Common positive stabilisation of open book decompositions

TL;DR

This paper proves that any two open book decompositions supporting isotopic contact structures can be stabilized positively to share a common stabilization, advancing understanding of the Giroux correspondence.

Contribution

It establishes that a common positive stabilization exists for any two open books supporting isotopic contact structures, refining the Giroux correspondence.

Findings

Any two open book decompositions supporting isotopic contact structures admit a common positive stabilization.

The result strengthens the connection between open book decompositions and contact structures.

It provides a new tool for comparing open book decompositions in contact topology.

Abstract

The Giroux Correspondence states that two open book decompositions supporting the same contact structure are related by a sequence of positive open book stabilisations and destabilisations. In this note we show that any two open book decompositions supporting isotopic contact structures admit a common positive stabilisation.