Lefschetz Fibrations on 4-Manifolds

Abstract

In the light of recent results it turns out that both closed symplectic 4-manifolds and Stein surfaces admit a purely topological description in terms of Lefschetz fibrations and Lefschetz pencils. In this chapter we give the necessary definitions and sketch this topological descriptions of symplectic and Stein manifolds. In the discussion we include achiral Lefschetz fibrations as well; these more general objects are useful in viewing open book decompositions as boundaries of certain achiral Lefschetz fibrations. The chapter concludes with some applications of Lefschetz fibrations in various low dimensional problems.