The group-cohomology tag has no usage guidance.

**5**

votes

**1**answer

151 views

### For which rings R is SL_n(R) a virtual duality group

A famous theorem of Borel and Serre says that if $R$ is the ring of integers in an algebraic number field, then $\text{SL}_n(R)$ satisfies virtual Bieri-Eckmann duality. In other words, there exists ...

**8**

votes

**1**answer

484 views

### Which information can be obtained from Poincaré series ?

If $A= \bigoplus_{i\ge 0}A_i$ is a graded commutative Noetherian algebra over a field, its Poincaré series is given by $P(t) = \sum_{i\ge 0} \dim(A_i)t^i$. Although the definition of $P(t)$ only ...

**12**

votes

**1**answer

961 views

### Interpretation of universal coefficients theorem for group cohomology

Suppose $G$ and $A$ are abelian groups (I'm setting $G$ abelian to keep the discussion simple, though there are analogues for non-abelian $G$) with $G$ acting trivially on $A$. By the universal ...

**2**

votes

**2**answers

320 views

### Whether such an algebra has to be the Group algebra

Let $\mathbb C$ be the field of the complex numbers, $\mathbb Q$ the field of the rational numbers.
Let $G$ be an additive subgroup of $\mathbb Q$.
$R$ is an commutative algebra over $\mathbb C$, ...

**3**

votes

**2**answers

391 views

### Cohomological dimension of finitely presented group

I have a group of cohomological dimension 2 generated by two elements. Is it possible to deduce that the group is commutative or, more generally, does $\mathrm{cd}\ G=2$ imply anything about the ...

**0**

votes

**0**answers

97 views

### Isomorphisms of group extensions arising from antisymmetric forms

Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...

**1**

vote

**1**answer

381 views

### Finiteness theorems for profinite groups

Let $G$ be a profinite group which fits in the following short exact sequence:
$$
1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1
$$
Assume that $N$ is a pro-$p$ group and that $K$ is ...

**17**

votes

**2**answers

939 views

### Proofs of the Stallings-Swan theorem

It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...

**9**

votes

**1**answer

309 views

### Second cohomology of group of $S_n$

Hello,
Let $k$ be a field of characteristic different from $2$.
Let $n\geq 1$ be an integer, and let $T$ be the maximal torus of the $k$-algebraic group $PGL_n$, namely the quotient of diagonal ...

**5**

votes

**1**answer

377 views

### Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?

The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...

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vote

**0**answers

380 views

### Profiniteness Condition for Hochschild-Serre Spectral Sequence?

This question may seem elementary to experts but I am quite confused about it:
According to the entry of Lyndon–Hochschild–Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to ...

**2**

votes

**2**answers

524 views

### Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective

I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...

**6**

votes

**1**answer

914 views

### How to Compute Transgressions in a Serre Spectral Sequence?

For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...

**7**

votes

**2**answers

1k views

### First group homology with general coefficients

When $G$ acts trivially on $M$, the first homology group is just the abelianisation of $G$ tensored with $M$, i.e. $H_1(G;M)=(G/[G,G])\otimes_\mathbb Z M$.
Is there any similar statement when $G$ ...

**1**

vote

**0**answers

147 views

### Twisted homology of free products

Let $G_1$ and $G_2$ be groups and let $M$ be a vector space equipped with actions of $G_1$ and $G_2$. The free product $G_1 \ast G_2$ thus acts on $M$. How can one compute the twisted group homology ...

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**1**answer

518 views

### Second homology group of free nilpotent p-group

Let $F_n$ be a free group on $n$ generators. Fix a prime $p$. Let $\gamma_k^p(F_n)$ be the mod $p$ lower central series, i.e. the inductively defined series
$$\gamma_0^p(F_n) = F_n \quad \text{and} ...

**1**

vote

**1**answer

161 views

### Homology of abelian groups and their finite-index subgroups

Fix some $1 \leq k \leq n$. I'm looking for finite-dimensional vector spaces $M_{n,k}$ over $\mathbb{Q}$ on which $\mathbb{Z}^n$ acts such that the natural map $H_k(\ell \mathbb{Z}^n,M_{n,k}) ...

**11**

votes

**1**answer

473 views

### The semidihedral group of order 16 and ko

Let $\mathcal{A}(1)$ denote the subalgebra of the $\mathrm{mod}\ 2$ Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$.
The cohomology with $\mathbf{F}_2$ coefficients of the ...

**9**

votes

**2**answers

595 views

### Cohomological dimension of a homomorphism

Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that ...

**9**

votes

**4**answers

430 views

### Can one do without a classifying space when showing vanishing of cohomology

Let $G$ be a discrete group and $A$ an abelian group, then $H^n (G,A)$ can be defined as $$ H^n (G,A) = H^n (B_G, A)$$
Where $B_G$ is the classifying space of $G$, i.e. $B_G = E_G / G$ where $E_G$ is ...

**22**

votes

**3**answers

1k views

### Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$G_0 > G_1 > ... > ...

**27**

votes

**1**answer

744 views

### Is there a Kan-Thurston theorem for fibrations ?

Given a fibration $F \to X \to B$ with all spaces path-connected. Is there a (discrete) group $G$ with normal subgroup $H$ such that
$$H^\ast(BG;\mathcal{A}) = H^\ast(X;\mathcal{A})$$
...

**4**

votes

**1**answer

325 views

### Cohomology $H^*(G,K)$ of wreath products

Let $G = Sym(a) \wr Sym(b)$ be a wreath product of symmetric groups - I'm particularly interested in the Weyl group of type $B$, $Sym(2) \wr Sym(n)$. Let $k$ be a field of characteristic $p$.
What is ...

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**0**answers

165 views

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

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

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votes

**0**answers

159 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 ...

