Relative sectional category revisited

TL;DR

This paper revisits the concept of relative sectional category, exploring its properties and demonstrating how it unifies various homotopic numerical invariants such as topological complexity and homotopic distance.

Contribution

It provides a comprehensive investigation of relative sectional category and shows its role in unifying multiple homotopic invariants in algebraic topology.

Findings

Unifies several homotopic invariants under the framework of relative sectional category.

Establishes properties and relationships of the relative sectional category.

Connects concepts like topological complexity and homotopic distance through this unified approach.

Abstract

The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its properties. We seek to uncover how the relative sectional category unifies several homotopic numerical invariants found in recent literature. These include the topological complexity of maps according to Murillo-Wu or Scott, relative topological complexity as defined by Farber, and homotopic distance for continuous maps in the sense of Mac\'{\i}as-Virg\'os and Mosquera-Lois, among others.