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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Group cohomology reference

I'm interested in studying group cohomology (for discrete groups). Are there accessible (lecture) notes that give a nice overview of the basics for group cohomology and develop the categorical ...
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The torsion subgroup of the coinvariants for a $G$-module.

Let $G$ be a finite group and $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. Consider the functor $$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
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Proving that $S^1 \vee S^1 \to S^1$ induces a projection on homology groups

I am trying to compute the homology of the Klein bottle using the Eilenberg-Steenrod axioms and after a number of steps, I reached a part where I need to show that the map $f:S^1 \vee S^1\to S^1$, ...
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Group Cohomology as a Left Derived Functor

The $n$th group cohomology is the $n$th right derived functor of the left exact $M \mapsto M^G$ functor. Using the equivalence $\mathbb Z[G]$-$\mathrm{Mod} \cong G$-$\mathrm{Mod}$ we can show that $\...
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Compute the ext functor of an abelian torsion group.

Let $G$ be an abelian torsion group. I have to compute the group $$ \text{Ext}^{1}_{\mathbb{Z}}(G, \mathbb{Z}) $$ and to prove the isomorphism $$ \text{Ext}^{1}_{\mathbb{Z}}(G, \mathbb{Z}) \simeq \...
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Find the number of group extensions of $\mathbb{Z}_{3}$ by $\mathbb{Z}_{6}$.

We have that the number of group extensions of $\mathbb{Z}_{3}$ by $\mathbb{Z}_{6}$ is: $$\textrm{Ext}_{\mathbb{Z}}^{1}(\mathbb{Z}_{3},\mathbb{Z}_{6}) \simeq \mathbb{Z}_{d} $$ where $d = \gcd(3,6) = 3 ...
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Taking $π:C^{n+1} \setminus \{ 0 \}→CP^n$, does the continuous application $q:CP^n→C^{n+1} \setminus \{ 0 \}$ exist such that $π∘q = Id$?

I'm trying to show that the continuous application $q:CP^n→C^{n+1} \setminus \{ 0 \}$ exists such that $π∘q = Id$ in $CP^n$. To show they are homotopy equivalent, I have been trying to use the lifting ...
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Exact $G-$sequences of $\mathbb{Z}G$

We have a exact sequence of $G-$modules $$0\to I_G \to \mathbb{Z}G\to \mathbb{Z}\to 0$$ here $\varepsilon: \mathbb{Z}G \to \mathbb{Z}: \sigma\to 1 \;\forall \sigma \in G$ and $I_G=\ker\varepsilon$ or ...
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Proving Hochschild-Serre spectral sequence (of group (co)homology) via Grothendieck spectral sequence

Let $G$ be a group and $N$ be its normal subgroup. Let $A$ be a $G$-module, then we have the Hochschild-Serre spectral sequence $$ E_2^{p,q} = H^p(G/N, H^q(N,A)) \Rightarrow H^{p+q}(G,A). $$ and $$ E_{...
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A cyclic group extension "on both sides"

I have two extensions of finite groups, letting $C_m$ denote the cyclic group of order $m$: $$1 \to C_n\to G\to C_3\to 1 \tag{1} $$ $$1 \to C_3\to G\to H\to 1 \tag{2} $$ where $n$ is some composite ...
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Stuck at $Conjugacy$ part in Schur-Zassenhaus Theorem.

I am writing the Conjugacy part of Schur-Zasssenhaus theorem, using the Robinson book. in case of $G/N$ soluble, I have considered, if there is a normal subgroup $N_0$ such that $\{e\}\neq N_0 \leq N$ ...
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What's the group extension corresponding to the sum of 2 classes of $H^2(G,A)$

I'm learning Galois/Group Cohomology and i've just seen that the second cohomology group $H^2(G,A)$ (constructed as the factor systems quotient splitting factors) classifies the group extensions of $G$...
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Second group cohomology $H^2(\mathbb Z^n, \mathbb C^*)=?$

Regard $\mathbb Z^n$ as an abelian group, and let $\mathbb C^* = \mathbb C-\{0\}$. Question: What is the group cohomology $H^2(\mathbb Z^n, \mathbb C^*)$? More specific question is as follows. I have ...
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Group Cohomology and Pontryagin Duality

My question is related to this question, which I tried to post an answer. I think it is better to ask a question directly. My question is from the book "Foundations of Quantum Theory: from ...
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The tensor product over $R$ or $M$?

Here is the Kunneth formula I found for cohomology on the internet here https://topospaces.subwiki.org/wiki/Kunneth_formula_for_cohomology: But I am confused about if the tensor product over $R$ or $...
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Quotient of group cohomology by $G$-module automorphisms?

Let $G$ be a finite group and $M$ a finite-dimensional $G$-module over $\mathbb{F}_2 = \{0, 1\}$. Let $f\in Z^1(G, M)$ be a $1$-cocycle, and let $\phi:M\mapsto M$ be a $G$-module automorphism. Do we ...
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what is the homology groups of a contracting circle in a torus to a point?

