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

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: ${\text{Dist}}_{W^{1/2,2}}({\mathcal{E}}_{d_1},{\mathcal{E}}_{d_2})=2\pi |d_2-d_1{|}^{1/2}$.