On the Giroux correspondence

This dissertation explores the correspondence between open books and contact structures on three-manifolds. We begin the thesis with background information necessary to describe the correspondence. After defining both open books and contact structures, we outline the technical results that describe the notions of compatibility, the Murasugi sum, and stabilization equivalence. We then provide a survey of the theorems comprising the correspondence, from a classical result of J. Alexander on the existence of open books, to recent results of Emmanuel Giroux on contact structures. Our result comprises one of the final parts of the proof of the correspondence. It involves a certain compact surface, called the ribbon, associated to an embedded Legendrian graph, as well as Giroux’s construction of a cell decomposition adapted to the contact structure. We prove a series of theorems detailing the relationship between these objects and open book decompositions.