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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.

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  • $\begingroup$ Could you please clarify what you mean by the loop group $L$ of a general manifold? $\endgroup$ – Mark Grant Mar 16 '14 at 18:32
  • $\begingroup$ @ThomasRot That is the spirit, but not the same. In particular, is there a version of the Chern-Weil homomorphism that takes value in non-Abelian cohomology groups? $\endgroup$ – geodude Mar 16 '14 at 18:33
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    $\begingroup$ While your general point of view is valid, it doesn't really offer anything new. Groups and spaces are essentially equivalent objects (modulo some discrepancies), because for any group we can take its delooping (classifying space BG) and it induces an equivalence between groups and connected pointed spaces. Also, while it is possible to choose a group structure on a (homotopy equivalent to the) space of loops (see Kan loop group), it is a very complex and intractable object. $\endgroup$ – Anton Fetisov Mar 16 '14 at 20:49
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    $\begingroup$ Note that the central extension you're talking about is with Gl (V), not V. This fact is the same as "V-bundles over X equal $ H^1 (X, BGl (V) ) $", which should be familiar. $\endgroup$ – Anton Fetisov Mar 16 '14 at 20:50
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    $\begingroup$ One thing we have of course is the classification of flat principal $G$-bundles by group homomorphisms $\pi_1(X) \to G$. For bundles with non-flat connections, and $G$ abelian, there is an analogous description with $\pi_1(X)$ replaced by the thin homotopy 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
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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.

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