# Trisections of 4-manifolds via Lefschetz fibrations

## TL;DR

This paper introduces a new technique for constructing trisection diagrams of closed 4-manifolds, especially those with Lefschetz fibrations, enabling explicit visualizations and classifications of complex 4-manifolds.

## Contribution

It develops a gluing method for relative trisection diagrams and applies it to produce explicit diagrams for various 4-manifolds, including those with Lefschetz fibrations and certain complex surfaces.

## Key findings

- Constructed trisection diagrams for 4-manifolds with Lefschetz fibrations.
- Provided explicit diagrams for S^2-bundles over surfaces.
- Reproved that all closed 4-manifolds admit a trisection using contact geometry.

## Abstract

We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold which admits a Lefschetz fibration over $S^2$ equipped with a section of square $-1$, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed $4$-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented $S^2$-bundle over any closed surface and in particular we draw the corresponding diagrams for $T^2 \times S^2$ and $T^2 \tilde{\times} S^2$ using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed $4$-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.