Homology, Homotopy and Applications

Volume 17 (2015)

Number 2

On cohomology theory of (di)graphs

Pages: 383 – 398

DOI: https://dx.doi.org/10.4310/HHA.2015.v17.n2.a18

Authors

An Huang (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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 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.

Keywords

igraph cohomology, CW complex, homotopy, sheaf cohomology for (di)graphs

2010 Mathematics Subject Classification

05C10, 55P65

Published 3 December 2015