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

Let $Sigma$ a closed $n$-dimensional manifold, $mathcal{N} subset mathbb{R}^M$ be a closed manifold, and $u in W^{s,frac ns}(Sigma,mathcal{N})$ for $sin(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if $pi_n(mathcal{N})={0}$ then there exists a minimizing $W^{s,frac ns}$-harmonic map homotopic to $u$. If $pi_n(mathcal{N}) eq {0}$, 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 $frac{n}{s} eq 2$ 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.