Parametrized topological complexity of sphere bundles

TL;DR

This paper investigates the parametrized topological complexity of sphere bundles, providing bounds and explicit computations, revealing that it can vary widely depending on the bundle's properties.

Contribution

It introduces bounds for parametrized topological complexity in sphere bundles and computes it explicitly in various cases, highlighting its potential to be arbitrarily large.

Findings

Upper bounds involve sectional categories of fibrations

Lower bounds are based on characteristic classes

Parametrized topological complexity can be arbitrarily large

Abstract

Parametrized motion planning algorithms have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.