zbMATH — the first resource for mathematics

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).

57T20 Homotopy groups of topological groups and homogeneous spaces
53C17 Sub-Riemannian geometry
Full Text: DOI arXiv
[1] Ambrosio, L.; Kirchheim, B., Rectifiable sets in metric and Banach spaces, Math. Ann., 318, 527-555, (2000) · Zbl 0966.28002
[2] D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry, Graduate Studies in Mathematics, Vol. 33. American Mathematical Society, Providence (2001). · Zbl 0981.51016
[3] Balogh, Z.M.; Fässler, K.S., Rectifiability and Lipschitz extensions into the Heisenberg group, Math. Z., 263, 673-683, (2009) · Zbl 1177.53032
[4] N. DeJarnette, P. Hajłasz, A. Lukyanenko, and J. Tyson. On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target. (2011), arXiv:1109.4641. · Zbl 1316.46034
[5] Freudenthal, H., Über die klassen der sphärenabbildungen I, Große Dimensionen. Compositio Math., 5, 299-314, (1938) · Zbl 0018.17705
[6] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, Progr. Math., Vol. 144, Birkhäuser, Basel (1996), pp. 79-323. · Zbl 0864.53025
[7] Guth, L., Contraction of areas vs. topology of mappings, Geom. and Func. Anal., 23, 1804-1902, (2013) · Zbl 1283.55003
[8] P. Hajłasz, A. Schikorra, and J. Tyson. Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups. To appear in Geom. and Func. Anal. arXiv:1301.4978. · Zbl 1332.55007
[9] P. Hajłasz and J. Tyson. Hölder and Lipschitz Peano cubes and highly regular surjections between Carnot groups. preprint.
[10] R. Kaufman. A singular map of a cube onto a square. J. Differential Geom. (4)14 (1979), 593-594 (1981). · Zbl 0463.57011
[11] Kirchheim, B., Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121, 113-123, (1994) · Zbl 0806.28004
[12] Magnani, V., Unrectifiability and rigidity in stratified groups, Arch. Math. (Basel), 83, 568-576, (2004) · Zbl 1062.22019
[13] P. Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) (1)129 (1989), 1-60. · Zbl 0678.53042
[14] S. Rigot and S. Wenger. Lipschitz non-extension theorems into jet space Carnot groups. Int. Math. Res. Not. IMRN (18) (2010), 3633-3648. · Zbl 1203.53028
[15] Wenger, S., Characterizations of metric trees and Gromov hyperbolic spaces, Math. Res. Lett., 15, 1017-1026, (2008) · Zbl 1162.53034
[16] G.W. Whitehead. Elements of homotopy theory. Graduate Texts in Mathematics, Vol. 61. Springer, New York (1978). · Zbl 0406.55001
[17] Wenger, S.; Young, R., Lipschitz extensions into jet space Carnot groups, Math. Res. Lett., 17, 1137-1149, (2010) · Zbl 1222.53037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.