sigma_k$ be its classifying space. How to prove:
for any $n\geq 1$ and the $n$-skeleton $sk_n (B\Sigma_k)$, there exists a finite dimensional $CW$-...
18
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1
answer
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Is the moduli space of graphs simply connected?
The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to ...
5
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1
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Transgression in terms of k-invariant for chain complexes
I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ ...
5
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0
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Second homology of mapping class group via universal coefficient theorem
Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$.
It is known ...
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What's the name of the cohomology class associated to a projective representation?
Suppose $\rho : G \to PGL_n(k)$ is a projective representation of a group $G$ over a field $k$. It's classical that the obstruction to lifting this to a linear representation $G \to GL_n(k)$ is a ...
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0
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Homology of spherical braid groups
By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
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0
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Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$
What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated.
For the definition of "cohomological dimension of a group ...
4
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1
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Obstruction for two subgroups to be conjugated by an automorphism
Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me
Take $p$ a prime number and $P$ an abelian finite $p$-group. Let $...
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2
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Groups with trivial rational homology and their finite index subgroups
For a short exact sequence $0 \to G \to H \to K \to 0$ of (discrete) groups with $K$ finite we have, as a consequence of the Hochschild-Serre spectral sequence, that $H^{\ast}(H;\mathbb Q) = H^{\ast}(...
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2
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How to compute second homology of a group given by presentation with two relators
I am interested in calculating of $H_2(G,\mathbb{Z})$, where $G$ is a group given by presentation with two relators $\langle a,b| r_1 = r_2 = 1\rangle$.
Moreover, I am interested in such ...
3
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2
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mod p cohomology ring of alternating groups
Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...
6
votes
1
answer
311
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Schur Multiplier of Tarski Monsters
Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?
3
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2
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Do quasi-isometric groups have the same rational cohomology?
Let $G_1$ and $G_2$ be two finitely generated groups which are quasi-isometric in the sense of geometric group theory.
Are their rational cohomology rings $H^{\ast}(G_i; \mathbb Q)$ necessarily ...
4
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0
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Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?
Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
8
votes
3
answers
723
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Computations in modular cohomology of finite groups
Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...
0
votes
1
answer
189
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cohomology ring of the fundamental group of unordered configuration space
From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...
8
votes
1
answer
251
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Obstructions to Picard-graded groups of maps
Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For $\...