Path homology of digraphs without multisquares and its comparison with homology of spaces

TL;DR

This paper develops a basis for path homology of digraphs without multisquares, compares it with space homology, and shows differences depending on the field characteristic, highlighting limitations of topological space models.

Contribution

It constructs explicit bases for path homology of certain digraphs and analyzes how the Euler characteristic varies with field characteristic, revealing fundamental differences from classical space homology.

Findings

Constructed bases for path homology of digraphs without multisquares.

Demonstrated Euler characteristic dependence on field characteristic.

Proved no topological space can match path homology over both Z and Z/2Z.

Abstract

For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $\Omega_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $\Omega_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $\chi^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $\Omega_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $\chi^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic depends on whether the characteristic of $\mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$