## Dissertations, Theses, and Student Research Papers in Mathematics

Alex Zupan

Mark Brittenham

#### Date of this Version

8-2020

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Alex Zupan and Mark Brittenham. Lincoln, Nebraska: August, 2020

#### Abstract

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 $\Sigma_g$ denote the closed, connected, orientable surface of genus $g$. In this thesis, we show that the direct product $\Sigma_g\times\Sigma_h$ 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 $2-\chi+2h$, where $h$ is the genus of the fiber and $\chi$ is the Euler characteristic of the base, these trisections are again minimal.

Advisor: Alex Zupan and Mark Brittenham

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