Questions tagged [group-homology]
The group-homology tag has no usage guidance.
15
questions
7
votes
1
answer
452
views
Computing homology groups with GAP
I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
2
votes
0
answers
76
views
Reference request: Lower group (co)homology of linear groups
I am finding references about the following general question:
What is the group (co)homology $H_{i}(G,\mathbb{Z})$ for a linear group $G$?
In my case, I'm particularly interested in the special case ...
7
votes
1
answer
339
views
How can I detect the homology image of a unipotent group in the general linear group?
Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements.
Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
4
votes
1
answer
263
views
The third homology stability of general linear groups over finite fields
Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...
12
votes
3
answers
503
views
Small simplicial set models for BG
Let $F$ be a finite group.
Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?
For example the Bar construction has the ...
2
votes
0
answers
73
views
Homology of linear groups over integral domains and their field of fractions
Let $A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $F$. Consider the natural injections $i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$, $j_n:GL_n(F)\...
6
votes
0
answers
125
views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
13
votes
0
answers
256
views
Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$
Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
7
votes
1
answer
373
views
Homology of symplectic groups in the unstable range
Let $Sp(2n,{\mathbb R})$ be the symplectic group and $H_3(Sp(2n,{\mathbb R});{\mathbb Z})$ its 3rd group homology (i.e., for the group with the discrete topology).
It is known that $$H_3(Sp(2n,{\...
3
votes
1
answer
260
views
Homology of solvable Lie groups made discrete
In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.
It is well-known how to compute the homology of abelian groups, ...
4
votes
0
answers
175
views
Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?
Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
4
votes
0
answers
84
views
Methods for showing homology of a subgroup survives to the larger group
Suppose we have an inclusion of groups $G_1<G_2$. I am curious about what methods there are out there for analyzing the map $H_k(G_1;\mathbb Q)\to H_k(G_2;\mathbb Q)$. In particular, what are tools ...
1
vote
1
answer
455
views
Cup-products and Transgression maps.
This question is related to Lyndon-Hochschild-Serre spectral sequence and cup products.
I have the followin result by J.S Milne in his book Arithmetic duality theorems pg 105.
Let $$0 \rightarrow C \...
1
vote
0
answers
292
views
Explicit transgression maps for Group homology in LHS
This question is related to another question of mine (Here: Lyndon-Hochschild-Serre spectral sequence and cup products).
I'm trying to figure out if some diagram commutes an one of the maps involved ...
2
votes
1
answer
1k
views
Transgression maps in group cohomology and group homology / duality of spectral sequences
I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem.
Let $G$ be a group, $H$ a normal subgroup of $...