Lipschitz homotopy groups of the Heisenberg groups

TL;DR

This paper explores the properties of Lipschitz and horizontal maps into Heisenberg groups, revealing differences between Lipschitz and smooth homotopy groups and establishing conditions for Lipschitz null-homotopy.

Contribution

It demonstrates that Lipschitz and smooth horizontal homotopy groups can differ and provides conditions under which Lipschitz maps are null-homotopic.

Findings

Lipschitz and horizontal maps are abundant from n-dimensional spaces to Heisenberg groups.

Some maps have Lipschitz fillings, contrasting with previous results on smooth fillings.

Any Lipschitz map from a sphere to  factors through a tree and is null-homotopic for dimensions .

Abstract

Lipschitz and horizontal maps from an $n$-dimensional space into the $(2n+1)$-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj{\l}asz-Lukyanenko-Tyson constructed horizontal maps from $S^k$ to $\H^n$ 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 $S^k\to \H^1$ factors through a tree and is thus Lipschitz null-homotopic if $k\ge 2$.