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

Abstract

The present paper presents a counterexample to the sequentially 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 that, if $p<\dim M $ is not an integer and the $[p]$-th homotopy group $\pi_{[p]}(N)$ of $N$ is not trivial, $[p]$ denoting the largest integer less then $p$, then smooth maps are not sequentially weakly dense in $W^{1, p} (M, N)$ for the strong convergence. On the other, 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 $\pi_{p}(N)\not = 0$. This is the case for instance for $M= \mathbb B^m$, the standard ball in $\mathbb R^m$, and $N=\mathbb S^p$ the standard sphere of dimension $p$, for which $\pi_{p}(N) =\mathbb Z$. 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 $p=3$, $\dim M\geq 4$ and $N=\mathbb S^2$, for which the homotopy group $\pi_3(\mathbb S^2)=\mathbb Z$ is related to the Hopf fibration.