  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 $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the
distributional Jacobian in case $N$ is a sphere. 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, $N$-valued maps with respect to Sobolev norms, extending a result by
Pakzad and Rivi\`ere. 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.
