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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38 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)+...
3
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0answers
63 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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1answer
15 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
70 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
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1answer
37 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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31 views
Simple interesting applications of group cohomology?
I am reading Neukirch's CFT because I'm interested in CFT.
Regardless, the first 60 pages are about group cohomology, and it's hard finding motivation to tread through them, there are so many ...
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32 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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67 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) \...
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1answer
21 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
34 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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2answers
42 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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85 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)$ ...
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3answers
90 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
11 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
65 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
200 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
40 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
23 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
44 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 ...
4
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1answer
73 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
21 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
55 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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1answer
38 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 $...
2
votes
1answer
33 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
65 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
112 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
42 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:
$$ \left( \frac{a}{p} \right)\...
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28 views
Does $H^{\bullet}(G, \mathbb{Z})$ have a coalgebra structure?
Here are two well-known facts:
Let $X$ be a topological space. We always have the diagonal map $\Delta :X\to X\times X$ and this induces a map $H^{\bullet}(X)\otimes H^{\bullet}(X) \simeq H^{\bullet}(...
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1answer
52 views
Cohomology in groups
I'm trying to find the Rham cohomology of the groups $SU(2)$ and $U(2)$. I know that $SU(2)$ is isomorphic to $S^3$ but I don't know what is $U(2)$ isomorphic to. My question is: if $SU(2) \simeq S^3$ ...
2
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1answer
69 views
Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence
For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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1answer
59 views
Homology of an equivariant product
I am struggling with checking the following fact:
$$
H_*(E\Sigma_n\times_{\Sigma_n}X^{\times n})\cong H_*(\Sigma_n;C_*(X)^{\otimes n}).
$$
But I am not sure how to start. It seems correct and I ...
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0answers
71 views
Find isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent
I am asking for help on the following exercise:
Find two isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent.
as taken from D. Robinson, A Course in ...
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1answer
47 views
Isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent
The following exercise from D. Robinson, A Course in the Theory of Groups confuses me:
Find two isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent.
...
2
votes
1answer
84 views
What's the ring structure of $H^*(C_2,\mathbb{Z})$ with a nontrivial group action?
Right now, I am trying to understand better the cup product structure. I am interested in deriving the ring structure of the group cohomology $H^*(C_2,\mathbb{Z})$ with a nontrivial group action. The ...
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1answer
24 views
Dualizing module and finiteness hypothesis
Serre, in his Galois Cohomology, states:
Proposition 17. Let $n$ be an integer $\geq 0$. Assume:
(a) $\text{cd}(G) \leq n$
(b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite.
...
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0answers
52 views
Stiefel Whitney classes on the simplex or the simplicial complex
The Stiefel Whitney classes of the base manifold $M$ are characteristic class as
$$
w_j(M) \in H^j(M,\mathbb{Z}_2),
$$
Puzzle: How do we write
$$
w_1(M) \in H^1(M,\mathbb{Z}_2)
$$
$$
w_2(M) \...
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0answers
109 views
Neukirch ANT - proving reciprocity map is multiplicative
I am learning class field theory from the famous ANT book by Neukirch, where I stuck at the middle of a long proof, whose goal is to prove multiplicativity of reciprocity map.
Defining various ...
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0answers
22 views
Comparing the cohomology rings of two central extensions
Consider two groups $G$ and $G'$, where $G$ is the direct product of groups $A$ and $B$, with $B$ abelian, and $G'$ is a nontrivial central extension of $A$ by $B$. Suppose that as groups, $H^1(G,M) \...
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34 views
spectral sequence and a comparison theorem
I am wondering how the comparison theorem can be useful for resolving extension problem. Here is a quote from the book An Introduction to Homological Algebra by Weibel.
(Weibel) Comparison Theorem ...
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1answer
33 views
Could we define injective modules or projective modules for topological modules?
Topological Modules are defined here Wikipedia. My question is can we define notions like injective modules or projective modules? Can we define $Tor$ and $Ext$ functors? I have tried hard to find ...
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1answer
31 views
The extension of a connected, simply-connected Lie group by $\mathbb{Z}$ is trivial.
I am reading a paper by Calvin Moore on group extensions. On page 54, there is a statement saying that when a Lie group is connected and simply-connected, the the extension of a connected, simply-...
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1answer
131 views
Long exact sequence of cohomology group “without” Snake lemma
Let a short exact sequence
$$
0 \to L \to M \to N \to 0
$$
is a short exact sequence of $G$-modules, then a long exact sequence is induced:
$$
0\longrightarrow L^G \longrightarrow M^G \...
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0answers
19 views
Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.
It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
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0answers
16 views
If the first cohomology is zero, can higher cohomologies be non-zero?
Let $G$ be a group and $M$ a $G$-module such that $H^1(G,M)=0$. Does this follow that $H^i(G,M)=0$ for all $i>0$? I couldn't come up with an example so far. Thanks!
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21 views
Cup Products and Group Extensions
Let $G$ be a group and let $A$ be a $G$-module. Suppose we're given a short exact sequence:
$$1 \to A \to X \to G \to 1$$
It is well known that there exists a bijection between $H^2(G, A)$ and ...
1
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0answers
35 views
Group actions on spectral sequences of group cohomology
Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
2
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0answers
29 views
$\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ v.s. $H^4(K(\mathbb{Z}_2,2),U(1))$: Cocycle form
The $SO$ bordism group of Eilenberg–MacLane space $K(\mathbb{Z}_2,2)$ is
$\Omega_4^{SO}(K(\mathbb{Z}_2,2))=\mathbb{Z}_4$.
The cohomology group of $K(\mathbb{Z}_2,2)$ with $\mathbb{R}/\mathbb{Z}=U(1)$...
2
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0answers
48 views
Motivation for cochains, cocycles and coboundaries
I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem ...
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0answers
14 views
Virtual finite cohomology implies finite cohomology
Let $G$ be a torsion-free pro-$p$ group, and let $H$ be an open subgroup of $G$. Suppose both $H$ and $G$ have finite cohomological dimension. What I want to show is:
If all groups $H^i(H, \mathbb{...
