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Given a compact manifold $N^n \subset \mathbb{R}^\nu$, $s \ge 1$ and $1 \le p< \infty$, we prove that the class of smooth maps on the cube with values into$N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we proveweak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$simply connected. The proofs are based on the existence of a retraction of$\mathbb{R}^\nu$ onto $N^n$ except for a small subset of $N^n$ and on apointwise estimate of fractional derivatives of composition of maps in $W^{s,p} \cap W^{1, sp}$.

Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$