Trisecting the 9-vertex complex projective plane

TL;DR

This paper provides a straightforward proof that Kuehnel's 9-vertex triangulation accurately represents the complex projective plane by analyzing its trisection structure within a symmetry-breaking subdivision.

Contribution

It offers a direct proof confirming the triangulation's correctness and details the construction process, enhancing understanding of minimal triangulations of complex surfaces.

Findings

Confirmed Kuehnel's triangulation as the complex projective plane

Demonstrated the trisection structure within the triangulation

Provided a detailed construction method

Abstract

In this paper we will give a short and direct proof that Wolfgang Kuehnel's 9-vertex triangulation of the complex projective plane really is the complex projective plane. The idea of our proof is to recall the trisection of the complex projective plane into 3 bi-disks and then to see this trisection inside a symmetry-breaking subdivision of the triangulation. Following the basic proof, we will elaborate on the construction.