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2022 Lipschitz Homotopy Groups of Contact 3-Manifolds
Daniel Perry
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Real Anal. Exchange 47(1): 75-96 (2022). DOI: 10.14321/realanalexch.47.1.1598582300


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 Carnot-Carathéodory 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. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of $\mathbb H^1$ 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 $K(\pi,1)$-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.


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Daniel Perry. "Lipschitz Homotopy Groups of Contact 3-Manifolds." Real Anal. Exchange 47 (1) 75 - 96, 2022.


Published: 2022
First available in Project Euclid: 13 June 2022

Digital Object Identifier: 10.14321/realanalexch.47.1.1598582300

Primary: 53C17 , 57K33
Secondary: 28A75 , 53D10 , 55Q70

Keywords: contact manifolds , geometric measure theory , Heisenberg group , Lipschitz homotopy groups , sub-Riemannian manifold , unrectifiability

Rights: Copyright © 2022 Michigan State University Press


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Vol.47 • No. 1 • 2022
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