Lefschetz fibrations and symplectic structures

Abstract

In this thesis, we study a relation between symplectic structures and Lefschetz ?brations to shed some light on 4-manifold theory. We introduce symplectic manifolds and state some results about them. We then introduce Lefschetz fi?brations, which are a generalization of ?fiber bundles, and discuss them briefly to obtain an intuitive understanding. The main theorem of this thesis is a result obtained by Gompf. It provides a way to construct a symplectic structure on a general Lefschetz ?fibration with homologeously nonzero fi?ber. We also discuss a generalization of this, achieved by Gompf, that generalizes this result to arbitrary even dimensions.