Simplicial complexes, stellar moves, and projective amalgamation

Abstract

We explore connections between stellar moves on simplicial complexes (these are fundamental operations of combinatorial topology) and projective Fra{\"i}ss{\'e} limits (this is a model theoretic construction with topological applications). We identify a class of simplicial maps that arise from the stellar moves of welding and subdividing. We call these maps weld-division maps. The core of the paper is the proof that the category of weld-division maps fulfills the projective amalgamation property. This gives an example of an amalgamation class that substantially differs from known classes. The weld-division amalgamation class naturally gives rise to a projective Fra{\"i}ss{\'e} class. We compute the canonical limit of this projective Fra{\"i}ss{\'e} class and its canonical quotient space. This computation gives a combinatorial description of the geometric realization of a simplicial complex and an example of a combinatorially defined projective Fra{\"i}ss{\'e} class whose canonical quotient space has topological dimension strictly bigger than $1$. The method of proof of the amalgamation theorem is new. It is not geometric or topological, but rather it consists of combinatorial calculations performed on finite sequences of finite sets and functions among such sequences. Set theoretic nature of the entries of the sequences is crucial to the arguments.