Higher Analogues of Discrete Topological Complexity

TL;DR

This paper introduces the n-th discrete topological complexity, explores its properties and relations with other topological invariants, and establishes bounds with concrete examples.

Contribution

It defines higher discrete topological complexity and analyzes its properties and bounds, extending classical topological complexity concepts to discrete settings.

Findings

Lower bound of n-discrete topological complexity established

Relation between discrete and classical topological complexity demonstrated

Example provided for strictness of the lower bound

Abstract

In this paper, we introduce the n-th discrete topological complexity and study its properties such as its relation with simplicial Lusternik-Schnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of $n-$discrete topological complexity which is given by the n-th usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.