Small Dehn surgery and SU(2)

Baldwin, John A¹ ; Li, Zhenkun² ; Sivek, Steven³ ; Ye, Fan⁴

¹ Department of Mathematics, Boston College, Chestnut Hill, MA, United States
² Department of Mathematics, Stanford University, Stanford, CA, United States, School of Mathematics and Statistics, University of South Florida, Tampa, FL, United States
³ Department of Mathematics, Imperial College London, London, United Kingdom
⁴ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom, Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 28 (2024) no. 4, pp. 1891-1922.

Résumé

We prove that the fundamental group of $3$–surgery on a nontrivial knot in $S^3$ always admits an irreducible $\mathrm{SU}(2)$–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 $[3,5)$.

Détail

DOI : 10.2140/gt.2024.28.1891

Keywords: instanton, $\mathrm{SU}(2)$, Floer homology, representations, knots, fundamental group

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