Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

TL;DR

This paper introduces a new invariant for hyperelliptic Lefschetz fibrations that counts Dirac braids within the monodromy, aiding in classifying fibrations up to stable isomorphism.

Contribution

It defines the invariant $w$ for hyperelliptic Lefschetz fibrations and proves its role in classifying fibrations up to stable isomorphism, especially for odd genus.

Findings

The invariant $w$ counts Dirac braids in the monodromy.

Two fibrations are stably isomorphic iff they share the same singular fiber counts and $w$ (for odd genus).

Examples show fibrations with identical singular fibers but different $w$ are not stably isomorphic.

Abstract

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.