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Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$,$0\textless{}s\textless{}\infty$ and $1\le p\textless{}\infty$. We prove that$C^\infty(\overline\Omega\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ;{\mathbb S}^1)$ except when $1\le sp\textless{}2$ and $n\ge 2$. The mainingredient is a new approximation method for $W^{s,p}$-maps when$s\textless{}1$. With $0\textless{}s\textless{}1$, $1\le p\textless{}\infty$and $sp\textless{}n$, $\Omega$ a ball, and $N$ a general compact connectedmanifold, we prove that $C^\infty(\overline\Omega \, ; N)$ is dense in$W^{s,p}(\Omega \, ; N)$ if and only if $\pi\_{[sp]}(N)=0$. This supplementsanalogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce andVan Schaftingen when $s=2,3,\ldots$ [General domains $\Omega$ have been treatedby Hang and Lin when $s=1$; our approach allows to extend their result to$s\textless{}1$.] The case where $s\textgreater{}1$, $s\not\in{\mathbb N}$, isstill open.

Density in $W^{s,p}(\Omega ; N)$