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

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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 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 $$\mathscr {N}$$ -well potentials.