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 $0 < s < +\infty$, $1 \leq p < +\infty$, $Q^{m}$ 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 $0 < s < +\infty$: the space $\mathcal{C}^{\infty}(\overline{Q}^{m};\mathcal{N})$ 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 $\Omega$, in the setting studied by Hang and Lin when $s = 1$.