Magnitude homology and path homology

Abstract

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials MH k ℓ ( G ) ⟶ MH k − 1 ℓ − 1 ( G ) $\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between magnitude homologies of a digraph G $G$ , which make them chain complexes. Then we show that its homology MH k ℓ ( G ) $\mathcal {MH}^{\ell }_k(G)$ 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 MH k k ( G ) $\mathcal {MH}^{k}_k(G)$ is isomorphic to the reduced path homology H ∼ k ( G ) $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology MH k ℓ ( G ) $\operatorname{MH}^{\ell }_k(G)$ , and the second page is isomorphic to our homology MH k ℓ ( G ) $\mathcal {MH}^{\ell }_k(G)$ . As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that H ∼ k ( g ) = 0 $\tilde{H}_k(g) = 0$ for k ⩾ 2 $k \geqslant 2$ and H ∼ 1 ( g ) ≠ 0 $\tilde{H}_1(g) \ne 0$ if any edges of an undirected graph g $g$ is contained in a cycle of length ⩾ 5 $\geqslant 5$ .