Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ℍn are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from S k to ℍn 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 Sk → ℍH1 factors through a tree and is thus Lipschitz null-homotopic if k ≥ 2.
ASJC Scopus subject areas
- Geometry and Topology