2022/09/28 by Yasuhiko Asao · 4 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · doi:10.1112/blms.12734
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 ) MHℓ k(G) \longrightarrow MHℓ -1k-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 MHℓ 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 MHkk(G) is isomorphic to the reduced path homology H ∼ k ( G ) Hk(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 ) MHℓ k(G) , and the second page is isomorphic to our homology MH k ℓ ( G ) \mathcal MHℓ 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 Hk(g) = 0 for k ⩾ 2 k \geqslant 2 and H ∼ 1 ( g ) ≠ 0 H1(g) ≠ 0 if any edges of an undirected graph g g is contained in a cycle of length ⩾ 5 \geqslant 5 .