Characteristic classes

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not. In other words, characteristic classes are global invariants which 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 Stiefel and Whitney about vector fields on manifolds.

Let G be a topological group, and for a topological space X, write b_G(X) for the set of isomorphism classes of principal G-bundles. This b_G is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f to the pullback operation f*.

A characteristic class c of principal G-bundles is then a natural transformation from b_G to a cohomology functor H*, regarded also as a functor to Set.

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