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Lipschitz homotopy groups of the Heisenberg groups. (English) Zbl 1310.57045
Let $$X$$ be a sub-Riemannian manifold and $${\pi }^ H _k (X)$$ the smooth horizontal groups and $${\pi }^ {Lip} _k (X)$$ the Lipschitz homotopy groups.
This paper answers two questions posed in [N. Dejarnette et al., “On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target”, Conform. Geom. Dyn. 18, 119–156 (2014; Zbl 1316.46034)] in the negative (see Theorems 1 and 2). The authors show that the Lipschitz and smooth horizontal homotopy groups of a space may differ. Conversely, they show that any Lipschitz map $$S^k \to H^1$$, $$\mathit{H}^ 1$$ the Heisenberg group, factors through a tree and is thus Lipschitz null-homotopic if $$k\geq 2$$ (see Theorems 3 and 5).

##### MSC:
 57T20 Homotopy groups of topological groups and homogeneous spaces 53C17 Sub-Riemannian geometry
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