Partitioning the projective plane and the dunce hat

TL;DR

This paper proves that any finite triangulation of the real projective plane or dunce hat can be partitioned, using novel gluing techniques that reduce complex problems to simpler subcomplexes.

Contribution

Introduces new gluing tools and methods to establish partitionability of complex topological surfaces like the projective plane and dunce hat.

Findings

All finite triangulations of the projective plane are partitionable.

Partitionability of complex surfaces can be reduced to simpler subcomplexes.

New techniques enable lifting partitionability through quotient maps.

Abstract

We show that any finite triangulation of the real projective plane or the dunce hat is partitionable. To prove this, we introduce simple yet useful gluing tools that allow us to reduce partitionability of a given complex to that of smaller constituent relative subcomplexes such as the disk or the open M\"obius strip. The gluing process generates partitioning schemes with a distinctive shelling/constructible flavor. We also give a tool to lift partitionability of relative simplicial complexes to that of the preimages of certain simplicial quotient maps.