We study the question: When are Lipschitz mappings dense in the Sobolev space
$W^{1,p}(M,H^n)$? Here $M$ denotes a compact Riemannian manifold with or
without boundary, while $H^n$ denotes the $n$th Heisenberg group equipped with
a sub-Riemannian metric. We show that Lipschitz maps are dense in
$W^{1,p}(M,H^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz
maps are not dense in $W^{1,p}(M,H^n)$ if $\dim M \ge n+1$ and $n\le p<n+1$.
The proofs rely on the construction of smooth horizontal embeddings of the
sphere $S^n$ into $H^n$.