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.

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Mathematical Subject Classification 2010

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