Strong density in Sobolev spaces to manifolds

In striking contrast with what happens to classical Sobolev spaces, the space of smooth maps with values into a compact manifold $N$ does not need to be dense in the space of $N$-valued $W^{s,p}$ maps.

In this talk, I will review the history of this problem, culminating with Bethuel's theorem for $W^{1,p}$ and its extensions to $W^{s,p}$, which gives a necessary and sufficient condition on the topology of the target manifold $N$ in order to smooth maps to be dense as well as a suitable class of almost smooth maps that is always dense.

I will finish with a new result about an improved dense class in Sobolev spaces to manifolds.