FROM HOMOLOGY TO DE RHAM COHOMOLOGY: An Expository Journey through the Topology of Smooth Manifolds

This exposition traces a path from the geometric intuition of shapes and holes to the analytic framework of de Rham cohomology. Beginning with the classical motivations of Euler, Poincaré, and Gauss-Bonnet, we see how algebraic invariants emerged to capture global topological features invisible to local geometry. After a brief review of smooth manifolds, orientation, and Stokes' theorem, we develop the language of homology theory: simplicial complexes, chains, cycles, and boundaries. These ideas lead naturally to cohomology, where algebraic duality and the cup product reveal a richer structure that connects topology to broader areas of mathematics. The second half of the exposition introduces differential forms and the generalized Stokes' theorem, culminating in de Rham's insight that analytic invariants defined by closed and exact forms coincide with topological invariants defined by singular cohomology. The de Rham theorem, Mayer-Vietoris sequence, and functorial properties are explored in detail, alongside explicit computations on familiar manifolds such as spheres, tori, punctured Euclidean space, projective space, and surfaces of higher genus. By weaving together geometry, algebra, and analysis, this work aims to present de Rham cohomology not as an isolated construction, but as a natural culmination of the search for invariants that bridge local calculus and global topology. The result is a coherent framework in which classical results find their proper home, while new insights emerge into the structure of smooth manifolds.