Hubbert linearization - Revision history

en>Jonesey95: Fixing citation authorlink parameter error using AutoEd

2014-06-20T14:18:09Z

Fixing citation authorlink parameter error using AutoEd

← Older revision		Revision as of 16:18, 20 June 2014
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	~~{{Confusing\|date=May 2007}}{{Refimprove\|date=December 2008}}~~		She is known by the name of Myrtle Shryock. For many years I've been operating as a payroll clerk. Minnesota is exactly where he's been living for years. Body developing is what my family members and I enjoy.<br><br>Also visit my web page; [http://youulike.com/profile/116412 at home std testing]

	~~In [[algebraic topology]], a branch of [[mathematics]], an ''orientation character'' on a [[group (mathematics)\|group]] <math>\pi</math> is a [[group homomorphism]]~~
	~~:<math>\omega\colon \pi \to \left\{\pm 1\right\}</math>. This notion~~ is of ~~particular significance in [[surgery theory]]~~.

	~~==Motivation==~~
	~~Given~~ a [[manifold]] ''M'', one takes <math>\pi=\pi_1 M</math> (the [[fundamental group]]), and then <math>\omega</math> sends an element of <math>\pi</math> to <math>-1</math> if and only if the class it represents is orientation-reversing.

	~~This map <math>\omega</math>~~ is ~~trivial if and only if~~ '~~'M'' is [[orientable]]~~.

	~~The orientation character~~ is ~~an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing~~ and ~~which are orientation preserving~~.

	~~==Twisted group algebra==~~
	~~The orientation character defines a twisted involution ([[*-ring]] structure) on the [[group ring]]~~ <~~math~~>~~\mathbf{Z}[\pi]~~<~~/math~~>~~, by <math>g \mapsto \omega(g)g^{-1}</math> (i.e., <math>\pm g^{-1}</math>, accordingly as <math>g</math> is orientation preserving or reversing). This is denoted <math>\mathbf{Z}[\pi]^\omega</math>.~~

	~~==Examples==~~
	*In [[real projective space]]s, the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.

	~~==Properties==~~
	~~The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.~~

	~~==See also==~~
	* [[Whitney characteristic class]]
	* [[Local system]]
	* [[Twisted Poincaré duality]]
	~~== External links ==~~
	*[http://~~www~~.~~map.mpim-bonn.mpg.de~~/~~Orientation_character Orientation character]~~ at ~~the Manifold Atlas~~
	~~[[Category:Geometric topology]~~]

en>Gravuritas: recently? 2005?

2013-09-20T19:59:27Z

recently? 2005?

← Older revision		Revision as of 21:59, 20 September 2013
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	~~Hi there. Allow me start by introducing the author~~, ~~her title~~ is ~~Sophia Boon but she never truly favored that name~~. ~~Office supervising~~ is ~~my occupation~~. ~~Alaska is exactly where he~~'~~s usually been living. It~~'~~s not a typical factor but what I like performing is~~ to ~~climb but I don't have~~ the ~~time lately~~.<br><br>~~my web blog~~ ..~~. good psychic~~ ([~~https:~~//~~www~~.~~machlitim~~.~~org.il~~/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 machlitim~~.~~org~~.il])		{{Confusing\|date=May 2007}}{{Refimprove\|date=December 2008}}

			In [[algebraic topology]], a branch of [[mathematics]], an ''orientation character'' on a [[group (mathematics)\|group]] <math>\pi</math> is a [[group homomorphism]]
			:<math>\omega\colon \pi \to \left\{\pm 1\right\}</math>. This notion is of particular significance in [[surgery theory]].

			==Motivation==
			Given a [[manifold]] ''M'', one takes <math>\pi=\pi_1 M</math> (the [[fundamental group]]), and then <math>\omega</math> sends an element of <math>\pi</math> to <math>-1</math> if and only if the class it represents is orientation-reversing.

			This map <math>\omega</math> is trivial if and only if ''M'' is [[orientable]].

			The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving.

			==Twisted group algebra==
			The orientation character defines a twisted involution ([[*-ring]] structure) on the [[group ring]] <math>\mathbf{Z}[\pi]</math>, by <math>g \mapsto \omega(g)g^{-1}</math> (i.e., <math>\pm g^{-1}</math>, accordingly as <math>g</math> is orientation preserving or reversing). This is denoted <math>\mathbf{Z}[\pi]^\omega</math>.

			==Examples==
			*In [[real projective space]]s, the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.

			==Properties==
			The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.

			==See also==
			* [[Whitney characteristic class]]
			* [[Local system]]
			* [[Twisted Poincaré duality]]
			== External links ==
			*[http://www.map.mpim-bonn.mpg.de/Orientation_character Orientation character] at the Manifold Atlas
			[[Category:Geometric topology]]

en>Wavelength: Wavelength moved page Hubbert Linearization to Hubbert linearization: revising letter case—WP:MOS#Article titles

2012-06-24T20:37:12Z

Wavelength moved page Hubbert Linearization to Hubbert linearization: revising letter case—WP:MOS#Article titles

New page

Hi there. Allow me start by introducing the author, her title is Sophia Boon but she never truly favored that name. Office supervising is my occupation. Alaska is exactly where he's usually been living. It's not a typical factor but what I like performing is to climb but I don't have the time lately.<br><br>my web blog ... good psychic ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 machlitim.org.il])