Discrete geometry and de-Rham cohomology of graphs

Abstract

We develop a theory of discrete spaces which are built by gluing coordinate patches together into an atlas, much as a manifold is built from charts. Graphs are the main examples, and an atlas on a graph gives it a geometric structure analogous to that of a manifold. This structure leads to a cohomology theory, which we show to be a discrete analogue of de Rham cohomology. We define a "discrete coordinatized space" as a cubical set equipped with the Alexandrov topology, on which differential forms become sheaves. When the atlas is regular, the de Rham cohomology agrees with the sheaf cohomology of the constant sheaf, just as in the smooth case. We compute it for several examples, including a torus graph and a graph resembling a sphere, and recover the cohomology of the corresponding surfaces.