On the topological complexity of directed parametrized motion planning
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915649874296832 |
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| author | Datta, Sutirtha Daundkar, Navnath Sarkar, Abhishek |
| author_facet | Datta, Sutirtha Daundkar, Navnath Sarkar, Abhishek |
| contents | We introduce and study a parametrized analogue of the directed topological complexity, originally developed by Goubault, Farber, and Sagnier. We establish the fibrewise basic dihomotopy invariance of directed parametrized topological complexity and explore its relationship with the parametrized topological complexity. In addition, we introduce the concept of the directed Lusternik-Schnirelmann (LS) category, prove its basic dihomotopy invariance, and investigate its connections with both directed topological complexity and directed parametrized topological complexity. We further investigate additional properties of our invariant and examine its connections with several other invariants that arise naturally in the context of topological robotics. Moreover, we compute the directed parametrized topological complexity of the Hopf fibrations and the Fadell-Neuwirth fibrations having specific directed fibration structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the topological complexity of directed parametrized motion planning Datta, Sutirtha Daundkar, Navnath Sarkar, Abhishek Algebraic Topology 55M30, 55S40, 55R80 We introduce and study a parametrized analogue of the directed topological complexity, originally developed by Goubault, Farber, and Sagnier. We establish the fibrewise basic dihomotopy invariance of directed parametrized topological complexity and explore its relationship with the parametrized topological complexity. In addition, we introduce the concept of the directed Lusternik-Schnirelmann (LS) category, prove its basic dihomotopy invariance, and investigate its connections with both directed topological complexity and directed parametrized topological complexity. We further investigate additional properties of our invariant and examine its connections with several other invariants that arise naturally in the context of topological robotics. Moreover, we compute the directed parametrized topological complexity of the Hopf fibrations and the Fadell-Neuwirth fibrations having specific directed fibration structures. |
| title | On the topological complexity of directed parametrized motion planning |
| topic | Algebraic Topology 55M30, 55S40, 55R80 |
| url | https://arxiv.org/abs/2504.06049 |