Inductive construction of path homology chains and the structure of $Ω_3(G;R)$

Path homology plays a central role in digraph topology and GLMY theory more generally. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $\Omega_n(G;R)$ from elements in the preceding two dimensions. This proceeds via the formation of what we call upper and lower extensions, that are parametrised by certain labelled multigraphs which we introduce and call face multigraphs. The inductive elements we construct generate $\Omega_*(G;R)$ when $R$ has characteristic $2$. With characteristic $0$ coefficients, the inductive elements at least generate $\Omega_i(G;R)$ for $i=0,1,2,3$. In low dimensions, the inductive elements coincide with the natural generators, and when the digraph contains no multisquares, the inductive elements coincide with the basis elements produced by Fu and Ivanov. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct explicit generators of $\Omega_3(G;R)$ for a ring $R$ of characteristic $0$ or $2$, answering an open question posed by Grigor'yan. Several universal coefficient statements for path homology are obtained as a byproduct.