Questions tagged [group-cohomology]
a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).
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Cap product of infinite cyclic group in degree one
I have a question about the cap-product in group cohomology. Maybe someone can help me out or clarify a mistake in my mind.
Suppose we have an infinite cyclic group $G$ with fixed generator $g$. Then ...
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The second chomology group $H^{2}(Z_{3}.M_{22},V)$, where $V$ is the 6-dimensional vectors space over GF(4)
Is there is anyone that could help me in evaluating $H^{2}(Z_{3}.M_{22},V)$, where $V$ is the 6-dimensional vector space over GF(4)?
Thank you very much in advance.
Goranac
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Notation in the Tate-Nakayama Theorem
In ch. IX, §8 of Serre’s Local Fields, we find the Tate–Nakayama theorem, an essential lemma for class field theory:
Theorem 14. Let $G$ be a finite group, $A$ a $G$-module, and $a\in H^2(G,A)$. Let $...
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Relationship between restriction and corestriction for Galois cohomology groups involving number fields and local fields
Let $L/K$ be an extension of number fields. Let $v \in M_K$ be a place of $K$ and $\nu \in M_L$ be a place of $L$ dividing $v$. Let $L_\nu/K_v$ be the extension of the corresponding local fields. Let $...
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Relations between the three different descriptions of 2nd cohomology group in Group Cohomology
I am coming at the 2nd cohomology group in Group Cohomology from the perspective of the Group Extension Problem (or rather the group central extension problem, which perhaps more closely related to ...
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Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Leray Spectral Sequence for Sheaf Cohomology?
Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$.
I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, ...
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Does every extension of a finite group by $\mathbb{R}^n$ split?
Suppose $G$ is a topological group containing a closed normal subgroup $N$ isomorphic to $(\mathbb{R}^n, +)$ such that $G/N$ is finite. Is $N$ a semidirect factor? Equivalently, does $G$ contain a ...
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Two group cohomology are isomorphic to each other if they have the same coefficient.
Suppose $M$ and $N$ are $\mathbb{F}[G]$-modules over a field $\mathbb{F}$, and $G$ is a finite group.
Claim: Suppose there is a map $f: M \rightarrow N$ such that $f$ is a $\mathbb{F}$-vector space ...
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Exercise 6.1.14 in Weibel's "homological algebra" on cohomology cross product
Let $G,H$ be two groups.
In $6.1.14$ of [Weibel], the Cohomology Cross Product is defined as follows.
Let $P\to \mathbb{Z}$ be a free $\mathbb{Z}G$-resolution of the trivial module $\mathbb{Z}$.
Let $...
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Cohomology of symmetric group of order $p!$ with coefficient tensor $p$ times of a graded vector space.
Suppose $\Sigma_p$ is the symmetric group of order $p!$. Let $V$ be a graded $\mathbb{F}_p$-vector space such that $V_i$ is non-trivial for $i$ arbitrary large at even and order degree.
Claim: $H^j(\...
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Restriction map in semilocal Galois cohomology
Let $F/K$ be a finite extension of number fields, let $v$ be a prime of $K$ that is totally split in $F$. Call $w_1,\dots, w_n$ the primes of $F$ above $v$. Let $T$ be a $Gal(\bar{K}/K)$-module. Then ...
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Group Cohomology, Module Extensions, and Group Extensions, and $Ext^2_{\mathbb{Z}G}(\mathbb{Z},A)$
I've read that for some $G$-module $A$, group cohomology can be defined as $$H^{n}(G,A)=Ext^{n}_{\mathbb{Z} G}(\mathbb{Z},A).$$
I've also read that for two $R$-modules $C,D,$ $Ext^{n}_{R}(C,D)$ can be ...
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Group Homology of a Sylow $p$-Subgroup
I am trying to understand the proof of the following proposition:
Proposition Let $G$ be a finite group and let $P$ be a normal Sylow $p$-subgroup of $G$. Then for all $n$ there is an isomorphism
$$
...
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second group cohomology of finite groups and bicharacters
Let $G$ be a finite abelian group. This is a product of cyclic groups. I need some facts and proof on the explicit cocycles generating its cohomology with $U(1)$ (or equivalently $\mathbb{C}^*$) ...
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Computing Tate cohomology using computer
$\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\coker}{coker}
\newcommand{\Z}{{\mathbb Z}}
$I want to use GAP in order to compute (on a computer) an example that I cannot compute by hand, see ...
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Group cohomology cannot be defined until we define what topology and action is?
