Volume 21, issue 6 (2021)

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Trisections of $3$–manifold bundles over $S^1$

Dale Koenig

Algebraic & Geometric Topology 21 (2021) 2677–2702
Abstract

Let X be a bundle over S1 with fiber a 3–manifold M and with monodromy φ. Gay and Kirby showed that if φ fixes a genus g Heegaard splitting of M then X has a genus 6g + 1 trisection. Genus 3g + 1 trisections have been found in certain special cases, such as the case where φ is trivial, and it is known that trisections of genus lower than 3g + 1 cannot exist in general. We generalize these results to prove that there exists a trisection of genus 3g + 1 whenever φ fixes a genus g Heegaard surface of M. This means that φ can be nontrivial, and can preserve or switch the two handlebodies of the Heegaard splitting. We additionally describe an algorithm to draw a diagram for such a trisection given a Heegaard diagram for M and a description of φ.

Keywords
trisection, 4–manifold
Mathematical Subject Classification 2010
Primary: 57M50, 57R45, 57R65
References
Publication
Received: 5 December 2017
Revised: 3 May 2020
Accepted: 18 October 2020
Published: 22 November 2021
Authors
Dale Koenig
Rapyuta Robotics
Tokyo
Japan