Let $n\ge 2$, $s$>$0$ and $p\ge 1$ be such that $1\le sp$<$2$. We prove that for each map $u\in W^{s,p}({\mathbb S}^n ; {\mathbb S}^1)$ one can find some $\varphi\in W^{s,p}({\mathbb S}^n ; {\mathbb R})$ and some $v\in W^{sp, 1}({\mathbb S}^n ; {\mathbb S}^1)$ such that $u=e^{\imath\varphi}\, v$. This yields a decomposition of $u$ into a part, $e^{\imath\varphi}$, that has a lifting in $W^{s,p}$, and a map, $v$, "smoother" than $u$ but which need not have a lifting within $W^{s,p}$. Our result generalizes a previous one of Bourgain and Brezis (J. Amer. Math. Soc. 2003), which corresponds to $s=1/2$ and $p=2$. As a consequence of the above factorization $u=e^{\imath\varphi}\, v$, we find an intuitive proof of the existence of the Jacobian $J u$ of maps $u\in W^{s, p}({\mathbb S}^n ; {\mathbb S}^1)$, result originally due to Bourgain, Brezis and the author (Comm. Pure Appl. Math. 2005). By completing a result of Bousquet (J. Anal. Math. 2007), we characterize the distributions of the form $J u$.