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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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


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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Additional Information
  • Steven Sivek
  • Affiliation: Department of Mathematics, Imperial College London, London, United Kingdom
  • MR Author ID: 781378
  • ORCID: 0000-0003-0230-8087
  • Email:
  • Raphael Zentner
  • Affiliation: Fakultät für Mathematik, Universität Regensburg, Germany
  • MR Author ID: 775700
  • Email:
  • 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:
  • MathSciNet review: 4402664