Let $E$ be a $G$-module. $1$st Group cohomology $H^1(G,E)$ is defined as a set of representative of continuous map $G \to E$ which satisfies cocycle relations modulo boundary relation.
This definition ...
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Why discrete modules?
The profinite group cohomology of discrete modules can be defined by right derived functors. Its application includes Galois cohomology, Brauer groups etc. These facts demonstrate that defining ...
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Induced $2$-cocycles on a normal subgroup
Let $G$ be a finite group acting trivially on an abelian group $A$. Let $\alpha$, $\beta \in Z^2(G,A)$ such that $\alpha\beta^{-1}\in B^2(G,A)$ . Let $S$ be a normal subgroup of $G$ and take $\alpha'=\...
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Inflation-restriction sequence for profinite groups
I recently learnt about profinite group cohomology to do class field theory and I am looking for a proof of the profinite version of the inflation-restriction sequence which hopefully just uses the ...
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Find functions $f(m,n)$ to make $a_m\times a_n= f(m,n)a_{m+n}$ associative
Let $A$ be a free Abelian monoid generated by the elements $\{a_n| n\in\mathbb{Z},n\geq 0\}$, i.e. a generic element of $A$ is a formal (finite) linear combination of $\{a_n| n\in\mathbb{Z},n\geq 0\}$ ...
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Second cohomology group of direct product of cyclic groups
We have the following theorem.
Let $G_1$ and $G_2$ be finite groups and let $n_1$ and $n_2$ be the exponent of $G_1/G_1'$ and $G_2/G_2'$ respectively, where $G_1'$ and $G_2'$ are commutator subgroups. ...
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How do we incorporate actions into the topological method of computing group (co)homology?
When computing the (co) homology of a group $G$, we can think about it in two related ways.
We define the homology of $G$ with coefficients in a $G$-module $M$ by constructing a resolution of $M$ ...
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Spectral sequences and equivariant cohomology
Let us consider a $G-$space M, let $M_G:=EG\times M/\simeq$ ($/\simeq$ is the quotient by the diagonal action) be the homotopy quotient and let us consider the corresponding fibration
$$
M\...
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A map in group cohomology from $H^2(G,\widehat{G})$ to $H^3(G,U(1))$
Let $G$ be an abelian finite group, and denote by $\widehat{G}$ its Pontriajin dual, which is (non-canonically) isomorphic to $G$. I found the statement that there exists a map in group cohomology
\...
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Group Cohomology and Change of Coefficients
Let $G$ be a group and $V$ be a representation of $G$ over a field $K$. Let $L$ be any field containing $K$. Consider the natural morphism $$\varphi:H^i(G,V)\otimes_KL\rightarrow H^i(G,V\otimes_KL).$$ ...
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Induced Euler short exact sequence on wedge product
From any short exact sequence $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, we can construct the following induced exact sequence on wedge product (following apendix A):
$$ ...
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Group action on co-induced module
$\newcommand\mb{\mathbb}\DeclareMathOperator{\Mor}{Mor}\DeclareMathOperator{\coInd}{coInd}$ Let $G$ be group, then we can define for an abelian group $N$ the co-induced module $\coInd^G(N) := \Mor_{\...
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Is there any calculation result for quandle homology group?
I'm investigating the calculation result for the quandle homology group of a dihedral quandle.
First of all, I know that the result for odd order(for odd n in $H_k^Q(R_n;\mathbb{Z})$) is well known, ...
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GAP Software: ComplementIntMatWithTransforms v/s ComplementIntMat
I am trying to implement the GAP algorithm given in https://arxiv.org/pdf/1708.06538.pdf to compute the group cohomology of finite groups. One of the functions they use is "...
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When do the group-homomorphism "separate points"?
Today, I asked myself the following question: Suppose I have some group $G$ and I look at the homomorphism $\mathrm{Hom}(G, \mathbf{Q})$ (or the first cohomology group, if you want to call it that). ...
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Naturality of Lyndon-Hochschild-Serre
I am currently trying to show that the Lyndon-Hochschild-Serre spectral sequence converges naturally in the following sense: Suppose we are given a $\mathbb{Z}G'$-module $A$ and a commutative diagram ...
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About conjugation map in second cohomology group
I am reading the chapter Second cohomology groups of Continuation of the Notas de Matemàtica.
On page 13, the author defines conjugation map.
Let $G$ act on an abelian group $A$ and $H$ be a subgroup ...
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Projective resolution induced by subgroup $H\subset G$ in group cohomology
I'm studying group homology and cohomology following the book An introduction to homological algebra from Weibel and I have difficulties in understanding a statement regarding projective resolutions.
