On diagrams of simplified trisections and mapping class groups

Abstract

A simplified trisection is a trisection map on a 4--manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3--tuple of systems of simple closed curves in a surface to be a diagram of a simplified trisection in terms of mapping class groups. As an application of this criterion, we show that trisections of spun 4--manifolds due to Meier are diffeomorphic (as trisections) to simplified ones. Baykur and Saeki recently gave an algorithmic construction of a simplified trisection from a directed broken Lefschetz fibration. We also give an algorithm to obtain a diagram of a simplified trisection derived from their construction.