2024/05/08 by Sergei O. Ivanov, Lev Mukoseev, Ivanov, Sergei O. +1 · 1 voice
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paper · pdf · doi:10.48550/arxiv.2405.04748
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 MH2,k(G). This allows us to define a combinatorial condition, denoted by (V_ℓ), which is equivalent to the vanishing of \rm MH2,k(G,ℤ) for all k>ℓ. In particular, diagonal digraphs satisfy (V2). 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 σG is Koszul over any field, and if and only if G satisfies (V2) and the path cochain algebra Ω^\bullet(G) is Koszul over any field. To provide a source of examples of digraphs, we study the extended Hasse diagram GK 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 GK in terms of the homology of M. As a corollary, we obtain that if K is a triangulation of a closed manifold M, then GK is diagonal if and only if M is a homology sphere.