Persistence and Topological Complexity
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866918125805502464 |
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| author | Mémoli, Facundo Zhou, Ling |
| author_facet | Mémoli, Facundo Zhou, Ling |
| contents | Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Persistence and Topological Complexity Mémoli, Facundo Zhou, Ling Algebraic Topology 55M30, 55N31, 53C23 Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology. |
| title | Persistence and Topological Complexity |
| topic | Algebraic Topology 55M30, 55N31, 53C23 |
| url | https://arxiv.org/abs/2506.17888 |