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](http://en.wikipedia.org/wiki/Holonomy#Holonomy_of_a_connection_in_a_principal_bundle) 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 $G$. 



Can a homotopy inverse of the map from a Lie group to loops on its classifying space be given by holonomy? 
Yes, certainly. The model for BG as you described it has a canonical, universal connection
for its $SO(n)$ bundle: just the induced Riemannian connection from Euclidean space.


As you
move an $n$-dimensional plane in $\mathbb E^{n+m}$, the induced connection
is the limit of compositions
 of orthogonal projections between nearby planes. In the limit, these become isometries.
It doesn't matter what is the dimension of the ambient space, as long as the projection is
defined. So, a loop in the Grassmanian gives an element of $G$. One example: if you
hold a bicycle wheel by its axle, and move it around in a loop, when it comes back, it has
rotated by some angle. That's the map.


It's a homotopy inverse of the map going the other way, namely, the classifying map for
the bundle obtained by the suspension of an element of $G$. The suspension of a homomorphism
has a canonical connection; its holonomy is $G$, so one composition of these two maps
equals the identity. The other composition is homotopic to the identity, basically
because the space of connections is contractible, and the bundle $EG$ over this model
of $BG$ has a universal connection: Every connection on every $SO(n)$ bundle over 
a $CW$ complex $X$ is induced from a map $X \rightarrow BG$ (and this is also true
in a relative form). This is a standard fact;
 I don't have a handy reference, but the proof is "soft". 


However, to get a more immediate classifying space for connections that works
for all Lie groups, just make
a simplicial complex whose simplices are connections for a $G-bundles$ over the simplex.
The $G$-bundle is specified by in terms of a trivialization associated with each vertex;
the data needed is the chart-transition cocyle. In addition, give a connection;
this amounts to specifying a connection form. Glue these simplices with $G$-bundles
and connections together, to make a model for $BG$. Since the space of connections
is contractible, this has the same homotopy type as the more usual model for $BG$ where just
the cocycle is specified.


The same construction works to give an explicit homotopy inverse
in a much more general context, e.g. the group of diffeomorphisms for a manifold.


