2020/08/16 by Daniel J. Perry, Perry, Daniel
Computer Science · Mathematics · #53C17 #53D10 (Secondary) #55Q70 #57K33 (Primary) 28A75 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2008.06928
openalex publication_date 2020/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (2n+1)-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group ℍn with its \cc metric. Then each contact (2n+1)-manifold endowed with a sub-Riemannian structure is purely k-unrectifiable for k>n. We then extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young (arXiv:1210.6943 [math.GT]) on the Lipschitz homotopy groups of ℍ1 to an arbitrary contact 3-manifold endowed with a \cc 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.