On cohomology theory of (di) graphs

An Huang Shing-Tung Yau

Algebraic Topology and General Topology mathscidoc:1912.43693

arXiv preprint arXiv:1409.6194
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.
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  title={On cohomology theory of (di) graphs},
  author={An Huang, and Shing-Tung Yau},
  booktitle={arXiv preprint arXiv:1409.6194},
An Huang, and Shing-Tung Yau. On cohomology theory of (di) graphs. In arXiv preprint arXiv:1409.6194. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205026580515257.
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