  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.
