Projectivities in simplicial complexes and colorings of simple polytopes

Abstract.

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