Formal aspects on parametrized topological complexity and its pointed version

TL;DR

This paper extends the concept of parametrized topological complexity to more general fiberwise spaces, introduces a pointed version, and explores conditions under which they coincide, enhancing the theoretical framework of topological complexity.

Contribution

It generalizes parametrized topological complexity to non-Hurewicz fiberwise spaces and introduces a pointed variant, providing conditions for their equivalence.

Findings

Extended parametrized topological complexity to broader fiberwise spaces.

Introduced the pointed version of parametrized topological complexity.

Provided conditions under which the two notions agree.

Abstract

The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fibrewise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally we give sufficient conditions so that both notions agree.