General Theory of Characteristic Classes

Abstract

Using vector bundles over a space X, we are able to associate with X various sets which reflect some of the topological properties of X, for example, Vect _F (X), the semigroup of isomorphism classes of F-vector bundles; Vect _F ⁿ(X), the set of isomorphism classes of n-dimensional vector bundles over X; and K _F (X), the group completion of Vect _F (X). We view a characteristic class as a morphism defined on one of the confunctors Vect _F , Vect _F ⁿ, or K _F with values in a cohomology cofunctor. In several important cases, we are able to give a complete description of all characteristic classes. We conclude with a discussion of properties of the Chern character.