Surgery structure set - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T22:17:46Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 00:17, 26 September 2014
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	~~{{unreferenced\|date=November 2010}}~~		Hi, everybody! My name is Tracee. <br>It is a little about myself: I live in Italy, my city of Spigno Monferrato. <br>It's called often Eastern or cultural capital of AL. I've married 4 years ago.<br>I have two children - a son (Rachelle) and the daughter (Emely). We all like Computer programming.<br><br>Also visit my webpage: [http://www.examiner.com/article/toothache-remedies-home-remedy-for-toothache Home Remedy for Toothache]
	~~In [[mathematics]], '''assembly maps''' are an important concept in [[geometric topology]]. From the [[homotopy]]-theoretical viewpoint~~, ~~an assembly map~~ is a [[Universal property\|universal]] approximation of a homotopy invariant [[functor]] by a [[homology theory]] from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

	Assembly maps for [[algebraic K-theory]] and [[L-theory]] play a central role in the topology of high-dimensional [[manifold]]s, since their [[homotopy fiber]]s have a direct geometric interpretation. [[Equivariant map\|Equivariant]] assembly maps are used to formulate the [[Farrell–Jones conjecture]]s in K- and L-theory.

	~~==Homotopy-theoretical viewpoint==~~
	~~It is a classical result that for any generalized [[homology theory]]~~ <~~math~~>~~h_*</math> on the [[category of topological spaces]] (assumed to be homotopy equivalent to [[CW-complex]]es), there~~ is a ~~[[spectrum (homotopy theory)\|spectrum]] <math>E</math> such that~~
	:~~<math>h_(X)\cong \pi_(X_+\wedge E)~~,~~</math>~~
	~~where <math>X_+:=X\coprod \{*\}</math>.~~

	~~The functor <math>X\mapsto X_+ \wedge E</math> from spaces to spectra has the following properties:~~
	* It is homotopy-invariant (preserves homotopy equivalences). ~~This reflects the fact that~~ <~~math~~>~~h_*</math> is homotopy-invariant.~~
	* It ~~preserves homotopy co-cartesian squares. This reflects that fact that <math>h_*</math> has [[Mayer-Vietoris sequence]]~~s~~, an equivalent characterization~~ of ~~excision~~.
	* It preserves arbitrary [[coproduct]]s. This reflects the disjoint-union axiom of <math>h_*</math>.
	~~A functor from spaces to spectra fulfilling these properties is called '''excisive''~~'.

	~~Now suppose that~~ <~~math~~>~~F</math> is a homotopy~~-~~invariant, not necessarily excisive functor. An assembly map is~~ a ~~[[natural transformation]] <math>\alpha\colon F^\%\to F</math> from some excisive functor <math>F^\%</math> to <math>F</math> such that <math>F^\%~~()~~\to F~~()~~</math> is a homotopy equivalence.~~

	If we denote by <math>h_:=\pi_\circ F^\%</math> the associated homology theory, it follows that the induced natural transformation of graded [[abelian group]]s <math>h_\to \pi_\circ F</math> is the universal transformation from a homology theory to <math>\pi_\circ F</math>, i.e. any other transformation <math>k_\to\pi_\circ F</math> from some homology theory <math>k_</math> factors uniquely through a transformation of homology theories <math>k_\to h_</math>.

	~~Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.~~

	~~==Geometric viewpoint==~~
	As a consequence of the [[Mayer-Vietoris sequence]], the value of an excisive functor on a space <math>X</math> only depends on its value on 'small' subspaces of <math>X</math>, together with the knowledge how these small subspaces intersect. ~~In a cycle representation of the associated homology theory, this means that~~ all ~~cycles must be representable by small cycles~~. ~~For instance, for [[singular homology]], the excision property is proved by subdivision of [[simplex\|simplices]], obtaining sums of small simplices representing arbitrary homology classes.~~

	In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of [[controlled topology]]. In this picture, assembly maps are 'forget-control' maps, i.e. they are induced by forgetting the control conditions.

	~~==Importance in geometric topology==~~
	~~Assembly maps are studied in geometric topology mainly for the two functors~~ <~~math~~>~~L(X)~~<~~/math~~>~~, algebraic [[L-theory]] of <math>X</math>, and <math>A(X)</math>, [~~[algebraic K-theory]] of spaces of <math>X</math>. In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when <math>X</math> is a compact topological manifold. Therefore knowledge about the geometry of compact topological manifolds may be obtained by studying <math>K</math>- and <math>L</math>-theory and their respective assembly maps.

