Invariants of real Lefschetz fibrations

TL;DR

This paper introduces real Lefschetz chains as invariants for real Lefschetz fibrations, proving their completeness for genus greater than 1 and enhancing them with binary decorations for genus 1 cases.

Contribution

It defines real Lefschetz chains and proves their effectiveness as invariants, also providing an improved invariant for genus 1 fibrations.

Findings

Real Lefschetz chains are complete invariants for genus > 1.

Additional binary decoration makes the invariant complete for genus 1.

The invariants distinguish different real Lefschetz fibrations effectively.

Abstract

In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber genus is 1, real Lefschetz chains are not sufficient to distinguish real Lefschetz fibrations. We show that by adding a certain binary decoration to real Lefschetz chains, we get a complete invariant.