Projectivities in Simplicial Complexes and Colorings of Simple Polytopes

For each strongly connected finite-dimensional (pure) simplicial complex ∆ we construct a finite group Π(∆), the group of projectivities of ∆, which is a combinatorial but not a topological invariant of ∆. 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 manifolds.