  We study the question: when are Lipschitz mappings dense in the Sobolev space
$W^{1,p}(M,\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with
or without boundary, while $\mathbf{H}^n$ denotes the $n$th Heisenberg group
equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in
$W^{1,p}(M,\mathbf{H}^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that
Lipschitz maps are not dense in $W^{1,p}(M,\mathbf{H}^n)$ 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 $\mathbf{H}^n$. We provide two such
constructions, one arising from complex hyperbolic geometry and the other
arising from symplectic geometry. The nondensity assertion can be interpreted
as nontriviality of the $n$th Lipschitz homotopy group of $\mathbf{H}^n$. We
initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.

Functional Analysis