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Minimal Path and Acyclic Models in the Path Complex

Xinxing Tang, Shing-Tung Yau

TL;DR

Tang and Yau study the path complex of a digraph under the strongly regular condition, introducing minimal paths as acyclic-models for path (co)homology. They prove a structure theorem that decomposes minimal paths into canonical s-regular components with a controlled boundary, and establish an acyclicity result for the path homology of the path-suppport digraphs via Mayer–Vietoris arguments. These results enable a clean derivation of the cup product skew-symmetry and support broader cohomological constructions (e.g., Künneth-type formulas) within the GLMY path-homology framework. The work bridges combinatorial path structures with topological tools, advancing digraph (co)homology and its algebraic structures.

Abstract

In this paper, firstly, we will study the structure of the path complex $(Ω_*(G;\Z),\partial)$ of a digraph $G$ via the $\Z$-generators of $Ω_*(G,\Z)$ under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.

Minimal Path and Acyclic Models in the Path Complex

TL;DR

Tang and Yau study the path complex of a digraph under the strongly regular condition, introducing minimal paths as acyclic-models for path (co)homology. They prove a structure theorem that decomposes minimal paths into canonical s-regular components with a controlled boundary, and establish an acyclicity result for the path homology of the path-suppport digraphs via Mayer–Vietoris arguments. These results enable a clean derivation of the cup product skew-symmetry and support broader cohomological constructions (e.g., Künneth-type formulas) within the GLMY path-homology framework. The work bridges combinatorial path structures with topological tools, advancing digraph (co)homology and its algebraic structures.

Abstract

In this paper, firstly, we will study the structure of the path complex of a digraph via the -generators of under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal -paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.
Paper Structure (27 sections, 21 theorems, 197 equations, 40 figures)

This paper contains 27 sections, 21 theorems, 197 equations, 40 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main theorems
  4. Organizations
  5. Path Complex of Digraphs
  6. Minimal Path
  7. Definitions and basic properties
  8. The structure theorem
  9. Proof of Theorem \ref{['structurethm']}
  10. The case $|E_1|=1$
  11. The case $|E_1|=2$
  12. Induction on $|E_1|$
  13. Minimal $3$-paths and some properties
  14. Homotopy theory of digraphs
  15. Examples: minimal $3$-paths
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $P\in\Omega_n(G;\mathbb Z)$ be a minimal path with the starting vertex $S$ and ending vertex $E$, $\mathop{\mathrm{Supp}}\nolimits(P)$ be its supporting digraph and $d_S$, $d_E$ be two distance functions (see the definition in Subsection defbasicpropsubsection). (1) Let $S_1=d_S^{-1}(1)$ and $E_ where (2) For any $v\in d_E^{-1}(k)\cap\mathop{\mathrm{Supp}}\nolimits(P)$, in $\mathop{\mathrm{Su

Figures (40)

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  • ...and 35 more figures

Theorems & Definitions (76)

  • Theorem 1.1: Theorem \ref{['structurethm']}
  • Theorem 1.2: Theorem \ref{['acyclicresult']}
  • Theorem 1.3: Theorem \ref{['skewsymm']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • ...and 66 more