Surgery obstructions and character varieties

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 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 S U ( 2 ) SU(2) character variety of the fundamental group, which for these manifolds is particularly simple: they are all S U ( 2 ) SU(2) -cyclic, meaning that every S U ( 2 ) SU(2) representation has cyclic image. Our analysis relies essentially on Gordon-Luecke’s classification of half-integral toroidal surgeries on hyperbolic knots, and other classical 3-manifold topology.