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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?

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    $\begingroup$ The holonomy does not provide such a bijection. It only provides a bijection between the space of flat connections on the trivial bundle and a connected component of of the representation variety. Often the representation variety will have more components. If $M$ is a compact manifold and $G$ is a complex reductive group, then this bijection is a homeomorphism. In fact if $M$ is a complex projective algebraic variety, the bijection is a complex analytic isomorphism. $\endgroup$ Sep 27, 2013 at 12:26
  • $\begingroup$ I've edited the question. $\endgroup$
    – ThiKu
    Jan 25, 2014 at 4:31
  • $\begingroup$ Does one really need the bundle to be trivial? Doesn't it work in general? $\endgroup$ Jan 25, 2014 at 16:43

1 Answer 1

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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 for a nice exposition on this topic with references for further reading.

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