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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

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in S3 . 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 S3 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.

prism manifold, Dehn surgery, changemaker
Mathematical Subject Classification 2010
Primary: 57M25, 57R65
Received: 17 May 2018
Revised: 6 June 2019
Accepted: 24 June 2019
Published: 23 April 2020
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