Surgery obstructions and character varieties

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 previously known examples, our proofs require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety group, for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every representation has cyclic image.