The h-principle for broken Lefschetz fibrations

TL;DR

This paper proves that a set of modifications called homotopies can generate all broken Lefschetz fibrations within a homotopy class on smooth 4-manifolds, establishing a form of the h-principle.

Contribution

It demonstrates that these modifications are sufficient to produce any broken fibration in a given homotopy class, generalizing the construction of such fibrations.

Findings

Modifications can generate all broken fibrations within a homotopy class.

Adding a projection move suffices to produce all broken fibrations.

The results have implications for understanding the topology of 4-manifolds.

Abstract

It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken fibration in a given homotopy class of smooth maps. One notable application is that adding an additional "projection" move generates all broken fibrations, regardless of homotopy class. The paper ends with further applications and open problems.