Abstract and classical Hodge-de Rham theory

In previous work, with Bartholdi and Schick [1], the authors developed a Hodgede Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L ² complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian. © 2012 World Scientific Publishing Company.