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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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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146 views

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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59 views

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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Show that $\dim H^1(\Gamma(11)) = 3$

Let $\Gamma(11) \subset \text{SL}(\mathbb{Z})$ be the invertible $2 \times 2$ matrices with one of the entries $0 \pmod {11}$. A congruence group is defined by: $$ \Gamma(11) = \left\{ \left( \begin{...
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27 views

Dimension of homology of groups

I am troubles when understanding why the following holds: Let $G$ be a group and $p\colon G \mapsto H$ a surjective homomorphism. Then, if $H_{1}(G, \mathbb{Q})$ is finite dimensional, so is $H_{1}(H,\...
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Is $\Omega^p(M)$ always $H^*(G,-)$-acyclic?

Suppose $M$ is a manifold on which the discrete group $G$ acts freely and properly discontinuously. Is it then true that $\Omega^p(M)$ is an acyclic $G$-module, in the sense that $H^k(G,\Omega^p(M)) = ...
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A special case of group extensions, other then the semidrect product.

It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extension of $G$ by $A$ is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation $\bullet$ ...
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1answer
29 views

Why does $H_{n}(\operatorname{Hom}_{R}(M,I_{\ast}(N))=H_{n}(\operatorname{Hom}_{R}(P_{\ast}(M),N))$?

I'm trying to show that the two definitions for the $\operatorname{Ext}$ functor are the same whether obtained via injective or projective resolutions. I understand I need to show the following but ...
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25 views

Group homology in degrees zero and one using the standard resolution

I'm trying to show using the standard resolution that for any group $G$, $$ H_0(G) = \mathbb{Z} \text{ and } \quad H_1(G) = \frac{G}{[G,G]}. $$ (I already know this holds, as by definition it is $...
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Why is $ \sigma (g) = \phi (g) g $ for some $ \phi \in \mathrm{Hom} (G,Z) $?

Let $ G $ be a group. Let $ Z $ be the center of $ G $ which is not finite, such that $ G/Z $ is a finite group. If $ \sigma \in \mathrm{Aut} ( G ) $, then $ \sigma ( Z ) = Z $, hence, we obtain ...
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20 views

Automorphism induces morphism between group homologies

I am thinking about this problem. Given $G=C_{n}$ be a cyclic group of order $n$, we have the next automorphism $\alpha: C_{n}\rightarrow C_{n}$ given by $\alpha(t)=t^{m}$ with $m$ and $n$ coprimes. ...
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62 views

$H^2(A,\mathbb{Z}_{(p)})$

This is not familiar ground to me, so I apologise in advance for possible inaccuracies or if the question is trivial. Let $\mathbb{Z}_{(p)}$ be the localisation of the integers at some $(p)$, viewed ...
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1answer
34 views

Cohomology of classifying space

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces: 1) $ H^\ast (BSU(2), \mathbb{Z}) $ 2) $ H^\ast (BO(3), \...
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116 views

$H^1(SO(2),T(2))$, or finding all subgroups of the group of motions of the plane which are complementary to the translation subgroup

My question is motivated by this part of 'Basic Notions of Algebra' from Igor R. Shafarevich. He mentions the group of motions of the plane in the following context. Let denote $SE(2)$ the 2-...
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Cocompact lattice

We have a complex 1-form $Ω$.suppose class of real part $ [Ω_{R}] $and imagine part $ [Ω_{I}] $ are linearly independent in $ \in H^1(M,\mathbb{R}) $ and There is a closed form $Ω^{\prime}\in Ω^1(Μ,\...
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32 views

First group homology for trivial module

Let $G$ be a group and $A$ be a $G$-module. I use $H^i(G,A)$ for group cohomology, and $H_i(G,A)$ for group homology. It is well known that if $A$ is a trivial module, then $$ H^1(G,A) \cong \...
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24 views

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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280 views

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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36 views

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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57 views

Group cohomology of symmetric group

Let $p \neq 2$ be a prime, $\Sigma_p$ be the symmetric group on $p$ elements, and $Z_{(p)}$ be the integers localized at $p$. Let $G$ be a finite group with Sylow $p$-subgroup $\mathbb{Z}/p$ and $\...
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1answer
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Stuck in a step of the deduction of cocycle identity

In the proof of the cocycle identity, how does the red arrow valid? I'd thought about it for a long while, but can't see the reason. PS: The author denote the operation in $G$ as $+$ for convenience (...
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36 views

Understanding the proof of a criterion of equivalent group extensions

I stuck in a step of the proof of the equivalence of two group extensions for 30 mins. In the place of red arrow, how does the previous line deduce the next? PS: The author denote the operation in $G,...
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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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24 views

What does “Realizes the operator” really mean?

