m-Contiguity Distance

TL;DR

This paper introduces the m-contiguity distance as a new discrete invariant for analyzing homotopical complexity in simplicial complexes, with applications to Lusternik-Schnirelmann category and topological complexity.

Contribution

It develops the theoretical framework of m-contiguity distance, establishing its properties and demonstrating its use in defining discrete homotopical invariants.

Findings

Established properties of m-contiguity distance under subdivision and composition.

Defined m-simplicial Lusternik-Schnirelmann category as a special case.

Proved the natural emergence of m-discrete topological complexity from the theory.

Abstract

In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. The fundamental properties of $m$-contiguity distance are established, including its behaviour under barycentric subdivision, under compositions, and a categorical product inequality. As applications of this theory, we define the $m$-simplicial Lusternik-Schnirelmann category and the $m$-discrete topological complexity, proving that each arises naturally as a special case of $m$-contiguity distance.