On the distance between homotopy classes in W^1/p,p(S¹; S¹)

For every p ∈ (1, ∞) there is a natural notion of topological degree for maps in W^1/p,p(S¹; S¹) which allows us to write that space_⋃as a disjoint union of classes, (Formula presented) For every pair d₁, d₂ ∈ Z, we show that the distance (Formula presented) equals the minimal W^1/p,p-energy in ε_d1−d2. In the special case p = 2 we deduce from the latter formula an explicit value: Dist_W1/2,2(ε_d1, ε_d2) = 2π|d₂ − d₁ |^1/2.