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

Given a compact manifold $N^n\subset {\mathbb{R}}^{\nu }$, $s\ge 1$ and $1 \le p < \infty$,weprovethattheclassofsmoothmapsonthecubewithvaluesinto$N^n$isstronglydenseinthefractionalSobolevspace$W^{s, p}(Q^m; N^n)$when$N^n$is$\lfloor sp \rfloor$simplyconnected\mathrm{.}For$sp$ integer, we prove weak 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 a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.