Discrete homotopic distance between Lipschitz maps
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917781962752000 |
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| author | Hoseinzadeh, Elahe Mirebrahimi, Hanieh Torabi, Hamid Babaee, Ameneh |
| author_facet | Hoseinzadeh, Elahe Mirebrahimi, Hanieh Torabi, Hamid Babaee, Ameneh |
| contents | In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance $D_r$ generalizes key concepts such as the discrete Lusternik-Schnirelmann category $\text{cat}_r$ and the discrete topological complexity $\text{TC}_r$. Furthermore, we prove that $D_r$ is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying $s$-Lipschitz maps, loops, and paths based on the choice of $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14376 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete homotopic distance between Lipschitz maps Hoseinzadeh, Elahe Mirebrahimi, Hanieh Torabi, Hamid Babaee, Ameneh Algebraic Topology In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance $D_r$ generalizes key concepts such as the discrete Lusternik-Schnirelmann category $\text{cat}_r$ and the discrete topological complexity $\text{TC}_r$. Furthermore, we prove that $D_r$ is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying $s$-Lipschitz maps, loops, and paths based on the choice of $r$. |
| title | Discrete homotopic distance between Lipschitz maps |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2409.14376 |