I need to compute the homology groups of the space $X$ obtained by contracting a circle in a torus $T$ to a point. I think $H_0(X)= \mathbb{Z}$ since $X$ is a connected. For $H_1(X)$ I think we need ...
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What is the most straight forward way to compute $H^{2}(\mathbb{Z}_{n},\mathbb{Z}_{k})$?

I'm trying to wrap my head around group cosmology, and I wanted to inquire what would be the most straight forward way to compute $H^{2}(\mathbb{Z}_{n},\mathbb{Z}_{k})$? Would it be to use the fact ...
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2 votes
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Relationship between group homology and group cohomology when the group is finite

Let $G$ be a finite group. If you consider $H^{n+1}(G,\mathbb{Z})$ and $H_{n}(G,\mathbb{Z})$ (with trivial actions) you will get that these are isomorphic. One can do this by considering instead the ...
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On a map induced by multiplication on cohomology

Studying group cohomology I landed on a paper with a lot of induced maps. However, I stumbled on something which seems to be a mistake, to me, although the author seems very confident with what he ...
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Group Cohomology as a Derived Functor

Let $G$ be a (finite, say) group acting on module $M$. I've been trying to understand how the (standard?) construction of group cohomology $H^i(G,M)$ given by wikipedia relates to the general ...
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Profinite Cohomology of $\hat{\mathbb{Z}}$: Abstract proof using $\delta$-functors

Lets $G=\hat{\mathbb{Z}}$ be the profinite completion of the integers, let $T$ be the topological generator of $G$. I'm interested in proving $$H^i(G,M)=\begin{cases} M^G & i= 0\\ M/(T-1)M &i=...
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$0\to \text{Hom}(P_0,G)\to \text{Hom}(P_1,G)\to \text{Hom}(P_2,G)\to\dots$ is an exact sequence. [duplicate]

I am reading ''Algebraic number theory'' by Cassels and Fröhlich and in the chapter IV.4 it says the following: If $\dots\to P_2\to P_1\to P_0\to\mathbb{Z}\to 0$ is a projective resolution and $G$ is ...
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2 votes
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Why the torus has open subsets diffeomorphic to a cylinder?

If I take the torus $T$ of revolution in $\mathbb{R}^3$ is very intuitive for me that there is an open subset of $T$ that is diffeomorphic to a cylinder, for example $T$ minus a circle on the $xz$ ...
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Adjoint representation of a Lie group in homology of its lie algebra

It is known that a (Lie) group acts trivially (by identity) in its homology. It is also known that Lie algebra acts trivially (by zero endomorphisms) in its homology. Can we say that a Lie group acts ...
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Can we use the algebraic definition of group cohomology for $H(BG, A)$ for compact Lie groups $G$?

For any discrete group $G$, the classifying space $BG$ is a $K(G, 1)$ and can be constructed as a simplicial set with $|G|^i$ $i$-simplices. Accordingly, elements of $H^i(BG, A)$ can be represented by ...
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If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character, why is $H^1(G,A)=0$?

If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character and $M=A^1$, where $A$ is a ring, why is $H^1(G,M)=0$ if $G$ acts via $\phi$? This is essentially the claim that Wiles makes on p. 465 ...
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Definition of Galois Cohomology

For Galois cohomology, one uses the cohomology constructed for profinite groups instead of the usual group cohomology. In other words, one also uses/takes into account the Krull topology we have on ...
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Another question on 1-cocycles (explicite computation of 1-cocycles)

I want to calculate 1-cocycles, as they are described in Neukirch's Class field theory. In what follows the Galois group of $L/\mathbb{Q}$ is acting on $L^{\times}$. The first comment in this question,...
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A very simple problem on 1-cocycles

Edit: According to the comment of Mindlack this special question is solved, but yet I have no idea if my calculations lead to another number $r\neq 0$. And in the case $r\neq 0$, I don't know how ...
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Compatibility and consistency between two definitions of cohomology in two books (about coboundary operators and 1-cocycles and computing cohomology)

I was reading cohomology from Neukirch's book, and there he referenced to Hall's book. The two approaches are almost the same (are they not?), and they should give us the same results (cohomology ...
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5 votes
2 answers
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Proving Schur-Zassenhaus Theorem, with added assumption that $G/H$ is cyclic

Schur-Zassenhaus Theorem: If there exists normal Hall-subgroup $H$ of finite group $G$, then there exists complement $K$ of $H$ in $G$. So if $\exists$ H $\unlhd$ G s.t. |H| is coprime to [G:H] then ...
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Weil–Châtelet group of a real elliptic curve is isomorphic to $\Bbb Z/2\Bbb Z$ when $\Delta>0$

This is a question from Silverman's The arithmetic of elliptic curves, exercise 10.7. Prove $WC(E/ \Bbb R)$ is isomorphic to $\Bbb Z/2\Bbb Z$ when discriminant $Δ>0$ and $0$ when $Δ<0$. From ...
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Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

