Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi }_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, ${\pi }_m(S^n)$. As an application we provide a new simplified proof of the fact that ${\pi }_n^{Lip}(H_n)\mathrm{\ne }0$, $n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, and we prove a new result that ${\pi }_{4n-1}^{Lip}(H_{2n})\mathrm{\ne }0$ for $n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$ The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $W^{1,p}(M,H_{2n})$ when $dimM\ge 4n$ and $4n-1\le p<4n$.