Gromov-Hausdorff distances from simply connected geodesic spaces to the circle
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913313767555072 |
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| author | Martín, Saúl Rodríguez |
| author_facet | Martín, Saúl Rodríguez |
| contents | We prove that the Gromov-Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than $\fracπ{4}$. We also prove that this bound is tight through the construction of a simply connected geodesic space $\mathrm{E}$ which attains the lower bound $\fracπ{4}$. We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov-Hausdorff distance between two geodesic metric spaces $(X, d_X)$ and $(Y, d_Y )$ has to be in order for $π_1(X)$ and $π_1(Y)$ to be isomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gromov-Hausdorff distances from simply connected geodesic spaces to the circle Martín, Saúl Rodríguez Metric Geometry Algebraic Topology Differential Geometry We prove that the Gromov-Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than $\fracπ{4}$. We also prove that this bound is tight through the construction of a simply connected geodesic space $\mathrm{E}$ which attains the lower bound $\fracπ{4}$. We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov-Hausdorff distance between two geodesic metric spaces $(X, d_X)$ and $(Y, d_Y )$ has to be in order for $π_1(X)$ and $π_1(Y)$ to be isomorphic. |
| title | Gromov-Hausdorff distances from simply connected geodesic spaces to the circle |
| topic | Metric Geometry Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2404.05153 |