All Questions
Tagged with spectral-sequences gr.group-theory
15 questions
12
votes
3
answers
1k
views
Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup
Is there an analogue of the Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup?
What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
What is the best ...
- 385
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
11
votes
2
answers
656
views
$G$ cocycle split and trivialized to a coboundary in $J$, given a group homomorphism $J \overset{r}{\rightarrow} G$
Consider a generic nontrivial 3-cocycle $\omega_3^G(g_1,g_2,g_3) \in H^3(G,U(1))$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\...
- 10.5k
9
votes
0
answers
421
views
Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
- 163
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
506
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
7
votes
1
answer
663
views
Explicit 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$
I would like to know the explicit expression of 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ with $U(1)\equiv \mathbb{R}/\mathbb{Z}$ coefficient, or namely $H^2[...
- 10.5k
4
votes
1
answer
349
views
The computation of $d_2$ in the Hochschild-Serre spectral sequence
I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups:
$1\to N\to G\to G/N\to 1$
where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$.
...
4
votes
1
answer
394
views
$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?
I was trying to understand this interesting question by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
- 755
4
votes
0
answers
207
views
Trivial action in the Hochschild-Serre spectral sequence
I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...
3
votes
0
answers
148
views
Group cohomology with coefficients in a graded module
I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows:
Let $G = C_4 = \langle \sigma \rangle$ be the ...
- 283
3
votes
0
answers
174
views
Induced Homomorphism on Cohomology of Symmetric Group 3
For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
- 39
3
votes
0
answers
120
views
Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary
Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle
$\omega_3^G$ of a ...
- 10.5k
2
votes
1
answer
264
views
Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$
Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
- 755
0
votes
1
answer
143
views
Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755