Trisections of 3–manifold bundles over S1

Koenig, Dale

doi:10.2140/agt.2021.21.2677

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

Dale Koenig

Algebraic & Geometric Topology 21 (2021) 2677–2702

DOI: 10.2140/agt.2021.21.2677

Abstract

Let $X$ be a bundle over $S^{1}$ 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 $6 g + 1$ trisection. Genus $3 g + 1$ 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 $3 g + 1$ cannot exist in general. We generalize these results to prove that there exists a trisection of genus $3 g + 1$ 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

Volume 21, issue 6 (2021)