A new obstruction to the extension problem for Sobolev maps between manifolds

The main result of the present paper, combined with earlier results of Hardt and Lin settles the extension problem for $W^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal M$ and $\mathcal N$ are compact riemannian manfolds, $\mathcal M$ having non-empty smooth boundary and assuming moreover that $\mathcal N$ is simply connected. The main question which is studied is the following: Given a map in the trace space $W^{1-\frac{1}{p}, p} (\partial \mathcal M, \mathcal N)$, does it possess an extension in $W^{1,p}(\mathcal M, \mathcal N)$? We show that the answer is negative in the case $\mathfrak p_{c} +1\leq p<m={\rm dim} \, \mathcal M$, where the number $\mathfrak p_{c}$ is related to the topology of $\mathcal N$. We also adress the case $\mathcal N$ is not simply connected, providing various results and rising some open questions. In particular, we stress in that case the relationship between the extension problem and the lifting problem to the universal covering manifold.