Small Dehn surgery and SU(2)

TL;DR

This paper proves that 3-surgeries on nontrivial knots in the 3-sphere always have irreducible SU(2) representations, resolving a question related to the Property P conjecture using Floer homology techniques.

Contribution

It establishes the existence of irreducible SU(2) representations for all nontrivial knot surgeries in S^3, extending to infinitely many slopes, and links instanton and symplectic Floer homologies.

Findings

Fundamental groups of 3-surgeries admit irreducible SU(2) representations.

Answer to a longstanding question of Kronheimer and Mrowka.

Extension of results to infinitely many surgery slopes in [3,5).

Abstract

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).