Characteristic class: Difference between revisions

Latest revision as of 19:34, 18 May 2008

The article on this topic in the Topology 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 $c(b_{g}^{*}(f)(P))=H^{*}(f)(c(P))$ .

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+{{topospaces version at|characteristic class}}
 ==Definition==
-A '''characteristic class''' is a function that associates, to every principal <math>G</math>-bundle <math>P \to X</math> an element <math>c(P)</math> in the cohomology algebra <math>H^*(X)</math>, such that if <math>f:Y \to X</math> is a continuous map, then <math>c(f^*P) = f^*(cP)</math> where the <math>f^*</math> on the left is the usual pullback of the bundle, and the <math>f^*</math? on the right is the induced map on the cohomology.
+Let <math>G</math> be a [[topological group]]. A '''characteristic class''' of principal <math>G</math>-bundles is a [[natural transformation]] from the contravariant functor <math>b_G</math> (which sends any topological space to the set of isomorphism classes of principal <math>G</math>-bundles on it) to the cohomology functor.
+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>.