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Small Dehn surgery and SU(2)

John A. Baldwin, Zhenkun Li, Steven Sivek, Fan Ye

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).

Small Dehn surgery and SU(2)

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).

Paper Structure

This paper contains 10 sections, 24 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Other small surgeries
  4. Organization
  5. Acknowledgements
  6. Genus two L-space knots
  7. $SU(2)$-abelian surgeries and branched double covers
  8. Branched double covers and Floer simple knots
  9. Floer simple knots and Theorem \ref{['thm:main']}
  10. Other small surgeries and $SU(2)$

Key Result

Theorem 1.2

$S^3_3(K)$ is not $SU(2)$-abelian for any nontrivial knot $K\subset S^3$.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 40 more