Lipschitz Homotopy Groups of Contact 3-Manifolds

TL;DR

This paper investigates the Lipschitz homotopy groups of contact 3-manifolds, establishing biLipschitz equivalences, unrectifiability, and the structure of their homotopy groups, revealing their complex topological and geometric properties.

Contribution

It extends known results on Lipschitz homotopy groups from the Heisenberg group to all contact 3-manifolds, showing they are $K( ext{uncountably generated } \pi_1)$ spaces.

Findings

First Lipschitz homotopy group of contact 3-manifolds is uncountably generated.

Higher Lipschitz homotopy groups of contact 3-manifolds are trivial.

Open subsets induce uncountable subgroups of the first Lipschitz homotopy group.

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.