论文标题
Lipschitz同型联系人组3个manifolds
Lipschitz Homotopy Groups of Contact 3-Manifolds
论文作者
论文摘要
我们使用亚riemannian的几何形状和几何测量理论的技术研究了3个脉冲，特别是建立了其Lipschitz同型组的特性。我们证明了Darboux定理的Bilipschitz版本：接触$（2N+1）$ - 具有亚riemannian结构的歧管是本地Bilipschitz，与Heisenberg Group $ \ Mathbb {H}^n $等同于Heisenberg Group $ \ Mathbb {H}^n $。然后，每个接触$（2N+1）$ - 具有子侵蚀结构的歧管纯粹是$ k $ - $ k> n $。然后，我们扩展了Dejarnette等人的结果。 (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所有较高的Lipschitz同型组都是微不足道的。因此，从Lipschitz同型组的意义上讲，触点3个manifold是$ k（π，1）$ - 空间，并具有无数生成的第一个同型组。在此过程中，我们证明，纯粹的2-不可分割的亚riemannian歧管之间的每个开放分布都嵌入在相关的第一Lipschitz同型基团上。因此，触点3-manifold的每个开放子集都决定了第一个Lipschitz同型组的无数子组。
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 $K(π,1)$-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.
