Cohomologia em variedades e o teorema de de Rham

Abstract

This work presents two algebraic approaches to the study of differentiable manifolds: de Rham cohomology and singular cohomology. Initially, based on chain complex theory, the abstract concepts of homological and cohomological theories are introduced. Next, de Rham cohomology - a tool based on the quotient between closed and exact differential forms - and singular cohomology, constructed as the algebraic dual of singular homology, are presented. In the first approach, the de Rham groups of known manifolds are rigorously calculated. In the second, theoretical construction is prioritized, given the complexity inherent in the direct calculation of examples in this theory. Finally, de Rham’s theorem is stated and proven, which establishes that the de Rham cohomology groups of a manifold are isomorphic to its singular cohomology groups, integrating the two perspectives and enabling the obtaining of topological results via analytical tools.