The group-cohomology tag has no wiki summary.

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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 ...

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355 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 ...

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887 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 ...

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343 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 ...

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224 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 ...

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909 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 ...

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609 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?

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### 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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461 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).
A while ago someone posted a question on this ...

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**1**answer

336 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 ...

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**1**answer

335 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 ...

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### 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 ...

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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 ...

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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 ...

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**1**answer

303 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 $H_3(BG)$ ...

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383 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 ...

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**1**answer

318 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?

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703 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 ...

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**1**answer

480 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 ...

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**2**answers

680 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 ...

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**1**answer

368 views

### Finitely generated subgroups of a product of free groups

Is it true that a finitely generated subgroup of a cartesian product of free groups has a finite cohomological dimension?
The same question about pro-$p$ groups:
Is it true that a finitely generated ...

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**1**answer

386 views

### Tamagawa numbers of abelian varieties and torsion.

Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there ...

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**1**answer

421 views

### corestriction and transversals

Is the corestriction map from a subgroup H to a group G, on the first Tate cohomology group H^1, dependent on the chosen transversal of H is G?

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312 views

### Cup products and the transfer map

Let $G_1$ be a finite-index subgroup of $G_2$. Let $i : H^{\ast}(G_2) \rightarrow H^{\ast}(G_1)$ be the induced map of rings. There is then a transfer homomorphism $\tau : H^{\ast}(G_1) \rightarrow ...

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### Why isn't the orbifold cohomology of $pt/G$ equal to the cohomology of $BG$?

The classifying space of a group $G$ is given by taking a contractible space $E$ equipped with a free $G$-action, and looking at the quotient, which we dub $BG$. The homotopy type of this space (and ...

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### How to compute the cohomology of the general linear group with integral entries

Q: So how does one compute the cohomology groups $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$?
First note that $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$ is isomorphic to $H_B^*(Y/GL_n(\mathbf{Z}),\mathbf{Z})$ (Betti ...

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### An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?

Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...

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282 views

### Cohomologies associated to residually torsion-free nilpotent groups

This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...

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### Nontrivial finite group with trivial group homologies?

I stumbled across this question in a seminar-paper a long time ago:
Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace ...

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247 views

### Cocyles for abelian extensions

Suppose we have an abelian extension of Hopf algebras,
$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$
According to the general theory there is a left action of $F$ on $G$ and a ...

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442 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 ...

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664 views

### A question about finite groups.

Let $k$ be a positive integer. Is it true that any finite group $H$ of cardinal $4k+2$ whose center contains an element $h$ of order $2$ is isomorphic to the direct product ...

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754 views

### Transfer homomorphisms with coefficients

In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^* (H;M) \to H^*(G;M)$.
In homotopy theory, there is a transfer ...

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### Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials

Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus ...

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**1**answer

361 views

### Tate-Farrell cohomology of a circle

I know my question is very imprecise. I am trying to understand Tate-Farrell cohomology of the infinite Lie group $S^1$ (say, with coefficients in $\mathbb C$). I would expect that the answer is ...

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670 views

### Another group cohomology cup product question

I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following:
Let $G=F/R$ be a finitely presented group ...

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**1**answer

447 views

### Group cohomology and cohomology in non-abelian categories

One defines the $H^n(G,M)$ where $M$ is a $\mathbb{Z}[G]$ module as $Ext^n_{\mathbb{Z}[G]}(\mathbb{Z},M)$ where $\mathbb{Z}$ is viewed as a trivial $\mathbb{Z}[G]$-module.
Is this part of a general ...

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102 views

### High-dimensional analogue of Skew

Suppose G and A are abelian groups. Suppose f is a 2-cocycle for the trivial group action of G on A. In other words, we have that:
$$f(g_1,g_2 + g_3) + f(g_2,g_3) = f(g_1 + g_2,g_3) + f(g_1,g_2)$$
...

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### Where can I easily look up / calculate (abelian) group cohomology?

For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...

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380 views

### Injection of Ext into H^2

Let $G$ be an abelian group, $A$ a trivial $G$-module. We know that $\text{Ext}(G,A)$ classifies abelian extensions of $G$ by $A$, whereas $H^2(G,A)$ classifies central extensions of $G$ by $A$. So ...

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### Classification of $p$-groups of order $p^n$ with rank $n-1$

Hello,
i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form
$ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow ...

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622 views

### Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?

Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness ...

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### 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+ ...

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### Duflot-type theorem for Hopf algebras ?

In group cohomology Duflot's theorem states that the depth of the mod p cohomology ring of a finite group is greater than or equal to the p-rank of the center of a Sylow p-subgroup.
Is there a ...

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### Why aren't there more classifying spaces in number theory?

Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested ...

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### Under what conditions does the second cohomology preserve injectivity?

Suppose G, A, and B are abelian groups with $i:A \to B$ an injective homomomorphism. Consider the groups $H^2(G,A)$ and $H^2(G,B)$ for the trivial action of G on A and B. i induces an injective ...

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### The splitting for the cohomology version of Kunneth formula

Suppose $G_1,G_2$ and A are abelian groups. Consider the cohomology groups for trivial action:
$$H^2(G_1 \times G_2,A), H^2(G_1,A), H^2(G_2,A)$$
We have projection maps $G_1 \times G_2 \to G_1$ and ...

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### Describe the second cohomology group $H^2(Z_n \times Z_n. k^*)$.

I would like to write down explicitly the generating cocycles of this second cohomology group, $H^2(Z_n \times Z_n, k^*)$. Here $k$ is an algebraically closed field of characteristic zero and $Z_n$ ...

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**1**answer

547 views

### Is it useful to consider cohomology of group representations?

In group representation theory, one attempts to explain and classify (some of) the modules over the group ring $k[G]$, for some field $k$. In group cohomology, one develops the machinery of the ...

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310 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 ...