  Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$, a
Riemannian covering $\pi : \smash{\widetilde{\mathcal{N}}} \to \mathcal{N}$ by
a noncompact covering space $\smash{\widetilde{\mathcal{N}}}$, $1 < p < \infty$
and $0 < s < 1$, the space of liftings of fractional Sobolev maps in
$\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathcal{N})$ is characterized when $sp >
1$ and an optimal nonlinear fractional Sobolev estimate is obtained when
moreover $sp \ge \dim \mathcal{M}$. A nonlinear characterization of the sum of
spaces $\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathbb{R}) + \smash{\dot{W}^{1,
sp}} (\mathcal{M}, \mathbb{R})$ is also provided.
