The Künneth Formula and Applications

The de Rham cohomology of a manifold is a homotopy invariant that expresses basic topological information about smooth manifolds. The q-th de Rham cohomology of the n-dimensional Euclidean space is the vector space defined by the closed q-forms over the exact q-forms. Furthermore, the support of a continuous function f on a topological space X is the closure of the set on which f is nonzero. The result of restricting the definition of the de Rham cohomology to functions with compact support is called the de Rham cohomology with compact support, or the compact cohomology. The concept of cohomology can also be expanded to general manifolds through constructions such as the Mayer-Vietoris Sequence. The Künneth Formula in differential topology relates the cohomology of the product of two manifolds to the cohomologies of the individual manifolds through the tensor product. In this project, we provide a proof of the Künneth Formula both for de Rham cohomology and compact cohomology and then show several applications