The group-cohomology tag has no usage guidance.

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### 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!
...

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311 views

### The relation between group cohomology and the cohomology of the classifying space

We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal ...

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**1**answer

117 views

### on lifting extensions

Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...

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**1**answer

261 views

### Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back.
Let $GH^*(X,Z)$ be the subgroup of ...

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**1**answer

380 views

### discrete group cohomology vs continuous group cohomology for profinite groups

Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map ...

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322 views

### Non-smooth function with all differences of translates smooth?

Suppose $f:\mathbb{R} \to \mathbb{R}$ has the property that for every fixed $t\in\mathbb{R}$ the function
$$
g_t : x \mapsto f(x) - f(x-t)
$$
is $C^\infty(\mathbb{R})$. Does it follow that $f$ is ...

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187 views

### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

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132 views

### Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...

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1k views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

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73 views

### Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...

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1k views

### $H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...

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**2**answers

572 views

### The sum of the carries when adding and multiplying two numbers in base p

Let $\sigma_p(m,n)$ (resp. $\pi_p(m,n)$) denote the sum of the carries when adding
(resp. multiplying) the numbers $m=\sum_{k\ge0}m_kp^k$ and $n=\sum_{k\ge0}n_kp^k$ using base-$p$ arithmetic where ...

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**1**answer

292 views

### Classes in $H^3(G; \mathbb{Z})$ that restrict to zero on abelian subgroups

Let $G$ be a finite $p$-group. Is it possible to have a nonzero class in $H^3(G; \mathbb{Z})$ that restricts to zero in $H^3(A; \mathbb{Z})$ for every abelian subgroup $A \subset G$? If so, what is a ...

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**1**answer

231 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...

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159 views

### Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...

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**1**answer

179 views

### Successive Schur covers

Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?

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**1**answer

356 views

### vanishing higher cohomology group for property T group?

Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology,
Is it known that $H^n(G, ...

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**4**answers

350 views

### The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...

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**1**answer

395 views

### Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological ...

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423 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 ...

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### when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?

Let $G$ be a finite group with a subgroup $K$. Given $[\beta]\in H^2(K,\mathbb{C}^*)$ is the an obstruction which checks whether or not $[\beta]$ is the restriction of some $[\alpha]\in H^2(G, ...

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113 views

### (Co)Homology of groups vs. Lie algebras: polynomial rings

For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be ...

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109 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 ...

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votes

**1**answer

219 views

### Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of
transfers for Bloch groups and scissors congruence groups/pre-Bloch
groups.
To fix notation and recall definitions:
From the ...

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201 views

### extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

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325 views

### stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

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**1**answer

131 views

### Explicit reference on generator of $H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ identified with second Chern class of standard representation

It is extremely well-known that $H^*(BQ_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and ...

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**1**answer

227 views

### spectral sequence with non-trivial action on coefficients

Set-up:
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}_2\xrightarrow{p}\mathbb{Z}_2\rightarrow 1$$
Define ...

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**1**answer

193 views

### Hochschild-Serre spectral sequence and non-trivial action on coefficients

Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action ...

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217 views

### The homology of the braid group with coefficients in the Burau representation

Let $B_n$ denote the braid group with $n$ braids. The Burau representation $B_n\to GL_n(\mathbb{Z}[t^{\pm1}])$ makes $(\mathbb{Q}[t^{\pm1}])^n$ a $B_n$-module. I am curious in knowing what $H_i(B_n, ...

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459 views

### Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...

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95 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
...

**0**

votes

**1**answer

121 views

### Known computations of certain 2-cohomology groups?

I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$.
Here ...

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246 views

### Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...

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**1**answer

208 views

### What are the cohomology classes $H^d(BSO_\infty,Z)$ and $H^d(BO_\infty,Z)$?

The Theorem 1.5 and 1.6 of
Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
give a general answer for $H^d(BSO_n,Z)$ ...

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362 views

### Introductory text on Group Cohomology [closed]

What are good introductory textbooks available on Cohomology of Groups?

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**1**answer

240 views

### Cohomology of elementary Abelian p-group

Let $E=(\mathbb{Z}/p\mathbb{Z})^n$, an elementary Abelian p-group.
Let $k$ be an algebraically closed field of characteristic 0.
There is a good description of $H^*(E,F^{\times})$ where $F$ is a field ...

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**1**answer

309 views

### What are the low-degree group cohomology of the mapping class group of a surface

Let $MCG_g$ be the mapping class group of genus $g$ closed surface.
(Say $MCG_1=SL(2,Z)$).
I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as ...

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votes

**1**answer

150 views

### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...

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2k views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

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404 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 ...

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186 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 ...

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273 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

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1k views

### Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...

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**1**answer

445 views

### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

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102 views

### Are finite-dimensional representations of groups of type $\text{FP}_{\infty}$?

Let $G$ be a group (possibly infinite) and $k$ be a field. A module $M$ over $k[G]$ is said to be of type $\text{FP}_{\infty}(k)$ if it has a projective resolution each of whose terms is finitely ...

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91 views

### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$?
Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...

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votes

**2**answers

429 views

### Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to ...

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### 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 ...

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votes

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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 ...