ABSTRACT AND CLASSICAL HODGE-DE RHAM THEORY

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge-de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale a. 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 a, 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.