Relative LS categories and higher topological complexities of maps

TL;DR

This paper introduces and compares new homotopy invariants related to the relative LS categories and higher topological complexities of maps, providing bounds and relationships with existing invariants.

Contribution

It defines the higher and weak higher topological complexities of a map and explores their properties, bounds, and relation to known topological complexities.

Findings

Introduced higher and weak higher topological complexities of maps.

Established bounds for these invariants.

Compared new invariants with existing topological complexities.

Abstract

In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the `higher topological complexity' and `weak higher topological complexity' of a map. Each of them are homotopy invariants. We discuss some lower and upper bounds of these in invariants and compare them with previously known `topological complexities' of a map.