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

We introduce an operator 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 submanifoldN of the Euclidean spaceRm, and coincides with the distributional Jacobian in caseN is a sphere. More precisely, the range ofS is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we useS to characterise strong limits of smooth, 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 N-well potentials.