Polytopal complexes: maps, chain complexes and... necklaces

TL;DR

This paper introduces a simple definition of polytopal maps between complexes, extends chain complex mappings, and provides a combinatorial proof of Alon's splitting necklace theorem, opening new research directions.

Contribution

It defines polytopal maps extending simplicial and cubical maps, and applies this to prove a key combinatorial theorem in a novel way.

Findings

Defined polytopal maps between complexes

Constructed induced chain maps for these polytopal maps

Provided the first combinatorial proof of Alon's splitting necklace theorem

Abstract

The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem.