Magnitude homology and Path homology

TL;DR

This paper explores the relationship between magnitude homology and path homology in digraphs, establishing chain complex structures, homotopy invariance, and spectral sequences, with applications to understanding graph topology.

Contribution

It introduces differentials making magnitude homology a chain complex, proves homotopy invariance, and links diagonal magnitude homology to reduced path homology, advancing the homology theory of digraphs.

Findings

Diagonal magnitude homology is isomorphic to reduced path homology.

Constructed a spectral sequence connecting magnitude and our homology.

Proved triviality of reduced path homology for certain graphs.

Abstract

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\mathcal MH}^{\ell}_k(G)$ is non-trivial and homotopy invariant in the context of `homotopy theory of digraphs' developed by Grigor'yan--Muranov--S.-T. Yau et al (G-M-Ys in the following). It is remarkable that the diagonal part of our homology ${\mathcal MH}^{k}_k(G)$ is isomorphic to the reduced path homology $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology ${\mathrm MH}^{\ell}_k(G)$, and the second page is isomorphic to our homology ${\mathcal MH}^{\ell}_k(G)$. As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that $\tilde{H}_k(g) = 0$ for $k \geq 2$ and $\tilde{H}_1(g) \neq 0$ if any edges of an undirected graph $g$ is contained in a cycle of length $\geq 5$.