Projectivities in Simplicial Complexes and Colorings of Simple Polytopes

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.