Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds:

$$BG \simeq \underset{m \to \infty}{\lim} SO(m+n)/SO(m) \times SO(n)$$

$$EG \simeq \underset{m \to \infty}{\lim} SO(m+n)/SO(m)$$

with the evident $G$-bundle projection $EG \to BG$. One may give an explicit map $i: G \to \Omega BG$ which is a weak homotopy equivalence, by comparison of long exact homotopy sequences. Since $G$ and $\Omega BG$ have the homotopy types of CW complexes, the map $i$ is in fact a homotopy equivalence.

I am interested in whether an explicit homotopy inverse $h: \Omega BG \to G$ to $i$ can be given by taking holonomy of loops with respect to some suitably chosen connection on the universal bundle. Naturally the "manifold" $BG$ is infinite-dimensional, but I'm thinking it would suffice to work with the finite-dimensional $G$-bundles $V_m \to G_m$ (Stiefel to Grassmann) which approximate to the direct limits above, provided that the connections on each of the approximating bundles are compatible with respect to inclusion into the next, "compatible" here having an obvious sense in terms of $G$-valued holonomy.

Has anyone seen this idea worked out? Naturally I'm curious also about whether a similar idea works out for a general compact Lie group $G$.

`ancient' - see my birth certificate! though it's more subtle than for smooth bundles with connection see my now updated version of`

parallel transport revisited' at the n-lab so what is it you'ld like to do with it $\endgroup$ – Jim Stasheff Jan 24 '11 at 1:44