All Questions
15 questions
0
votes
1
answer
201
views
Are these two natural cohomology classes of a manifold constructed from a 1-cochain and a group extension equal?
Let $X$ be a manifold, $G$ and $A$ finite abelian groups and $\epsilon \in H^2(G,A)$ a group cohomology class (for the moment I am assuming there is no action of $G$ on $A$). Given $\alpha \in H^1(X,G)...
- 813
1
vote
0
answers
146
views
Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
- 813
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
2
votes
1
answer
356
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
14
votes
2
answers
6k
views
Group cohomology of the cyclic group
It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...
- 183
1
vote
0
answers
268
views
On normalized 2-cocycle
Let $G$ be a group acts trivially on an abelian group $A$. Let
$\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume
that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times
...
- 181
9
votes
1
answer
804
views
Known results in the Cohomology of finite groups
I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
- 529
3
votes
2
answers
601
views
mod p cohomology ring of alternating groups
Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...
- 2,223
4
votes
1
answer
759
views
cohomology ring of symmetric group of order $3$
Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring
$$
H^*(S_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S_n$.
For $n=1$, $S_1$ is trivial. ...
- 1,990
2
votes
0
answers
345
views
Ring structure on cohomology of groups
Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
- 441
5
votes
0
answers
330
views
Cocycle condition for 2-groups
I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
- 113
13
votes
2
answers
737
views
Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?
Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness let'...
- 28.1k
1
vote
0
answers
263
views
When can a 2-cocycle on a subgroup can be extended?
This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?
I am asking this as a new question as I already asked that user but got no ...
- 265
4
votes
0
answers
131
views
When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?
Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$.
By Example of a Schur-nontrivial group with no abelian subgroup ...
2
votes
1
answer
83
views
How to claculate the $T$-stable subgroup of second cohomology group
Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \...
- 265