Topological singular set of manifold-valued maps weakly approximable 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 $\mathcal{N}$. The target manifold is required to satisfy suitable topological conditions; in particular, the action of $\pi_1(\mathcal{N})$ over the $\pi_p(\mathcal{N})$ must be trivial. However, we do not assume that $\mathcal{N}$ is $(p-1)$-connected. Using tools from geometric measure theory -- namely, flat chains with coefficients in~$\pi_p(\mathcal{N})$ -- we associate to each map $u$ in the weak sequential closure of smooth maps an object that captures its point singularities. The vanishing of this object characterizes local strong approximability by smooth maps.