Characteristic class

In mathematics, the idea of characteristic class is one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. The theory explains, in very general terms, why fiber bundles cannot always have sections. In other words, characteristic classes are global invariants which measure the deviation of a local product structure from a product structure.

Definition

Let G be a group, and for a topological space X, write bG(X) for the set of isomorphism classes of principal G-bundles. This is a functor from Top to Set, sending a map f to the pullback operation f*. A characteristic class c of principal G-bundles is then a natural transformation from bG to a cohomology functor H*, regarded also as a functor to Set.

Related Topics:
Group - Topological space - Isomorphism class - Principal ''G''-bundles - Functor - Pullback - Natural transformation - Cohomology

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In other words, we want to associate to any principal G-bundle P → X an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f *P) = f *c(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.

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