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- Title
LIPSCHITZ HOMOTOPY GROUPS OF CONTACT 3-MANIFOLDS.
- Authors
Perry, Daniel
- Abstract
We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group H n with its Carnot-Carathéodory metric. Then each contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is purely k -unrectifiable for k > n. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of H 1 to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a K (π, 1) -space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.
- Subjects
HOMOTOPY groups; GEOMETRIC measure theory
- Publication
Real Analysis Exchange, 2022, Vol 47, Issue 1, p75
- ISSN
0147-1937
- Publication type
Article
- DOI
10.14321/realanalexch.47.1.1598582300