Thin Position through the lens of trisections of 4-manifolds

TL;DR

This paper introduces a new concept of thin position for 4-manifolds based on handle decompositions, explores its properties, and relates it to trisection diagrams and branched covers, extending ideas from 3-manifold topology.

Contribution

It defines the width and thin position for 4-manifolds, characterizes manifolds with minimal width, and connects these concepts to trisection diagrams and branched covers.

Findings

Characterized all manifolds with width equal to 1.

Established relations between the width of a manifold and its double.

Constructed specific trisection diagrams for sphere bundles over surfaces.

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 $\{1,\dots, 1\}$, and give a relation between the width of $M$ and its double $M\cup_{id_\partial} \overline M$. 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.