Trisections of Flat Surface Bundles over Surfaces

A trisection of a smooth 4-manifold is a decomposition into three simple pieces with nice intersection properties. Work by Gay and Kirby shows that every smooth, connected, orientable 4-manifold can be trisected. Natural problems in trisection theory are to exhibit trisections of certain classes of 4-manifolds and to determine the minimal trisection genus of a particular 4-manifold. Let Σg denote the closed, connected, orientable surface of genus g. In this thesis, we show that the direct product Σg × Σh has a ((2g + 1)(2h + 1) + 1;2g + 2h)-trisection, and that these parameters are minimal. We provide a description of the trisection, and an algorithm to generate a corresponding trisection diagram given the values of g and h. We then extend this construction to arbitrary closed, flat surface bundles over surfaces with orientable fiber and orientable or non-orientable base. If the fundamental group of such a bundle has rank 2 - χ + 2h, where h is the genus of the fiber and χ is the Euler characteristic of the base, these trisections are again minima