Sequential parametrized topological complexity and related invariants

Farber, Michael¹ ; Oprea, John²

¹ School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
² Department of Mathematics, Cleveland State University, Cleveland, OH, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1755-1780

Résumé

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.

Détail

DOI : 10.2140/agt.2024.24.1755

Keywords: topological complexity, sectional category, fibration