SIMPLIFIED BROKEN LEFSCHETZ FIBRATIONS AND TRISECTIONS OF SMOOTH 4-MANIFOLDS (Intelligence of Low-dimensional Topology)

Abstract

We present a survey on explicit algorithms for simplifying the topology of indefinite fibrations on 4 manifolds, which include broken Lefschetz fibrations and indefinite generic maps, from the viewpoint of singularity theory. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations on all closed oriented 4-manifolds, and a theorem of Auroux-Donaldson-Katzarkov on the existence of simplified broken Lefschetz pencils on nearsymplectic 4-manifolds. We moreover show the existence of simplified trisections on all 4-manifolds and establish a correspondence between broken Lefschetz fibrations and trisections of 4-manifolds.