The “good” concepts from the theory of manifolds translate naturally to the discrete framework of graphs, or abelian groups viewed as “discrete vector spaces”. In this article, we focus on a discrete analog of manifolds and their de Rham cohomology.

Starting from the well-known algebraic topologists version of cohomology of finite abelian groups, we recall the abstract algebraic framework. This is reminiscent of the Möbiusinversion in the convolution algebra, dual to the group ring used to define the above cohomology. The usual interpretation as the fundamental theorem of calculus, in terms of finite differences and sums, allows to present it as a discrete analog of de Rham cohomology, in a geometric framework.

The extension of the theory from Cayley graphs of abelian groups to graphs provides a discrete version of manifold theory and associated de Rham theory of differential forms.

de Rham cohomology, finite abelian group, Cayley graph.