Different Types of Topological Complexity on Higher Homotopic Distance

TL;DR

This paper explores advanced concepts in topological complexity and homotopic distance, providing new inequalities and estimates relevant to robot motion planning and higher homotopic structures.

Contribution

It introduces generalized versions of relative topological complexity using homotopic distance and establishes inequalities linking these concepts with classical topological invariants.

Findings

Derived inequalities between topological complexity and Lusternik-Schnirelmann category.

Established estimates for parametrized topological complexity in higher settings.

Extended the framework of topological complexity to include homotopic distance and generalized pairs.

Abstract

We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and the homotopic distance. With these concepts, we give some inequalities including the topological complexity and the Lusternik-Schnirelmann category, the most important parts of the study of robot motion planning in topology. Finally, by defining the parametrised topological complexity via the homotopic distance, we present some estimates on the higher setting of this concept.