A complete answer to the strong density problem in Sobolev spaces with values into compact manifolds

We consider the problem of strong density of smooth maps in the Sobolev space $W^{s,p}(Q^m;\mathcal{N})$, where $0<s<+\mathrm{\infty }$, $1\le p<+\mathrm{\infty }$, $Q^m$ is the unit cube in ${\mathbb{R}}^m$, and $\mathcal{N}$ is a smooth compact connected Riemannian manifold without boundary. Our main result fully answers the strong density problem in the whole range $0<s<+\mathrm{\infty }$: the space ${\mathcal{C}}^{\mathrm{\infty }}({\bar{Q}}^m;\mathcal{N})$ is dense in $W^{s,p}(Q^m;\mathcal{N})$ if and only if ${\pi }_{[sp]}(\mathcal{N})=\{0\}$. This completes the results of Bethuel ($s=1$), Brezis and Mironescu ($0<s<1$), and Bousquet, Ponce, and Van Schaftingen ($s=2$, $3$, …). We also consider the case of more general domains $\mathrm{\Omega }$, in the setting studied by Hang and Lin when $s=1$.