Real open books and real contact structures

TL;DR

This paper introduces the concept of real open book decompositions on real 3-manifolds, demonstrating their compatibility with real contact structures and proposing a real Giroux correspondence, supported by examples in lens spaces.

Contribution

It establishes that real open books support real contact structures and conjectures a one-to-one correspondence between them, extending Giroux's correspondence to real manifolds.

Findings

Every real open book supports a real contact structure.

Two real contact structures supported by the same real open book are equivariantly isotopic.

Examples of real open books and Heegaard decompositions in lens spaces are provided.

Abstract

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open book carries a real contact structure and two real contact structures supported by the same real open book decomposition are equivariantly isotopic. We also show that every real contact structure on a closed 3-dimensional real manifold is supported by a real open book. Finally, we conjecture that two real open books on a real contact manifold supporting the same real contact structure are related by positive real stabilizations and equivariant isotopy and that the Giroux correspondence applies to real manifolds as well namely that there is a one to one correspondence between the real contact structures on a real 3-manifold up to equivariant contact isotopy and the real open books up to positive real stabilization. Meanwhile, we study some examples of real open books and real Heegaard decompositions in lens spaces.