**3**

votes

**2**answers

316 views

### Sign in the product of the LHS spectral sequence

Given an extension of groups
$$ 1 \to H \to G \to Q \to 1,$$
there is a spectral sequence
$$E^{ip}_2(M) = H^i(Q,H^p(H,M)) \Rightarrow H^{i+p}(G,M).$$
I understand that the composition of the cup ...

**3**

votes

**0**answers

209 views

### Identifying projective representations using “gauge-invariant” traces tr[V_g V_h V_k … ]

Background
A projective representation $V_g\in \mathrm{GL}_n(\mathbb{C})$ of a group $G$ is characterized by $V_gV_h=\omega(g,h)V_h$, where $\omega(g,h)\in\mathrm{U}(1)$ is a 2-cocycle. Changing the ...

**5**

votes

**2**answers

668 views

### The Norm Map in (group) cohomology via classifying spaces

The well-known transfer map in group (co)homology can be defined with only homological algebra, or with algebraic topology via classifying spaces (group cohomology of $G$ is isomorphic to ordinary ...

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votes

**6**answers

2k views

### Characterization of the transfer map in group theory

Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ ...

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votes

**2**answers

705 views

### Is super-vector spaces a “universal central extension” of vector spaces?

Is there some sense in which the category $sVect$ of super-vector spaces is the "maximal non-trivial extension" of $Vect$ as a symmetric monoidal category?
Is the $\mathbb Z/2$ that shows up in the ...

**1**

vote

**1**answer

218 views

### Dimension of fixed points of Galois group actions

I have a question about fixed points of Galois group actions.
I am hoping that this is easy for the experts.
Let $k$ be a field of characteristic $0$. Let $K$ be a finite
Galois extension of $k$ ...

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votes

**3**answers

2k views

### Why is BG infinite dimensional for G finite ?

If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$
then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This
can be found, for example, there:
...

**2**

votes

**1**answer

399 views

### Inseparable Galois Cohomology

First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...

**5**

votes

**3**answers

705 views

### Continuous cohomology of semi-simple Lie group.

Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_{cont}^q(G,M)$ vanish?
A ...

**10**

votes

**4**answers

735 views

### Examples of Tate cohomology rings

If $G$ is a finite group with periodic cohomology then the Tate cohomology ring can be easily computed to be the localization $\hat{H}^\ast(G,\mathbb{Z}) = H^\ast(G,\mathbb{Z})_{(z)}$ where $z$ is a ...

**1**

vote

**1**answer

535 views

### Group cohomology with $Z_2$ coefficient

I would like to know what are the group cohomology classes $H^d[Z_n, Z_2]$, $H^d[U(1), Z_2]$, $H^d[SO(n), Z_2]$, $H^d[SU(n), Z_2]$, etc. Thanks!
(Here the group cohomology $H^d[G, M]$ for a group ...

**3**

votes

**1**answer

246 views

### Large modules with non-trivial cohomology

Let $p$ be a prime and $F$ algebraic closer of $F_p$.
I want to know if it is possible to construct family of groups $\{G_i\}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of ...

**3**

votes

**1**answer

420 views

### Hirsch length and cohomological dimension

It seems to be known that Hirsch length and cohomological dimension agree for (torsion-free, finitely generated) polycyclic groups.
If we drop the assumption "torsion-free", then cd is of course ...

**6**

votes

**1**answer

2k views

### Kuenneth-formula for group cohomology with nontrivial action on the coefficient

For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G_1 \times G_2; M) \cong
[\oplus_{i= 0}^n H^i(G_1;M) \otimes_M H^{n-i}(G_2;M)]
\oplus ...

**3**

votes

**3**answers

1k views

### Group cohomology of compact Lie group with integer coeffient

It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology ...

**5**

votes

**2**answers

843 views

### Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...

**2**

votes

**0**answers

177 views

### A local-global question on group representations

Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$
...

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votes

**2**answers

672 views

### Invariants and base change

Suppose $R$ is a Noetherian commutative ring, and $M$ a finite free $R$-module, with an action of a finitely generated discrete group $G$ by $R$-linear maps.
Is there any homological condition on ...

**5**

votes

**2**answers

552 views

### Subobject-poset (co-)homology

Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and ...

**13**

votes

**3**answers

2k views

### The relationship between group cohomology and topological cohomology theories

I was recently trying to learn a little bit about group cohomology, but one point has been confusing me. According to wikipedia (http://en.wikipedia.org/wiki/Group_cohomology and some other sources on ...

**2**

votes

**1**answer

412 views

### Why does the Cheeger--Chern--Simons class descend to H_3(G/B)?

Cheeger and Simons here defined a family of characteristic classes for principal $G$-bundles ($G$ a Lie group). I'm interested in the special case of $\hat c_2:H_3(SL(2,\mathbb C);\mathbb ...

**5**

votes

**1**answer

187 views

### Annulators for minimal primes in group cohomology

Let $G$ be a finite group and $p$ be an odd prime. It's known by work of Quillen that the minimal primes of $H^{2\ast}(G;\mathbb{F}_p)$ are in one-to-one correspondence with the maximal elementary ...

**8**

votes

**0**answers

370 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.
...

**7**

votes

**3**answers

475 views

### Tabulation of known unstable rational homology of moduli space?

Does there exist a tabulation of the known rational homology of mapping class groups of genus $g$ with $m$ punctures? I'm most interested in the case when punctures can be permuted.

**1**

vote

**1**answer

408 views

### Equivalence of central extensions of Abelian groups

Background: For a projective representation of $G$ on a Hilbert space there is
a 2-cocycle $c:G\times G \to \mathbb T$ where the cocycle condition $\delta c=0$ reads $c(f,g)c(fg,k) =c(f,hk)c(h,k)$ ...