|Dept.||Korea Institute for Advanced Study|
In this lecture series we introduce recent developments in the theory of topological Lefschetz fibrations on 4-manifolds. Specifically, after explaining some introductory material about Lefschetz fibrations, branched coverings, ribbon surfaces and braided surfaces, and their connections, we will give the moves for Lefschetz fibrations that fix the total 4-dimensional 2-handlebody up to the so-called 2-equivalence (special kinds of diffeomorphisms which are compatible, in some sense, with the given handlebody decomposition). We will give also a sketch of the proof, pointing out the main ideas and difficulties. This involves the branched covering representation of Lefschetz fibrations, by means of colored braided surfaces, and it is based on the Bobtcheva-Piergallini moves for branched coverings of the 4-ball.
We give also a talk on universal Lefschetz fibrations, which are defined in analogy with the classical universal vector bundles. After proving their existence in the bounded case, we will go a step ahead to the closed case and this leads to a more general definition of Lefschetz fibrations over manifolds of arbitrary dimensions. We will give some applications to multisections of Lefschetz fibrations over closed surfaces.