A real open book not fillable by a real Lefschetz fibration

TL;DR

The paper demonstrates that certain real open books on 3- or 4-manifolds cannot be filled by real Lefschetz fibrations, highlighting limitations in the compatibility of real structures with these fibrations.

Contribution

It provides the first example of a real open book that cannot be filled by a compatible real Lefschetz fibration, despite being fillable by non-real Lefschetz fibrations.

Findings

Existence of real open books not fillable by real Lefschetz fibrations

Distinction between real and non-real Lefschetz fibration fillability

Examples illustrating limitations of real structures in Lefschetz fibrations

Abstract

A real 3- or 4-manifold has by definition an orientation preserving smooth involution acting on it. We consider Lefschetz fibrations of 4-dimensional manifolds-with-boundary and open book decompositions on their boundary in the existence of a real structure. We prove that there is a real open book which cannot be filled by a real Lefschetz fibration, although it is filled by non-real Lefschetz fibrations.