The prism manifold realization problem

Ballinger, William; Hsu, Chloe; Mackey, Wyatt; Ni, Yi; Ochse, Tynan; Vafaee, Faramarz

doi:10.2140/agt.2020.20.757

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The prism manifold realization problem

William Ballinger, Chloe Ching-Yun Hsu, Wyatt Mackey, Yi Ni, Tynan Ochse and Faramarz Vafaee

Algebraic & Geometric Topology 20 (2020) 757–816

DOI: 10.2140/agt.2020.20.757

Abstract

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in $S^{3}$ . In recent years, the realization problem for C–, T–, O– and I–type spherical manifolds has been solved, leaving the D–type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as $P (p, q)$ for a pair of relatively prime integers $p > 1$ and $q$ . We determine a list of prism manifolds $P (p, q)$ that can possibly be realized by positive integral surgeries on knots in $S^{3}$ when $q < 0$ . Based on the forthcoming work of Berge and Kang, we are confident that this list is complete. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.

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Keywords

prism manifold, Dehn surgery, changemaker

Mathematical Subject Classification 2010

Primary: 57M25, 57R65

References

Publication

Received: 17 May 2018

Revised: 6 June 2019

Accepted: 24 June 2019

Published: 23 April 2020

Authors

William Ballinger
Department of Mathematics Princeton University Princeton, NJ United States

Chloe Ching-Yun Hsu
Department of Mathematics California Institute of Technology Pasadena, CA United States

Wyatt Mackey
Department of Mathematics Stanford University Stanford, CA United States

Yi Ni
Department of Mathematics California Institute of Technology Pasadena, CA United States

Tynan Ochse
Department of Mathematics University of Texas, Austin Austin, TX United States

Faramarz Vafaee
Department of Mathematics Duke University Durham, NC United States

Volume 20, issue 2 (2020)