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?