Question

Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation group acting on $A$ has nontrivial stable subgroup. I want to see that the only flat connection in $R$ is the product connection.

Answer 1

I think the statement is that in a principal $SU(2)$-bundle $P$ over a connected manifold $M$ with $H_1(M;\mathbb{Z})=0$, every reducible flat connection has trivial holonomy and therefore trivializes $P$.

For a Lie group $G$, gauge transformations $u$ that stabilize a $G$-connection $A$ are covariant-constant, and therefore determined by their value $u(x)$ at a point $x\in M$. Since $u$ commutes with the parallel transport defined by $A$, it centralizes the holonomy group $H_A\subset G$ (the image of the holonomy representation $\rho\colon \pi_1(M,x)\to G$). This sets up an isomorphism between the stabilizer of $A$ and $C_G(H_A))/Z(G)$, where $C_G(H_A)$ is the centralizer of $H_A$ in $G$ and $Z(G)$ the center ($\pm I$ for $G=SU(2)$). See Donaldson-Kronheimer's book.

The centralizer of a non-central element in $SU(2)$ is a circle $\mathbb{T}$ (to see this, observe that any element is conjugate to a diagonal element, or else consider rotations in $SO(3)$). The centralizer of any non-abelian subgroup of $SU(2)$ lies in the intersection of two different circle-subgroups, and is therefore just the center.

When $H_1(M)=0$, every non-trivial homomorphism $\pi_1(M)\to SU(2)$ has non-abelian image, since the abelian ones factor through $H_1$. For any flat connection $A$ with non-trivial holonomy, the centralizer therefore equals the center, which makes $A$ irreducible.

Answer 2

The following is probably well known. For example, one can find it in Fukaya's notes on Floer homology, $A_\infty$-categories and topological field theory.

Let $P$ be the trivial principal $SU(2)$-bundle on $M$ and let $\mathcal{A}(M)$ (resp. $\mathcal{A}^{\textrm{flat}}(M)$) denote the space of connections (resp. flat connections) on $P$. For $a\in\mathcal{A}(M)$ let $\mathcal{G}^a$ denote the stabilizer of $a$ under the action of the gauge group $\mathcal{G}(M)$. Then $\mathcal{G}^a=\pm 1, U(1)$ or $SU(2)$.

One can elaborate on a proof of this later. May be from this one can deduce that there are no flat connections with stabilizers $U(1)$ and the ones that have full stabilizers are the ones that you're looking for. On the other hand, following what Dan had said before, one has the following :

$\mathcal{A}^{\textrm{flat}}(M)/\mathcal{G}(M)\cong \textrm{Hom}(\pi_1(M),SU(2))/SU(2)$.

Here the right hand quotient is to be interpreted as all homomorphisms up to conjugation. Again, this may lead to some proof but it's been a while since I have seen these things. I'll put up something once I think it through a bit.