Surgery obstructions and character varieties

TL;DR

This paper constructs infinitely many rational homology 3-spheres with specific fundamental groups that cannot be obtained via Dehn surgery on knots, using character varieties instead of gauge theory or Floer homology.

Contribution

It introduces a new method leveraging $SU(2)$ character varieties to identify manifolds not arising from knot surgeries, avoiding gauge theory techniques.

Findings

Constructed infinitely many such 3-spheres

Demonstrated these manifolds are $SU(2)$-cyclic

Provided a new approach to study surgery obstructions

Abstract

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety of the fundamental group, which for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every $SU(2)$ representation has cyclic image.