Polyhedron Cross-section And Small Cover Can Be Directed,

Small cover is an n dimensional manifold Mn endowed with a (Z2)n action whose orbit space is a simple convex polytope F.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 orientability of the section.Let 7r:Mnâ†'Pn be a small cover of Pn, S is an n-1 dimensional section of Pn, such thatÏ€-1(S) is an n-1 dimensional closed submanifoldWe give a necessary and sufficient condition in whichÏ€-1(S) is orientable.