Projectivities in Simplicial Complexes and Colorings of Simple Polytopes

TL;DR

This paper introduces a finite group called the group of projectivities for simplicial complexes, explores its properties for manifolds and polytopal spheres, and applies these findings to a coloring problem in toric algebraic varieties.

Contribution

It constructs a new combinatorial invariant, the group of projectivities, and applies it to solve a coloring problem related to polytopes in algebraic geometry.

Findings

Defined the group of projectivities for simplicial complexes.

Analyzed the group for combinatorial manifolds and polytopal spheres.

Applied the group to a coloring problem in toric algebraic varieties.

Abstract

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.