Surgery obstructions and character varieties

Sivek, Steven; Zentner, Raphael

doi:10.1090/tran/8596

Surgery obstructions and character varieties
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by Steven Sivek and Raphael Zentner PDF

Trans. Amer. Math. Soc. 375 (2022), 3351-3380 Request permission

Abstract:

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. 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.

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