Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

Piotr Hajłasz, Armin Schikorra, Jeremy T. Tyson

Research output: Contribution to journalArticle

Abstract

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 (Formula presented), and we prove a new result that (Formula presented) 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 dim M ≥ 4n and 4n-1 ≤ p < 4n.

Original languageEnglish (US)
Pages (from-to)245-268
Number of pages24
JournalGeometric and Functional Analysis
Volume24
Issue number1
DOIs
StatePublished - Feb 2014

Keywords

  • 55Q25
  • 55Q40
  • Primary 53C17
  • Secondary 46E35

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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