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

Abstract

For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $\hat{I}{\text{c◯}}_n(G;\mathbb{F})$ for $n\ge 0$, generalising the basis of $\hat{I}{\text{c◯}}_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $\ddot{I}{‡}^{\mathbb{F}}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $\hat{I}{\text{c◯}}_{\bullet }(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $\ddot{I}{‡}^{\mathbb{F}}(G)$. 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 $H_{\ast }(X;\mathbb{K})\cong {\mathrm{PH}}_{\ast }(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$