The orientability of small covers and coloring simple polytopes

Abstract

Small Cover is an $n$-dimensional manifold endowed with a $\mathbf{Z}_2^n$ action whose orbit space is a simple convex polytope $P$. It is known that a small cover over $P$ is characterized by a coloring of $P$ which satisfies a certain condition. In this paper we shall investigate the topology of small covers by the coloring theory in combinatorics. We shall first give an orientability condition for a small cover. In case $n=3$, an orientable small cover corresponds to a four colored polytope. The four color theorem implies the existence of orientable small cover over every simple convex $3$-polytope. Moreover we shall show the existence of non-orientable small cover over every simple convex $3$-polytope, except the $3$-simplex.