Low-slope Lefschetz fibrations

TL;DR

This paper constructs genus-g Lefschetz fibrations over the sphere with slopes approaching 2, showing the range of possible slopes and that extremal slopes are not realized.

Contribution

It introduces new constructions of Lefschetz fibrations with slopes arbitrarily close to 2 and analyzes the bounds of possible slopes.

Findings

Constructed fibrations with slopes approaching 2

Total spaces can be minimal and simply connected

Extremal slopes are not realized

Abstract

For $g\geq 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes all Lefschetz fibrations are not realized as slopes.