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?



Representation variety vs. space of flat connections
The answer to your question is yes, the bijection between the Betti moduli space and the de Rham moduli space is a homeomorphism.


See [here](http://arxiv.org/pdf/1404.5025v1.pdf) for a nice exposition on this topic with references for further reading.


