Trisections of Lefschetz pencils

Abstract

Donaldson [J. Differential Geom. 53 (1999) 205–236] showed that every closed symplectic $4$–manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby [Geom. Topol. 20 (2016) 3097–3132] showed that every closed $4$–manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a $4$–dimensional analog of the following trivial $3$–dimensional result: for every open book decomposition of a $3$–manifold $M$, there is a decomposition of $M$ into three handlebodies, each of which is a regular neighborhood of a page.