  Let $\pi : \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering,
with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ 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 $u = \pi \, \circ \, \widetilde{u}$ for some mapping
$\widetilde{u} \in W^{s, p} (\mathcal{M}, \widetilde{\mathcal{N}})$. 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., $u =
\pi \, \circ \, \widetilde{u}$ holds in this range provided $u$ is in the
strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$. 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 $C^\infty({\mathcal{M}}, \mathcal{N})$, such that $u = \pi \, \circ
\, \widetilde{u}$ does not hold for any $\widetilde{u} \in W^{s, p}
({\mathcal{M}}, \widetilde{\mathcal{N}} )$.
