On the distance between homotopy classes of maps between spheres

Certain Sobolev spaces of maps between manifolds can be written as a disjoint union of homotopy classes. Rubinstein and Shafrir [Israel J. Math. 160 (2007), 41–59] computed the distance between homotopy classes in the spaces W1,p(S1, S1) for different values of p, and in the space W1,2(Ω, S1) for certain multiply connected two-dimensional domains Ω. We generalize some of these results to higher dimensions. Somewhat surprisingly we find that in W1,p(S2, S2), with p > 2, the distance between any two distinct homotopy classes equals a universal positive constant c(p). A similar result holds in W1,p(Sn, Sn) for any n ≥  2 and p > n.