# 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.

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### Applications of one of Serre's Theorems

This theorem is due to Serre:
Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no
element of order $p$ and let $H \leq G$ be an open subgroup. Then
$cd_p(G) = cd_p(H)$.
Where $cd_p(...

**1**

vote

**0**answers

76 views

### When is the restriction map in cohomology an isomorphism?

When is the restriction map in cohomology an isomorphism? Let $G$ be a group, $H$ one of its normal subgroups and $M$ a $G$-module. If $H^1(G,M)\simeq H^1(H,M)^{G/H}$, can we conclude that $H=G$?

**11**

votes

**2**answers

191 views

### Actions of locally compact groups on the hyperfinite $II_1$ factor

Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.
(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
(2) If so, how does one ...

**2**

votes

**0**answers

107 views

### Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...

**3**

votes

**1**answer

200 views

### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

**4**

votes

**0**answers

74 views

### Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...

**2**

votes

**0**answers

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### Singular homology: Lifting simplices gives map in homology

Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$.
Then the ...

**11**

votes

**1**answer

215 views

### Quasimorphisms and Bounded Cohomology: Quantitative Version?

Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...

**3**

votes

**1**answer

93 views

### Group cohomology with coefficients in a chain complex

Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each $g \in G$ I have a self ...

**6**

votes

**0**answers

92 views

### Second bounded cohomology and normal subgroups

It may be a naive question, but:
If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups?
But "many", ...

**7**

votes

**1**answer

223 views

### Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$

In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...

**5**

votes

**1**answer

205 views

### Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...

**8**

votes

**1**answer

155 views

### Second Bounded Cohomology of a Group: Interpretations

Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...

**4**

votes

**1**answer

139 views

### The computation of $d_2$ in the Hochschild-Serre spectral sequence

I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups:
$1\to N\to G\to G/N\to 1$
where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$.
...

**4**

votes

**0**answers

90 views

### Second homology of finitely presented group with free abelianisation

It is known that for a presented group $G=F/N$ we have
$$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$
In general, the right side seems to be difficult to calculate. I am in the special ...