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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 ...

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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, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
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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
votes
1answer
78 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
103 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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80 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 ...
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167 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 ...
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114 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 $n$) $$\Omega_d(BG)_p.$$ ...
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47 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) \...
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1answer
231 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 ...
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180 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 $M'={\rm Hom} (M,\mathbb{Z})$ ...
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1answer
168 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
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1answer
300 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
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1answer
222 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 ...
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1answer
300 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 ...
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344 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
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1answer
91 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 ...
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1answer
149 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
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83 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
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1answer
105 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....
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1answer
118 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 ...
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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}\...
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98 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? ...
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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
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1answer
136 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 $...
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106 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, ...
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1answer
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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}\...
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420 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 ...
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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 ...
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1answer
133 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 ...
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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) ...
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1answer
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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 \...
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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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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 $...
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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\\ @...
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2answers
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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
150 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 ...
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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 $H^n(W_0, X^\vee)$ ?
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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^*(...
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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 ...
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71 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 $\...
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381 views

A functor on the category of commutative rings, algebras or Banach algebras

Edit: According to the comments of abx and Yemon Choi I revise the question as follows: Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...
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A possible generalization of “Group Cohomolgy”

The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the ...
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1answer
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Group (Co)Homology of Symmetric Group

The question concerns the group homology or group cohomology of symmetric groups. The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$. groupprops....
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Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring

Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" component is a ...
5
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1answer
286 views

How to compute the group cohomology of $\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ with coefficient in a trivial module?

The group cohomology of cyclic groups can be computed easily due to the periodity. Now how can one compute the group cohomology $H^r(\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z},M)$? As least ...
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1answer
85 views

cohomology of finite groups of lie type with coefficients in the adjoint module

Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with ...
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105 views

Use GAP program to obtain explicit cocycles in group cohomology

I'm trying to compute group cohomology $H^n(G,\mathbb{Z})$ of some crystal groups $G$ which are infinite but finitely generated groups. I succeed in obtaining cohomology groups using projective ...
5
votes
2answers
330 views

A finite group that splits and does not split

Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup) such that there exists an epimorphism $$\tau \colon ...
3
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1answer
90 views

Equivalence of finiteness of $spliG$ and periodicity isomorphisms being induced by cup product

I am trying to prove that the following are equivalent for a group $G$ with periodic cohomology with period $q$ after $k$ steps: $(i)\ spliG<\infty$ (where $spliG$ is the supremum of injective ...
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2answers
296 views

Kazhdan constant and finite index subgroups

I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups? Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...