Stiefel-Whitney class - Revision history

Vipul: 7 revisions

2008-05-11T19:59:06Z

7 revisions

← Older revision	Revision as of 19:59, 11 May 2008
(No difference)

Vipul at 22:21, 24 December 2007

2007-12-24T22:21:05Z

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	* [[Chern class]]		* [[Chern class]]
	* [[Euler class]]		* [[Euler class]]

			==Facts==

			The ring of characteristic classes on [[real vector bundle]]s with <math>\mathbb{Z}_2</math> coefficients, is a polynomial ring in the Stiefel-Whitney classes.

Vipul at 22:18, 24 December 2007

2007-12-24T22:18:51Z

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	==Related notions==		==Related notions==

			* [[First Stiefel-Whitney class]]
	* [[Chern class]]		* [[Chern class]]
	* [[Euler class]]		* [[Euler class]]

Vipul at 22:14, 24 December 2007

2007-12-24T22:14:03Z

← Older revision		Revision as of 22:14, 24 December 2007
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	# <math>w_i(f^(E)) = f^(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is part of the definition of a [[characteristic class]])		# <math>w_i(f^(E)) = f^(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is part of the definition of a [[characteristic class]])
	# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math> (this is ~~equivalent to the~~ [[Whitney sum formula]])		# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math> where <math>\smile</math> denotes the [[cap product]] (this is a [[Whitney sum formula]])
	# <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>		# <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>
	* For the canonical line bundle <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>		* For the [[canonical real line bundle]] <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>

	<math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.		<math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.

			==Related notions==

			* [[Chern class]]
			* [[Euler class]]

Vipul at 22:05, 24 December 2007

2007-12-24T22:05:26Z

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			{{characteristic class}}

	==Definition==		==Definition==

Vipul at 22:04, 24 December 2007

2007-12-24T22:04:44Z

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	==Definition==		==Definition==

	The Stiefel-Whitney class is a [[characteristic class]] for the topological group <math>GL(n,\R)</math> with coefficients mod 2. It can be defined axiomatically as follows.		The Stiefel-Whitney class is a [[characteristic class]] (or collection of characteristic classes in different dimensions) for the topological group <math>GL(n,\R)</math> with coefficients mod 2. It can be defined axiomatically as follows.

	To each [[real vector bundle]] <math>p:E \to B</math>, an element <math>w \in H^*(B;\mathbb{Z}_2)</math> such that if <math>w_i</math> denotes the component of <math>w</math> in <math>H^i(B;\mathbb{Z}_2)</math>, we have:		To each [[real vector bundle]] <math>p:E \to B</math>, an element <math>w \in H^*(B;\mathbb{Z}_2)</math> such that if <math>w_i</math> denotes the component of <math>w</math> in <math>H^i(B;\mathbb{Z}_2)</math>, we have:

Vipul at 22:03, 24 December 2007

2007-12-24T22:03:56Z

← Older revision		Revision as of 22:03, 24 December 2007
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	# <math>w_i(f^(E)) = f^(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is part of the definition of a [[characteristic class]])		# <math>w_i(f^(E)) = f^(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is part of the definition of a [[characteristic class]])
	# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math>		# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math> (this is equivalent to the [[Whitney sum formula]])
	# <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>		# <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>
	* For the canonical line bundle <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>		* For the canonical line bundle <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>

	<math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.		<math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.

Vipul at 22:03, 24 December 2007

2007-12-24T22:03:03Z

New page

==Definition==

The Stiefel-Whitney class is a [[characteristic class]] for the topological group <math>GL(n,\R)</math> with coefficients mod 2. It can be defined axiomatically as follows.

To each [[real vector bundle]] <math>p:E \to B</math>, an element <math>w \in H^*(B;\mathbb{Z}_2)</math> such that if <math>w_i</math> denotes the component of <math>w</math> in <math>H^i(B;\mathbb{Z}_2)</math>, we have:

# <math>w_i(f^*(E)) = f^*(w_i(E))</math> where <math>f^*</math> denotes the pullback (this is the condition for being a [[natural transformation]], and is part of the definition of a [[characteristic class]])
# <math>w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)</math>
# <math>w_i(E) = 0</math> if <math>i</math> is greater than the dimension of <math>E</math>
* For the canonical line bundle <math>E \to \R P^\infty</math>, <math>w_1(E)</math> is a generator of <math>H^1(\R P^\infty; \mathbb{Z}_2)</math>

<math>w</math> is termed the '''total Stiefel Whitney-class''' and <math>w_i</math> is termed the <math>i^{th}</math> Stiefel-Whitney class.