Vol. 313, No. 2, 2021

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Thin position through the lens of trisections of 4-manifolds

Román Aranda

Vol. 313 (2021), No. 2, 293–326
DOI: 10.2140/pjm.2021.313.293
Abstract

Motivated by M. Scharlemann and A. Thompson’s definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to {1,,1}, and give a relation between the width of M and its double M id M¯. In particular, we describe how to obtain genus 2g + 2 and g + 2 trisection diagrams for sphere bundles over orientable and nonorientable surfaces of genus g, respectively. Finally, we study the problem of describing relative handlebodies as cyclic covers of a 4-space branched along knotted surfaces from the width perspective.

Keywords
Heegaard splittings, handle decompositions, thin position, trisections of 4-manifolds, tunnel number of links
Mathematical Subject Classification 2010
Primary: 57M27, 57N13
Milestones
Received: 21 February 2020
Revised: 10 June 2021
Accepted: 16 June 2021
Published: 12 October 2021
Authors
Román Aranda
Department of Mathematical Sciences
Binghamton University
Binghamton, NY
United States