Parametrised topological complexity of group epimorphisms

TL;DR

This paper introduces a new invariant called parametrised topological complexity for group epimorphisms, extending existing bounds and providing applications such as alternative computations and homotopy invariance results.

Contribution

It extends bounds for topological complexity to parametrised cases and applies these to group epimorphisms, including new computational methods and invariance results.

Findings

Parametrised topological complexity is an invariant of group epimorphisms.

Bounds for topological complexity extend to parametrised cases.

Alternative computation method for planar Fadell–Neuwirth fibrations.

Abstract

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell--Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases.