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 the loop group $L$ of $M$ based at a point $p$.


The holonomy of a connection is a morphism of groups $\omega:L\to Aut(V)$, where $V$ is the fiber at $p$, that gives the parallel transport of a vector around the loop, and defines on $V$ an $L$-module structure. 


Therefore we can consider an extension:
$$
0 \rightarrow V \hookrightarrow G \rightarrow L \rightarrow 1.
$$


If the sequence is split, $G=V\rtimes L$ is made of elements in the form $(v, x)$ which compose in the following way:
$$
(v,x)(w,y) = (v+xw, xy),
$$


which can be seen as composing the loops, and summing the vectors after $w$ has been transported around the loop $x$.


I would think that the sequence is a direct product if and only if the bundle is trivial, because the action of $L$ on $V$ is trivial.
Going this way, I'm also tempted to think that a connection is linked with splittings of this exact sequence.


Can anybody enlighten me?


([I posted this before on Math.SE](https://math.stackexchange.com/questions/703999/can-group-cohomology-be-used-to-study-fiber-bundles) before, but I got no answers.)


EDIT: Since group cohomology (of low degrees) can be used to study central extensions, I wondered if, in turn, it could be used to study vector bundles.
For example, do $H^1(L,V)$ and $H^2(L,V)$ enter the picture in some way that could be linked to $H^\*(M,\mathbb{Z})$, and so to characteristic classes?


EDIT: By "loop group", I mean a vertex group of the [path groupoid](http://ncatlab.org/nlab/show/path+groupoid). Or, the set of loops, with their obvious [composition](http://ncatlab.org/nlab/show/path#concatenation), where elements in the form $ll^{-1}$ are considered equivalent to the identity under [thin homotopy](http://ncatlab.org/nlab/show/thin+homotopy).



Can group cohomology be used to study fiber bundles?
I presume that you are here referring to non-Abelian cohomology of a group, $G$. There is also the non-Abelian cohomology of spaces which relates directly to the classification of principal bundles. That requires the use of local (sometimes called twisted) coefficients. The link between them is via the classifying space $Ner \, G[1]$. Probably the best (but not the most rapid) way to procede is to look at some source on non-Abelian group cohomology (perhaps Serre's cohomologie galoisienne is a good place to start), and to fiddle around with the cocycles, etc. for a short while! Now look at the classification of principal bundles again keep it fairly classical for a while, and then try to figure out the link! 


The way forward from that point depends on your background. I like the simplicial approach and so Larry Breen's Notes on 1- and 2-gerbes,<http://arxiv.org/abs/math/0611317> are one good source. (I have a longer treatment of some of this in my Crossed Menagerie notes (see the n-Lab entry on that), but I think that may deviate from what you want.) Breen's papers on Schreier theory are also useful but less easy to read. Also his paper with Messing is important.


The main point is that the non-Abelian cohomology of spaces leads to gerbes and so does that of groups. The methods overlap and a comparison will perhaps give you what you want. As to you first question in the edit, try looking at bundle gerbes in the nLab. The section of the n-lab on non-Abelian cohomology is also very useful on this.