	In the case of <math>L</math>-theory, the homotopy fiber <math>L_\%(M)</math> of the corresponding assembly map <math> L^\%(M)\to L(M)</math>, evaluated at a compact topological manifold <math>M</math>, is homotopy equivalent to the space of block structures of <math>M</math>. Moreover, the fibration sequence
	:~~<math> L_\%(M)\to L^\%(M)\to L(M)<~~/~~math>~~
	~~induces a [[long exact sequence]] of homotopy groups which may be identified with the [[surgery exact sequence]] of <math>M<~~/~~math>~~. ~~This may be called the '''fundamental theorem of surgery theory''' and was developed subsequently by Browder, Novikov, Sullivan, Wall, Quinn, and Ranicki~~.

	~~For <math>A</math>-theory, the homotopy fiber <math>A_\%(M)<~~/~~math> of the corresponding assembly map is homotopy equivalent to the space of stable [[h-cobordism]]s on <math>M<~~/~~math>. This fact is called the '''stable parametrized h~~-~~cobordism theorem''', proven by Waldhausen~~-~~Jahren~~-~~Rognes. It may be viewed as a parametrized version of the classical theorem which states that equivalence classes of h~~-~~cobordisms on <math>M</math> are in 1~~-~~to-1 correspondence with elements in the [[Whitehead group]] of <math>\pi_1(M)</math>.~~
	~~[[Category:Surgery theory]]~~
	~~[[Category:K-theory]~~]

en>Mark viking: Put topic in bold per MOS:LEAD and added wl

2013-10-17T03:37:20Z

Put topic in bold per MOS:LEAD and added wl

← Older revision		Revision as of 05:37, 17 October 2013
Line 1:		Line 1:
	~~CMS provides the best platform to create websites that fulfill all the specifications of SEO. If you cherished this article and you would like to get much more facts relating to~~ [~~http://htxurl.info/wordpressbackup286444 wordpress dropbox backup~~] ~~kindly stop by the page~~. ~~You may discover this probably~~ the ~~most time~~-~~consuming part~~ of ~~building~~ a ~~Word - Press MLM website. I thought about what would happen~~ by ~~placing~~ a ~~text widget in~~ the ~~sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue~~. ~~All this is very simple, and~~ the ~~best thing is that it is totally free~~, ~~and you don~~'~~t need~~ a ~~domain name or web hosting~~. ~~<br><br>~~		{{unreferenced\|date=November 2010}}
			In [[mathematics]], '''assembly maps''' are an important concept in [[geometric topology]]. From the [[homotopy]]-theoretical viewpoint, an assembly map is a [[Universal property\|universal]] approximation of a homotopy invariant [[functor]] by a [[homology theory]] from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

	~~Choosing what kind~~ of ~~links you'll be using~~ is a ~~ctitical aspect~~ of ~~any linkwheel strategy~~, ~~especially since~~ there ~~are several different types of links~~ that ~~are assessed by search engines~~. ~~WPTouch~~ is ~~among~~ the ~~more well known Word~~ - ~~Press smartphone plugins which is currently in use by thousands~~ of ~~users~~. This ~~plugin allows a blogger get more Facebook fans on~~ the ~~related fan page~~. E-~~commerce websites are meant to be buzzed with fresh contents~~, ~~graphical enhancements, and functionalities~~. ~~Moreover, many Word - Press themes need~~ to ~~be purchased and designing your own WP site can be boring~~. <br><br>~~Digital photography~~ is a ~~innovative effort~~, ~~if you removethe stress~~ to ~~catch every position and~~ viewpoint of a ~~place~~, ~~you free yourself up to be more innovative and your outcomes will be much better. The nominee in each category~~ with the ~~most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as~~ the ~~best product~~, ~~tip or place in~~ that ~~category~~. ~~Are you considering getting your website redesigned~~. ~~Our skilled expertise~~, ~~skillfulness and excellence have been well known all across~~ the ~~world~~. ~~Purchase these from our site~~, ~~or bring your own~~, ~~it doesn't matter~~, ~~we will still give you free installation~~ and ~~configuration.~~ <br><br>~~Whether your Word~~ - ~~Press themes is premium or not~~, ~~but nowadays every theme~~ is ~~designed with widget~~-~~ready~~. ~~High Quality Services: These companies help you in creating high quality Word~~ - ~~Press websites. However~~, ~~you may not be able~~ to ~~find~~ a ~~theme that~~ is ~~in sync with your business~~. ~~Fast Content Update - It's easy~~ to ~~edit or add posts~~ with ~~free Wordpress websites. The Pakistani culture is in demand~~ of ~~a main surgical treatment.~~ <br><br>~~He loves sharing information regarding wordpress~~, ~~Majento~~, ~~Drupal~~ and ~~Joomla development tips & tricks~~. ~~Visit our website~~ to ~~learn more about how you can benefit~~. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. ~~Extra investment in Drupal must~~ be ~~bypassed~~ as ~~value for money is what Drupal provides. As with~~ a ~~terminology, there~~ are ~~many methods~~ to ~~understand how to use~~ the ~~terminology~~.		Assembly maps for [[algebraic K-theory]] and [[L-theory]] play a central role in the topology of high-dimensional [[manifold]]s, since their [[homotopy fiber]]s have a direct geometric interpretation. [[Equivariant map\|Equivariant]] assembly maps are used to formulate the [[Farrell–Jones conjecture]]s in K- and L-theory.

