  We introduce an equivalence relation on the space $W^{1,1}(\Omega;{\mathbb
S}^1)$ which classifies maps according to their "topological singularities". We
establish sharp bounds for the distances (in the usual sense and in the
Hausdorff sense) between the equivalence classes. Similar questions are
examined for the space $W^{1,p}(\Omega;{\mathbb S}^1)$ when $p>1$.
