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4
votes
1answer
228 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
votes
1answer
989 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
3answers
645 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?
8
votes
2answers
850 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 ...
3
votes
0answers
502 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
1answer
347 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
1answer
343 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 ...
14
votes
4answers
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
4answers
1k views

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
votes
7answers
2k views

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
1answer
308 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
votes
3answers
413 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
vote
1answer
330 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?
16
votes
3answers
747 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
1answer
527 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 ...
6
votes
2answers
719 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 ...
9
votes
1answer
385 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 ...
4
votes
1answer
431 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 ...
2
votes
1answer
477 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?
3
votes
2answers
320 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 ...
8
votes
1answer
1k views

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 ...
10
votes
2answers
1k views

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 ...
10
votes
2answers
438 views

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 ...
2
votes
0answers
296 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 \ ...
56
votes
1answer
3k views

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 ...
3
votes
1answer
253 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 ...
13
votes
0answers
457 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 ...
3
votes
2answers
679 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 ...
6
votes
2answers
831 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 ...
15
votes
1answer
1k views

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 ...
2
votes
1answer
373 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 ...
2
votes
3answers
689 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 ...
5
votes
1answer
476 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 ...
3
votes
0answers
103 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: $$f(g_1,g_2 + g_3) + f(g_2,g_3) = f(g_1 + g_2,g_3) + f(g_1,g_2)$$ ...
8
votes
1answer
1k views

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 ...
8
votes
2answers
385 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 ...
5
votes
1answer
585 views

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 ...
12
votes
2answers
629 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 ...
4
votes
0answers
439 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+ ...
1
vote
0answers
139 views

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 ...
24
votes
3answers
2k views

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 ...
0
votes
1answer
332 views

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 ...
1
vote
1answer
442 views

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 ...
2
votes
1answer
297 views

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 $Z_n$ ...
0
votes
1answer
605 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 ...
6
votes
0answers
314 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 ...
1
vote
2answers
428 views

Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...
2
votes
1answer
719 views

Relation between Sheaf and Group Cohomology

Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...
14
votes
2answers
763 views

Geometric model for classifying spaces of alternating groups

The classifying space of the nth symmetric group $S_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S_n$ have related models. For ...
5
votes
1answer
152 views

How large a subset do you need to uniquely determine a 2-cocycle?

Suppose A and B are abelian groups. I want to find subsets D of $A \times A$ such that any 2-cocycle $c:A \times A \to B$ for the trivial action is uniquely determined by what it does on D. ...