Formal manifolds: foundations, function spaces, and Poincaré's lemma

This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the notion of formal manifolds in the context of differential geometry, inspired by the notion of formal schemes in algebraic geometry. We develop the basic theory for formal manifolds, including a generalization of the theory of vector-valued distributions and generalized functions on smooth manifolds to the setting of formal manifolds. Additionally, we establish Poincaré's lemma for de Rham complexes with coefficients in formal functions, formal generalized functions, compactly supported formal densities, or compactly supported formal distributions.