Magnitude homology and path homology
Yasuhiko Asao
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 $\geqslant 5$ .