Let $K,~Q$ be two groups, $K$ is abelian and $K$ is also a $\Bbb Z[Q]$-module. Then a group extension of $K$ by $Q$: $0\to K\to G\to Q\to 1$ realizes the operator if the scalar multiplication $Ck=\ell(...
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37 views

$H^2(G;\mathbb Z) \cong H^1(G;\mathbb {C}^*)$.

$H^2(G;\mathbb Z) \cong H^1(G;\mathbb {C}^*)$. Where $G$ is a finite group and $G$ acts trivially on $\mathbb Z, \mathbb C^*$. I have really tried hard but still I couldn't solve it , any help will ...
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Inverse limit of projective profinite groups is projective

I'm trying to prove the following (H.W question): Let $ G $ be inverse limit of projective profinite groups. Prove that $G$ is projective group. Projective group means "cohomological dimension ...
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Help understanding a theorem about group homology $H_0$

I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in ...
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Rank of a free Abelian subgroup gives a lower bound for vcd

Let $G$ be any group. The cohomological dimension (cd) of $G$ is the smallest integer $n$ such that $\mathbb{Z}$ admits a projective resolution of length $n$ over the group ring $\mathbb{Z}G$. Serre ...
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1answer
62 views

group cohomology of abelianization

Is it true that, for a finite or compact group $G$, $$H^3(G,\mathbb{Z})=H^3(G/[G,G],\mathbb{Z})\times H^3([G,G],\mathbb{Z})~?$$ It is clearly true for abelian and perfect $G$. I have checked a few ...
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1answer
53 views

Cohomological dimension of a topological group with torsion

I'm interested in a proof (or counter-example) of the following: Let $G$ be a topological group. If $G$ contains torsion then $H^n(BG)\neq 0$ for infinitely many $n$. I know this is true for ...
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Higher cup-1 product of coboundaries is also a coboundary?

In the cohomology or the group cohomology theory, suppose $\mu_1$ and $\mu_2$ are coboundaries of arbitrary dimensions, $$ \mu_1=\delta \eta_1 $$ $$ \mu_2=\delta \eta_2 $$ where $\eta_1$ and $\eta_2$ ...
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Cohomology generalized Quaternions

Good day to everyone I have a doubt about where I can find about the cohomology of the generalized Quaternions. I managed to find something on the book "Homological Algebra" by Henri Cartan and ...
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72 views

Generalisation of Hilbert's 90 Theorem

Let $L/K$ be a finite Galois extension of fields with Galois group $G = Gal(L/K)$. According to the famous Hilbert's 90 we know that the first cohomology vanish: $$H^1(G, L^*)=\{1\}$$ My question is ...
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79 views

A consequence of Schanuel's lemma

In Carlson's Cohomology and representation theory, the author states Schanuel's lemma, and then derives a consequence that I cannot understand. They define, for a $kG$ module $M$, $\Omega (M)$ to be ...
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23 views

group cohomology with coefficients in a complex

I am reading Brown's "cohomology of groups" when he introduces the group homology and cohomology with coefficients in a chain complex $C_*$. (pp 168) . The homology is defined as $H_*(G, C_*) = H(...
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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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91 views

How to compute (co)homology group of the Eilenberg-Maclane space $K(\pi,1)$

It is well-known that the homology group and cohomology group of the Eilenberg-Maclane space $K(\pi,1)$ are: $$H_n(K(\pi,1))=\mathrm{Tor}^{\mathbf{Z}\pi}_n(\mathbf{Z},\mathbf{Z}),\quad H^n(K(\pi,1))...
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group cohomology equivalent to topological singular cohomology

Let $G=<\sigma>$ be a cyclic group of order $n$. For any $\mathbb{Z}[G]$ module $M$ it is known that the group cohomology $$ H^i(G, M) = \begin{cases} M^G &\text{ if } i = 0 \\ M^G/NM &...
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1answer
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A spectral sequence with only one index in Atiyah's paper?

I would like to read Atiyah's paper Characters and cohomology of finite groups; but when I started reading the introduction, Atiyah mentions that he will prove that there is a "spectral sequence $\{E^...
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164 views

Why are principal crossed homomorphisms coboundaries?

According to Wikipedia (and to many other sources): The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) $f : G \to M$ satisfying $f(ab)=f(a)+...
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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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1answer
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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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1answer
85 views

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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1answer
45 views

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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38 views

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.
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Intuition behind the Connecting Morphism

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) \...
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1answer
24 views

Misunderstanding in definition of homology of groups

I am following Brown's 'Cohomology of groups' and the homology is defined as follows: Let $\cdots \rightarrow F_{n}\rightarrow F_{n-1}\rightarrow \cdots \rightarrow F_{1}\rightarrow F_{0}\rightarrow \...