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Questions tagged [group-cohomology]

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of ...

8
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1answer
152 views

The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
11
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161 views

sequences in non-abelian group cohomology

In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
4
votes
0answers
83 views

Is the following variant of the Universal Coefficient Theorem valid?

A version of the Universal Coefficient Theorem that relates the integer cohomology of a group $G$ to its cohomology with coefficients in an abelian group $M$ is as follows: $H^n(G,M) = H^n(G,\mathbb ...
3
votes
1answer
82 views

Convergence of the Lyndon-Hochschild-Serre spectral sequence as an algebra

Consider a short (not necessarily split) exact sequence of groups $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ and suppose we wish to find the cohomology of $G$ with coefficients in a ...
9
votes
0answers
132 views

Hochschild-Serre spectral sequence via explicit filtration

Let $$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$ be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
26
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476 views

Cohomology of symmetric groups and the integers mod 12

When $n \ge 4$, the third homology group $H_3(S_n,\mathbb{Z})$ of the symmetric group $S_n$ contains $\mathbb{Z}_{12}$ as a summand. Using the universal coefficient theorem we get $\mathbb{Z}_{12}$ ...
3
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0answers
110 views

Induced Homomorphism on Cohomology of Symmetric Group 3

For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
6
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1answer
253 views

Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point. Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...
30
votes
2answers
543 views

Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
8
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92 views

Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?

Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$. Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...
2
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94 views

Pontryagin square on spin and non-spin manifold

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
6
votes
1answer
264 views

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
4
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1answer
91 views

Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...
3
votes
1answer
107 views

Is it possible to classify extensions $G$ of an abelian $A$ by an abelian $N$ such that the map $G\rightarrow A$ is abelianization?

Suppose we have a given action $\varphi : A\rightarrow\text{Aut}(N)$ with $A,N$ abelian groups. Is it possible describe the isomorphism classes of extensions $G$ of $A$ by $N$ realizing $\varphi$ such ...
2
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0answers
83 views

Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence $$ 1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1. $$ The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...
9
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0answers
198 views

Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...
8
votes
1answer
260 views

Spin cobordism v.s. KO theory in low or in any dimensions

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension. If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$ ...
3
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0answers
48 views

Comparing the cohomology rings of two central extensions

Consider two groups $G$ and $G'$, where $G$ is the direct product of groups $A$ and $B$, with $B$ abelian, and $G'$ is a nontrivial central extension of $A$ by $B$. Suppose that as groups, $H^1(G,M) \...
5
votes
1answer
241 views

Interesting properties in $…\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to …$

Let $K(G,n)$ be the Eilenberg Maclane space. Consider the map from $$ K(\mathbb{Z}_2,1) \to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to \dots, $$ It ...
4
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184 views

Periodicity of cohomology

Let $G$ be a finite group, and $M$ a $\mathbb{Z} [G]$-module. Assume that given an sequence $$0\to M'\to F\to M\to 0 ,$$ where $F$ is a free $\mathbb{Z} [G]$-module and $M'={\rm Hom} (M,\mathbb{Z})$ ...
11
votes
1answer
216 views

Is there a kind of Poincare duality for Borel equivariant cohomology?

Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian ...
14
votes
1answer
302 views

$p$-groups with trivial $H^3$

Let $Q_8$ be the group of quaternions of order $8$. It is a non-abelian $2$-group such that $H^3(Q_8,\mathbb{Z})=0$, where $\mathbb{Z}$ has the trivial action. For a proof, see the book "Homological ...
7
votes
1answer
248 views

Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence

I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given ...
5
votes
1answer
316 views

Kunneth formula for semidirect product

I wonder if the following Kunneth formula for semidirect product is valid $$ H^n(N\rtimes_\phi G;\mathbb{Z}) = \sum_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$ where $H^*$ is the group cohomology and $G$ has ...
12
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0answers
347 views

The third (co)homology group

I need to prove that the third (co)homology group of a certain finitely presented group is not finitely generated. The group is not an amalgamated product or an HNN extension, and it does not act ...
6
votes
1answer
94 views

Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$

Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e. \begin{equation} D(g) D(h) = e^{i \omega(g,h)} D(gh) \end{equation} These can be classified by the equivalence ...
6
votes
1answer
153 views

Trivializing unitary cocycles in abelian von Neumann algebras that are uniformly close to the trivial one

Suppose $M$ is an abelian von Neumann algebra, carrying a (point-ultraweakly) continuous action $G\curvearrowright M$ of a locally compact, second-countable group. Let $y: G\to\cal U(M)$ be an ...
5
votes
0answers
87 views

Restricting projective representations of Lie groups to lattices

Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
2
votes
1answer
110 views

Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?

Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal Levi subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2....
4
votes
1answer
123 views

Trivial cohomology with free module coefficient

Let $G$ be a group and $M$ be a free $\mathbb{Z} G$-module. Then $H^2(G,M)=0$. Is this statement correct? I know that if $M$ is injective module, then $H^n(G,M)=0$ for all $n\geq 1$. But I have no ...
4
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0answers
71 views

Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
2
votes
0answers
103 views

Description of Shapiro lemma by cochain

Let $H\to G$ be a subgroup of a finite group, let $M$ be an $H$-module, $N$ be the induced $G$-module, by Shapiro lemma we have $H^i(G,N)\cong H^i(H,M)$. What is the map on the level of cochains? ...
9
votes
1answer
250 views

Finite group representation as $\mathrm{Aut}(\Gamma)$ action $H^1(\Gamma,\mathbb{Z})$ of graph?

Let $\Gamma$ be a finite graph, then $H^1(\Gamma,\mathbb{Z})\cong \mathbb{Z}^{g(\Gamma)}$ can be viewed as a $\mathrm{Aut}(\Gamma)$ module. Conversely, given a finite group $G$, and a $G$-module $\...
3
votes
1answer
137 views

When is a sequence of group extensions associative?

Suppose I have groups $A,B$ and $C$ for which the following information is given: 1) The group $G_{AB}$ is a central extension of $B$ by $A$, where the abelian group $B$ acts trivially 2) The group $...
7
votes
0answers
109 views

Small modules over finite group with large cohomology

Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here. Using Lyndon resolvent, ...
12
votes
1answer
167 views

Example of group cohomology not annihilated by exponent of $G$?

Is there an example of a finite group $G$ and an action on $M=\mathbb{Z}^n$ such that $H^2(G,M)$ has exponent greater than the exponent of $G$? (Especially, can we have $G=\mathbb{Z}/2\mathbb{Z}\...
18
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2answers
424 views

primary decomposition for nonabelian cohomology of finite groups

Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$). If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...
4
votes
0answers
117 views

Interpreting $H^n(BG,\mathbb Z)$ when $G$ is an infinite discrete group

Suppose $G$ is a two-dimensional space group, for example a semidirect product of $\mathbb Z^2$ with a crystallographic point group such as $\mathbb Z_2$, where the action of $\mathbb Z_2$ on $\mathbb ...
6
votes
1answer
140 views

Calculation of the Schur multiplier of $\mathbb Z^2$

Consider a projective representation of $\mathbb Z^2$ with $U(1)$ coefficients. I would like to find the covering group corresponding to this representation. For this, one needs to find the ...
4
votes
0answers
67 views

Contracting the rational bar cocomplex for a finite group G

Let $G$ be a finite group and define $B^p(G,\mathbb{Q}) = {\rm Functions}(G^p,\mathbb{Q})$. These $\mathbb{Q}$-vector spaces assemble into a cochain complex with differential $$d \sigma(g_0,\dots,g_p) ...
9
votes
1answer
135 views

Projective resolutions of finite-dimensional representations of infinite groups

Let $G$ be a group and let $V$ be a finite-dimensional complex representation of $G$. Question: Under what circumstances can I find a projective resolution $$ \cdots \longrightarrow P_3 \...
2
votes
0answers
100 views

How to show the equality of two descriptions for the cohomology of a non-finite group

I am learning about group cohomology. For a group $G$ and a $G$-mod $A$, we can define $X^n(G,A)=Map(G^{n+1},A)$, and get a resolution $0\to A\to X^\cdot$ and then define cohomology groups $H^n(G,A)$...
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0answers
63 views

Number of surjections of a given complexity

Definition: The complexity of a surjection $\{1,\ldots,n+k\}\rightarrow \{1,\ldots,n\}$ is defined in the following way. First think of this map as the tuple $(f(1),\ldots,f(n+k))$. For two numbers $...
1
vote
0answers
122 views

Isomorphism in homology

I asked this question on Mathematics SE three days ago, but didn't get the answer. $\require{AMScd}$Let $G, H, K$ be groups and suppose that we have a diagram $$\begin{CD} G @>f_1>> H\\ @...
25
votes
2answers
840 views

Is the cohomology ring of a finite group computable?

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
3
votes
1answer
157 views

Computation of group homology $H_2 ((\mathbb{Z}/3\mathbb{Z}) \rtimes (\mathbb{Z}/4\mathbb{Z}),\mathbb{Z})$

In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we ...
3
votes
0answers
111 views

Cohomology of root data

Let $(X,\Phi,X^\vee,\Phi^\vee)$ be a semisimple root datum (in the sense of SGAIII), and $W_0$ its (finite) Weyl group. What is known about the cohomology groups $H^n(W_0, X^\vee)$ ?
1
vote
0answers
212 views

A cohomology associated to a Riemannian manifold

Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(...
31
votes
0answers
2k views

Homology of $\mathrm{PGL}_2(F)$

Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
5
votes
0answers
80 views

Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients: An oriented, (assumed here to be smooth) manifold $M^n$ A finite group $G$ (and a field, chosen to be $\...