Topological point singularities of manifold-valued maps and approximability by smooth maps

Given a positive integer~$p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\NN$. The target manifold is required to satisfy a suitable topological condition, that is, the action of the fundamental group over the~$p$-th homotopy group must b