Equivariant parametrized topological complexity

TL;DR

This paper introduces an equivariant version of parametrized topological complexity, extending existing concepts to symmetric settings and providing bounds and computations for specific equivariant fibrations.

Contribution

It defines and studies an equivariant analogue of parametrized topological complexity, extending known results to the equivariant context and computing it for certain fibrations.

Findings

Established fibrewise equivariant homotopy invariance.

Derived bounds involving equivariant category.

Computed equivariant topological complexity for specific fibrations.

Abstract

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