Lifting of fractional Sobolev mappings to noncompact covering space

Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$, a Riemannian covering $\pi :\tilde{\mathcal{N}}\to \mathcal{N}$ by a noncompact covering space $\tilde{\mathcal{N}}$, $1<p<\mathrm{\infty }$ and $0<s<1$, the space of liftings of fractional Sobolev maps in ${\dot{W}}^{s,p}(\mathcal{M},\mathcal{N})$ 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 ${\dot{W}}^{s,p}(\mathcal{M},\mathbb{R})+{\dot{W}}^{1,sp}(\mathcal{M},\mathbb{R})$ is also provided.