[Topology and Geometry Seminar] On diagrams of simplified trisections and mapping class groups

A trisection, due to Gay and Kirby, is a decomposition of a 4-manifold into three 4-dimensional handlebodies, which can be considered as a 4-dimensional counterpart of a Heegaard splitting of a 3-manifold. We can obtain a trisection of a 4-manifold from a stable map to the plane with a specific configuration of the critical value set, which we call a trisection map. 
In studying smooth maps from 4-manifolds to surfaces with stable and Lefschetz singularities, Baykur and Saeki introduced simplified trisections, which are trisection maps such that in their critical value sets, there are no double points and cusps only appear in triples on innermost fold circles, and gave an algorithm to construct a simplified trisection from a broken Lefschetz fibration. In this talk, we first review basic properties of trisections, and then explain how to obtain a diagram of a simplified trisection derived from Baykur and Saeki's construction.