Given a smooth manifold M, we can talk about the de Rham cohomology and cohomology with compact support of M. There is a third kind of cohomology that can be defined for any vector bundle, called Compact Vertical Cohomology. In this talk, we’ll define the compact vertical cohomology of vector bundles and explore its relation with the cohomology of M to get the Thom isomorphism and Thom Class, which then becomes the main ingredient to prove the existence of Euler Class, the most relevant characteristic class for oriented vector bundles.
