The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
1answer
206 views

Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for: A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...
12
votes
2answers
439 views

The Teichmüller's algebraic interpretation of $H^3$ in group cohomology

In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an algebraic ...
0
votes
0answers
155 views

Descent for group actions

Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$. Finally, suppose I have an action $\sigma$ of $G$ on a ...
8
votes
2answers
341 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 ...
3
votes
0answers
123 views

non-Abelian inflation-restriction sequence?

Let $1 \to H \to G \to G/H \to 1$ be a group extension. The long exact sequence that arises from Hochschild-Serre spectral sequence for this extension relates objects that are clasified by cohomology ...
7
votes
0answers
277 views

Albrecht Fröhlich's text `Groupoids, groupoid spaces and cohomology' (1965)

I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). In this text Fröhlich defines cohomology of a group with coefficients in a groupoid (this was ...
3
votes
0answers
320 views

Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...
2
votes
0answers
97 views

Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
0
votes
0answers
121 views

cohomology with coefficients in a $p$-adic representation

I am cross-posting this question which I posted in math.stackexchange.com since I realized that there are people in Mathoverflow who are not signed-up there. Edit: For a topological group $G$ and a ...
1
vote
1answer
100 views

“Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...
10
votes
2answers
311 views

H*(braid group, irrep of symmetric group) = ?

As in the title, say $\lambda$ is some irrep of the symmetric group $S_n$, and $Br_n$ the braid group on $n$ strands, What is $H^*(Br_n, \lambda)$?
5
votes
0answers
226 views

Group cohomology in dimension $-1$

This may seem like a pie-in-the-sky speculation question, but I have good reasons for asking this. Is there any sense in which $H^{-1}(G;M)$ is defined for a group $G$ and a $G$-module $M$? The ...
2
votes
1answer
155 views

A sort of “group-ring” construction on coefficient systems in group homology (+ special case involving GL(n,Z))

Let $G$ be a discrete group and $M$ be an $RG$-module for some ring $R$ (I'm happy to assume that $R = \mathbb{Q}$). Define $R[M]$ to be the set of $R$-linear combinations of formal symbols of the ...
2
votes
1answer
114 views

Resolution of coefficient system in group homology

Let $G$ be a discrete group and let $M$ be a $G$-module. Assume that I have a resolution $$\cdots \rightarrow M_1 \rightarrow M_0 \rightarrow M \rightarrow 0$$ of $M$ by $G$-modules (with no further ...
4
votes
3answers
514 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 ...
1
vote
0answers
72 views

$H^2(K, Q_p(1))$

In Tate's local duality theorem we find the isomorphism $H^2(K, Q_p(1)) \cong Q_p$ where $K$ is a finite extension of $Q_p$. I haven't found any reference where this isomorphism is given explicitly ...
3
votes
1answer
147 views

Non-degeneracy of cup products on Tate-cohomology groups

I am working on a paper of R.P Langlands called "Representations of abelian algebraic groups", available here: http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf Now on page ...
9
votes
2answers
308 views

Cohomological dimension of knit products

Let $G$ be a group with complementary subgroups $A$ and $B$ (meaning $A\cap B=1$ and $AB=G$). If $A$ and $B$ are both normal in $G$, then $G\cong A\times B$ is a direct product. If $A$ is normal, ...
2
votes
1answer
147 views

transgression in terms of cup product in case of non-trivial action of the group on the coeffecients module

It is well known that given a short exact sequence $1\to H \to G \to G/H \to 1$ the transgression map $$ H^{p-1}(G/H, H^1(H,A)) \to H^{p+1}(G/H,A^H) $$ in the inflation-restriction sequence is in fact ...
4
votes
1answer
242 views

p-adic Lie group vs Lie algebra cohomology with mod p coefficients

My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$). Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action. Lazard ...
4
votes
1answer
148 views

A group 3-cocycle, trivial on a pair of generating subgroups?

I'm looking for an example of the following situation: A group $G$ generated by finite subgroups $H$ and $K$, a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$ such that the ...
6
votes
1answer
493 views

Can group cohomology be interpreted as an obstruction to lifts?

The standard way to view the first and second group cohomologies is this: The Standard Story Let $G$ be a group, and let $M$ be a commutative group with a $G$-action. Then the first cohomology has ...
1
vote
1answer
216 views

Cup-products and Transgression maps.

This question is related to Lyndon-Hochschild-Serre spectral sequence and cup products. I have the followin result by J.S Milne in his book Arithmetic duality theorems pg 105. Let $$0 \rightarrow C ...
13
votes
1answer
562 views

Second nonabelian group cohomology: cocycles vs. gerbes

In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title. In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in ...
6
votes
0answers
373 views

Lyndon-Hochschild-Serre spectral sequence and cup products

First here is my setup: Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particular we get on action ...
5
votes
1answer
220 views

Truncation of BG?

