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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29 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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0answers
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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0answers
43 views
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 ...
2
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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 ...
2
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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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0answers
59 views
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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0answers
14 views
$H^2(G,A)$ 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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0answers
19 views
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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1answer
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 ...
5
votes
1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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1answer
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:
$$H_n(K(\pi,1))=\mathrm{Tor}^{\mathbf{Z}\pi}_n(\mathbf{Z},\mathbf{Z}),\quad H^n(K(\pi,1))...
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47 views
group cohomology equivalent to topological singular cohomology
Let $G=<\sigma>$ be a cyclic group of order $n$. For any $\mathbb{Z}[G]$ module $M$ it is known that the group cohomology
$$ H^i(G, M) = \begin{cases} M^G &\text{ if } i = 0 \\ M^G/NM &...
2
votes
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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0answers
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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votes
0answers
69 views
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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vote
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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votes
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 ...
3
votes
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.
4
votes
0answers
69 views
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) \...
1
vote
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 \...
0
votes
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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votes
2answers
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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0answers
91 views
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)$ ...
4
votes
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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0answers
12 views
(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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2answers
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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votes
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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0answers
45 views
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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0answers
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$-...
5
votes
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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0answers
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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vote
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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votes
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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0answers
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 $$h_k:\Omega^k(\mathbb{R}^n)\to \Omega^{k-1}(\mathbb{R}^n)$$
...
5
votes
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:
$$ \left( \frac{a}{p} \right)\...
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0answers
33 views
Does $H^{\bullet}(G, \mathbb{Z})$ have a coalgebra structure?
Here are two well-known facts:
Let $X$ be a topological space. We always have the diagonal map $\Delta :X\to X\times X$ and this induces a map $H^{\bullet}(X)\otimes H^{\bullet}(X) \simeq H^{\bullet}(...
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votes
1answer
53 views
Cohomology in groups
I'm trying to find the Rham cohomology of the groups $SU(2)$ and $U(2)$. I know that $SU(2)$ is isomorphic to $S^3$ but I don't know what is $U(2)$ isomorphic to. My question is: if $SU(2) \simeq S^3$ ...
2
votes
1answer
82 views
Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence
For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
2
votes
1answer
66 views
Homology of an equivariant product
I am struggling with checking the following fact:
$$
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 ...
