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2
votes
0answers
72 views

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup ...
0
votes
0answers
133 views

Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
2
votes
1answer
122 views

Coboundary for the Cohomology of free groups

Let $G$ be a group. Let $\Bbbk$ be a field of char. $0$. We denote with $C^{n}(G, \Bbbk)$ the set of maps $f\: : \: G^{n}\to \Bbbk$ and with $\partial_{G}\: : \: C^{n-1}(G, \Bbbk)\to C^{n}(G, \Bbbk)$ ...
6
votes
3answers
413 views

Solvable irreducible subgroups of the $\mathbf{GL}_n$ of $\mathbf{F}_p$ ($p$ prime)

I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $...
6
votes
1answer
462 views

Non-vanishing of the Tate-Shafarevich kernel in group cohomology

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$). We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...
3
votes
0answers
70 views

Can a finite group have its third cohomology determined by a quotient group?

Does there exist a finite group $G$ satisfying either of the following? $G$ admits a non-trivial, proper, central subgroup $A$ such that every (normalized) 3-cocycle of $G$ is cohomologous to a (...
12
votes
2answers
526 views

1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
4
votes
2answers
201 views

Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

Say we have a number field $K$. Let $G_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G_K$-module. We know that $H^1(K, M) := H^1(G_K, M)$, i.e. profinite group cohomology. For each place $v$ ...
2
votes
0answers
88 views

Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group $G$, and we pick two elements $g,h \in G$, could we find the order of the element $g \wedge h \in G \wedge G$? The best thing I could find is Theorem 1.1 in Ellis' Book (...
2
votes
0answers
94 views

Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
9
votes
0answers
301 views

Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions $G$ and $H$ are finite groups and $K$ an infinite group. there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...
15
votes
0answers
272 views

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 ...
5
votes
1answer
323 views

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 ...
0
votes
1answer
63 views

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
votes
1answer
137 views

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 ...
5
votes
1answer
219 views

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 ...
10
votes
2answers
311 views

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 ...
2
votes
1answer
170 views

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 ...
5
votes
0answers
79 views

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 \...
2
votes
1answer
188 views

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}...
21
votes
0answers
303 views

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 ...
13
votes
1answer
338 views

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 ...
3
votes
0answers
120 views

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
votes
0answers
67 views

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 $H$ ...
1
vote
1answer
120 views

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,...
5
votes
2answers
189 views

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 ...
19
votes
0answers
350 views

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 ...
25
votes
0answers
457 views

“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 ...
13
votes
2answers
240 views

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
votes
2answers
421 views

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 ...
5
votes
0answers
143 views

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?
4
votes
2answers
132 views

$2$-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 $T$-...
5
votes
0answers
157 views

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,...
4
votes
0answers
76 views

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 ...
5
votes
0answers
102 views

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 ...
2
votes
2answers
247 views

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}...
2
votes
0answers
118 views

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 ...
0
votes
0answers
73 views

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 ...
1
vote
1answer
51 views

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 \...
5
votes
1answer
106 views

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} \...
9
votes
0answers
197 views

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 ...
14
votes
0answers
301 views

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
votes
2answers
521 views

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 ...
3
votes
2answers
253 views

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 $CW$-...
16
votes
1answer
517 views

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 ...
4
votes
1answer
170 views

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
votes
0answers
107 views

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 ...
0
votes
0answers
99 views

When can the restriction map be zero or onto?

Let $G$ be a finite $p$ group of order $p^n$ and $N$ be a subgroup of $G$ of index $p$. Let $res:H^2(G,A) \rightarrow H^2(N,A)$ be the restriction map on cohomology, where $A$ is a trivial $G$ module. ...
7
votes
0answers
194 views

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 ...
1
vote
0answers
99 views

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 ...