Equivariant parametrized topological complexity

Abstract

In this paper, we define and study an equivariant analogue of Cohen, Farber andWeinberger's parametrized topological complexity. We show that several results inthe non-equivariant case can be extended to the equivariant case. For example, weestablish the fibrewise equivariant homotopy invariance of the sequential equivariantparametrized topological complexity. We obtain several bounds on sequentialequivariant topological complexity involving the equivariant category. We also obtainthe cohomological lower bound and the dimension-connectivity upper bound on thesequential equivariant parametrized topological complexity. In the end, we use theseresults to compute the sequential equivariant parametrized topological complexity ofequivariant Fadell-Neuwirth fibrations and some equivariant fibrations involvinggeneralized projective product spaces.