Characteristic class: Difference between revisions

Revision as of 21:19, 24 December 2007

The article on this topic in the Differential Geometry Wiki can be found at: characteristic class

Definition

Let $G$ be a topological group. A characteristic class of principal $G$ -bundles is a natural transformation from the contravariant functor $b_{G}$ (which sends any topological space to the set of isomorphism classes of principal $G$ -bundles on it) to the cohomology functor.

For a given topological space $X$ , a characteristic class of principal $G$ -bundles associates, to every principal $G$ -bundle $P \to X$ , an element $c (P) \in H^{*} (X)$ , such that if $f : X \to Y$ is a continuous map, then $c (b_{g}^{*} (f) (P)) = H^{*} (f) (c (P))$ .

When we talk of characteristic classes of vector bundles, we are implicitly thinking of characteristic classes for the associated principal $G L (n)$ -bundle.

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	For a given topological space <math>X</math>, a characteristic class of principal <math>G</math>-bundles associates, to every principal <math>G</math>-bundle <math>P \to X</math>, an element <math>c(P) \in H^(X)</math>, such that if <math>f:X \to Y</math> is a [[continuous map]], then <math>c(b_g^(f)(P)) = H^*(f)(c(P))</math>.		For a given topological space <math>X</math>, a characteristic class of principal <math>G</math>-bundles associates, to every principal <math>G</math>-bundle <math>P \to X</math>, an element <math>c(P) \in H^(X)</math>, such that if <math>f:X \to Y</math> is a [[continuous map]], then <math>c(b_g^(f)(P)) = H^*(f)(c(P))</math>.

			When we talk of characteristic classes of vector bundles, we are implicitly thinking of characteristic classes for the associated principal <math>GL(n)</math>-bundle.