An improved dense class in Sobolev spaces to manifolds

We consider the strong density problem in the Sobolev space $W^{s,p}(Q^{m};\mathscr{N})$ of maps with values into a compact Riemannian manifold $\mathscr{N}$. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by $\mathscr{N}$-valued maps that are smooth outside of a finite union of $(m -\lfloor sp \rfloor - 1)$-planes. Our main result establishes the strong density in $W^{s,p}(Q^{m};\mathscr{N})$ of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu. In the special case where $\mathscr{N}$ has a sufficiently simple topology and for some values of $s$ and $p$, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of $s$ and $p$ in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique.