Parametrized topological complexity of sphere bundles

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