Characteristic class - Revision history

Vipul: 3 revisions

2008-05-18T19:34:23Z

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Vipul at 23:37, 13 December 2007

2007-12-13T23:37:55Z

← Older revision		Revision as of 23:37, 13 December 2007
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			{{topospaces version at\|characteristic class}}

	==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~~.		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>.

Vipul: /* Definition */

2007-07-07T07:22:06Z

Definition

← Older revision		Revision as of 07:22, 7 July 2007
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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.

Vipul at 04:53, 14 May 2007

2007-05-14T04:53:11Z

New page

==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.