The group-cohomology tag has no wiki summary.

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### 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

730 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

314 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

164 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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**0**answers

156 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

308 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

198 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

639 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 ...

**15**

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 $H$ ...

**20**

votes

**2**answers

692 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 $k$ ...

**16**

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

389 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

664 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**

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**4**answers

697 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

522 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

245 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

400 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 ...

**5**

votes

**1**answer

1k 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**

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**2**answers

817 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 ...

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**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 $V_l=V\otimes\mathbb Q_l.$
...

**3**

votes

**2**answers

664 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

544 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 ...

**12**

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**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 ...

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**1**answer

393 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 ...

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**1**answer

185 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 ...

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359 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**

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**3**answers

466 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.

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**1**answer

382 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 $c(f,g)c(fg,k) =c(f,hk)c(h,k)$ ...

**32**

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**1**answer

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### H^4 of the Monster

The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$.
Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding ...

**1**

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**3**answers

368 views

### Is $\mathbb{Z}_p$ flat $\mathbb{Z}_pG$-module for a finite $p$-group $G$?

Hello?
I have a simple question.
Is $\mathbb{Z}_p$ flat $\mathbb{Z}_pG$-module for a finite $p$-group $G$?
Here, $p$ is prime and $\mathbb{Z}_p$ means the integers localized at $(p)$.
If not, is it ...

**16**

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**0**answers

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### 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 ...

**6**

votes

**1**answer

369 views

### Posets of cosets and contractibility

For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset ...

**4**

votes

**1**answer

238 views

### When is a Massey product the image of a Bockstein operator?

I have a discrete group $G$ and classes $x,y\in H^1(G;\mathbb{Q})$ (group cohomology with coefficients in the rationals viewed as a trivial $G$-module) such that the Massey product $$\alpha:=\langle ...

**17**

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**1**answer

1k views

### Grothendieck's question on the Brauer group for groups

Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear ...

**1**

vote

**3**answers

678 views

### Group cohomology of an abelian group with nontrivial action

How do I compute the group cohomology $H^2(G,A)$ if G is a finite abelian group acting nontrivially on a finite abelian group A?

**10**

votes

**2**answers

973 views

### How do I get the correct long exact sequence for relative group cohomology in terms of derived functors?

Background:
I want to consider relative group cohomology: the construction is as follows. I have a subgroup $H\subseteq G$ (and note that I don't want to assume that $H$ is normal in $G$), and a ...

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**0**answers

508 views

### Untwisting the Cohomology with Twisted Coefficients

This question is set on a finite $2$-group $G$ and a subgroup $H$ of index $2$ (but perhaps the question could be answered for arbitrary orders/indexes).
It was asked here on MO whether ...

**4**

votes

**1**answer

349 views

### Examples of p-groups exhibiting isomorphic mod-p cohomology rings.

Hello,
given 2 different finite $p$-groups $G$ and $H$, $|G|=|H|=p^n$.
It has been shown by Ian Leary that the mod-p cohomology rings do not determine the groups $G$ and $H$. In fact he gave an ...

**7**

votes

**1**answer

346 views

### Cohomology of the infinite loop space of the affine grassmanian (as in the generalized Mumford conjecture)

I've been reading Hatcher's survey "A short exposition of the Madsen-Weiss theorem". In it, he outlines a nice proof of the "generalized Mumford conjecture", which asserts that the stable cohomology ...

**15**

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**4**answers

2k views

### Non-vanishing of group cohomology in sufficiently high degree

Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as $ K(BG)$, gives bunch of corollaries of ...

**13**

votes

**4**answers

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### metaplectic group does not split

I'm trying to understand the Weil representation and hope there are some experts around who can set me straight. Let $F$ be a non-Archimedean local field (I don't mind assuming that the characteristic ...

**15**

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**7**answers

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### universal cover of SL2(R): does it admit central extensions?

Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective action of ...

**6**

votes

**1**answer

314 views

### Third bordism group of BG, where G is an arbitrary compact Lie group.

Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...

**7**

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**3**answers

438 views

### Embedding groups into groups with some vanishing homology groups

Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H_i(\Gamma;\mathbb{Z})=0$ for all $i \in ...

**1**

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**1**answer

332 views

### On the group homology

Let $G$ be a group satisfying
$H_1(G;\mathbb{Z})$ is free abelian group and $H_i(G;\mathbb{Z})=0$ for $i\geq 2$.
Is it true that $G$ is free group?

**17**

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781 views

### central extensions of Diff(S^1) and of the semigroup of annuli

$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are ...

**9**

votes

**1**answer

570 views

### A torsionfree group with infinite cohomological dimension and no infinitely generated free abelian subgroup

Recently I've been reading about cohomological finiteness conditions for groups, my main source being Brown's book "Cohomology of Groups".
One of the first things one learns is that a group with ...

**8**

votes

**3**answers

838 views

### Group cohomology with compact support

Let $\Gamma$ be a discrete group, $V$ a left $\Gamma$-module. One can define the
groups $H^i(\Gamma,V)$ ($i=0,1,2,\dots$) in many ways, and then prove their equivalence: as derived functors
of the ...