The (𝑛+1)-Lipschitz homotopy group of the Heisenberg group ℍⁿ

Hajłasz, Piotr

doi:10.1090/proc/13811

The ($n+1$)-Lipschitz homotopy group of the Heisenberg group $\mathbb {H}^n$
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Proc. Amer. Math. Soc. 146 (2018), 1305-1308 Request permission

Abstract:

We prove that for $n\geq 2$, the Lipschitz homotopy group $\pi _{n+1}^\textrm {{Lip}}(\mathbb {H}^n)$ $\neq 0$ of the Heisenberg group $\mathbb {H}^n$ is nontrivial.

References

Luigi Ambrosio and Bernd Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527–555. MR 1800768, DOI 10.1007/s002080000122
Zoltán M. Balogh and Katrin S. Fässler, Rectifiability and Lipschitz extensions into the Heisenberg group, Math. Z. 263 (2009), no. 3, 673–683. MR 2545863, DOI 10.1007/s00209-008-0437-z
Zoltán M. Balogh, Piotr Hajłasz, and Kevin Wildrick, Weak contact equations for mappings into Heisenberg groups, Indiana Univ. Math. J. 63 (2014), no. 6, 1839–1873. MR 3298724, DOI 10.1512/iumj.2014.63.5411
Noel Dejarnette, Piotr Hajłasz, Anton Lukyanenko, and Jeremy T. Tyson, On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target, Conform. Geom. Dyn. 18 (2014), 119–156. MR 3226622, DOI 10.1090/S1088-4173-2014-00267-7
Tobias Ekholm, John Etnyre, and Michael Sullivan, Non-isotopic Legendrian submanifolds in $\Bbb R^{2n+1}$, J. Differential Geom. 71 (2005), no. 1, 85–128. MR 2191769
Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823
Piotr Hajłasz, Armin Schikorra, and Jeremy T. Tyson, Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups, Geom. Funct. Anal. 24 (2014), no. 1, 245–268. MR 3177382, DOI 10.1007/s00039-014-0261-z
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
Valentino Magnani, Unrectifiability and rigidity in stratified groups, Arch. Math. (Basel) 83 (2004), no. 6, 568–576. MR 2105335, DOI 10.1007/s00013-004-1057-4
P. Pansu, Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin, preprint, 2016, arXiv:1604.06333.
Stefan Wenger and Robert Young, Lipschitz extensions into jet space Carnot groups, Math. Res. Lett. 17 (2010), no. 6, 1137–1149. MR 2729637, DOI 10.4310/MRL.2010.v17.n6.a12
Stefan Wenger and Robert Young, Lipschitz homotopy groups of the Heisenberg groups, Geom. Funct. Anal. 24 (2014), no. 1, 387–402. MR 3177387, DOI 10.1007/s00039-014-0264-9