The group-cohomology tag has no usage guidance.

**25**

votes

**0**answers

449 views

### “Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...

**21**

votes

**0**answers

301 views

### Does the Tate construction (defined with direct sums) have a derived interpretation?

Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...

**19**

votes

**0**answers

350 views

### A cohomology class associated with a complex representation of a group

$\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex ...

**17**

votes

**0**answers

1k views

### Is the equivariant cohomology an equivariant cohomology?

Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology).
$\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...

**15**

votes

**0**answers

267 views

### Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...

**14**

votes

**0**answers

292 views

### In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?

First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...

**13**

votes

**0**answers

511 views

### Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $...

**13**

votes

**0**answers

440 views

### To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...

**10**

votes

**0**answers

176 views

### Stable range of some classifying spaces and iterated loop spaces

Galatius (in his talk) has made very interesting remarks about the stable range of some classifying spaces of groups. To be more concrete, I will mention two examples to illustrate his (?) point of ...

**9**

votes

**0**answers

296 views

### Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
$G$ and $H$ are finite groups and $K$ an infinite group.
there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...

**9**

votes

**0**answers

196 views

### Homological stability for orthogonal groups

In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal ...

**9**

votes

**0**answers

422 views

### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

**9**

votes

**0**answers

384 views

### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

**8**

votes

**0**answers

430 views

### Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...

**8**

votes

**0**answers

115 views

### Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to K_3^{\operatorname{...

**8**

votes

**0**answers

192 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 ...

**7**

votes

**0**answers

193 views

### What's the name of the cohomology class associated to a projective representation?

Suppose $\rho : G \to PGL_n(k)$ is a projective representation of a group $G$ over a field $k$. It's classical that the obstruction to lifting this to a linear representation $G \to GL_n(k)$ is a ...

**7**

votes

**0**answers

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 ...

**7**

votes

**0**answers

462 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 $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...

**7**

votes

**0**answers

340 views

### Relation between group representations and elements of group cohomology groups

Having already seen group cohomology, I was just introduced to the formula $U \otimes Ind W = Ind(Res(U) \otimes W)$ from representation theory. This seems oddly like the formula $\mathrm{Cor}(u) \cup ...

**6**

votes

**0**answers

136 views

### vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a $G$-...

**6**

votes

**0**answers

136 views

### Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial

Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...

**6**

votes

**0**answers

516 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 ...

**6**

votes

**0**answers

127 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 ...

**5**

votes

**0**answers

79 views

### Random pro-p groups via iterated uniformly random central extensions

Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...

**5**

votes

**0**answers

143 views

### Continuous cohomology via model category

Is it possible to formulate notion of continuous cohomology in terms of model categories?
If yes, then is there a reference for this?

**5**

votes

**0**answers

101 views

### Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$

$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...

**5**

votes

**0**answers

105 views

### Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$.
It is known ...

**5**

votes

**0**answers

149 views

### Example of a torsionfree group satisfying a cohomological condition

Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind ...

**5**

votes

**0**answers

108 views

### (Eichler-Shimura Isomorphism) Proving c(f) is not a boundary

I have seen a couple of questions related to the Eichler-Shimura Isomorphism, but almost all of them have to do with hodge theory (things I am unfamiliar with) and seem, to me, different/unrelated.
...

**5**

votes

**0**answers

189 views

### Third cohomology of mapping class group

I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...

**5**

votes

**0**answers

237 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 ...

**5**

votes

**0**answers

291 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 ...

**5**

votes

**0**answers

237 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.)
...

**4**

votes

**0**answers

151 views

### Is central extension of a group equivalent to a bundle with gauge field?

Let $\tilde G$ be a central extension of a group $G$ by $U(1)$.
One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$
However,...

**4**

votes

**0**answers

76 views

### Generalization of a 1D unitary representation

Probably a very naive question, but I'd be grateful for any input. Consider 1D representations of finite group $G$: $\chi(g) \chi(h) = \chi(g h)$ with $\chi(g)\in \text{U}(1)$, and $\chi(1)=1$. The ...

**4**

votes

**0**answers

145 views

### Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...

**4**

votes

**0**answers

62 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...

**4**

votes

**0**answers

99 views

### Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet

I have a question on the ranks of rational cohomology groups of
$S$-arithmetic groups over function fields. To fix the situation, $G$
is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite
...

**4**

votes

**0**answers

90 views

### On restriction to centralizers of all elements in group cohomology

If $G$ is a finite group, what do we know of the natural «restriction» map $$H^\bullet(G,\mathbb Z)\to\left(\bigoplus_{g\in G}H^\bullet(Z(g),\mathbb Z)\right)^G,$$
with $Z(g) $ the centralizer of $g $....

**4**

votes

**0**answers

169 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**

votes

**0**answers

167 views

### Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...

**4**

votes

**0**answers

463 views

### Second cohomology group

Dear Forum,
Let A be an associative division algebra (i.e. a skew field), G a subgroup of the multiplicative group of A and E an extension of the additive group A+ of A by G such that G acts on A+ ...

**3**

votes

**0**answers

70 views

### Can a finite group have its third cohomology determined by a quotient group?

Does there exist a finite group $G$ satisfying either of the following?
$G$ admits a non-trivial, proper, central subgroup $A$ such that every (normalized) 3-cocycle of $G$ is cohomologous to a (...

**3**

votes

**0**answers

118 views

### Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...

**3**

votes

**0**answers

67 views

### Have locally principal crossed homomorphisms been studied?

Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...

**3**

votes

**0**answers

96 views

### Schur Multiplier of Tarski Monsters

Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?

**3**

votes

**0**answers

85 views

### Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a $\mathbb{C}[[z]]_1$-...

**3**

votes

**0**answers

97 views

### Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...

**3**

votes

**0**answers

165 views

### torsion free for the 2nd cohomology group?

Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T),
My question is:
is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free?
Thanks in advance!
...