We provide infinitely many rational homology 3-spheres with weight-
one fundamental groups which do not arise from Dehn surgery on knots in S3. 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. Our analysis relies essentially on Gordon-Luecke’s classification of half-integral toroidal surgeries on hyperbolic knots, and other classical 3-manifold topology.