The holonomy provides a bijection from
- the space of flat $G$-connections (modulo gauge equivalence) on a trivial $G$-bundle over $M$
to
- a connected component of the representation variety $Hom(\pi_1M,G)/G$.
Is this a homeomorphism for the $C^\infty$-topology on the space of connections?
If not, what can be said? Does it preserve path components?