Existence of a small cover over a 15-colorable simple 4-polytope

Abstract

The chromatic number for properly colouring the facets of a combinatorial simple n-polytope Pn that is the orbit space of a quasitoric manifold satisfies the inequality n≤Pn≤2n−1. The inequality is sharp for n=2 but not for n=3 due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals 15 and deduce that the predicted upper bound is attained for n=4. Analogous results are verified for the case of oriented small covers in dimensions 4 and 5.