Let $G$ be a topological group. In some cases, e.g. when $G$ is discrete or when the spaces $G^n$ are locally contractible and the coefficients are discrete, the cohomology of the classifying space ...
7
votes
1answer
160 views

Exponent of the cohomology of a product of groups

Suppose $G$, $H$ are finite groups and $M$ is a module over $G\times H$. Question: Is the exponent of $H^i(G\times H,M)$ a divisor of $lcm(|G|,|H|)$ for $i> 0$ ? The Künneth formula answers the ...
1
vote
1answer
136 views

gluing gerbes over a spectrum of a field

A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar statements hold in other ...
8
votes
2answers
541 views

H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
3
votes
1answer
254 views

Cohomological dimension of groups & number of generators

I have a torsion-free non-abelian nilpotent group $\Gamma$ of cohomological dimension $n$. Is it possible to say anything about the number of generators of $\Gamma$ in a minimal presentation? Can I ...
2
votes
1answer
297 views

Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)

We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies ...
1
vote
0answers
115 views

clarify a question in group cohomology

In page 43 of Kenneth S.Brown's book "Cohomology of Groups", GTM 87, we have a proposition: If $G=F(S)/R$ then there is an exact sequence $0\to R_{ab}\overset{\theta}{\to} ...
3
votes
2answers
241 views

Homology groups of divisible and powered (nilpotent) groups

(1) Suppose $\pi$ is a set of primes and $G$ is a $\pi$-divisible nilpotent group, i.e., for any $g \in G$ and $p \in \pi$, there exists $x \in G$ such that $x^p = g$. Is it necessary that all the ...
2
votes
1answer
206 views

Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces

I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, ...
3
votes
1answer
162 views

Explicit formula for Bockstein hom in group cohomology of elementary abelian p-groups

Suppose $G$ is an elementary abelian $p$-group of rank n (for simplicity we can assume n=1). Denote by $\beta$ the well-known Bockstein boundary map from $H^1(G,\mathbb F_p)$ to $H^2(G,\mathbb F_p)$. ...
3
votes
0answers
229 views

The second homology of a group G and presentation complex of G

Let $G$ be a finitely presentable group. If we assume $H_2(G,Z/pZ) =0$, $p$ is a prime, then can we always find a finite presentation $\mathcal{P}$ of $G$ so that its presentation complex ...
8
votes
1answer
262 views

Generalizations and limitations of Quillen's F-isomorphism theorem

Quillen proved in 1971 ("The Spectrum of an Equivariant Cohomology Ring: I,II") for a large class of groups $G$ including compact Lie groups groups of finite virtual cohomological dimension ...
4
votes
0answers
168 views

geometric interpretation of the transgression map

Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G$, so ...
6
votes
1answer
198 views

Mod-p cohomology of $GL(n,p^d)$

In the classic paper On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field, Quillen proved (Theorem 6): $H^i(GL(n,p^d),\mathbb{F}_p)=0$ for $0 < i < d(p-1)$ and ...
6
votes
2answers
238 views

Taking invariants under pro-p-group is exact?

Let $l$, $p$ be primes. Is it true that the functor of taking invariants under pro-$p$-group $P$ of finite-dimensional $\mathbb Q_l$-vector spaces ($l\neq p$) is an exact functor? Thanks! NOTE 1: I ...
2
votes
1answer
682 views

Which limits does group cohomology commute with?

For a discrete group G, if $M$ is a direct/inverse limit of $M_i$, is $H^i(G, M)$ the direct/inverse limit of the $H^i(G, M_i)$? Of course, cohomology commutes with finite direct sums, but how about ...
4
votes
2answers
587 views

Group cohomology of orthogonal groups with integer coefficient

I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group: $H^*(BSO(n);\mathbb{Z}) = $ ? (for example for ...
3
votes
3answers
379 views

Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite. Let us assume we are given a group $G$ and a ...
0
votes
1answer
370 views

Transfer map for group homology.

I'm trying to figure out what the transfer map looks like in a specific case. Here's the set up Let $G$ be a group and $H$ a subgroup of finite index, and let $h_{i}$ for $i=1,..,n$, be coset ...
1
vote
1answer
522 views

Transgression maps in group cohomology and group homology / duality of spectral sequences

I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem. Let $G$ be a group, $H$ a normal subgroup of ...
7
votes
1answer
190 views

Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups

I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions. $G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ ...
3
votes
1answer
240 views

Computations of cup products in Serre's Local Fields

I have been reading the appendix in Serre's Local fields, to do with explicit computations of cup products (pg 176), but I'm stuck on one bit of lemma 4. It goes as follows Let B be a $G$-module, $u: ...
13
votes
3answers
478 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 ...
2
votes
0answers
250 views

galois cohomology over finite field

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$. Do we have a formula to ...
3
votes
1answer
228 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 ...