Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Let G be a topological group, and for a topological space , write for the set of isomorphism classes of principal G-bundles over . This is a contravariant functor from Top (the of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map to the pullback operation . A characteristic class c of principal G-bundles is then a natural transformation from to a cohomology functor , regarded also as a functor to Set. In other words, a characteristic class associates to each principal G-bundle in an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(fP) = fc(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology. Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers. Some important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class , and a G-bundle with characteristic classes , one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into .