Different types of topological complexity based on higher homotopic distance

Abstract

We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduce the generalized version of the relative topological complexity of a topological pair with respect to 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. Later, by defining the parametrized topological complexity via the homotopic distance, we present some estimates on the higher setting of this concept. Finally, we give some important examples of the parametrized topological complexities of fiber bundles with their fibers.