Parametrized topological complexity of bundles of real projective spaces, I

Analysis of motion algorithms for autonomous systems operating under variable external conditions leads to the concept of parametrized topological complexity \cite{CFW}. In \cite{CFW}, \cite{CFW2} the parametrized topological complexity was computed in the case of the Fadell - Neuwirth bundle which is pertinent to algorithms of collision free motion of many autonomous systems in ${\Bbb R}^d$ avoiding collisions with multiple obstacles. The parametrized topological complexity of sphere bundles was studied in detail in \cite{FW}, \cite{FW2}, \cite{FP}. In this paper we make the next step by studying parametrized topological complexity of bundles of real projective spaces which arise as projectivisations of vector bundles. This leads us to new problems of algebraic topology involving theory of characteristic classes and geometric topology. We establish sharp upper bounds for parametrized topological complexity ${\sf TC}[p:E\to B]$ improving the general upper bounds. We develop algebraic machinery for computing lower bounds for ${\sf TC}[p:E\to B]$ based on the Stiefel - Whitney characteristic classes. Combining the lower and the upper bounds we compute explicitly many specific examples.