Magnitude homology and path homology

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