Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations

TL;DR

This paper advances methods to compute and analyze the trisection genus of both orientable and non-orientable 4-manifolds using triangulations, providing new bounds and explicit constructions.

Contribution

It extends existing algorithms to non-orientable 4-manifolds, offers new lower bounds based on Betti numbers, and constructs minimal genus trisections from triangulations.

Findings

Computed trisection genus for all standard simply connected PL 4-manifolds.

Constructed minimal genus trisections for specific non-orientable 4-manifolds.

Extended algorithms to non-orientable cases and triangulation-based constructions.

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 $\mathbb{R}P^4$.