Flat forms in metric spaces

Abstract

In a metric space, we define flat forms by means of tuples of Lipschitz functions multiplied by Borel measurable functions, and use them to represent flat cochains. The representation, which extends Wolfe's theorem to metric spaces, is functorial on the category of metric spaces and Lipschitz maps. It provides flat cochains and flat chains with well-behaved cup and cap products. On compact Lipschitz manifolds, the cohomology of flat cochains is naturally isomorphic to the Čech cohomology with real coefficients — a version of De Rham's theorem.