en>R'n'B: Help needed: which Definability?

2013-08-17T23:08:37Z

Help needed: which Definability?

← Older revision		Revision as of 01:08, 18 August 2013
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	~~The author is called Irwin Wunder but it~~'~~s not~~ the ~~most masucline title out there~~. ~~To do aerobics~~ is a ~~factor~~ that I'~~m completely addicted to~~. ~~Years~~ in the ~~past~~ we ~~moved to North Dakota~~. ~~My working day occupation~~ is a ~~meter reader~~.<br><br>~~Review my web site~~; ~~home std test (~~[~~http:~~//~~www~~.~~zinepal~~.~~com/feed/std~~-~~test~~-~~guidestd~~-~~test~~-~~guide Suggested Internet site~~])		[[Image:Dehn twist.png\|thumb\|A positive Dehn twist applied to a cylinder about the red curve ''c'' modifies the green curve as shown.]]

			In [[geometric topology]], a branch of [[mathematics]], a '''Dehn twist''' is a certain type of [[homeomorphism\|self-homeomorphism]] of a [[surface]] (two-dimensional [[manifold]]).

			==Definition==
			Suppose that ''c'' is a [[curve\|simple closed curve]] in a closed, [[Orientability\|orientable]] surface ''S''. Let ''A'' be a [[tubular neighborhood]] of ''c''. Then ''A'' is an [[Annulus (mathematics)\|annulus]] and so is [[homeomorphic]] to the [[Cartesian product]] of

			:<math>S^1 \times I,</math>

			where ''I'' is the [[unit interval]]. Give ''A'' coordinates (''s'', ''t'') where ''s'' is a complex number of the form

			:<math>e^{{\rm{i}} \theta}</math>

			with

			:<math>\theta \in [0,2\pi],</math>

			and ''t'' in the unit interval.

			Let ''f'' be the map from ''S'' to itself which is the identity outside of ''A'' and inside ''A'' we have

			:<math>\displaystyle f(s,t) = (s e^{{\rm{i}} 2 \pi t}, t). </math>

			Then ''f'' is a '''Dehn twist''' about the curve ''c''.

			Dehn twists can also be defined on a non-orientable surface ''S'', provided one starts with a [[2-sided]] simple closed curve ''c'' on ''S''.

			==Mapping class group==
			[[Image:Lickorish Twist Theorem.svg\|thumb\|350px\|The 3''g'' − 1 curves from the twist theorem, shown here for ''g'' = 3.]]
			It is a theorem of [[Max Dehn]] that maps of this form generate the [[mapping class group of a surface\|mapping class group]] of [[homotopy\|isotopy]] classes of orientation-preserving homeomorphisms of any closed, oriented [[genus (mathematics)\|genus]]-<math>g</math> surface. [[W. B. R. Lickorish]] later rediscovered this result with a simpler proof and in addition showed that Dehn twists along <math>3g-1</math> explicit curves generate the mapping class group (this is called by the punning name "[[W. B. R. Lickorish\|Lickorish]] twist theorem"); this number was later improved by [[Stephen P. Humphries]] to <math>2g+1</math>, for <math>g > 1</math>, which he showed was the minimal number.

			Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "[[Y-homeomorphism]]s."

			==See also==
			* [[Lantern relation]]

			==References==

			* [[Andrew J. Casson]], Steven A Bleiler, ''Automorphisms of Surfaces After Nielsen and Thurston'', [[Cambridge University Press]], 1988. ISBN 0-521-34985-0.
			* Stephen P. Humphries, ''Generators for the mapping class group'', in: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, [[Springer Verlag\|Springer]], Berlin, 1979. {{MR\|0547453}}
			* [[W. B. R. Lickorish]], ''A representation of orientable combinatorial 3-manifolds.'' Ann. of Math. (2) 76 1962 531—540. {{MR\|0151948}}
			* W. B. R. Lickorish, ''A finite set of generators for the homeotopy group of a 2-manifold'', Proc. Cambridge Philos. Soc. 60 (1964), 769–778. {{MR\|0171269}}

			[[Category:Geometric topology]]
			[[Category:Homeomorphisms]]

en>Abradoks: uk and ru interwikis are wrong

2012-08-08T13:23:47Z

uk and ru interwikis are wrong

New page

The author is called Irwin Wunder but it's not the most masucline title out there. To do aerobics is a factor that I'm completely addicted to. Years in the past we moved to North Dakota. My working day occupation is a meter reader.<br><br>Review my web site; home std test ([http://www.zinepal.com/feed/std-test-guidestd-test-guide Suggested Internet site])

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en>R'n'B: Help needed: which Definability?

en>Abradoks: uk and ru interwikis are wrong