Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes

Let $\mathrm{\Sigma }$ a closed $n$-dimensional manifold, $$\mathcal{N}$ \subset $\mathbb{R}$^M$beaclosedmanifold,and$u \in W^{s,\frac ns}(\Sigma,$\mathcal{N}$)$for$s\in(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if ${\pi }_n(\mathcal{N})=\{0\}$ then there exists a minimizing $W^{s,\frac{n}{s}}$-harmonic map homotopic to $u$. If ${\pi }_n(\mathcal{N})\mathrm{\ne }\{0\}$, then we prove that there exists a $W^{s,\frac{n}{s}}$-harmonic map from ${\mathbb{S}}^n$ to $\mathcal{N}$ in a generating set of ${\pi }_n(\mathcal{N})$. Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when $\frac{n}{s}\mathrm{\ne }2$ one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing $W^{s,\frac{n}{s}}$-maps into manifolds.