Let $B_{k}(S_{g}),$ $MCG(S_{g};k)$ and $MCG(S_{g}))$ are Braid group, Mapping class group (relative to $k$) and Mapping class group of orientable surface $S_{g}$, respectively. For $g\geq3,$ we have ...
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Finiteness of group cohomology

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$. Then if $M,N$ are $k[G]$-modules which are finitely generated as $k$-module, then $$\text{Ext}^i(M,N)=\...
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The question about the proof isomorphism between simplicial and singular homology group

I want mainly to ask the two question while reading Theorem 2.27 the Hatcher's textbook Theorem 2.27) The homomorphisms between $H^{\Delta}_n(X,A) \to H_n(X,A) $ are isomorphism for all n and all $\...
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3 votes
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Computing Group Extension and $2$-cocycle in GAP

It is a well known theorem that $2$-cocycle are in bijection with Group extensions. Suppose if I have group extension, say, for example, $$1\rightarrow C_2 \rightarrow D_{16}\rightarrow D_8\rightarrow ...
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4 votes
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Interpretation of vanishing of cohomology groups

Let $G$ be a group and $M$ a left $G-$module. It is well know that for some conditions all the cohomology groups $H^{i}(G,M)$, $i=0,1,2,.....$ vanish. The same can be do for the Hochschild cohomology ...
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2-cocycles in Lie group vs Lie algebra cohomology (context of projective reps)

I'm confused by the relationship between the cocycle condition in Lie algebras vs Lie groups. For Lie groups, a 2-cocycle is defined (e.g. here) as a map $\Phi : G \times G \rightarrow \mathbb{F}$ ...
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What Percentage of Type F Groups are not Poincare Duality Groups?

What "percentage" (in a meaningful sense of the term) of Type F groups are not Poincare duality groups (over any coefficient ring, in particular, over $\mathbb{F}_2$)? Of course, it's an ...
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On pairing and cup product in group cohomology

I'm reading the wiki page of local Tate duality. https://en.wikipedia.org/wiki/Local_Tate_duality#cite_note-2 Let $K$ be a non-archimedean local field, let $K^s$ denote a separable closure of $K$, and ...
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Properties of 3-cocycles

Let $G$ be a finite group, $H$ a subgroup of $G$, and $A$ an abelian group. Assume given a 3-cocycle $\omega:G\times G\times G\rightarrow A$ such that $\omega|_{H\times H\times H}$ is coboundary, say $...
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Question on Group cohomology of $C_p \times C_p$

Let $G= C_p \times C_p$, where the cyclic group in the first factor is generated by $a$ and the second factor by $b.$ Let $H_j$ be a subgroup of $G$ generated by $(a, b^j)$ where $0 \le j \le p-1.$ ...
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Free action of Group $G$ on $S^n$ gives free resolution of $\mathbb{Z}$ as a $\mathbb{Z}[G]$-module.

Suppose I have a group ${ G }$ acting freely on the sphere ${ S^n }$ for ${ n \geq 2 }$. Does this somehow get me a free resolution of ${ \mathbb{Z} }$ as a ${ \mathbb{Z}[G] }$-module? Forgive me if ...
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What is the action on $H^1$ in Galois cohomology?

This is a question that I haven't been able to find answers for in the books I've looked at, so I wanted to ask it here to get some clarification. Given an abelian group $A$ and a Galois group $G$ ...
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4 votes
1 answer
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Bijection between isomorphism class of torsors and first cohomology set

Let $G$ be a profinite group and let $A$ be a $G$-group. A torsor over $A$ is a non-empty $G$-set $P$ endowed with a simply transitive right action $\star$ of $A$ which is compatible with the left ...
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2 votes
1 answer
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The order of $G$ is invertible in $\mathbb{Q}$, and so we have that $H^n(G,\mathbb{Q})=0$ for $n≥1$, why is so?

Group Cohomology: Question $2$ I am learning group cohomology. In Wikipedia, I need to understand the section $\text{Higher cohomology groups are torsion}$. The discussion says the cohomology groups $...
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What is the second cohomology group $H^2(\mathbb{R},U(1))$?

What is the second cohomology group $H^{2}(\mathbb{R},U(1))$? I am not necessarily looking for a derivation, but a reference where I can found this result which I assume it is known.
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2 votes
1 answer
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Notation in Group cohomology and cochains

Group Cohomology: Question $1$ I am learning group cohomology. In the Wikipedia, I couldn't understand few terminologies. For example, in the section $$\text{The functors $\text{Ext}^n$ and formal ...
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1 vote
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number of Schur covering groups

Let $G$ be a finite group with derived subgroup $G'$ and Schur multiplier $M(G)$. A Schur cover of $G$ is a group $H$ with a subgroup $Z\le H'\cap Z(H)$ such that $H/Z\cong G$ and $Z\cong M(G)$. Let $...
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