On the distance between homotopy classes in $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)$

TL;DR

This paper investigates the metric structure of homotopy classes of maps from the circle to itself within fractional Sobolev spaces, establishing explicit formulas for distances between classes based on their topological degree.

Contribution

It introduces a natural notion of topological degree in fractional Sobolev spaces and characterizes the distances between homotopy classes in terms of minimal energy.

Findings

Distance between classes equals minimal energy in the difference class.

Explicit formula for the distance when p=2: 2π|d2 - d1|^{1/2}.

Homotopy classes form a disjoint union with quantifiable separation.

Abstract

For every $p\in(1,\infty)$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)=\bigcup_{d\in{\mathbb Z}}\mathcal{E}_d$. For every pair $d_1,d_2\in {\mathbb Z}$, we show that the distance $\text{Dist}_{W^{1/p,p}}({\mathcal E}_{d_1}, {\mathcal E}_{d_2}):=\sup_{f\in{\mathcal E}_{d_1}}\ \inf_{g\in{\mathcal E}_{d_2}}\ d_{W^{1/p,p}}(f, g)$ equals the minimal $W^{1/p,p}$-energy in $\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce from the latter formula an explicit value: $\text{Dist}_{W^{1/2,2}}({\mathcal E}_{d_1}, {\mathcal E}_{d_2})=2\pi|d_2-d_1|^{1/2}$.