Geometric and Functional Analysis

, Volume 24, Issue 1, pp 387–402 | Cite as

Lipschitz Homotopy Groups of the Heisenberg Groups



Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group \({\mathbb{H}^n}\) are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from Sk to \({\mathbb{H}^n}\) which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map \({S^k \to \mathbb{H}^1}\) factors through a tree and is thus Lipschitz null-homotopic if \({k \geq 2}\) .


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© Springer Basel 2014

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de FribourgFribourgSwitzerland
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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