Foliated open books

Licata, Joan E; Vértesi, Vera

doi:10.2140/agt.2024.24.3139

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Foliated open books

Joan E Licata and Vera Vértesi

Algebraic & Geometric Topology 24 (2024) 3139–3197

DOI: 10.2140/agt.2024.24.3139

Abstract

This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation $ℱ_{ξ}$ on its boundary. These foliated open books offer a finer tool for studying contact manifolds with convex boundary than existing models, as the boundary foliation carries more data than the dividing set. In addition to establishing fundamental results about the uniqueness and existence of foliated open books, we carefully examine their relationship with the partial open books introduced by Honda, Kazez, and Matić. Foliated open books have user-friendly cutting and gluing properties, and they arise naturally as submanifolds of classical open books for closed three-manifolds. We define three versions of foliated open books (embedded, Morse, and abstract), and we prove the equivalence of these models as well as a Giroux Correspondence which characterizes the foliated open books associated to a fixed triple $(M, ξ, ℱ)$ .

Keywords

contact structure, open book, partial open book, open book foliation, characteristic foliation, gluing

Mathematical Subject Classification

Primary: 57K33

Secondary: 37D15

References

Publication

Received: 1 May 2020

Revised: 23 June 2022

Accepted: 4 September 2022

Published: 7 October 2024

Authors

Joan E Licata
Mathematical Sciences Institute The Australian National University Canberra, ACT Australia

Vera Vértesi
Department of Mathematics University of Vienna Vienna Austria

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Volume 24, issue 6 (2024)