On cohomology theory of (di)graphs

TL;DR

This paper constructs a CW complex from digraphs to establish a canonical isomorphism with their path cohomology, revealing new insights into their topological and algebraic properties.

Contribution

It introduces a CW complex model for digraph cohomology, proving independence of homotopy type from basis choice and connecting it with sheaf theory and Poincare lemma.

Findings

CW complex is canonically isomorphic to digraph path cohomology

Homotopy type of the CW complex is basis-independent

Simple formula for cup product on digraphs

Abstract

To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expected but perhaps combinatorially subtle properties of the digraph cohomology and homotopy proved very recently \cite{GLMY2}. Furthermore, one gets a very simple expected formula for the cup product of forms on the digraph. On the other hand, we present an approach of using sheaf theory to reformulate (di)graph cohomologies. The investigation of the path cohomology from this framework, leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex.