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Sequential parametrized topological complexity and related invariants

Michael Farber and John Oprea

Algebraic & Geometric Topology 24 (2024) 1755–1780

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 TCr[p: E 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 2. We analyze the dependence of the complexity TCr[p: E B] on an initial bundle with structure group G and on its fibre X viewed as a G–space. Our main results estimate TCr[p: E 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 secatf[p: E B] which plays an important role in the study of sectional category of towers of fibrations.

topological complexity, sectional category, fibration
Mathematical Subject Classification
Primary: 55M30
Received: 5 September 2022
Revised: 6 January 2023
Accepted: 16 January 2023
Published: 28 June 2024
Michael Farber
School of Mathematical Sciences
Queen Mary University of London
United Kingdom
John Oprea
Department of Mathematics
Cleveland State University
Cleveland, OH
United States

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