Representation variety vs. space of flat connections

Question

The holonomy provides a bijection from

the space of flat $G$-connections (modulo gauge equivalence) on a trivial $G$-bundle over $M$

to

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?

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. — Tony Pantev, Commented Sep 27, 2013 at 12:26
Does one really need the bundle to be trivial? Doesn't it work in general? — Vít Tuček, Commented Jan 25, 2014 at 16:43

Sean Lawton · Accepted Answer · 2016-06-20 22:05:11Z

4

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.

answered Jun 20, 2016 at 22:05

Sean Lawton

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