Geometric and Functional Analysis

, Volume 24, Issue 1, pp 245–268 | Cite as

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

  • Piotr Hajłasz
  • Armin Schikorra
  • Jeremy T. Tyson


We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, \({\pi_m^{\rm Lip}(\mathbb{H}_n)}\), in terms of properties of the classical homotopy group of the sphere, \({\pi_m(\mathbb{S}^n)}\). As an application we provide a new simplified proof of the fact that \({\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}\), and we prove a new result that \({\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}\) 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 \({W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}\) when \({\dim \mathcal{M} \geq 4n}\) and 4n−1 ≤  p < 4n.

Mathematics Subject Classification (2000)

Primary 53C17 Secondary 46E35 55Q40 55Q25 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Piotr Hajłasz
    • 1
  • Armin Schikorra
    • 2
  • Jeremy T. Tyson
    • 3
  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Max-Planck Institut MiS LeipzigLeipzigGermany
  3. 3.Department of MathematicsUniversity of Illinois atUrbana-ChampaignUrbanaUSA

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