Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups

Piotr Hajlasz, Armin Schikorra, and Jeremy T. Tyson

Contact the author: Please use for correspondence this email.
Submission date: 16. Aug. 2013
published in: Geometric and functional analysis, 24 (2014) 1, p. 245-268 
DOI number (of the published article): 10.1007/s00039-014-0261-z
Bibtex
MSC-Numbers: 53C17, 46E35

Abstract:
We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, π_m^Lip(H_n), in terms of properties of the classical homotopy group of the sphere, π_m(Sⁿ). As an application we provide a new simplified proof of the fact that π_n^Lip(H_n)≠0, n = 1,2,... and we prove a new result that π_4n-1^Lip(H_2n)≠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 dimM ≥ 4n and 4n- 1 ≤ p < 4n.