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Abstract:We study magnitude homology of digraphs, with a particular focus on diagonal digraphs, i.e., digraphs whose magnitude homology is concentrated on the diagonal. For any digraph $G$, we provide a complete description of the second magnitude homology ${\rm MH}_{2,k}(G)$. This allows us to define a combinatorial condition, denoted by $(\mathcal{V}_\ell)$, which is equivalent to the vanishing of ${\rm MH}_{2,k}(G,\mathbb{Z})$ for all $k>\ell$. In particular, diagonal digraphs satisfy $(\mathcal{V}_2)$. As a corollary, we obtain that the 2-dimensional CW-complex obtained from a diagonal undirected graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $\sigma G$ is Koszul over any field, and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $\Omega^\bullet(G)$ is Koszul over any field. To provide a source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$ of a pure simplicial complex $K$. For a triangulation $K$ of a topological manifold $M$, we express the non-diagonal part of the magnitude homology of $\hat G_K$ in terms of the homology of $M$. As a corollary, we obtain that if $K$ is a triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere.

Submission history

From: Sergei Ivanov Olegovich [view email]
[v1] Wed, 8 May 2024 01:29:57 UTC (39 KB)
[v2] Sun, 28 Jul 2024 13:18:34 UTC (40 KB)
[v3] Tue, 30 Jul 2024 01:35:32 UTC (40 KB)
[v4] Sun, 1 Dec 2024 08:58:38 UTC (35 KB)
[v5] Sat, 18 Jul 2026 09:14:49 UTC (36 KB)