Homology, Homotopy and Applications
Volume 17 (2015)
On cohomology theory of (di)graphs
Pages: 383 – 398
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 that was studied recently. The homotopy type of the CW complex turns out to be independent of the choice of basis. The construction is functorial, and it makes many of the recently proved properties of digraph cohomology and homotopy manifest. Furthermore, one gets an expected formula for the cup product of forms on a digraph. On the other hand, we present an approach using sheaf theory to reformulate (di)graph cohomologies. The investigation of the digraph path cohomology from this sheaf theory framework leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex.
igraph cohomology, CW complex, homotopy, sheaf cohomology for (di)graphs
2010 Mathematics Subject Classification