Fixed-point free maps of Euclidean spaces

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.

💡 Research Summary

The paper investigates the relationship between fixed‑point‑free continuous self‑maps of Euclidean spaces and a combinatorial property known as “colorability.” A map f : ℝⁿ → ℝⁿ is called colorable if ℝⁿ can be partitioned into finitely many closed sets C₁,…,C_k such that for each i, the image f