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Brezis and Mironescu have announced several years ago that for a compactmanifold $N^n \subset \mathbb{R}^\nu$ and for real numbers $0 < s < 1$ and $1\le p < \infty$ the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on thecube with values into $N^n$ is dense with respect to the strong topology in theSobolev space $W^{s, p}(Q^m; N^n)$ when the homotopy group $\pi_{\lfloor sp\rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of thisbeautiful result is long and rather involved. Under the additional assumptionthat $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a shorter proof oftheir result. Our proof for $sp \ge 1$ is based on the existence of aretraction of $\mathbb{R}^\nu$ onto $N^n$ except for a small subset in thecomplement of $N^n$ and on the Gagliardo-Nirenberg interpolation inequality formaps in $W^{1, q} \cap L^\infty$. In contrast, the case $sp < 1$ relies on thedensity of step functions on cubes in $W^{s, p}$.

Strong approximation of fractional Sobolev maps