Sequential parametrized topological complexity and related invariants
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911949013385216 |
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| author | Farber, Michael Oprea, John |
| author_facet | Farber, Michael Oprea, John |
| contents | Parametrized motion planning algorithms \cite{CFW} have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions. The latter are viewed as parameters and constitute part of the input of the algorithm. The concept of sequential parametrized topological complexity ${\sf TC}_r[p:E\to B]$ is a measure of the complexity of such algorithms. It was studied in \cite{CFW, CFW2} for $r=2$ and in \cite{FP} for $r\ge 2$. In this paper we analyse the dependence of the complexity ${\sf TC}_r[p:E\to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$-space. Our main results estimate ${\sf TC}_r[p:E\to B]$ in terms of certain invariants of the bundle and the action on the fibre. Moreover, we also obtain estimates depending on the base and the fibre. Finally, we develop a calculus of sectional categories featuring a new invariant ${\sf secat}_f[p:E\to B]$ which plays an important role in the study of sectional category of towers of fibrations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_01990 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sequential parametrized topological complexity and related invariants Farber, Michael Oprea, John Algebraic Topology 55M30 Parametrized motion planning algorithms \cite{CFW} have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions. The latter are viewed as parameters and constitute part of the input of the algorithm. The concept of sequential parametrized topological complexity ${\sf TC}_r[p:E\to B]$ is a measure of the complexity of such algorithms. It was studied in \cite{CFW, CFW2} for $r=2$ and in \cite{FP} for $r\ge 2$. In this paper we analyse the dependence of the complexity ${\sf TC}_r[p:E\to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$-space. Our main results estimate ${\sf TC}_r[p:E\to B]$ in terms of certain invariants of the bundle and the action on the fibre. Moreover, we also obtain estimates depending on the base and the fibre. Finally, we develop a calculus of sectional categories featuring a new invariant ${\sf secat}_f[p:E\to B]$ which plays an important role in the study of sectional category of towers of fibrations. |
| title | Sequential parametrized topological complexity and related invariants |
| topic | Algebraic Topology 55M30 |
| url | https://arxiv.org/abs/2209.01990 |