Computing trisections of 4-manifolds

TL;DR

This paper introduces an algorithm to construct trisections of 4-manifolds from their pentachoron decompositions, providing explicit bounds on the trisection genus based on the number of simplices.

Contribution

It presents the first explicit algorithmic method to derive trisections from a 4-manifold's triangulation, linking complexity to the number of pentachora.

Findings

Algorithmically constructs trisections from pentachoron decompositions.

Provides explicit bounds on the trisection genus.

First to connect triangulation complexity with trisection genus.

Abstract

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.