LIPSCHITZ HOMOTOPY GROUPS OF CONTACT 3-MANIFOLDS

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 H-n with its Carnot-Caratheodory 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 H-1 to an arbitrary contact 3-manifold endowed with a Carnot-Caratheodory 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 subRiemannian 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.