Lipschitz Homotopy Groups of Contact 3-Manifolds

DOI10.14321/REALANALEXCH.47.1.1598582300arXiv2008.06928MaRDI QIDQ6347171

Daniel Perry

Publication date: 16 August 2020

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 $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $mathbb{H}^n$ 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 $mathbb{H}^1$ 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 $K(pi,1)$-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.

Mathematics Subject Classification ID

Length, area, volume, other geometric measure theory (28A75) Contact manifolds (general theory) (53D10) Sub-Riemannian geometry (53C17) Homotopy groups of special types (55Q70) Contact structures in 3 dimensions (57K33)

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