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

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