Sequential parametrized topological complexity and related invariants

Farber, Michael; Oprea, John

doi:10.2140/agt.2024.24.1755

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Sequential parametrized topological complexity and related invariants

Michael Farber and John Oprea

Algebraic & Geometric Topology 24 (2024) 1755–1780

DOI: 10.2140/agt.2024.24.1755

Abstract

Parametrized motion planning algorithms 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 ${T C}_{r} [p : E \to B]$ is a measure of the complexity of such algorithms. It was studied by Cohen, Farber and Weinberger (2021, 2022) for $r = 2$ and by Farber and Paul (2022) for $r \geq 2$ . We analyze the dependence of the complexity ${T C}_{r} [p : E \to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$ –space. Our main results estimate ${T C}_{r} [p : E \to B]$ 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 ${s e c a t}_{f} [p : E \to B]$ which plays an important role in the study of sectional category of towers of fibrations.

Keywords

topological complexity, sectional category, fibration

Mathematical Subject Classification

Primary: 55M30

References

Publication

Received: 5 September 2022

Revised: 6 January 2023

Accepted: 16 January 2023

Published: 28 June 2024

Authors

Michael Farber
School of Mathematical Sciences Queen Mary University of London London United Kingdom

John Oprea
Department of Mathematics Cleveland State University Cleveland, OH United States

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Volume 24, issue 3 (2024)