A note on sections of broken Lefschetz fibrations

TL;DR

This paper demonstrates that simplified broken Lefschetz fibrations can have infinitely many homotopy classes of sections and sections with non-negative square, contrasting with properties of classical Lefschetz fibrations, and provides conditions for spin structures.

Contribution

It shows that properties of Lefschetz fibrations do not extend to broken Lefschetz fibrations and characterizes when their total space admits a spin structure.

Findings

Existence of simplified broken Lefschetz fibrations with infinitely many homotopy classes of sections.

Existence of simplified broken Lefschetz fibrations with sections of non-negative square.

Necessary and sufficient condition for the total space to admit a spin structure.

Abstract

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above conditions. Smith proved that every Lefschetz fibration has only finitely many homotopy classes of sections, and Smith and Stipsicz independently proved that a Lefschetz fibration is trivial if it has a section with non-negative square. So our results indicate that there are no generalizations of the above results to broken Lefschetz fibrations. We also give a necessary and sufficient condition for the total space of a simplified broken Lefschetz fibration with a section admitting a spin structure, which is a generalization of Stipsicz's result on Lefschetz fibrations.