Handlebody argument for modifying achiral Lefschetz singularities

TL;DR

This paper provides a handlebody proof demonstrating how to modify achiral Lefschetz singularities into broken Lefschetz fibrations, establishing their existence on all closed smooth oriented 4-manifolds.

Contribution

It introduces a handlebody argument for converting achiral Lefschetz singularities into broken Lefschetz fibrations, extending previous results to all closed smooth oriented 4-manifolds.

Findings

Handlebody proof of existence of broken Lefschetz fibrations

Method for modifying achiral Lefschetz singularities

Extension of Gay and Kirby's work to all 4-manifolds

Abstract

This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topology 13 (2009), 312-317; the references are updated herein.