			==Homotopy-theoretical viewpoint==
			It is a classical result that for any generalized [[homology theory]] <math>h_*</math> on the [[category of topological spaces]] (assumed to be homotopy equivalent to [[CW-complex]]es), there is a [[spectrum (homotopy theory)\|spectrum]] <math>E</math> such that
			:<math>h_(X)\cong \pi_(X_+\wedge E),</math>
			where <math>X_+:=X\coprod \{*\}</math>.

			The functor <math>X\mapsto X_+ \wedge E</math> from spaces to spectra has the following properties:
			* It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that <math>h_*</math> is homotopy-invariant.
			* It preserves homotopy co-cartesian squares. This reflects that fact that <math>h_*</math> has [[Mayer-Vietoris sequence]]s, an equivalent characterization of excision.
			* It preserves arbitrary [[coproduct]]s. This reflects the disjoint-union axiom of <math>h_*</math>.
			A functor from spaces to spectra fulfilling these properties is called '''excisive'''.

			Now suppose that <math>F</math> is a homotopy-invariant, not necessarily excisive functor. An assembly map is a [[natural transformation]] <math>\alpha\colon F^\%\to F</math> from some excisive functor <math>F^\%</math> to <math>F</math> such that <math>F^\%()\to F()</math> is a homotopy equivalence.

			If we denote by <math>h_:=\pi_\circ F^\%</math> the associated homology theory, it follows that the induced natural transformation of graded [[abelian group]]s <math>h_\to \pi_\circ F</math> is the universal transformation from a homology theory to <math>\pi_\circ F</math>, i.e. any other transformation <math>k_\to\pi_\circ F</math> from some homology theory <math>k_</math> factors uniquely through a transformation of homology theories <math>k_\to h_</math>.

			Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.

			==Geometric viewpoint==
			As a consequence of the [[Mayer-Vietoris sequence]], the value of an excisive functor on a space <math>X</math> only depends on its value on 'small' subspaces of <math>X</math>, together with the knowledge how these small subspaces intersect. In a cycle representation of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for [[singular homology]], the excision property is proved by subdivision of [[simplex\|simplices]], obtaining sums of small simplices representing arbitrary homology classes.

			In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of [[controlled topology]]. In this picture, assembly maps are 'forget-control' maps, i.e. they are induced by forgetting the control conditions.

			==Importance in geometric topology==
			Assembly maps are studied in geometric topology mainly for the two functors <math>L(X)</math>, algebraic [[L-theory]] of <math>X</math>, and <math>A(X)</math>, [[algebraic K-theory]] of spaces of <math>X</math>. In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when <math>X</math> is a compact topological manifold. Therefore knowledge about the geometry of compact topological manifolds may be obtained by studying <math>K</math>- and <math>L</math>-theory and their respective assembly maps.

			In the case of <math>L</math>-theory, the homotopy fiber <math>L_\%(M)</math> of the corresponding assembly map <math> L^\%(M)\to L(M)</math>, evaluated at a compact topological manifold <math>M</math>, is homotopy equivalent to the space of block structures of <math>M</math>. Moreover, the fibration sequence
			:<math> L_\%(M)\to L^\%(M)\to L(M)</math>
			induces a [[long exact sequence]] of homotopy groups which may be identified with the [[surgery exact sequence]] of <math>M</math>. This may be called the '''fundamental theorem of surgery theory''' and was developed subsequently by Browder, Novikov, Sullivan, Wall, Quinn, and Ranicki.

			For <math>A</math>-theory, the homotopy fiber <math>A_\%(M)</math> of the corresponding assembly map is homotopy equivalent to the space of stable [[h-cobordism]]s on <math>M</math>. This fact is called the '''stable parametrized h-cobordism theorem''', proven by Waldhausen-Jahren-Rognes. It may be viewed as a parametrized version of the classical theorem which states that equivalence classes of h-cobordisms on <math>M</math> are in 1-to-1 correspondence with elements in the [[Whitehead group]] of <math>\pi_1(M)</math>.
			[[Category:Surgery theory]]
			[[Category:K-theory]]

en>ChrisGualtieri: /* Examples */General fixes / CHECKWIKI fixes using AWB

2012-08-30T23:27:15Z

Examples: General fixes / CHECKWIKI fixes using AWB

New page

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Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. High Quality Services: These companies help you in creating high quality Word - Press websites. However, you may not be able to find a theme that is in sync with your business. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. The Pakistani culture is in demand of a main surgical treatment. He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. Visit our website to learn more about how you can benefit. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. As with a terminology, there are many methods to understand how to use the terminology.