Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups

Abstract

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 $\pi_n^{Lip}(H_n)\neq 0$, $n=1,2,...$, and we prove a new result that $\pi_{4n-1}^{Lip}(H_{2n})\neq 0$ for $n=1,2,...$ 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 $dim M\geq 4n$ and $4n-1\leq p<4n$.