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 𝓁 𝑘 (𝐺) ⟶ MH 𝓁−1 𝑘−1 (𝐺) between magnitude homologies of a digraph 𝐺, which make them chain complexes. Then we show that its homology  𝓁 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  𝑘 𝑘 (𝐺) is isomorphic to the reduced path homology H𝑘 (𝐺) also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology MH 𝓁 𝑘 (𝐺), and the second page is isomorphic to our homology  𝓁 𝑘 (𝐺). As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that H𝑘 (g) = 0 for 𝑘 ⩾ 2 and H1 (g) ≠ 0 if any edges of an undirected graph g is contained in a cycle of length ⩾ 5.

Causal order complex and magnitude homotopy type of metric spaces

“…Then the Euler characteristic χ(X) is determined by the number of critical cells of each dimension. This yields (1). By Proposition 4.14, we also have (2).…”

“…It has been established that for closed sets in Euclidean space, the magnitude homology reflects properties such as convexity [21] and the diameter of a hole [15]. In recent years, the notion of magnitude homology group has been studied in relation to various research topics such as path homology of graphs [1], random graphs [2], topological invariants of point clouds [10,24], magnitude cohomology [13] etc. However, the information captured by the magnitude homology for general metric spaces, including finite ones, is not well understood.…”

Categorifying connected domination via graph überhomology

“…We can reinterpret the magnitude homology groups as follows -see also [Asa22,Section 2] for the case of directed graphs. Given a non-negative integer k, let Λ k (G; R) := R (v 0 , .…”

“…Rather than on magnitude itself, here we focus on its categorification: magnitude (co)homology, as defined by Hepworth and Willerton [HW17,Hep22], see also [LS21]. As with magnitude, also its categorification has attracted some attention -see, for instance, the recent papers [Gu18,AHK21,SS21,Asa22]. In this context, categorification means to associate to a numerical (or polynomial) invariant a whole homology theory, whose Euler characteristic recovers the original invariant.…”

“…This concludes the proof. From Remark 3.4 and [Asa22] follows that (reduced) path homology, as introduced in [GLMY20], appears as the diagonal in the second page of a spectral sequence whose 0-th page features magnitude chain groups. Shifting to later pages in the spectral sequence is obtained by subsequent subquotients.…”