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 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. Or, the set of loops, with their obvious composition, where elements in the form $ll^{-1}$ are considered equivalent to the identity under thin homotopy.

thinhomotopy group $\pi_1^1(X)$ (that's the group defined in this nlab page; in this group, thin homotopic loops are identified, making concatenation strictly associative). Relevant papers are here and here. $\endgroup$ – Konrad Waldorf Mar 17 '14 at 1:59