NAT SMALE
Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA
STEVE SMALE
University Distinguished Professor.
City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Received 8 January 2011
Accepted 10 January 2011

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 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 L² 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.

Hodge theory; de Rham theory; cohomology
AMSC numbers: 58A12, 58A14