All Questions
37 questions
4
votes
0
answers
75
views
Alternating bihomomorphism is a skew 2-cocycle
It seems to be a well-known fact that every alternating bihomomorphism $G\times G\to\mathbb{C}^\times$ for a finite abelian group $G$ is the skew of some 2-cocycle (see for instance Symmetric analogue ...
- 41
1
vote
1
answer
232
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
1
vote
0
answers
120
views
Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
- 1,013
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"). ...
- 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 ...
- 1,759
8
votes
1
answer
227
views
Non-finitely presented FP groups with cohomological dimension $2$
In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
- 15.8k
1
vote
0
answers
175
views
Cochains with multilinear differentials
Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.
We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
1
vote
0
answers
92
views
Group structure extension
Let $G$ be a finite group and $X$ a finite $G$-set. Let $H$ be the set-theoretical cartesian product of $G$ and $X$.
Is there an homological theory controlling all possible group structure on $H$ (...
- 2,384
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
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.
...
- 6,881
5
votes
1
answer
429
views
Cohomology of linear algebraic groups
Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...
user39380
19
votes
0
answers
604
views
How is this group theoretic construct called?
Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...
user6671
5
votes
1
answer
390
views
mod p (odd) cohomology of dihedral groups
I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal ...
- 115
9
votes
2
answers
571
views
Algorithm for group cohomology
Let $G$ be a finite group, and let $0\to M_1\xrightarrow{\iota} M_2\xrightarrow{\pi} M_3\to 0$ be a short exact sequence of $G$-modules (finitely generated over $\mathbb Z$, not necessarily free). I ...
- 161
15
votes
1
answer
352
views
$p$-groups with trivial $H^3$
Let $Q_8$ be the group of quaternions of order $8$. It is a non-abelian $2$-group such that $H^3(Q_8,\mathbb{Z})=0$, where $\mathbb{Z}$ has the trivial action. For a proof, see the book "Homological ...
- 153
3
votes
0
answers
365
views
Coinflation in cohomology
Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index ...
- 423
7
votes
0
answers
229
views
Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$
Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ?
I have looked at the very few books on cohomology of groups ...
- 1,766
6
votes
1
answer
486
views
Fourth cohomology of the modular group
Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}_0$ and $\mathrm{Co}_1$.
- 35.5k
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,097
3
votes
1
answer
372
views
Reference for real and complex projective representation of finite group
I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
- 977
3
votes
0
answers
113
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$ ...
- 11.3k
4
votes
2
answers
704
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}...
- 51
5
votes
2
answers
332
views
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde ...
- 2,851
3
votes
0
answers
222
views
torsion free for the 2nd cohomology group?
Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T),
My question is:
is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free?
Thanks in advance!
...
- 1,528
1
vote
0
answers
223
views
Reference on calculation of 2nd cohomology group
Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...
- 1,528
3
votes
1
answer
708
views
vanishing higher cohomology group for property T group?
Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology,
Is it known that $H^n(G, l^2(G))=...
- 1,528
17
votes
1
answer
575
views
Group cochains invariant under the action of the symmetric group
Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
9
votes
1
answer
290
views
Calculations of nonabelian group cohomology of R^n
I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
- 35.5k
12
votes
2
answers
523
views
A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
- 123
9
votes
3
answers
3k
views
Reference for Ring Structure on Group Cohomology
As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning ...
- 4,920
17
votes
3
answers
815
views
Does this subgroup of "even braids" have a name?
The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...
- 35.9k
3
votes
1
answer
426
views
Naturality of the transfer in group cohomology
Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module ...
- 35.9k
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
22
votes
2
answers
2k
views
Proofs of the Stallings-Swan theorem
It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...
- 35.9k
8
votes
1
answer
739
views
Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?
The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...
- 83
7
votes
1
answer
499
views
Posets of cosets and contractibility
For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset $\mathcal{C}(...
- 1,680
11
votes
1
answer
3k
views
Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
- 54.6k