Upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus

TL;DR

This paper establishes upper bounds on the minimal number of singular fibers in Lefschetz fibrations over the torus by analyzing relations in the mapping class groups of surfaces.

Contribution

It introduces new relations in mapping class groups that lead to bounds on singular fibers in Lefschetz fibrations over the torus.

Findings

Upper bounds for singular fibers in Lefschetz fibrations over the torus

Relations in mapping class groups involving Dehn twists and commutators

Method to estimate minimal singular fibers using algebraic surface topology

Abstract

In this paper, we give some relations in the mapping class groups of oriented closed surfaces in the form that a product of a small number of right hand Dehn twists is equal to a single commutator. Consequently, we find upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus.