A note on sections of broken Lefschetz fibrations

For every g, we show that there exists a non-trivial genus-g simplified broken Lefschetz fibration that has infinitely many homotopy classes of sections, and construct a non-trivial genus-g simplified broken Lefschetz fibration that has a section with non-negative square. It is known that no Lefschetz fibrations satisfy either of the above conditions. Smith proved that every Lefschetz fibration has only finitely many homotopy classes of sections. Smith and Stipsicz independently proved that a Lefschetz fibration is trivial if it has a section with non-negative square. Thus, our results indicate that there are no generalizations of the above results to broken Lefschetz fibrations (the genus-1 simplified broken Lefschetz fibrations Baykur constructed also indicate our second result above, but our construction and the resulting fibrations are different from them). We also give a necessary and sufficient condition for the total space of a simplified broken Lefschetz fibration with a section to admit a spin structure, which is a generalization of Stipsicz's result on Lefschetz fibrations. As a corollary of this, we completely classify genus-1 simplified broken Lefschetz fibrations with spin structures.