On colorings of maps

A fixed-point free map f : X → X is said to be colorable with k colors if there exists a closed cover C of X consisting of k elements such that C ∩ f(C) = ∅ for every C in C. It is shown that every fixed-point free continuous selfmap of a compact space X with dim X ⩽ n can be colored with n + 3 colors. Similar results are obtained for finitely many maps. It is shown that every free Zp-actionon an n-dimensional compact space X has genus at most n + 1.