Up to one approximations of sectional category and topological complexity

TL;DR

This paper introduces relative and strong relative forms of sectional category and topological complexity, showing their relationships, invariance properties, and calculating these invariants for certain fibrations and suspensions.

Contribution

It defines new variants of sectional category, establishes their properties and relationships, and computes these invariants for specific cases like Ganea fibrations and suspensions.

Findings

Strong relative category differs from sectional category by at most 1.

Homotopy pushouts do not increase strong relative category.

Strong complexity of a suspension is at most 2.

Abstract

James' sectional category and Farber's topological complexity are studied in a general and unified framework. We introduce `relative' and `strong relative' forms of the category for a map. We show that both can differ from sectional category just by 1. A map has sectional or relative category less than or equal to $n$ if, and only if, it is `dominated' (in some sense) by a map with strong relative category less than or equal to $n$. A homotopy pushout can increase sectional category but neither homotopy pushouts, nor homotopy pullbacks, can increase (strong) relative category. This makes (strong) relative category a convenient tool to study sectional category. We completely determine the sectional and relative categories of the fibres of the Ganea fibrations. As a particular case, the `topological complexity' of a space is the sectional category of the diagonal map. So it can differ from the (strong) relative category of the diagonal just by 1. We call the strong relative category of the diagonal `strong complexity'. We show that the strong complexity of a suspension is at most 2.