The group-cohomology tag has no wiki summary.

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### 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**

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**1**answer

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

**1**answer

235 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 ...

**3**

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**1**answer

146 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 ...

**5**

votes

**1**answer

474 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

**1**answer

210 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**

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**1**answer

541 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 ...

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**0**answers

351 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**

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**1**answer

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**

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**1**answer

159 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 ...

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vote

**1**answer

134 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**

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**2**answers

539 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**

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**1**answer

253 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 ...

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**1**answer

279 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 ...

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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**

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237 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**

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**1**answer

198 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**

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**1**answer

159 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 $H^2(G,\mathbb F_p)$. ...

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222 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**

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**1**answer

259 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
...

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167 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 ...

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194 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 ...

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**2**answers

235 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 ...

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**1**answer

625 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**

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571 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 ...

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375 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 ...

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**1**answer

342 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 ...

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**1**answer

505 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 ...

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**1**answer

187 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**

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**1**answer

236 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: ...

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473 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 ...

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241 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 ...

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**1**answer

221 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 ...

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**1**answer

377 views

### An isomorphism between different Ext's coming from group cohomology

Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$.
On the other hand ...

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253 views

### Kuranishi map, group cohomology and the bar complex

Let $\pi$ be a group, $G$ a compact lie group with lie algebra $g$, $A:\pi\to G$ a representation which composes with the adjoint map to give $g$ a $\pi$-module structure. I want to construct a ...

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167 views

### Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...

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**1**answer

301 views

### Why do we use the diagonal for diagonal approximations ?

First recall how the cup product is defined for the cohomology of a group $G$:
Fix a projective resolution $P \to \mathbb{Z}$ over $\mathbb{Z}G$. Then $P \otimes P \to \mathbb{Z} \otimes \mathbb{Z} ...

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141 views

### Modified radical group of a Kummer extension

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I ...

**4**

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285 views

### Why is the transfer map Tate-dual to restriction ?

In one of their papers (before Theorem 7.2), Benson and Carlson state that the transfer map is Tate-dual to the restriction homomorphisms (also see Remark 1.3 of this recent paper).
More ...

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152 views

### Computing Slim Extensions representing Ext

Hey Everyone
Let $A$ be an algebra over a field (group rings $k[G]$ for group cohomology, the Steenrod Algebra). We want to compute, say, $Ext_A(k,k)$, so let $F_*\to k$ be an $A$-free resolution. ...

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1k views

### reference for (co)homology theories

Hi everyone,
Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it.
I was wondering if someone could recommend a ...

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**1**answer

111 views

### Dimension of the cohomology ring of an extension of groups

Given an extension $1 \to N \to P \to Q \to 1$ of p-groups. Is it true that
$$\dim H^\ast(P,\mathbb{F}_p) = \dim \text{im}(res^P_N) + \dim \text{im}(inf^P_Q)$$
where $\dim$ denotes the Krull ...

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221 views

### Eilenberg-Mac Lane spaces for surface group extensions.

(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.)
...

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256 views

### Does the following “symmetric” 2nd cohomology group of a finite group with coefficients in $Z_2$ always vanish?

Let $G$ be a finite group. Usually, a 2-cocycle on $G$ with values in $\mathbb{Z}\_2 = \{+1, -1\}$ is a collection of signs $\epsilon_{g,h} \in \{+1, -1\}$, $g,h \in G$, satisfying the cocycle ...

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529 views

### Mackey(also Green and Tambara) functors and Greenlees-May

This is somewhat related to a question that I asked on Math.SE but, sadly, received no response. I apologize ahead of time if this is not appropriate for MO. Feel free to vote to close if this is the ...

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338 views

### Can the Bockstein spectral sequence be used to compute cohomology rings ?

If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...

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103 views

### Extensions of orthogonal groups of torsion quadratic forms.

Hi. This is related to a question I asked earlier. The setup is:
Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = ...

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125 views

### Is it true that there are exactly two conjugacy classes of order two elements in Out(R)?

In the title, $R$ stands for the hyperfinite III1 factor.
An order two element $\alpha\in Out(M)$ ($M$ any factor) has an invariant $c(\alpha)\in H^3(\mathbb Z/2,S^1)=\mathbb Z/2$.
Q: Is $c$ the ...

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248 views

### What is a higher genus analogue of the Pontryagin product?

Given a compact oriented aspherical $3$--manifold $M$ with torus boundary $\partial M\simeq T^2$ (e.g. a knot complement), the condition that the images in $\pi_1 M$ of basis $x,y\in \pi_1 T^2$ under ...

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940 views

### When does a homomorphism factor through a free group?

Let $f\colon\thinspace G\to H$ be a surjective homomorphism of finitely generated groups. Are there any methods to decide whether $f$ factors through a free group? That is, does there exist a free ...