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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.10420 |
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| _version_ | 1866910334006067200 |
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| author | Perry, Daniel |
| author_facet | Perry, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$ Perry, Daniel Metric Geometry 53C17, 28A75 (Primary) 57K33, 54E35 (Secondary) 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. |
| title | The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$ |
| topic | Metric Geometry 53C17, 28A75 (Primary) 57K33, 54E35 (Secondary) |
| url | https://arxiv.org/abs/2402.10420 |