Questions tagged [nonabelian-cohomology]
The nonabelian-cohomology tag has no usage guidance.
19 questions
1
vote
0
answers
42
views
When is $B^G\backslash(B/A)^G$ finite?
Let $G$ be a locally compact group, let $A,B$ be (not necessarily abelian) connected reductive complex groups equipped with continuous actions of $G$ via algebraic automorphisms. Let $\phi:A\to B$ be ...
- 387
2
votes
1
answer
262
views
Connecting homomorphism in non-abelian cohomology
Let $G$ be a simply connected, semisimple algebraic group over $\mathbb{R}$ and let $X$ be a homogeneous space for $G$ with finite commutative stabilizer $\mu$. There is a connecting homomorphism from ...
- 177
0
votes
0
answers
120
views
$G\cdot H$ with $G,H$ non-Abelian finite simple
Can a non-split extension of one non-Abelian finite simple group by another exist?
- 2,773
3
votes
0
answers
308
views
Non-abelian group cohomology, additional information
Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module.
We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...
- 463
4
votes
1
answer
301
views
What extra structure does the group of automorphisms of a torsor carry?
Let $Y$ be a space, and $G$ a group. For simplicity we can take $G$ to be finite, $Y$ to be a point, if this avoids technical issues. Then for $X/Y$ a $G$ torsor, so a sheaf of sets with $G$ action on ...
- 1,949
6
votes
1
answer
300
views
Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology
First, consider group extensions with non-abelian kernel
$$1\to K\to G \to Q \to 1$$
It is well-known that these are classified by certain cohomological objects, specifically: Any such extension ...
- 9,650
6
votes
1
answer
450
views
Long exact sequence of cohomology from 2-groups
I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.
So following the reference there to Nikolaus-Waldorf tells us that given any (...
13
votes
2
answers
2k
views
Non-Abelian Hodge theory
Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli ...
- 1,061
6
votes
1
answer
343
views
Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?
Given a topological space $X$, and a cover $\mathcal{U} :=\cup_{\alpha \in I}U_{\alpha}$ of $X$, one can define a groupoid called Čech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\...
- 1,215
7
votes
2
answers
997
views
Non-abelian Ext functor and non-abelian $H^2$
Let $G$ be a group and
$$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$$
a short exact sequence of groups. Now these are abelian groups, if I want to show that $\text{Hom}(G,M)\rightarrow \...
- 4,418
12
votes
0
answers
272
views
sequences in non-abelian group cohomology
In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
- 273
2
votes
1
answer
439
views
Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?
Let $G$ be a connected,reductive group over a $p$-adic field $k$. Let $M_0$ be a minimal Levi subgroup of $G$, and define $M_0^{\operatorname{der}}$ to be $M_0 \cap G_{\operatorname{der}}$. Lemma 2....
- 6,180
18
votes
2
answers
592
views
primary decomposition for nonabelian cohomology of finite groups
Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).
If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...
- 35.9k
5
votes
0
answers
254
views
How can one concretely approximate a homotopy type by a scheme or (higher/derived) stack?
What methods are there to approximate an arbitrary homotopy type by an algebraic-geometric object in a concrete (read: computable) way?
I know this is an ambitious question, so maybe I should ...
- 1,635
10
votes
0
answers
347
views
The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals
I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme $...
- 359
3
votes
0
answers
65
views
Connection on 3-bundle given as triplet of forms
A connection on a bundle is given locally by a Lie algebra-valued 1-form.
Gauge transformations act in the usual way on the forms, and form a groupoid.
A connection on a 2-bundle is given locally by ...
- 2,129
8
votes
3
answers
2k
views
Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?
Is there a (non-Abelian) homology theory that realizes the following:
Let $X,Y$ be manifolds with complexes $C(X),C(Y)$.
Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...
- 2,129
7
votes
0
answers
303
views
Albrecht Fröhlich's text `Groupoids, groupoid spaces and cohomology' (1965)
I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). In this text Fröhlich defines cohomology of a group with coefficients in a groupoid (this was ...
- 14.2k
19
votes
3
answers
2k
views
Second nonabelian group cohomology: cocycles vs. gerbes
In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title.
In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in ...
- 14.2k