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 MH2,(G,Z) is a free abelian group for any , and to describe its rank. This allows us to give a condition, denoted by (V₂), equivalent to vanishing of MH2,(G,Z) for >2. Recall that a digraph is called diagonal, if its magnitude homology is concentrated in diagonal degrees. Using the condition ( V₂), 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 σ G is Koszul for any ground field; and if and only if G satisfies (V₂) and the path cochain algebra Ω^•(G) is Koszul for any ground field. Besides, we show that the path cochain algebra Ω^•(G) is quadratic for any $G.$ To obtain a source of examples of (non-)diagonal digraphs, we study the extended Hasse diagram GK 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 GK 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 GK is diagonal if and only if M is a homology sphere.