On the path homology theory of digraphs and Eilenberg–Steenrod axioms

Abstract

In the paper we continue the investigation of the path homology theory of digraphs that was constructed in our previous papers. We prove basic theorems that are similar to the theorems of classical algebraic topology and introduce several natural constructions of digraphs which are very helpful to investigate the path homology theory. We describe relation of our results to the Eilenberg-Steenrod axiomatic of homology theory.

Path homology theory of multigraphs and quivers

“…Since ε(v 0 ) = 1 = 0, we conclude that v 0 / ∈ Im ∂. For N = 1, the same line of arguments as in [14,Proposition 2.12] shows that v i − v 0 ∈ Im ∂, which proves (4.4) in the case N = 1.…”

“…and, hence, is isomorphic to Z⊕ ( n Z). Again, the same line of arguments as in [14,Proposition 2.12] shows that Im ∂ coincides with the subgroup of Ω 0 (Q) generated by…”

Magnitude homology and Path homology

“…Here we employ the notion of homotopy for digraphs (Definition 5.2) introduced in [2] and [3]. We also show non-triviality of MH ℓ k (G) for ℓ = k in Proposition 8.11.…”

“…On the other hand, Gregor'yan-Jimenez-Muranov-S.-T. Yau and Gregor'yan-Lin-Muranov-S.-T. Yau have introduced and developed the theory of path homology of directed graphs (digraphs) ( [2], [3]). It possesses homotopy invariance in a sense, and can be considered as a pivotal object to develop a homotopy theory of digraphs.…”

“…It possesses homotopy invariance in a sense, and can be considered as a pivotal object to develop a homotopy theory of digraphs. Indeed, it is shown that path homology satisfies an analogue of the Eilenberg-Steenrod axioms in [2]. We denote the k-th reduced path homology of a digraph G by Hk (G).…”