Topological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces

Abstract

We introduce an operator $\\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$.\n\t\t\t\t In particular, this operator is defined on maps that take values in a closed submanifold $\\mathcal{N}$ of the Euclidean space $\\mathbb{R}^m$, and coincides with the distributional Jacobian \n\t\t\t\t in case $\\mathcal{N}$ is a sphere. More precisely, the range of $\\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\\mathbf{S}$ to characterise strong limits of smooth, $\\mathcal{N}$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with $\\mathcal{N}$-well potentials.