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From a more topological viewpoint, I think it is worth reading the famous paper of Dennis Sullivan, Infinitesimal computations in topology (Publications Mathématiques de l'IHES, 1977), in which he builds algebraic models of manifolds from their differential forms. He explains how to construct recursively such models and gives several examples. He obtains from this construction deep results about homotopy types and diffeomorphism types of manifolds.