$\mathbb S^1$-Valued Sobolev maps

We describe the structure of the space $W^{s,p}(\mathbb S^n;\mathbb S^1)$, where $0<s<\infty$, $1\le p<\infty$. According to the values of $s$, $p$ and $n$, maps in $W^{s,p}(\mathbb S^n;\mathbb S^1)$ can either be characterised by their phases or by a couple (singular set, phase). Here are two examples: $W^{1/2,6}(\mathbb S^3;\mathbb S^1)=\{e^{\imath\varphi}\colon\varphi\in W^{1/2,6}+W^{1,3}\}$, $W^{1/2,3}(\mathbb S^2;\mathbb S^1)\approx D\times\{e^{\imath\varphi}\colon\varphi\in W^{1/2,3}+W^{1,3/2}\}$. In the second example, $D$ is an appropriate set of infinite sums of Dirac masses. The sense of $\approx$ will be explained in the paper.
The presentation is based on the papers of H.-M. Nguyen [22], of the author [20], and on a joint forthcoming paper of H. Brezis, H.-M. Nguyen, and the author [15].