Polytopal complexes: maps, chain complexes and... necklaces
Abstract
Description
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.
Presented at the TGGT 08 Conference, May 2008, Paris. The definition of a polytopal map has been modified
Presented at the TGGT 08 Conference, May 2008, Paris. The definition of a polytopal map has been modified