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10
votes
1answer
494 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
1answer
158 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
1answer
438 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
2answers
582 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
4answers
423 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
3answers
979 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 > ... > ...
22
votes
0answers
523 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
1answer
286 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 ...
1
vote
0answers
159 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 ...
4
votes
0answers
149 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
2answers
305 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
0answers
194 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
2answers
612 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 ...
14
votes
6answers
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
2answers
673 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
1answer
215 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 $k$ ...
15
votes
3answers
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
0answers
309 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
3answers
595 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
4answers
655 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
1answer
502 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
1answer
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
1answer
378 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
1answer
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
3answers
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
2answers
749 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
0answers
176 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
2answers
663 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
2answers
532 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
votes
3answers
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
1answer
354 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
1answer
183 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
0answers
342 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
3answers
452 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
1answer
330 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
votes
1answer
1k views

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
vote
3answers
361 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 ...
15
votes
0answers
961 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 ...
6
votes
1answer
355 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
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 ...
16
votes
1answer
956 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
637 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
812 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
500 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
345 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 ...
12
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
1k 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
306 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 $H_3(BG)$ ...