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m-Contiguity Distance

Nilay Ekiz Yazici, Nursultan Kuanyshov, Ayse Borat

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 invariance under barycentric subdivision, behavior 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.

m-Contiguity Distance

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 invariance under barycentric subdivision, behavior 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.
Paper Structure (11 sections, 22 theorems, 46 equations)

This paper contains 11 sections, 22 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Contiguity class
  4. Moore Paths
  5. The path complex
  6. $m$-homotopic distance
  7. $m-$ Contiguity Distance
  8. Categorical product
  9. $m$-dimensional sectional category (Svarc genus)
  10. $m$-Discrete Topological complexity
  11. Acknowledgement

Key Result

Theorem 2.1

FTGCMVV Let $K$ be an arbitrary simplicial complex. Then the simplicial map is a simplicial finite fibration, where the maps $\alpha$ and $\omega$ are those defined in Definition initialfinal.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.1
  • ...and 51 more