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4
votes
2answers
364 views

Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$

Using the exact sequence $$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$ it is easy to compute ...
10
votes
2answers
313 views

3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where $$ D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle $$ is the dihedral group of ...
1
vote
1answer
259 views

Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
4
votes
2answers
400 views

n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$. (1) What is the explicit forms of its cohomology group (see my definition) in a ...
2
votes
2answers
184 views

Differentials in the Lyndon-Hoschild-Serre Sequence for p=0

I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact ...
12
votes
1answer
292 views

Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$ C^k = \{f: G^k \to A\} $$ and the coboundary map $$ \delta : C^k \to C^{k+1} $$ $$ (\delta f)(g_1, \ldots, g_{k+1}) ...
8
votes
1answer
367 views

Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question: Is there some duality known between the quasi Hopf algebra $D^\omega(H)$ of a finite group $H$ to an orbifold model (such as SU(2)/$G$ or SO(3)/$G$ orbifold of some ...
9
votes
1answer
267 views

Calculations of nonabelian group cohomology of R^n

I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
5
votes
1answer
2k views

Double coset formulas for Orthogonal groups [Solved]

According to Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170 Theorem 3.11, we can compute the composition ...
11
votes
1answer
439 views

Do there exist non-isomorphic groups with the same cohomology?

For any group $G$, cohomology can be viewed as a functor $$ H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp}, $$ where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ...
-1
votes
1answer
166 views

one-cocycles over finite groups

Let $G$ be a finite group and let $S$ be a subgroup of $G$. Let $\alpha$ be a one-cocycle $G\to M$ for some $G$-module $M$, and suppose that $Res_{S}^{G}\alpha=0$. For $g\in G$, and $S^{g}=gSg^{-1}$, ...
9
votes
1answer
319 views

Obstruction to extension of non-abelian groups - finite example?

Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner ...
5
votes
1answer
403 views

Maps between classifying spaces

Let $G$ be a discrete group and let $BG \simeq K(G,1)$ be its classifying space. Let $H$ be a topological group with classifying space $BH$. In case $H$ is also discrete, it was pointed out in the ...
12
votes
1answer
372 views

Crossed modules and degree-3 group cohomology

It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, ...
0
votes
0answers
180 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
9
votes
1answer
422 views

Cohomology of Formal Groups

Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow ...
9
votes
1answer
647 views

A map from the coinvariants of the dual to the dual of the invariants for a G-module

Suppose $G$ is a group and $X$ is a $\mathbb{Z}[G]$-module. Recall that the augmentation ideal $I \subset \mathbb{Z}[G]$ is generated by elements of the form $g - 1$ for $g \in G$, the coinvariants ...
5
votes
1answer
221 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
469 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
173 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
366 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 ...
7
votes
0answers
281 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
361 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
103 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
132 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
105 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
324 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
234 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
161 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
116 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
708 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
162 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
324 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
157 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
287 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
165 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 ...
7
votes
1answer
638 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
241 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
746 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
501 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
237 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
176 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
151 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
559 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
280 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 ...
3
votes
1answer
344 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
116 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} ...
4
votes
2answers
270 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
273 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, ...