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).

597 questions
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Is this a finite extension?

Let $G$ be a group, $H$ a $2$-index subgroup of $G$ and $N$ a normal subgroup of $G$. I want to understand why if $H_{1}(N; \mathbb{Q})$ is infinite dimensional, so is $H_{1}(N\cap H; \mathbb{Q})$. ...
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Cocycles and group extensions

I'm trying to understand how elements in the second cohomology group with coefficients in some other group correspond to group extensions. This is what I understand: Suppose we have two (countable) ...
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Why does cocycle condition imply a group?

In group cohomology, the 2-cocycle condition emerges from associativity (see e.g. here). From the answer to a previous question we see (at least for normalized cocycles) that the cocycle condition ...
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A condition for normalized 2-cocycles from the existence of the inverse element in a group extension

It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extensions of $G$ by $A$ is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation $\bullet$...
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Group homology of action on left cosets

If $G$ is a group and $H$ a subgroup which is not normal. What is the homologies of the action of $G$ on the left coset space $G/H$? The action is multiplication from left. More precisely, $G/H$ is ...
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Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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Reference for Universal Coefficient Theorem

I am looking for a proof of the following fact: let $C$ be a chain complex of real vector spaces, $C^*$ the dual cochain complex. Then $H^n(C^*) \cong H_n(C)^*$. That is, taking homology commutes with ...
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Is my proof that $H^1(X,F) = 0$ for the skyscraper sheaf correct?

Let $X$ be compact connected. Let $F$ be the skyscraper sheaf over $p$ for a group $G$. Let $H^1(X,F)$ be the first Cech cohomology of the sheaf. We want to show $\ker(d_1) = im(d_0)$ is trivial. It ...
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Cohomology of colimit is limit of cohomology ? (group cohomology)

In Homotopy theoretic methods in group cohomology, Henn's part, section 1.2, the example following definition 1 has the following sentence "the cohomology $H^*(G,\mathbb{F}_p)$ of a group $G$, which ...
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Why is the Bockstein morphism a derivation?

I'm trying to understand the Bockstein morphism in cohomology, and one of the points is that $\delta : H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$ is a derivation that squares to $0$. I could ...
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Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
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Definitions of Group Cohomology

I am trying to understand group cohomology, and I have a very basic question. So as I understand it, let $\Gamma$ be a group, and $V$ be a $\Gamma$-module (which is essentially another abelian group ...
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Chain complexes question between free $K$-modules and almost zero chain.

I have this sentence from the article Resolutions for extensions of groups by C.T.C. Wall: Let $Z(K)$ denote the group ring of the group $K$ over the ring $Z$ of the integers. Let $\otimes_K$ ...
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Definition of the $0$-coboundary in group cohomology

I'm trying to learn, what is group cohomology. Since I'm not a matematician, the general definition is too abstract to me (at least for the time being), and requires too much category theory and ...
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Cohomology of free groups topologically

I'm trying to see an example of the topological interpretation of group cohomology, with the free group $F(S)$ on a set $S$ of generators, with coefficients in $\mathbb{Z}$ (on which we act trivially),...
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Borel subgroup of $SL_2(\mathbb{Z})$

As the title indicates, I want to ask what is the Borel subgroup of $SL_2(\mathbb{Z})$? I believe I read about it in one of James Milne's notes. But now I cannot find it. That is why I want to ask.
I have seen that a s.e.s of $G$-modules $$0\longrightarrow A \longrightarrow B\longrightarrow C \longrightarrow 0$$ gives rise to the following long exact sequence:  0 \rightarrow H^0(G,A) \...