Thin Position through the lens of trisections of 4-manifolds

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 non-orientable surfaces of genus g, respectively. By last, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective.