Abstract

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4–manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the simplicial structure of triangulations using the Budney-Burton census of closed triangulated 4-manifolds. These experiments include the construction of minimal genus trisections of the non-orientable 4-manifolds $S^3\tilde{\times }{S}^1$ and $\mathbb{R}{P}^4.$

摘要

Gay 和 Kirby 最近为任意光滑、有向的封闭 4 流形引入了三等分的概念，以及一个新的拓扑不变量，称为三等分属。本文改进并实现了 Bell、Hass、Rubinstein 和 Tillmann 使用三角剖分计算三等分的算法，并将其扩展到不可定向的 4 流形。三等分属的下界以 Betti 数给出，并用于确定所有标准简单连接的 PL 4 流形的三等分属。此外，我们使用封闭三角 4 流形的 Budney-Burton 人口普查直接从三角剖分的简单结构构造小属的三等分。这些实验包括构建不可定向的 4 流形的最小属三等分${\mathrm{小号}}^3\stackrel{～}{\times }{\mathrm{小号}}^1$和$\mathbb{R}{磷}^4.$