Questions tagged [group-homology]

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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
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0answers
107 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
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209 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 $\mathrm{Sym}^g(k^2)$...
6
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0answers
273 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
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1answer
241 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
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0answers
166 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
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0answers
80 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
1answer
338 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
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0answers
188 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 ...
1
vote
1answer
818 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 $...