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.References
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Additional Information
- Steven Sivek
- Affiliation: Department of Mathematics, Imperial College London, London, United Kingdom
- MR Author ID: 781378
- ORCID: 0000-0003-0230-8087
- Email: s.sivek@imperial.ac.uk
- Raphael Zentner
- Affiliation: Fakultät für Mathematik, Universität Regensburg, Germany
- MR Author ID: 775700
- Email: raphael.zentner@mathematik.uni-regensburg.de
- Received by editor(s): May 14, 2021
- Received by editor(s) in revised form: October 11, 2021, and October 12, 2021
- Published electronically: February 24, 2022
- Additional Notes: The second author was supported by the SFB “Higher invariants” (funded by the Deutsche Forschungsgemeinschaft (DFG)) at the University of Regensburg, and by a Heisenberg fellowship of the DFG
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 3351-3380
- MSC (2020): Primary 57K10, 57K31, 57K32, 57R65; Secondary 57K30
- DOI: https://doi.org/10.1090/tran/8596
- MathSciNet review: 4402664