All Questions
Tagged with cohomology group-cohomology
117
questions
1
vote
0
answers
60
views
How can we construct a non-trivial central extension of a Lie group
Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
- 277
4
votes
1
answer
369
views
Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
- 498
3
votes
0
answers
84
views
Explicit examples of 4-cocycles over finite 2-groups
By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...
- 2,921
2
votes
0
answers
74
views
G-modules vs. $\Delta(NG)$-modules
Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...
- 215
3
votes
1
answer
195
views
Subgroups of top cohomological dimension
Let $G$ be a geometrically finite group, i.e. there exists a finite CW complex of type $K(G,1)$.
By Serre's Theorem, every finite-index subgroup $H$ of $G$ satisfies $cd(H)=cd(G)$, but what about the ...
- 833
6
votes
1
answer
398
views
Is there a clear pattern for the degree $2n$ cohomology group of the $n$'th Eilenberg-MacLane space?
Let $G$ be a finite abelian group, and its higher classifying space is $B^nG=K(G,n)$. For $n=1$ it is well known that $H^2(B G, \mathbb{R}/\mathbb{Z}) \cong H^2(G,\mathbb{R}/\mathbb{Z})$ is isomorphic ...
- 761
3
votes
0
answers
127
views
Is there a simple explicit expression of the Pontryagin square in terms of the cup product on a spin 4-manifold?
$A$ a finite abelian group, and denote $\Gamma(A)$ its universal quadratic group. The Pontryagin square $\mathfrak{P}\in H^4(B^2A,\Gamma(A))\cong \text{Hom}(\Gamma(A),\Gamma(A))$ is the element ...
- 761
2
votes
1
answer
222
views
Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products
The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as
$$
\rho_1(a,b)=(b,a) \ .
...
- 761
0
votes
1
answer
199
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)...
- 761
3
votes
1
answer
250
views
Cohomology ring $H^*(\operatorname{SL}(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$
$\DeclareMathOperator\SL{SL}$In Soulé's paper "The cohomology of $\SL_3(\mathbb{Z})$" the cohomology ring $H^*(\SL(3,\mathbb{Z}),\mathbb{Z})_{(2)}$ is determined in Theorem 4.iv. I'm wanting ...
- 525
1
vote
0
answers
131
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 ...
- 761
3
votes
1
answer
111
views
What is this cochain complex about, whose $H^1 = \mathbb{R}$?
$\DeclareMathOperator\QEnd{QEnd}$Let $C^n$ be the set of functions $\mathbb{Z}^n \to \mathbb{Z}$, and $B^n$ the set of bounded such functions. For $a_1,...,a_{n+1} \in \mathbb{Z}$, the differential of ...
- 1,326
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