  We prove that the fundamental group of 3-surgery on a nontrivial knot in the
3-sphere always admits an irreducible SU(2)-representation. This answers a
question of Kronheimer and Mrowka dating from their work on the Property P
conjecture. An important ingredient in our proof is a relationship between
instanton Floer homology and the symplectic Floer homology of genus-2 surface
diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to
extend our main result to infinitely many surgery slopes in the interval [3,5).
