University of Arkansas Topology Seminar: 9/13/2018

Speaker: Nickolas Castro

Title: Trisections and Lefschetz fibrations

All smooth, compact 4–manifolds can be trisected into three diffeomorphic, codimension zero submanifolds whose pairwise intersections are 3–dimensional handlebodies and whose triple intersection is a surface. In the case of 4–manifolds with boundary, trisections induce an open book decomposition on the bounding 3–manifold(s). A Lefschetz fibration is a smooth map from a 4–manifold to a surface with isolated singularities that can be roughly thought of as a complex Morse function. Lefschetz fibrations over the disk also induce open book decompositions on the non-empty boundary of a 4–manifold. In this talk I will discuss the basic definitions of trisections and Lefschetz fibrations, giving plenty of examples of each, and show how to obtain a trisection diagram from the (topological) data of a Lefschetz fibration. Using this dictionary, together with the gluing theorem of relative trisection, we have an alternate proof of the existence of trisections of closed 4–manifolds. This work is joint with Burak Ozbagci.



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