Lefschetz fibrations on 4-manifolds

Abstract

Lefschetz pencils and fibrations were introduced for studying topological properties of smooth complex projective varieties. More recently, as an application of Donaldson’s asymptotically holomorphic methods [13], Lefschetz pencils have been found on all symplectic manifolds [14], [15]. Conversely, Gompf showed that a 4-manifold admitting a Lefschetz pencil/fibration carries a symplectic structure [23], [24]. Since symplectic 4-manifolds play a prominent role in modern low-dimensional topology, the Donaldson Gompf correspondence piqued interest in the study of Lefschetz fibrations. In addition, these structures provide a connection between symplectic topology and geometric group theory.

In this chapter we collect the most basic definitions, describe the fundamental results and explain some of the consequences of the correspondence mentioned above. In some cases we provide a full or partial proof of the result. Most of the time, however, we restrict ourselves to quoting the theorems, and try to put them in perspective. The only new result is stated in Corollary 5.16. We would like to point out that although most of the fundamental results hold in any (even) dimension, we restrict our attention to the case of symplectic 4-manifolds, in which case the Lefschetz fibrations have surface fibers, and therefore the theory of mapping class groups enters in an essential way.