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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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
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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“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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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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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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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 $...
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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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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
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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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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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
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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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. ...
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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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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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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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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
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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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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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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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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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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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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 ...
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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 $N$. ...
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$\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ 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 $\mathbb{R}/\mathbb{Z}=U(1)$...
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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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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{...