A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces

The present paper presents a counterexample to the sequential weak density of smooth maps between two manifolds M and N in the Sobolev space W 1,p (M, N), in the case p is an integer. It has been shown (see e.g. [6]) that, if p < dim M is not an integer and the [p]-th homotopy group π [p] (N) of N is not trivial, [p] denoting the largest integer less then p, then smooth maps are not sequentially weakly dense in W 1,p (M, N). On the other hand, in the case p < dim M is an integer, examples have been provided where smooth maps are actually sequentially weakly dense in W 1,p (M, N) with π p (N) = 0, although they are not dense for the strong convergence. This is the case for instance for M = B m , the standard ball in R m , and N = S p the standard sphere of dimension p, for which π p (N) = Z. The main result of this paper shows however that such a property does not holds for arbitrary manifolds N and integers p. Our counterexample deals with the case p = 3, dim M ≥ 4 and N = S 2 , for which the homotopy group π 3 (S 2) = Z is related to the Hopf fibration. We construct explicitly a map which is not weakly approximable in W 1,3 (M, S 2) by maps in C ∞ (M, S 2). One of the central ingredients in our argument is related to issues in branched transportation and irrigation theory in the case of the exponent is critical, which are possibly of independent interest. As a byproduct of our method, we also address some questions concerning the S 3-lifting problem for S 2-valued Sobolev maps