Motion Planning on One-Dimensional Peano Continua
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908614152683520 |
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| author | Brazas, Jeremy Pavesic, Petar |
| author_facet | Brazas, Jeremy Pavesic, Petar |
| contents | We study the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces. We define both invariants as lengths of suitable closed filtrations, as opposed to a more common definition based on open covers. Our main results provide a precise description of $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a 1-dimensional Peano continuum $X$ in terms of the wildness rank of $X$. A surprising consequence is that $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a general 1-dimensional space $X$ can be arbitrarily high, which is in stark contrast with the analogous results for 1-dimensional CW-complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22901 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Motion Planning on One-Dimensional Peano Continua Brazas, Jeremy Pavesic, Petar Algebraic Topology 55M30, 54F50 We study the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces. We define both invariants as lengths of suitable closed filtrations, as opposed to a more common definition based on open covers. Our main results provide a precise description of $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a 1-dimensional Peano continuum $X$ in terms of the wildness rank of $X$. A surprising consequence is that $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a general 1-dimensional space $X$ can be arbitrarily high, which is in stark contrast with the analogous results for 1-dimensional CW-complexes. |
| title | Motion Planning on One-Dimensional Peano Continua |
| topic | Algebraic Topology 55M30, 54F50 |
| url | https://arxiv.org/abs/2510.22901 |