On the relative Giroux correspondence

TL;DR

This paper explicates the relative Giroux correspondence, establishing a one-to-one relationship between partial open book decompositions and contact 3-manifolds with convex boundary, extending Giroux's work to the relative case.

Contribution

It provides a detailed exposition of the relative Giroux correspondence, clarifying the connection between partial open books and contact manifolds with boundary.

Findings

Established a one-to-one correspondence between partial open books and contact 3-manifolds with convex boundary.

Generalized Giroux's correspondence to the case of manifolds with boundary.

Clarified the structure of adapted partial open books in the relative setting.

Abstract

Recently, Honda, Kazez and Matic described an adapted partial open book of a compact contact 3-manifold with convex boundary by generalizing the work of Giroux in the closed case. They also implicitly established a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive stabilization and isomorphism classes of compact contact 3-manifolds with convex boundary. In this expository article we explicate the relative version of Giroux correspondence.