Characteristic class: Difference between revisions

Revision as of 07:22, 7 July 2007

Definition

A characteristic class is a function that associates, to every principal $G$ -bundle $P\to X$ an element $c(P)$ in the cohomology algebra $H^{*}(X)$ , such that if $f:Y\to X$ is a continuous map, then $c(f^{*}P)=f^{*}(cP)$ where the $f^{*}$ on the left is the usual pullback of the bundle, and the $f^{*}$ on the right is the induced map on the cohomology.

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	==Definition==		==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.		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.