The directed distance between homotopy classes of critical Sobolev self-maps of spheres equals an explicit constant times the difference in their Brouwer degrees.
Shafrir, On the distance between homotopy classes in W^ 1/p,p ( S^1; S^1) , Confluentes Math
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The distance between homotopy classes of Sobolev maps on spheres
The directed distance between homotopy classes of critical Sobolev self-maps of spheres equals an explicit constant times the difference in their Brouwer degrees.