In their celebrated proof of the Property P Conjecture and its sequel, Kronheimer and Mrowka proved that the fundamental group of r-surgery on a nontrivial knot in the 3-sphere admits an irreducible SU(2)-representation whenever r is at most 2 in absolute value (which implies in particular that surgery on a nontrivial knot is never a homotopy 3-sphere). They asked whether the same is true for other small values of r -- in particular, for r = 3 and 4 -- noting that it's false for r = 5 since 5-surgery on the right-handed trefoil is a lens space. I'll describe recent work which answers their question in the affirmative. Our proof involves Floer homology and also the dynamics of surface homeomorphisms. All of this work is joint with Steven Sivek, and significant parts are also joint with Zhenkun Li and Fan Ye.