The extension of traces for Sobolev mappings between manifolds

The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings ${\dot{W}}^{1,p}(\mathcal{M},\mathcal{N})$ to the fractional Sobolev-Slobodeckiĭ space ${\dot{W}}^{1-1\mathrm{/}p,p}(\mathrm{\partial }\mathcal{M},\mathcal{N})$ is surjective when $1<p<\dim \mathcal{M}$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p\ge \dim \mathcal{M}$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.