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

We introduce an operator \(\mathbf {S}\) on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold \(\mathscr {N}\) of the Euclidean space \(\mathbb {R}^m\), and coincides with the distributional Jacobian in case \(\mathscr {N}\) 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, \(\mathscr {N}\)-valued maps with respect to Sobolev norms, extending a result by Pakzad and Riviere. 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 \(\mathscr {N}\)-well potentials.