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