Questions tagged [group-cohomology]
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
891
questions
4
votes
1
answer
270
views
Poincaré duality and Mayer–Vietoris sequence
In his article, Davis states that if a $n$-dimensional Poincaré duality group $G$ splits along a subgroup $C$ then the cohomological dimension of $C$ must be $n-1$. I am struggling to understand why ...
- 61
2
votes
0
answers
87
views
Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect
In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...
- 51
5
votes
2
answers
130
views
Explicit $2$-cocycle for $2^{1+2n}_+$
Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions
$$0\to(\...
- 215
2
votes
0
answers
70
views
Action of $V$ on the homology of a subposet of the poset of affine subspaces of $V$
Let $(V,Q)$ be a pair, with $V=\mathbb{F}_2^{2n}$ ($n\geq 2$) and $Q$ a nondegenerate quadratic form on $V.$ We consider the poset $\mathcal{P}_n$ of affine totally isotropic (with respect to $Q$) ...
- 215
8
votes
1
answer
294
views
Decomposing the homology of a finite-index subgroup into isotypic components
$\newcommand\C{\mathbb{C}}$Let $\Gamma$ be a discrete group and let $M$ be a $\C[\Gamma]$-module. Let $G \lhd \Gamma$ be a finite-index normal subgroup with quotient $Q = \Gamma/G$. The conjugation ...
- 83
3
votes
0
answers
86
views
Homological criterion for finite generation
Let $G$ be a (discrete) group. Assume that for all finitely generated $\mathbb{Z}[G]$-modules $M$, the homology group $H_1(G;M)$ is finitely generated. Does it follow that $G$ is finitely generated?
...
- 31
7
votes
2
answers
824
views
Hilbert's Satz 90 for real simply-connected groups?
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...
- 1,606
5
votes
1
answer
308
views
Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$
Let $G \leq \mathrm{PGL}(2,\mathbb{C})$ be the subgroup of upper-triangular matrices. I am interested in the natural morphism on the Schur multiplier (i.e. group homology as discrete groups)
$H_{2}(G,...
- 285
2
votes
0
answers
107
views
Crossed homomorphism as morphism in the ambient category
Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
- 179
3
votes
1
answer
164
views
Difficulties in the proof of finiteness of n-Selmer group using cohomology
I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.
1st ...
- 75
0
votes
1
answer
188
views
Finiteness of Selmer group
I was reading the proof of finiteness of $S^n(E/\mathbb{Q})$ but I am unable to understand from the following lemma how it follows that $S^n(E/L)$ finite.
LEMMA 3.13 For any finte subset $T$ of $\...
- 75
11
votes
2
answers
803
views
H^2 of symmetric group
I'm a number theorist in need of some group cohomology lemmas, and I'm rather bewildered by the level of generality used in the literature. Specifically, the result I need is as follows: the ...
- 748
11
votes
0
answers
324
views
Interpretation of $H^3(\mathrm{Gal}(L/K),L^\times)$
During my work I came across the group $H^3(\mathrm{Gal}(L/K),L^\times)=H^3(L/K,L^\times)$ for certain (infinite) Galois extensions $L/K$ (for an arbitrary field $K$) and I wondered if there is an ...
- 61
4
votes
0
answers
61
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
2
votes
2
answers
105
views
Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
- 339