A Degree-Theoretic Proof of a Coarse Fixed Point Principle

TL;DR

This paper develops a coarse degree theory for Euclidean spaces and proves that all coarse maps from a Euclidean half-space to itself have a fixed point, extending classical fixed-point concepts to large-scale geometry.

Contribution

It introduces a coarse degree concept and proves a fixed-point theorem for coarse maps on Euclidean half-spaces, broadening fixed-point theory to coarse geometry.

Findings

Coarse degree theory for Euclidean spaces is established.

Every coarse map from a Euclidean half-space to itself has a fixed point.

The results extend classical fixed-point principles to large-scale settings.

Abstract

We introduce a large scale analogue of the classical fixed-point property for continuous maps, which shall apply to coarse maps. We also develop a coarse version of degree for coarse maps on Euclidean spaces. Then, applying a coarse degree-theoretic argument, we prove that every coarse map from a Euclidean half-space to itself has the coarse fixed-point property.