Poset splitting and minimality of finite models

TL;DR

This paper introduces poset splitting, a new technique that solves longstanding open problems about the minimal number of points in finite models of certain surfaces, and provides a complete classification of these models.

Contribution

The paper develops poset splitting, enabling the resolution of minimal finite models for surfaces like the real projective plane, torus, and Klein bottle, and derives new topological results.

Findings

No finite model of the real projective plane has fewer than 13 points.

Finite models of the torus must have at least 16 points.

Finite spaces with fewer than 13 points have torsion-free homology groups.

Abstract

We develop a novel technique, which we call poset splitting, that allows us to solve two open problems regarding minimality of finite models of spaces: the nonexistence of a finite model of the real projective plane with fewer than 13 points and the nonexistence of a finite model of the torus with fewer than 16 points. Indeed, we prove much stronger results from which we also obtain that there does not exist a finite model of the Klein bottle with fewer than 16 points and that the integral homology groups of finite spaces with fewer than 13 points are torsion-free, settling a conjecture of Hardie, Vermeulen and Witbooi. Furthermore, we also apply our technique to give a complete characterization of the minimal finite models of the real projective plane, the torus, and the Klein bottle. In addition, we show that the poset splitting technique has an intrinsic interest giving original topological results that can be obtained from its application, such as a generalization of Hurewicz's theorem for non-simply-connected spaces and a generalization of a result of R. Brown on the fundamental group of a space.