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

Abstract

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 be trivial; however, we do \emph{not} assume that $\NN$ is $(p-1)$-connected. Using tools from geometric measure theory --- namely, chains with coefficients in the $p$-th homotopy group of~$\NN$--- we associate to each map a new object that captures its topological point singularities. We provide a characterisation of strong and weak sequential closures of smooth maps in~$W^{1,p}$ in terms of their topological singularities.

V1 of this submission contained a mistake, which has been fixed in v2. The latter also contains new results