Trisections of Lefschetz Pencils

TL;DR

This paper establishes a direct construction of trisections of 4-manifolds from topological Lefschetz pencils, linking two fundamental structures in 4-dimensional topology and providing a new perspective on their relationship.

Contribution

It introduces a method to derive a trisection of a 4-manifold directly from a topological Lefschetz pencil, bridging two key theorems in 4-manifold topology.

Findings

Constructs a trisection from a Lefschetz pencil with each sector as a neighborhood of a fiber

Shows the relation between Lefschetz pencils and trisections in 4-manifolds

Provides a 4D analog of a classical 3D decomposition result

Abstract

Donaldson showed that every closed symplectic 4-manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby 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.