Generalizations of the "de Bruijn-Erdosh" Theorem and an Application
by
J. Vermeer
Technical Univ Delft

We have the following topological version of the "de Bruijn-Erdosh" theorem.

Theorem. Let X be a metrizable space with dimX <= n. Let f_i : X --> X (i = 1, ... , k) be fixed-point free homeomorphisms of X. Then f₁, ... , f_k can be colored by n+2k+1 colors.

As an application of this theorem we present the following:

Recall first that a map f : X --> X is called a free Z_p - action on X if for every x in X, f^p (x) = x and the set { x, f(x), ... , f^p-1(x) } has cardinality p.

Let f : X --> X be a free Z_p action on the compact X. A subset C subset X is called a set of first type if there exists a closed set A subset X such that C = \cup { f^j (A) : j = 0, 1, ... , p-1 } and A \cap f^j (A) = \emptyset, for j in { 1, 2, ... , p-1 }. The minimal number k = g(X, f) such that X is the union of k-many closed sets of first type is called the genus of (X, f).

It is a classical result of Krasnoselski that the genus of every free Z_p action f : Sⁿ --> Sⁿ on the n-dimensional sphere Sⁿ is equal to n+1. We show:

Theorem. Let X be a compact n-dimensional space and let f : X --> X be a free Z_p action on X. Then g(X, f) <= n+1.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-88.