Handlebody argument for modifying achiral Lefschetz singularities

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.