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

TL;DR

This paper explores advanced algebraic and homological invariants of directed graphs, introducing $r$-fundamental groups and a spectral sequence linking magnitude and path homology, revealing deeper structural insights.

Contribution

It introduces the $r$-fundamental groups and the magnitude-path spectral sequence, connecting them via classical theorems to enhance understanding of directed graph topology.

Findings

$r$-fundamental groups capture properties beyond the fundamental group.

The magnitude-path spectral sequence links magnitude and path homology.

Relations between these invariants are established through the Hurewicz and Seifert-van Kampen theorems.

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.