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

Surgery obstructions and character varieties