Charts, signatures, and stabilizations of Lefschetz fibrations

TL;DR

This paper introduces a chart-based method to describe monodromy in Lefschetz fibrations, generalizes signature formulas to arbitrary surfaces, and proves stabilization theorems for these fibrations.

Contribution

It develops a new chart-based framework for Lefschetz fibrations, extends signature formulas to more general cases, and establishes stabilization results under fiber summing.

Findings

Generalized signature formula for Lefschetz fibrations over any closed surface.

Proved stabilization theorems for Lefschetz fibrations via fiber summing.

Introduced a chart-based approach for describing monodromy in Lefschetz fibrations.

Abstract

We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.