Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups
Piotr Hajlasz, Armin Schikorra, and Jeremy T. Tyson
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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
MSC-Numbers: 53C17, 46E35
We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, πmLip(Hn), in terms of properties of the classical homotopy group of the sphere, πm(Sn). As an application we provide a new simplified proof of the fact that πnLip(Hn)≠0, n = 1,2,... and we prove a new result that π4n-1Lip(H2n)≠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 W1,p(M,H2n) when dimM ≥ 4n and 4n- 1 ≤ p < 4n.