Trace theory for Sobolev mappings into a manifold

We review the current state of the art concerning the characterization of traces of the spaces $W^{1,p}({\mathbb{B}}^{m-1}\times (0,1),\mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p<m$ is an integer and the homotopy group ${\pi }_p(\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group ${\pi }_1(\mathcal{N})$ is finite and $\pi_2 ($\mathcal{N}$) \simeq \dotsb \simeq \pi_{\lfloor p - 1 \rfloor} ($\mathcal{N}$) \simeq {0}$. We present several open problems connected to the extension problem.