Moduli space of flat connection over homology 3-sphere I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over 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?

*

*The trivial connection is the only reducible connection.


*$R$ is isolated generically (according to the perturbation of Chern-Simons functional)


*$R$ is compact.
 A: Let $M$ be your homology $3$-sphere. First, (as suggested by @MoisheKohan on MSE), note that your space is the quotient of $\mathrm{Hom}(\pi_1(M), SU(2))$ by the conjugation action of $SU(2)$. Indeed, every flat connection gives a holonomy representation, defined up to conjugation, which is invariant by Gauge transformation, from every representation $\rho$ you can build the flat $SU(2)$-bundle
$$\tilde M \times SU(2)/(x,g)\sim(\gamma \cdot x, \rho(\gamma)g)~.$$
Let me now answer your questions, by order of generality.
Question 3: Yes. This is simply because $\pi_1(M)$ is finitely generated, and $\mathrm{Hom}(\pi_1(M), SU(2))$ embeds in $SU(2)^S$ where $S$ is a generating set.
Question 1: Yes. Indeed, the holonomy of a reducible connection takes values (up to conjugation) in $U(1)\times U(1)$ which is abelian. It is thus trivial, since $H_1(X) = \{0\}$.
Question 2: I don't understand what you mean by "generically" but I think the answer is No.
The trivial representation is isolated because its algebraic tangent space in the character variety is $H^1(\pi_1(X),\mathfrak{su}(2)) = \{0\}$ since $X$ is a homology sphere.
But other representations might not be: take $X_1$ and $X_2$ two homology spheres admitting non-trivial flat $SU(2)$-bundles. You have irreducible representations $\rho_i: \pi_1(X_i) \to SU(2)$. Consider now the homology sphere $X= X_1 \sharp X_2$ (connected sum). We have $\pi_1(X) = \pi_1(X_1) \star \pi_1(X_2)$. For every $g\in SU(2)$, there is a representation
$$\rho_g: \pi_1(X) \to SU(2)$$
such that ${\rho_g}_{\vert \pi_1(X_1)} = \rho_1$ and
${\rho_g}_{\vert \pi_1(X_2)} = g\rho_2 g^{-1}$.
These form a pairwise non-conjugate $SU(2)$-family of representations of $\pi_1(X)$.
I wonder if $SU(2)$ representations of irreducible homology spheres are rigid, in which case the above construction would completely describe deformations of flat bundles on homology spheres. The only explicit examples I can think of are Brieskorn spheres, the $SU(2)$-representations of which are rigid (I think).
