Inductive construction of path homology chains

TL;DR

This paper introduces an inductive method to construct elements of the path homology chain complex for digraphs, enabling better understanding and computation of path homology across various coefficient rings.

Contribution

It presents a novel inductive approach using face multihypergraphs to construct path homology chains, applicable over different coefficient rings, and addresses an open question on path Euler characteristic.

Findings

Inductive elements generate the entire chain complex over finite fields.

Inductive elements generate low-dimensional chains over integers and rationals.

Path Euler characteristic can vary arbitrarily with coefficient field choice.

Abstract

Path homology plays a central role in digraph topology and GLMY theory more general. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $\Omega_n(G;R)$ from elements in the proceeding two dimensions. This proceeds via the formation of what we call upper and lower \emph{extensions}, that are parametrised by certain labeled multihypergraphs which we introduce and call \emph{face multihypergraphs}. When the coefficient ring $R$ is a finite field the inductive elements we construct generate $\Omega_*(G;R)$. With integral or rational coefficients, the inductive elements generate at least $\Omega_i(G;R)$ for $i=0,1,2,3$. Since in low dimensions the inductive elements extended over labeled multigraphs coincide with naturally occurring generating sets up to sign, they are excellent candidates to reduce to a basis. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of field coefficients. In particular, answering an open question posed by Fu and Ivanov.