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2 votes
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119 views

Crossed homomorphism as morphism in the ambient category

Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$) $\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
rick's user avatar
  • 199
7 votes
1 answer
415 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
  • 44.8k
5 votes
0 answers
171 views

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
  • 51
2 votes
0 answers
84 views

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
  • 21
3 votes
1 answer
199 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
6 votes
1 answer
312 views

A Tate resolution for $\Sigma_p$ - Reference request

Below I will describe a mod $p$ Tate resolution for the symmetric group $\Sigma_p$, i.e. a $\mathbb{Z}$-graded periodic acyclic chain complex $C^*$ of finitely generated modules over $\mathbb{F}_p[\...
Neil Strickland's user avatar
3 votes
1 answer
248 views

Identifying group extension from cohomology class of $D_8$

I have the following problem. It is well known that $H^\ast(D_8,\mathbb{Z}/2)\cong \mathbb{F}_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). ...
Igor Sikora's user avatar
  • 1,759
9 votes
1 answer
309 views

Comparing cohomology of a total complex with the cohomology of semidirect product

$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
Igor Sikora's user avatar
  • 1,759
5 votes
1 answer
512 views

What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...
John Depp's user avatar
  • 331
7 votes
2 answers
539 views

Injectivity of the cohomology map induced by some projection map

Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence $$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$ where $G_c$ is the normal subgroup which ...
Ye Weicheng's user avatar
2 votes
0 answers
65 views

Representing elements of $H_2$ of a group using the bar (or standard) chain complex [duplicate]

Let $G$ be a discrete group, and let $a_1,b_1,\ldots,a_g,b_g \in G$ be elements such that $$[a_1,b_1] \cdots [a_g,b_g] = 1$$ in $G$. These correspond to a map of a genus-$g$ surface group $\pi_1(\...
Leonard's user avatar
  • 21
13 votes
2 answers
795 views

Which finite groups have low-degree essential cohomology?

Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
Theo Johnson-Freyd's user avatar
4 votes
1 answer
207 views

Groups homology with coefficients fitting into filtration or exact sequence

Let $G$ be a group. I have two questions about the homology of $G$: Consider a finite exact sequence $$0 \rightarrow M_1 \rightarrow \cdots \rightarrow M_m \rightarrow 0$$ of $G$-modules. How are ...
Laura's user avatar
  • 43
0 votes
0 answers
194 views

Equivariant cohomology with discrete group action

As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
Light man's user avatar
8 votes
1 answer
414 views

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 $...
Mark Grant's user avatar
  • 35.9k
3 votes
0 answers
128 views

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 ...
Nicolas Boerger's user avatar
6 votes
1 answer
244 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;...
Linda's user avatar
  • 63
8 votes
0 answers
128 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 ...
Theo Johnson-Freyd's user avatar
8 votes
0 answers
238 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;...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
308 views

How small can the support of a nontrivial $\mathbb F_p$-cocycle on $C_p$ be?

Let $p$ be a prime, and let $\phi : C_p^n \to \mathbb F_p$ be an $\mathbb F_p$-valued $n$-cocycle on $C_p$ (the cyclic group of order $p$) which is not an $n$-coboundary, i.e. $\phi$ represents a ...
Tim Campion's user avatar
  • 63.9k
6 votes
1 answer
426 views

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^...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
288 views

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 ...
Jens Reinhold's user avatar
16 votes
1 answer
505 views

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 ...
Jens Reinhold's user avatar
5 votes
1 answer
384 views

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{...
Theo Johnson-Freyd's user avatar
0 votes
1 answer
676 views

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 ...
Hussain Kadhem's user avatar
13 votes
1 answer
289 views

Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
Sam Nariman's user avatar
  • 1,003
7 votes
2 answers
494 views

How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
Theo Johnson-Freyd's user avatar
12 votes
1 answer
522 views

Realizing inner automorphisms on Eilenberg-MacLane spaces

Let $G$ be a discrete group and let $(X,x_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi_1(X,x_0) = G$ and the universal cover $\widetilde{X}$ is contractible. ...
Lisa's user avatar
  • 225
12 votes
2 answers
583 views

Do there exist acyclic simple groups of arbitrarily large cardinality?

Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$. In ...
Tim Campion's user avatar
  • 63.9k
17 votes
1 answer
998 views

Where should I search for computations of group cohomology rings of not-too-complicated finite groups?

A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low degrees, and I'd like to determine where to search for preexisting computations. ...
Arun Debray's user avatar
  • 6,881
15 votes
1 answer
629 views

Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
Bob's user avatar
  • 439
4 votes
1 answer
1k views

First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....
user's user avatar
  • 323
3 votes
2 answers
291 views

How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?

I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}_p)^k \to 1$, where $G$ is a compact lie ...
mathstudent's user avatar
7 votes
1 answer
1k views

Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
Totoro's user avatar
  • 2,535
4 votes
0 answers
136 views

Second homology of finitely presented group with free abelianisation

It is known that for a presented group $G=F/N$ we have $$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$ In general, the right side seems to be difficult to calculate. I am in the special ...
FKranhold's user avatar
  • 1,623
8 votes
1 answer
519 views

The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
Francesco Polizzi's user avatar
9 votes
0 answers
420 views

Hochschild-Serre spectral sequence via explicit filtration

Let $$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$ be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
Laura's user avatar
  • 163
35 votes
3 answers
1k views

Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
YCor's user avatar
  • 63.9k
18 votes
2 answers
592 views

primary decomposition for nonabelian cohomology of finite groups

Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$). If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...
Mark Grant's user avatar
  • 35.9k
9 votes
1 answer
308 views

Projective resolutions of finite-dimensional representations of infinite groups

Let $G$ be a group and let $V$ be a finite-dimensional complex representation of $G$. Question: Under what circumstances can I find a projective resolution $$ \cdots \longrightarrow P_3 \...
Joan's user avatar
  • 91
11 votes
1 answer
167 views

A group of type F that is an extension of type F-by-type F

Let us first recall that a group of type $F$ is a group admitting a compact classifying space. Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
Janusz Przewocki's user avatar
22 votes
1 answer
719 views

What is the cohomological dimension of the commutator subgroup of the pure braid group?

I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature. Because $[P_n,P_n]$ ...
David Recio-Mitter's user avatar
7 votes
1 answer
197 views

Homology of a limit of semidirect products

Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...
2223's user avatar
  • 111
4 votes
0 answers
172 views

from 2-cocycle to classifying map

Let $A,E,G:\mathrm{Set}_*\to\mathrm{Grp}_*$ be functors from pointed sets to (discrete) groups ($*=1$) together with natural transformations $i:A\to E, \ p: E\to G$ such that for any set $X$ \begin{...
hsldfgh sfkdlhguh's user avatar
3 votes
0 answers
120 views

Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary

Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle $\omega_3^G$ of a ...
wonderich's user avatar
  • 10.5k
2 votes
1 answer
264 views

Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$

Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
miss-tery's user avatar
  • 755
4 votes
1 answer
394 views

$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?

I was trying to understand this interesting question by example. Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
miss-tery's user avatar
  • 755
7 votes
1 answer
506 views

$G$ cocycle split to a coboundary in $J$, via a group extension

Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
wonderich's user avatar
  • 10.5k
14 votes
1 answer
704 views

What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?

The symmetric group $S_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real ...
Theo Johnson-Freyd's user avatar
11 votes
2 answers
656 views

$G$ cocycle split and trivialized to a coboundary in $J$, given a group homomorphism $J \overset{r}{\rightarrow} G$

Consider a generic nontrivial 3-cocycle $\omega_3^G(g_1,g_2,g_3) \in H^3(G,U(1))$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\...
wonderich's user avatar
  • 10.5k