Sections, pseudosections, and multisections of Lefschetz fibrations

Abstract

We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.