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?