Branched Covers, Open Books and Contact Structures

In this thesis we study the relationship between branched covers, open book decompositions and contact structures in dimension three. The main result shown is the one to one correspondence between contact structures and open book decompositions up to positive stabi- lization, now known as “Giroux’s correspondence”. Then, we proceed to define contact branched covers and, using Giroux’s correspondence together with a theorem by Montesinos and Morton, we show that every closed contact 3-manifold can be obtained as a 3-fold simple branched cover with base manifold the standard contact 3-sphere and branch set a transverse link. Finally, we present a recent result by Etnyre and Casals concerning the existence of a 4-components transverse universal link. This is a transverse link in the standard contact 3–sphere such that all closed contact 3–manifolds are contact branched covers over this link