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A doubt in a theorem about group extensions
I am reading this chapter.
Part 1 in theorem 1.2 tells us that
Let $E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1$ be an extension of $A$ by $G$ let $t : G \to X$ be a section of $f$ i.e. $t(...
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Conjugation action of $D_m/C_m = C_2$ on group (co)homologies of $C_m$ with trivial coefficients
This is Example $6.7.10$ in Weibel's "An introduction to homological algebra", after Example $6.7.7$ in which Weibel explained how does $G$ act by "conjugation" on (co)homologies ...
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Coinvariants of rigid meromorphic function (on the p-adic upper half-plane)
Maybe someone can help me out. I am considering the p-adic upper half-plane $\mathcal{H}_p$ given on points by $\mathbb{P}^1(\mathbb{C}_p)\setminus \mathbb{P}^1(\mathbb{Q}_p)$, viewed as a rigid ...
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Cohomology rings of polyhedra's complemet
I was told, that there is an example of two polyhedra in $\mathbb{R}^n$, such that their complements have non-isomorphic coholomology rings, but can't come up with anything, because it seems a little ...
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Group cohomology of fundamental group and singular cohomology
Let $X$ be a nice complex variety and fix a basepoint $x_0$. Denote $\pi:=\pi_1(X,x_0)$. Let $\mathbb{V}$ be a local system on $X$, and consider the cohomology groups $H^i(\pi, \mathbb{V}_{x_0}), H^i(...
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Description of projective $\mathrm{SL}_{2}(\mathbb{Z})$-modules
I am working my way through Ken Browns book on the cohomology of groups, and in particular chapters 8 and 9 on finiteness conditions, and Euler characteristics.
Most of the concepts in chapter 9 (such ...
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Reconstructing a nilpotent Lie algebra from cohomology
Let $\mathfrak{g}$ be a nilpotent Lie algebra over a field $k$ of characteristic 0. I heard that it is possible to construct a presentation of $\mathfrak{g}$ using $H^1(\mathfrak{g},k)$ and $H^2(\...
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$2$-cocycles with values in non-abelian group
Let $G$ be a non-abelian group which acts trivially on an abelian group $A$. The abelian group $Z^{2}(G,A)$ denotes the set of
all $2$-cocycles $c$ on $G$ with values in $A$.
Now, let $c\in Z^{2}(G,A)$...
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Motivation for long exact sequence of group cohomology
Whenever people with weak abstract algebra backgrounds ask me to explain group cohomolgy, I typically say something along the lines of: "Taking G-invariants is a very natural process. The failure ...
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Action on group cohomology
Let $A$ be an abelian group with an action of $S_3$. Then one has the spectral sequence $H^i (\mathbb{Z}/2, H^j (\mathbb{Z}/3, A))$ converging to $H^{i+j} (S_3, A)$. I want to understand the action of ...
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Bott Tu, Chapter 14, Filtered complexes
I'm reading Differential Forms in Algebraic Topology from Bott,Tu and i confused in section of spectral sequence of filtered complexes. In this section book consider ungraded complexes They're just ...
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Recovering original definition of group cohomology from Ext definition
I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a ...
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Explicit cochain for Shapiro's lemma lift with trivial coefficients
I'm wondering the following: if we have a finite-index subgroup $H\subset G$, and a cocycle $[c]\in H^1(H,\mathbb{Z})$, is there any way to get an explicit cochain representing its image under the ...
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About relative Group Homology long exact sequence
In this article https://arxiv.org/abs/0905.0071, Jan Essert defines Relative Group Homology $H_*(G,G';M)$ of the groups $G'\leq G$ with coefficients in a $G$-module $M$ as the homology of quotient
$$
\...
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Map induced by inclusion in $5$-term exact sequence?
I am reading through "Cohomology of Groups" by Brown and have a question regarding corollary 6.4 on page 171, which goes as follows:
(6.4) Corollary: For any group extension $1 \to H \to G \...
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What is the second Chevalley-Eilenberg cohomology group of the Lie algebra of $U(1)$?
I am trying to understand a comment on one of my questions in Physics Stackexchange.
The comment itself is this:
$U(1)$ has the second Chevalley-Eilenberg cohomology group trivial, hence by Bargmann'...
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Understanding Hopf's Formula for continuous cohomology of profinite groups.
General idea
I am trying to understand a bijection between two 'flavours' of the second continuous cohomology group $H^2(\widehat Q; M)$ of the profinite completion $\widehat Q$ of a finitely ...
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