Small Heegaard genus and SU(2)

Baldwin, John A; Sivek, Steven

doi:10.2140/agt.2025.25.2369

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Small Heegaard genus and $\mathrm{SU}(2)$

John A Baldwin and Steven Sivek

Algebraic & Geometric Topology 25 (2025) 2369–2390

DOI: 10.2140/agt.2025.25.2369

Abstract

Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_{1} (Y; ℤ)$ has order $1$ , $3$ , or $5$ , then there is a representation $π_{1} (Y) \to SU (2)$ with nonabelian image. Similarly, if $H_{1} (Y; ℤ)$ has order $2$ then we find a nonabelian representation $π_{1} (Y) \to SO (3)$ . We also prove that a knot $K$ in $S^{3}$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $π_{1} (S^{3} ∖ K) \to SU (2)$ sending a fixed meridian to $diag (i, - i)$ .

Keywords

3-manifolds, character varieties, SU(2)-simple knots, instanton Floer homology

Mathematical Subject Classification

Primary: 57R58

Secondary: 57K31, 57M12

References

Publication

Received: 20 September 2023

Revised: 29 February 2024

Accepted: 23 May 2024

Published: 11 August 2025

Authors

John A Baldwin
Department of Mathematics Boston College Chestnut Hill, MA United States
https://sites.google.com/bc.edu/john-baldwin/
Steven Sivek
Department of Mathematics Imperial College London London United Kingdom
https://www.ma.imperial.ac.uk/~ssivek/

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Volume 25, issue 4 (2025)