Trisections of 4-manifolds via Lefschetz fibrations

Abstract

We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds. As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S-2 equipped with a section of square -1, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed 4-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented S-2-bundle over any closed surface and in particular we draw the corresponding diagrams for T-2 x S-2 and T-2(x) over tildeS(2) using our gluing technique. Furthermore, we provide an alternate proof of the fundamental result of Gay and Kirby which says that every closed 4-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.