Sections of surface bundles and Lefschetz fibrations

TL;DR

This paper characterizes the self-intersection bounds of sections in surface bundles and Lefschetz fibrations, and explores related properties in the mapping class group, revealing new bounds and unboundedness results.

Contribution

It establishes the adjunction bound as the universal limit for self-intersection numbers and calculates exact commutator lengths in the mapping class group.

Findings

The adjunction bound 2h-2 is the only universal bound for self-intersection numbers.

The stable commutator length of a boundary Dehn twist is 1/2.

No upper bound exists on the number of critical points for certain Lefschetz fibrations.

Abstract

We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h-2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g at least two.