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I'm trying to understand the space of flat connections over the trivial $SU(2)$ -bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it).

From now on I will just define the space of gauge equivalent classes of flat connections on it to be $R$.

Are the following facts correct and why?

  1. The trivial connection is the only reducible connection.

  2. $R$ is isolated generically (according to the perturbation of Chern-Simons functional)。 That is to say in the moduli space of gauge equivalent classes of connections, elements of $R$ are isolated. Generic here means specifically if we choose enough perturbation of Chern-Simons functional, its critical points will be considered as deformed flat connections, denoted as $R_h$ w.r.t. the perturbation $h$. $R_h$ is isolated.

  3. $R$ is compact.

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  • $\begingroup$ What do you mean by (2)? As written, it is totally unclear. What are your thoughts about the question and how much do you know about moduli spaces of flat $SU(2)$-connections and their relation to spaces of representations of the fundamental group to $SU(2)$? $\endgroup$
    – Moishe Kohan
    Commented Apr 14, 2022 at 18:13
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    $\begingroup$ Cross-posted on MO here. $\endgroup$
    – Moishe Kohan
    Commented Apr 14, 2022 at 18:20
  • $\begingroup$ @MoisheKohan Thanks for the comment! I've editted the question. I know loosely that a flat connection gives you a holonomy map, but I don't see how the quotient be group action comes from. $\endgroup$
    – user388493
    Commented Apr 14, 2022 at 20:16

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