1
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0answers
25 views

Kunneth formula for group homology

I'm trying to prove Kunneth formula for group homology. $$ 0 \to \bigoplus_p H_p(G,M)\otimes H_{n-p}(G',M') \to H_n(G\times G',M \times M') \to \bigoplus_p Tor_1^{\mathbb Z}(H_p(G,M),H_{n-p-1}(G',M')) ...
1
vote
1answer
128 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
0
votes
0answers
24 views

free resolutions of $\mathbb Z$ in Mod(G)

Lang Algebra, XX.2,3 I'm asked to show that $E_\bullet \cong F_\bullet$ are isomorphic free resolutions of $\mathbb Z$, in Mod(G), where $E_\bullet$ is the standard complex: $E_i$ is the free ...
1
vote
0answers
75 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
1
vote
2answers
57 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
1
vote
1answer
62 views

Cohomology of finite p-groups

Given a finite abelian $p$-group $A$ acted on by a finite $p$-group $G$. Under the assumption $\operatorname{H}^1(G,A_1)=0$, where $A_1$ is the set of elements of $A$ having order at most $p$, what ...
7
votes
1answer
129 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
0
votes
0answers
58 views

Exact Sequences and Cohomology.

I am currently working through exercises in "Cohomology of Groups" by Kenneth S. Brown. In particular I am struggeling with exercise 4 (d) on page 90. My question is concerning the last step in that ...
5
votes
1answer
244 views

The Zig Zag Lemma in Cohomology

I´m reading the Zig Zag lemma in Cohomology and i want to prove the exactness of cohomology sequence at $ H^k(A)$ and $H^k(B)$ : A short exact sequence of cochain complexes $ 0 \to A \ ...
0
votes
0answers
77 views

Extension of divisible abelian group

Let $A$, $G$ be abelian groups, $R \to F \to A$ be the standard free resolution of $A$, $\phi:R \to G$ be a morphism and $N:=\text{coker}(-\phi \times i:R \to G \otimes F)$. From MacLane - Homology ...
1
vote
1answer
49 views

Group extension reference request

I'm looking for a reference for the following "well known" result Let $C$ be an abelian group and $G$ a finite group, and let $$0 \rightarrow C \rightarrow W \rightarrow G \rightarrow 0$$ be a ...
1
vote
1answer
59 views

''Commutative'' 2-cocycles

Let ba $G$ an abelian group and $L$ is $G$-module. If $f$ is a 2-cocycle in $Z(G,L)$, is it true that $f(g,h)=f(h,g)$ for all $g,h \in G$? Or even for $\bar{f} \in H^{2}(G,L)$ is ...
6
votes
5answers
198 views

(Elementary) applications of group (co-)homology

I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology. My background is, ...
3
votes
1answer
126 views

Group extension: $H^n(\mathbb{Z},M)$

Let $M$ be any $\mathbb{Z}$-module. Find $H^n(\mathbb{Z}, M)$ for all $n$. Use different approaches for the case $n=2$. First approach: because $\mathbb{Z}$ is a free $\mathbb{Z}$-module, it is ...
6
votes
1answer
178 views

$\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$

I'm trying to read (part of) "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)). Here are some basic assertions I need ...
1
vote
1answer
139 views

Relation between group zero-cohomology and the dual of group zero-homology

Let $\Gamma$ be a group and $A$ be an abelian group and let's take group zero-homology and zero-cohomology, $H_0(\Gamma,A)$, $H^0(\Gamma,A)$. Is there any relation between $H^0(\Gamma,A)$ and ...
2
votes
0answers
163 views

Dual sequence and its exactness

I am reading Lang's Algebra and trying to fill in the gaps in my mathematical background while I train for the quals. So I came across the following exercise (chapter 20, ex. 26 in the third edition). ...
1
vote
1answer
83 views

Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions…

Let X be complex curve (complex manifold and $\dim X=1$). For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions vanish at the points $x_1$ and ...
1
vote
1answer
71 views

Explicit 3-Cocycles of $Z_2\times Z_2$ over $U(1)$

I know that $H^3(Z_2\times Z_2, U(1))=Z_2^3$, I'd like to know all the cocycles explicitly. Is there a systematical way to find the cocycles (I guess one can always try to solve the cocycle conditions ...
1
vote
2answers
181 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
2
votes
1answer
97 views

Other differentials for group cohomology other than the standard one.

In group cohomology, one defines $H^i(G;A)$ for $G$ a group and $A$ a $G$-module (an abelian group with a $G$-action) as the $i$-th right derived functor of the functor $$(-)^G: G-mod \rightarrow Ab, ...
1
vote
1answer
111 views

equivalence of definition of the first cohomology group

I've found different definitions of the same cohomology group and I would like to prove that they are equivalent. For $G$ a group and $A$ a $G$-module, Weibel defines in "An introduction to ...
9
votes
2answers
591 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
3
votes
1answer
127 views

Homology/Cohomology of Free Product

I recently completed an exercise showing that $$ H_1(G*H,A) \cong H_1(G,A) \oplus H_1(H,A) $$ for $A$ a trivial $G*H$-module, and also proved a similar statement for cohomology. This is exercise ...
4
votes
1answer
128 views

Cohomology of the trivial action of $\mathbb{Z}_p$ on $\mathbb{Z}$

I'm wondering if the next exercise in 'An introduction to homological algebra' by Weibel is correct: Let $G$ be the profinite group $\widehat{\mathbb{Z}}_p$. Show that $$H^i(G;\mathbb{Z}) = ...
4
votes
1answer
52 views

$C^n(G,A)$ with $G$ a profinite group and $A$ a discrete $G$-module as direct limit

Let $C^n(G,A)$ be the set of continuous functions $G^n \rightarrow A$ with $G$ a profinite group and $A$ a discrete $G$-module (these are the functions that are locally constant). I want to prove that ...
2
votes
2answers
147 views

Explicitly finding a cocycle in $H^3(S_3,\mathbb{Z}_3)$

I know that $H^3(S_3,\mathbb{Z}_3)\cong \mathbb{Z}_3$ (S_3 is the symmetric group for three elements). So this group is generated by any nontrivial cocycle. But I don't know how to explicitly find ...
4
votes
0answers
249 views

How to understand the diagonal approximation?

In the Brown's book “Cohomology of groups”, chapter 5.1, there is a concept diagonal approximation, maybe that is not a standard definition, I feel something hard to understand it. The book says that ...
3
votes
1answer
350 views

how can we compute the homology of these groups without using topology?

I'd like to know the homology of a free group and a free abelian group of rank 2. I know that they could be computed topologically, but I'm searching a proof purely algebraic, could you help me ...
4
votes
0answers
299 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
6
votes
1answer
627 views

Direct sum commuting with homology functor

I'm trying to understand a fact about commutation between homology functors and direct sums. In particular, let $G$ be a group of type $FP$ (i.e. there exists a projective resolution of finite length ...