Characteristic classes

Abstract

Many topological spaces exist as the total spaces of real vector bundles over some base spaces. Topological properties like Hausdorffness, connectedness, the first axiom of countability, path connectedness, local connectedness of the total space of a vector bundle can be studied by knowing these topological properties of the base space. We want to classify vector bundles up to vector bundle isomorphism. It is very difficult to classify vector bundles using topological properties. We would be using algebraic topology concepts like singular homology and singular cohomology of base space to classify vector bundles. We have used axioms of Stiefel-Whitney classes to classify some vector bundles.