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The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$

Daniel Perry

Abstract

The goal of this paper is to define and inspect a metric version of the universal path space and study its application to purely 2-unrectifiable spaces, in particular the Heisenberg group $\mathbb{H}^1$. The construction of the universal Lipschitz path space, as the metric version is called, echoes the construction of the universal cover for path-connected, locally path-connected, and semilocally simply connected spaces. We prove that the universal Lipschitz path space of a purely 2-unrectifiable space, much like the universal cover, satisfies a unique lifting property, a universal property, and is Lipschitz simply connected. The existence of such a universal Lipschitz path space of $\mathbb{H}^1$ will be used to prove that $π_{1}^{\text{Lip}}(\mathbb{H}^1)$ is torsion-free in a subsequent paper.

The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$

Abstract

The goal of this paper is to define and inspect a metric version of the universal path space and study its application to purely 2-unrectifiable spaces, in particular the Heisenberg group . The construction of the universal Lipschitz path space, as the metric version is called, echoes the construction of the universal cover for path-connected, locally path-connected, and semilocally simply connected spaces. We prove that the universal Lipschitz path space of a purely 2-unrectifiable space, much like the universal cover, satisfies a unique lifting property, a universal property, and is Lipschitz simply connected. The existence of such a universal Lipschitz path space of will be used to prove that is torsion-free in a subsequent paper.
Paper Structure (12 sections, 23 theorems, 74 equations, 2 figures)

This paper contains 12 sections, 23 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of the universal Lipschitz path space
  3. Background
  4. Metric structure on the universal path space
  5. Endpoint projection and an action of $\pi_{1}^{\text{Lip}}(M,p_0)$
  6. Unique lifting property and the universal path space
  7. Lifts of paths to the universal path space
  8. Unique path lifiting and unique lifting properties
  9. Unique lifting property and Lipschitz simply connected universal path space
  10. The universal property of the universal path space
  11. The universal path space of tree-like homotopy spaces
  12. Unique path lifting for the universal path space of a tree-like homotopy space

Key Result

Theorem 1.1

For a based purely 2-unrectifiable space $(M,p_0)$, there exists a based Lipschitz simply connected length space $(\mathcal{P}_{M},[p_0])$ and a based Lipschitz map $\pi:(\mathcal{P}_{M},[p_0])\rightarrow (M,p_0)$ that satisfies the unique lifting property.

Figures (2)

  • Figure 1: Image of $\beta_n^{1,j}$.
  • Figure 2: $\beta_n^j\simeq \beta_{n+1}\left|_{I_{n}^{j}}\right.$, which is comprised of $\beta_{n+1}^{2j-1}$ and $\beta_{n+1}^{2j}$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3: BiLipschitz Theorem of Darboux, Corollary 23 in perry2020lipschitz
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 43 more