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Trisections of Flat Surface Bundles over Surfaces
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 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 minimal
Subject Area
Mathematics
Recommended Citation
Williams, Marla, "Trisections of Flat Surface Bundles over Surfaces" (2020). ETD collection for University of Nebraska-Lincoln. AAI28086334.
https://digitalcommons.unl.edu/dissertations/AAI28086334