Parametrized topological complexity of bundles of real projective spaces, I

Abstract In this paper we study parametrized topological complexity of bundles of real projective spaces which arise as projectivisations of vector bundles. We develop algebraic machinery for computing lower bounds for the parametrized topological complexity of projective bundles based on theory of Stiefel - Whitney characteristic classes. We establish sharp upper bounds for the parametrized topological complexity of projective bundles improving the general upper bounds. Combining the lower and the upper bounds we compute explicitly many specific examples.