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6
votes
4answers
387 views
What do higher cohomologies mean concretely (in various cohomology theories)?
Superficially I think I understand the definitions of several cohomologies:
(1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't ...
14
votes
1answer
385 views
Group cohomology versus deRham cohomology with twisted coefficients
Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
5
votes
1answer
224 views
Torsion-free virtually-Z is Z
It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan...
Is there a simple cohomological proof of the fact that a torsion-free group ...
6
votes
1answer
158 views
$H^1$ of $\Bbb Z$ as a trivial $G$-module is the abelianization of $G$ [duplicate]
Let $G$ be a group and $\mathbb{Z}$ regarded as a trivial $G$-module. As title, I'm trying to prove that $H^1(G,\mathbb{Z})$ is isomorphic to $G/[G,G]$.
It is easy to see that $H^1$ is isomorphic to ...
45
votes
0answers
844 views
+50
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
8
votes
1answer
225 views
Finite groups with periodic cohomology
I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
6
votes
2answers
90 views
Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology
First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has ...
