In this work we showcase the concept of a winding number and potential fields from analysis and we develop the de Rham theory, which generalizes problems and concepts of this kind to smooth manifolds. The central objects are the cohomology vector spaces, which tell us in the special case of the classical vector calculus, for example, how much is an irrotational vector field on any particular domain also a potential one. The main result is the Poincaré duality, which intuitively says that for a closed orientable $n$-dimensional manifold, the dual of a cohomology space in dimension $k$ is again a cohomology space in dimension $n - k$. We begin with a quick review of the essential basics of smooth manifolds and with the definition of differential forms, which are the source of the rich algebraic structure at the heart of de Rham theory. We generalize the Riemann integral to integration over smooth manifolds and we prove the general Stokes’ theorem. We develop the basics of de Rham theory, where we introduce the Mayer-Vietoris sequence, which is a powerful computational tool and we also show that the theory is homotopy invariant. It is here that we compute the cohomology of the euclidean space and prove the Poincaré lemma. Using these results we prove the Brouwer fixed point theorem and in the end we prove the Poincaré duality.
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