Lefschetz fibrations, open books, and symplectic fillings of contact 3-manifolds

Abstract

Ever since Donaldson showed that every closed symplectic 4-manifold admits a Lefschetz pencil and Giroux proved that every closed contact 3-manifold admits an adapted open book decomposition, Lefschetz fibrations and open books have been used fruitfully to obtain significant results about the topology of symplectic 4-manifolds and contact 3-manifolds. In this expository article, we present the highlights of our contribution to the subject at hand based on joint work with several coauthors during the past twenty years.