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The fundamental group and the magnitude-path spectral sequence of a directed graph

Daisuke Kishimoto, Yichen Tong

TL;DR

This work builds a bridge between higher $r$-fundamental groupoids of directed graphs and the magnitude-path spectral sequence (MPSS) by establishing a Hurewicz-type correspondence between $\pi_1^r(X)$ and $H_1(F_r RC_*(X))$, and by analyzing the MPSS with $E^1_{p,q}\cong MH_{p+q}^p(X)$ and $E^2_{p,0}\cong PH_p(X)$. It develops a combinatorial and algebraic framework for $\Pi_1^r(X)$, extends the theory to $r=\infty$, and shows how the MPSS differentials are detected by the $oldsymbol{\nabla}_r$-based combinatorics, notably yielding a nontrivial $d^{r-1}$-differential. The Seifert–van Kampen theorem is extended to $r$-separable pairs and $r$-cofibrations, providing pushouts and Mayer–Vietoris sequences for the MPSS, thereby linking topological-like decompositions to spectral information in directed graphs. Overall, the paper deepens the connection between directed-graph homotopy, spectral sequences, and homology theories, enabling refined invariants that capture directional structure beyond the classical fundamental group.

Abstract

The fundamental group of a directed graph admits a natural sequence of quotient groups called $r$-fundamental groups, and the $r$-fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude-path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. In this paper, we study relations of the $r$-fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem.

The fundamental group and the magnitude-path spectral sequence of a directed graph

TL;DR

This work builds a bridge between higher -fundamental groupoids of directed graphs and the magnitude-path spectral sequence (MPSS) by establishing a Hurewicz-type correspondence between and , and by analyzing the MPSS with and . It develops a combinatorial and algebraic framework for , extends the theory to , and shows how the MPSS differentials are detected by the -based combinatorics, notably yielding a nontrivial -differential. The Seifert–van Kampen theorem is extended to -separable pairs and -cofibrations, providing pushouts and Mayer–Vietoris sequences for the MPSS, thereby linking topological-like decompositions to spectral information in directed graphs. Overall, the paper deepens the connection between directed-graph homotopy, spectral sequences, and homology theories, enabling refined invariants that capture directional structure beyond the classical fundamental group.

Abstract

The fundamental group of a directed graph admits a natural sequence of quotient groups called -fundamental groups, and the -fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude-path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. In this paper, we study relations of the -fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem.

Paper Structure

This paper contains 19 sections, 51 theorems, 101 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. $r$-Fundamental groupoid
  3. Directed graph
  4. Nerve
  5. Algebraic description
  6. Combinatorial description
  7. Path
  8. Combinatorial description
  9. Homotopy invariance
  10. Product formula
  11. Hurewicz theorem
  12. Magnitude-path spectral sequence
  13. Proof of Theorem \ref{['main 1']}
  14. Hurewicz map
  15. Differential in the MPSS
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(X,x_0)$ be a connected pointed directed graph. Then there is a commutative diagram \xymatrix{ \pi_1^{2}(X,x_0)\ar[r]\ar[d]^{h^{2}}&\pi_1^3(X,x_0)\ar[r]\ar[d]^{h^3}&\pi_1^4(X,x_0)\ar[d]^{h^4}\ar[r]&\cdots\\ E^2_{1,0}(X)\ar[r]&E^3_{1,0}(X)\ar[r]&E^4_{1,0}(X)\ar[r]&\cdots }such th which is identified with abelianization, where $\mathrm{RH}_*(X)$ denotes the reachability homology

Figures (7)

  • Figure 1: The directed graphs $X_1,X_2,X_3,X_4$
  • Figure 2: The directed graph $\Gamma_r$ for $r\ge 2$
  • Figure 3: $C$-homotopies
  • Figure 4: The directed graphs $X$ and $Y$
  • Figure 5: Deformation of $\tilde{f}_a$
  • ...and 2 more figures

Theorems & Definitions (114)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 104 more