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.

Contact surgery numbers

“…Lower bounds on surgery numbers can be given for example by the rank of the first homology or the fundamental group or induced from the linking pairing [Au97]. More advanced obstructions (that can yield lower bounds on homology spheres) can be obtained from gauge theory [Au97], from Heegaard-Floer homology [HKL16,HL18] or from the SU (2) character variety of the fundamental group [SZ19]. In general, the known lower bounds do not coincide with the number of components in explicit surgery diagrams and therefore it is in general hard to compute surgery numbers.…”