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

$H^2(G;\mathbb Z) \cong H^1(G;\mathbb {C}^*)$. 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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15 views

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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1answer
25 views

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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21 views

group homology of vector space. [closed]

Let $V$ be a $\mathbf{Q}$-vector space. what is $H_{\ast}^{group}(V,\mathbf{Z})$ the integral group homology of $V$ when seen as an abelian group?
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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
56 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
49 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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$H^2(G,A)$ is in bijection with the class of extensions of $A$ by $G$ - does this depend on the action of $G$ on $A$?

Let $G$ be a profinite group, and $A$ an Abelian group. Given an extension of $A$ by $G$, $0 \longrightarrow A \longrightarrow E \longrightarrow G \longrightarrow 1$, it is known that we can make $A$...
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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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60 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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78 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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16 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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38 views

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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31 views

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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32 views

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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66 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
60 views

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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127 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
18 views

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
77 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
43 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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35 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
23 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 \...
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1answer
38 views

cohomological dimension of groups vs cohomological dimension of subgroups

Let $\Gamma$ be a group and $\Gamma^\prime$ a subgroup. Then, $\text{cd }\Gamma^\prime \leq \text{cd } \Gamma$ because a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}\Gamma$ can also be ...
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44 views

Group cohomology of product with swapping (twisting) factors

Let $M$ be a $G$-module, where $G = \Bbb Z / 2 \Bbb Z$. Define a $G$-module structure on $A = M \oplus M$ by $g \cdot (a,b) = (g \cdot b, g \cdot a)$. What is the group cohomology $H^*(G, A)$ is terms ...
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Group cohomology topologically with simplicial sets

I have a question about the usual formula for the differential in the usual projective resolution of $\mathbb{Z}$ as a $G$-module for a finite group $G$ Recall that for a $G$-module $A$, $C^i(G,A)$ ...
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3answers
95 views

Group cohomology of the natural action of automorphism group on a finitely generated abelian group

It's well known that we can classify finitely generated abelian groups. Let $M$ be a finitely generated abelian group, in principle we can decide the group structure of $G=Aut(M)$ from $M$. What about ...
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(symmetric) generators for cohomology group of a del pezzo surface of degree 6

I'm working on the surface $X$ which is birational to the blowing-up of $\mathbb{P}^1\times \mathbb{P}^1$ at two points. When I consider its cohomology group $H^2(X,\mathbb{Z})$, I can use a basis as $...
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1answer
84 views

How does group cohomology behaves where coefficient is direct sum?

Let $G$ be a finite group and $A_1$ and $A_2$ be $\mathbb ZG$ modules. Is it always true that for any $n\in \mathbb N$, the cohomology group of $G$ with coefficients in $A_1\oplus A_2$ is the same as ...
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217 views

Cohomology ring of $H^*(A_5;\mathbb Z_2)$?

What is the cohomology ring of $H^*(A_5;\mathbb Z_2)$? Here $A_5$ is the alternating group on $5$ letters. I am comfortable with the Lyndon-Hochschild-Serre spectral sequence, and understand how to ...
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1answer
42 views

Showing that any central group extension of a finite group $G$ by the trivial $G$-module $\mathbb Q$ is a semidirect product

Suppose that $G$ is a finite group and view the additive group $\mathbb{Q}$ as a trivial $G$-module. I want a concrete way of understanding why the second cohomology group $H^2(G, \mathbb{Q})$ is ...
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1answer
24 views

Why does $\underline{cd}\:G\leq silp\:G\leq\underline{cd}+1$ hold?

I read the about text in a book but don't understand how or why this inequality is "straightforward" can anybody explain this to me?
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Proof of the main theorem on non-abelian Kummer extensions (following Lang)

I am trying to understand the proof of Theorem 11.1, Chapter VI from Lang’s Algebra and the conditions of Corollary 11.2. I have two specific questions: In the proof of 11.1, Lang says that the ...
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1answer
78 views

Computing cohomology of dihedral group in detail

So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, \Bbb Z)$. using Lyndon SS. I have obtained a few obstacles: My computation, using the fact that there is a $C_2$ action ...
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23 views

“Restriction” map in group homology, what was meant? Rotman

Def 1: $\alpha:G \rightarrow G'$, group homomorphism. If $A'$ is a $G'$ module $f:A \rightarrow A'$ is a $\Bbb Z$ map we call $(\alpha, f)$ a compatible pair if $f:A \rightarrow _{\alpha}A'$ is a $G$-...
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1answer
63 views

Computing Pontryagin Square

Suppose $v$ is a $\mathbb{Z}_2$ cochain on a four dimensional spin manifold $M$, i.e. $v\in H^1(M, \mathbb{Z}_2)$. I am interested in evaluating the quantity $$\exp \bigg(i \frac{\pi}{2}\int_M \...
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44 views

Axioms for the generalized cohomology

Here I would like to understand in the homotopy axiom what is the induced homomorphism on (co)homology? What it means and how is the induced homomorphism defined?
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1answer
39 views

Order of $Z^1(G,A)$, the group of $1$-cocycles, when both $G$ and $A$ are finite

I was reading about first cohomology group $H^1(G,A)$ for a group $G$ acting on an abelian group $A$. As one can see from the definition, $H^1(G,A)=Z^1(G,A)/B^1(G,A)$ where $Z^1(G,A)$ is the group of $...
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1answer
34 views

Extension that split for all p-Sylow

In my Group Cohomology class, the professor stated the following theorem If one takes an extension $1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1 $ with $A$ abelian and finite and $G$ ...
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67 views

De Rham cohomology groups of $\mathbb{R}^n$

I want to show that for each $1 \le k\le n$ we have $$ H_{dR}^k(\mathbb{R}^n)=0 $$ The strategy is to construct for each $k$ a linear map $$h_k:\Omega^k(\mathbb{R}^n)\to \Omega^{k-1}(\mathbb{R}^n)$$ ...
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1answer
116 views

Algebraic 1-cocycles and Galois gerbs

We have the following set up: $K/F$ is Galois, $D$ is an algebraic group of mult. type and $E$ is an extension of groups: $$1\to D(K)\to E\to Gal(K/F)\to 1$$ Now take a linear algebraic group G over F....
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1answer
43 views

Writting Legendre Symbol as an element of group cohomology of $\mathbb{Q}$

Is it possible to write the Legendre symbol as an element of the cohomology of some kind? We certainly have that it is multiplicative in both numerator and denominator: $$ \left( \frac{a}{p} \right)\...
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Does $H^{\bullet}(G, \mathbb{Z})$ have a coalgebra structure?

Here are two well-known facts: Let $X$ be a topological space. We always have the diagonal map $\Delta :X\to X\times X$ and this induces a map $H^{\bullet}(X)\otimes H^{\bullet}(X) \simeq H^{\bullet}(...
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1answer
53 views

Cohomology in groups

I'm trying to find the Rham cohomology of the groups $SU(2)$ and $U(2)$. I know that $SU(2)$ is isomorphic to $S^3$ but I don't know what is $U(2)$ isomorphic to. My question is: if $SU(2) \simeq S^3$ ...
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1answer
82 views

Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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1answer
66 views

Homology of an equivariant product

I am struggling with checking the following fact: $$ H_*(E\Sigma_n\times_{\Sigma_n}X^{\times n})\cong H_*(\Sigma_n;C_*(X)^{\otimes n}). $$ But I am not sure how to start. It seems correct and I ...