Fixed-point free maps of Euclidean spaces

TL;DR

The paper proves that all fixed-point free continuous maps on Euclidean spaces are colorable and extends this result to their Stone-Čech compactifications, providing new insights into fixed-point theory.

Contribution

It establishes that fixed-point free maps on Euclidean spaces are colorable and generalizes this property to their compactifications, offering new theoretical understanding.

Findings

Every fixed-point free continuous map of ${f R}^n$ is colorable.

Fixed-point freeness is preserved under the Stone-Čech compactification.

The paper provides examples illustrating the main results.

Abstract

Our main result states that every fixed-point free continuous self-map of ${\mathbb R}^{n}$ is colorable. This result can be re-formulated as follows: A continuous map $f: {\mathbb R}^{n}\to {\mathbb R}^{n}$ is fixed-point free iff $\widetilde f: \beta {\mathbb R}^{n}\to \beta {\mathbb R}^{n}$ is fixed-point free. We also obtain a generalization of this fact and present some examples.