All Questions
Tagged with cohomology gr.group-theory
55
questions
2
votes
0
answers
74
views
G-modules vs. $\Delta(NG)$-modules
Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...
- 215
3
votes
1
answer
195
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 ...
- 833
0
votes
2
answers
140
views
Examples of isomorphic non-equivalent twisted group algebras
Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
- 193
1
vote
1
answer
100
views
Cohomological variety in case that Sylow subgroup is elementary abelian
Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V_G$ of ...
- 1,077
5
votes
1
answer
172
views
Restriction vs. multiplication by $n$ in Tate cohomology
$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and ...
- 13.8k
3
votes
1
answer
423
views
Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?
Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout (i.e. amalgamated free product) $G_1 \sqcup_H G_2$. Suppose that we have ...
- 5,953
4
votes
1
answer
162
views
Cohomologically trivial module $M$ such that $M/pM$ is not cohomologically trivial for some $p\in\mathbb{N}$
I was reading Brown's Cohomology Theory of Finite groups and was wondering whether there's an example of the following.
Let $n\in \mathbb{N}$. Does there exist a $\mathbb{Z}_n$ module $M$ which is ...
- 143
5
votes
1
answer
234
views
Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\to H^2(G,V)$ is injective
Let $G$ be a group and $F$ a field. I am particularly interested in the case where $G$ is a uniform lattice in a Lie group and $F=\mathbb{F}_2$, or in finite groups $G$ where
$\operatorname{char} F$ ...
- 2,846
0
votes
1
answer
401
views
Coboundary operators, 1-cocycles and computing cohomology
My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago on MathSE
and I set a bounty on it, but I didn't ...
5
votes
0
answers
217
views
Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$
Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
- 689
6
votes
1
answer
431
views
Cohomological dimension of torsion-free groups and its subgroups
In this thesis by Martin Hamilton on
Finiteness Conditions in Group Cohomology there is on page 11 a reference to following result:
Theorem 1.2.14. If $G$ is a torsion-free group and $H$ is a
subgroup ...
- 5,780
1
vote
0
answers
110
views
Rack cohomology as derived functor cohomology
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
- 1,066
2
votes
1
answer
355
views
Non-trivial example of $H^2(G,M)$ where $M$ is a non-trivial G-representation
Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$.
Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism.
Let M be the $G$ representation $\mathbb{Z}...
- 229
3
votes
1
answer
261
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,441
2
votes
1
answer
388
views
An action of the symmetric group $S_n$ on group cohomology $H^n(G, A)$ of abelian groups
Let $H$, $A$ be discrete abelian groups, and for simplicity suppose $A$ is given the trivial $H$-action.
When considering the second cohomology group $H^2(H,A)$, it is natural to talk about the ...
- 915