Codimension one foliation related to contact topology in low-dimensional manifolds via Open Books

Abstract

In this study we have been trying to understand various aspects of Contact Geometry in a contact-3 manifold setting. We began by looking at the di↵erential topology aspects of contact manifolds and their relation to more topological objects like Knots and Braids in contact 3-manifolds, relation between foliations and contact structures. A contact structure is a non-integrable plane field on a 3-manifold. An “Open Book” is an important tool that serves as a bridge between the di↵erential geometric side of contact geometry and the cut-and-paste methods of low-dimensional topology. An “Open Book” is a topological decomposition of a 3-manifold that also specifies an equivalence class of contact structures on the manifold. Furthermore, when contact structures are viewed only as a homotopy classes of plane fields, we can consider foliations in the same class and explore their relations. We explore in details relation between contact structures and their relation to codimension 1 foliations, in particular the construction of a foliation close to any given contact structure. We study other related foliations and conclude whether it perturbs to a tight or overtwisted contact structure.