Simplotopal maps and necklace splitting

We show how to prove combinatorially the Splitting Necklace Theorem by Alon for any number of thieves. Such a proof requires developing a combinatorial theory for abstract simplotopal complexes and simplotopal maps, which generalizes the theory of abstract simplicial complexes and abstract simplicial maps. Notions like orientation, subdivision, and chain maps are defined combinatorially, without using geometric embeddings or homology. This combinatorial proof requires also a Z p -simplotopal version of Tucker’s Lemma.