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Discrete Topological Complexities of Simplical Maps

Melih İs, İsmet Karaca

Abstract

In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unravel the interplay between topological complexity, discrete geometric structures, and computational efficiency. Moreover, we examine the properties of the discrete TC number in higher dimensions and its relationship with scat. We also touch on the basic properties of the notion of higher contiguity distance, and show that it is possible to consider discrete TC computations in a simpler sense.

Discrete Topological Complexities of Simplical Maps

Abstract

In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unravel the interplay between topological complexity, discrete geometric structures, and computational efficiency. Moreover, we examine the properties of the discrete TC number in higher dimensions and its relationship with scat. We also touch on the basic properties of the notion of higher contiguity distance, and show that it is possible to consider discrete TC computations in a simpler sense.
Paper Structure (9 sections, 19 theorems, 40 equations, 2 figures)

This paper contains 9 sections, 19 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complex and Simplicial Homotopy
  4. Simplicial Fibration and Discrete TC Number
  5. Schwarz Genus Form and Higher Contiguity Distance
  6. Contiguity Distance Form
  7. Higher Discrete TC Of A Simplicial Fibration
  8. scat-Related Results
  9. Conclusion

Key Result

Proposition 2.3

TerCalMacVil:2021 Assume that $\varphi$, $\varphi^{'} : L \rightarrow L^{'}$ are two simplicial maps. Then $\varphi \sim \varphi^{'}$ if and only if there is at least one $m \geq 1$ and one simplicial map with the property $G(\sigma,0) = \varphi$ and $G(\sigma,m) = \varphi^{'}$ for any $\sigma \in L$.

Figures (2)

  • Figure 3.1: A simplicial map $\varphi : L \rightarrow L^{'}$.
  • Figure 3.2: The subcomplexes $L_{0}$ and $L_{1}$ of $L^{'}$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 3.1
  • Example 3.2
  • ...and 32 more