#### Vol. 313, No. 2, 2021

 Recent Issues Vol. 320: 1 Vol. 319: 1  2 Vol. 318: 1  2 Vol. 317: 1  2 Vol. 316: 1  2 Vol. 315: 1  2 Vol. 314: 1  2 Vol. 313: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
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 $\left\{1,\dots ,1\right\}$, 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 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