Chern Classes and Stiefel-Whitney Classes

Abstract

We consider Chern classes, Stiefel-Whitney classes, and the Euler class from an axiomatic point of view. The uniqueness of the classes follows from the splitting principle, and the existence is derived using the bundle of projective spaces associated with a vector bundle and the Leray-Hirsch theorem. These results could be obtained by using obstruction theory or the cohomology of the classifying space. Finally, we consider the relation between characteristic classes and the Thorn isomorphism.