Discrete homotopic distance between Lipschitz maps

TL;DR

This paper introduces a discrete homotopic distance for Lipschitz maps that generalizes classical topological invariants, providing a new tool for classifying maps in spaces with complex hole structures.

Contribution

It defines a discrete homotopic distance that extends key topological concepts and proves its invariance under discrete homotopy, offering a flexible classification framework.

Findings

Discrete homotopic distance generalizes Lusternik-Schnirelmann category and topological complexity.

The distance effectively classifies maps in spaces with many holes.

Invariance under discrete homotopy is established.

Abstract

In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance $D_r$ generalizes key concepts such as the discrete Lusternik-Schnirelmann category $\text{cat}_r$ and the discrete topological complexity $\text{TC}_r$. Furthermore, we prove that $D_r$ is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying $s$-Lipschitz maps, loops, and paths based on the choice of $r$.