On de Rham's theorem

Abstract

THE CLASSICAL theorem of de Rham states, briefly, that the (real singular) cohomology of a (compact, smooth) manifold is naturally isomorphic to the cohomology derived from differential forms. There are many proofs of this in the literature, notably that by de Rham himself [4], that by Weil [6] (“pre-sheaf theory”), that by Schwartz [5] (“axiomatic”), and that by sheaf theory (Leray, e.g., in Godement’s book [2]). The purpose of the following lines is to give a different proof, using the handle decomposition of a manifold.