en>Gareth Jones: remove unnecessary comma

2013-06-11T21:39:46Z

remove unnecessary comma

← Older revision		Revision as of 23:39, 11 June 2013
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	~~Friends call him Royal. Some time ago he selected~~ to ~~live in Kansas~~. ~~Bookkeeping~~ is ~~what I do for~~ a ~~living~~. ~~Bottle tops collecting~~ is the only ~~hobby his spouse doesn~~'~~t approve~~ of.<br><br>~~Here~~ is ~~my blog extended car warranty~~ ([http://~~srgame~~.co.kr/~~qna~~/~~12373 just click~~ the ~~next web site~~])		In [[topology]], a branch of mathematics, an '''aspherical space''' is a [[topological space]] with all [[homotopy groups]] π<sub>''n''</sub>(''X'') equal to 0 when ''n''>1.

			If one works with [[CW complex]]es, one can reformulate this condition: an aspherical CW complex is a CW complex whose [[universal cover]] is [[contractible]]. Indeed, contractibility of a universal cover is the same, by [[Whitehead's theorem]], as asphericality of it. And it is an application of the [[exact sequence of a fibration]] that higher homotopy groups of a space and its universal cover are same. (By the same argument, if ''E'' is a [[Connected space\|path-connected space]] and ''p'': ''E'' → ''B'' is any [[covering space\|covering map]], then ''E'' is aspherical if and only if ''B'' is aspherical.)

			Aspherical spaces are, directly from the definitions, [[Eilenberg-MacLane space]]s. Also directly from the definitions, aspherical spaces are [[classifying space]]s of their fundamental groups.

			==Examples==
			* Using the second of above definitions we easily see that all orientable compact [[surface]]s of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover).

			* It follows that all non-orientable surfaces, except the real [[projective plane]], are aspherical as well, as they can be covered by an orientable surface genus 1 or higher.

			* Similarly, a [[product topology\|product]] of any number of [[circle]]s is aspherical.

			* Any [[hyperbolic 3-manifold]] is, by definition, covered by the hyperbolic 3-space '''H'''<sup>3</sup>, hence aspherical.

			* Let ''X'' = ''G''/''K'' be a [[Riemannian symmetric space]] of negative type, and '''Γ''' be a [[lattice (mathematics)\|lattice]]{{dn\|date=May 2012}} in ''G'' that acts freely on ''X''. Then the [[locally symmetric space]] <math>\Gamma\backslash G/K</math> is aspherical.

			* The [[Bruhat-Tits building]] of a simple [[algebraic group]] over a field with a [[discrete valuation]] is aspherical.

			* The complement of a [[knot (mathematics)\|knot]] in '''S'''<sup>3</sup> is aspherical, by the [[sphere theorem (3-manifolds)\|sphere theorem]]

			* Metric spaces with nonpositive curvature in the sense of [[Aleksandr Danilovich Aleksandrov\|Aleksandrov]] (locally [[CAT(0) space]]s) are aspherical. In the case of [[Riemannian manifold]]s, this follows from the [[Cartan–Hadamard theorem]], which has been generalized to [[geodesic metric space]]s by [[Mikhail Gromov (mathematician)\|Gromov]] and Ballmann. This class of aspherical spaces subsumes all the previously given examples.

			* Any [[nilmanifold]] is aspherical.

			==Symplectically aspherical manifolds==
			If one deals with [[symplectic manifold]]s, the meaning of "aspherical" is a little bit different. Specifically, we say that a symplectic manifold (M,ω) is symplectically aspherical if and only if

			:<math>\int_{S^2}f^\omega=\langle c_1(TM),f_[S^2]\rangle=0</math>

			for every continuous mapping

			:<math>f\colon S^2 \to M,</math>

			where <math>c_1(TM)</math> denotes the first [[Chern class]] of an [[almost complex manifold\|almost complex structure]] which is compatible with ω.

			By [[Stokes' theorem]], we see that symplectic manifolds which are aspherical are also symplectically aspherical manifolds. However, there do exist symplectically aspherical manifolds which are not aspherical spaces.<ref>Robert E. Gompf, ''Symplectically aspherical manifolds with nontrivial π<sub>2</sub>'', Math. Res. Lett. 5 (1998), no. 5, 599–603. {{MR\|1666848}}</ref>

			Some references<ref>Jarek Kedra, [[Yuli Rudyak]], and Aleksey Tralle, ''Symplectically aspherical manifolds'', J. Fixed Point Theory Appl. 3 (2008), no. 1, 1–21. {{MR\|2402905}}</ref> drop the requirement on ''c''<sub>1</sub> in their definition of "symplectically aspherical." However, it is more common for symplectic manifolds satisfying only this weaker condition to be called "weakly exact."

			==See also==
			*[[Acyclic space]]
			*[[Essential manifold]]

			==Notes==
			<references/>

			==References==
			* Bridson, Martin R.; [[André Haefliger\|Haefliger, André]], ''Metric spaces of non-positive curvature''. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN 3-540-64324-9 {{MR\|1744486}}

			== External links ==
			* [http://www.map.him.uni-bonn.de/index.php/Aspherical_manifolds Aspherical manifolds] on the Manifold Atlas.

			[[Category:Algebraic topology]]
			[[Category:Homology theory]]
			[[Category:Homotopy theory]]

en>Brown motion: Added an intuitive proof of why the Milstein approximation works

2012-05-03T13:03:06Z

Added an intuitive proof of why the Milstein approximation works

New page

Friends call him Royal. Some time ago he selected to live in Kansas. Bookkeeping is what I do for a living. Bottle tops collecting is the only hobby his spouse doesn't approve of.<br><br>Here is my blog extended car warranty ([http://srgame.co.kr/qna/12373 just click the next web site])

Milstein method - Revision history

en>Gareth Jones: remove unnecessary comma

en>Brown motion: Added an intuitive proof of why the Milstein approximation works