Lifting in compact covering spaces for fractional Sobolev mappings

Let $\pi :\tilde{\mathcal{N}}\to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\tilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, $0<s<1$ and $2\le sp<m$, we prove that every mapping $u\in W^{s,p}(\mathcal{M},\mathcal{N})$ has a lifting in $W^{s,p}$, i.e., we have $u=\pi \circ \tilde{u}$ for some mapping $\tilde{u}\in W^{s,p}(\mathcal{M},\tilde{\mathcal{N}})$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result \emph{settles completely} the question of the lifting in Sobolev spaces over covering spaces. The proof relies on an a priori estimate of the oscillations of $W^{s,p}$ maps with $0<s<1$ and $sp>1$, in dimension $1$. Our argument also leads to the existence of a lifting when $0<s<1$ and $1<sp<2\le m$, provided there is no topological obstruction on $u$, i.e., $u = \pi , \circ , \widetilde{u}$holdsinthisrangeprovided$u$ is in the strong closure of $C^{\mathrm{\infty }}(\mathcal{M},\mathcal{N})$. However, when $0<s<1$, $sp=1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}(\mathcal{M},\mathcal{N})$ in the strong closure of $C^{\mathrm{\infty }}(\mathcal{M},\mathcal{N})$, such that $u = \pi , \circ , \widetilde{u}$doesnotholdforany$\widetilde{u} \in W^{s, p} ({$\mathcal{M}$}, \widetilde{$\mathcal{N}$} )$.