On diagonal digraphs, Koszul algebras and triangulations of homology spheres

We present the magnitude homology of a finite digraph $G$ as a certain subquotient of its path algebra. We use this to prove that the second magnitude homology group ${\rm MH}_{2,\ell}(G,\mathbb{Z})$ is a free abelian group for any $\ell$, and to describe its rank. This allows us to give a condition, denoted by $(\mathcal{V}_2)$, equivalent to vanishing of ${\rm MH}_{2,\ell}(G,\mathbb{Z})$ for $\ell>2.$ Recall that a digraph is called diagonal, if its magnitude homology is concentrated in diagonal degrees. Using the condition $(\mathcal V_2),$ we show that the GLMY-fundamental group of a diagonal (undirected) graph is trivial. In other words, the two-dimensional CW-complex obtained from a diagonal 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 for any ground field; and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $\Omega^\bullet(G)$ is Koszul for any ground field. Besides, we show that the path cochain algebra $\Omega^\bullet(G)$ is quadratic for any $G.$ To obtain a source of examples of (non-)diagonal digraphs, we study the extended Hasse diagram $\hat G_K$ of a simplicial complex $K$. For a combinatorial triangulation $K$ of a piecewise-linear manifold $M,$ we express the non-diagonal part of the magnitude homology of $\hat G_K$ via the homology of $M$. As a corollary we obtain that, if $K$ is a combinatorial triangulation of a closed piecewise-linear manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere.