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 $\{1,\dots ,1\}$, and give a relation between the width of $M$ and its double $M{\cup }_{i{d}_{\partial }}\overline{M}$. 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.

Version 3 has a new definition of width which considers 3-handles. Comments are welcome. 27 pages

References in corpus (5)

• Bridge trisections of knotted surfaces in $S^4$
• Classification of trisections and the Generalized Property R Conjecture
• Bridge trisections of knotted surfaces in 4--manifolds
• Simplifying indefinite fibrations on 4-manifolds
• Characterizing Dehn surgeries on links via trisections