Distances between homotopy classes of $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$

TL;DR

This paper studies the distances between homotopy classes of Sobolev maps from the sphere to itself, introducing a new equivalence relation based on topological degree and providing explicit formulas in certain cases.

Contribution

It introduces a novel equivalence relation on Sobolev maps involving topological degree and calculates distances between classes, including exact formulas in special cases.

Findings

Explicit formulas for distances between homotopy classes

Introduction of a new equivalence relation based on topological degree

Analysis of distances in both usual and Hausdorff senses

Abstract

We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special cases we even obtain exact formulas.