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In [[topology]], a branch of mathematics, a '''Dehn surgery''', named after [[Max Dehn]], is a construction used to modify [[3-manifold]]s. The process takes as input a 3-manifold together with a [[link (knot theory)|link]].  '''Dehn surgery''' can be thought of as a two stage process: '''drilling''' and '''Dehn filling'''.
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== Definitions ==
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* Given a [[3-manifold]] <math>M</math> and a [[link (knot theory)|link]] <math>L \subset M</math>, ''<math>M</math> drilled along <math>L</math>'' is obtained by removing an open [[tubular neighborhood]] of <math>L</math> from <math>M</math>. ''<math>M</math> drilled along <math>L</math>'' is also known as the [[link complement]], since if one removed the corresponding closed [[tubular neighborhood]] from <math>M</math> one obtains a manifold diffeomorphic to <math>M \setminus L</math>.
 
* Given a 3-manifold with torus boundary components, we may glue in a [[solid torus]] by a [[homeomorphism]] (resp. [[diffeomorphism]]) of its boundary to the torus boundary component <math>T</math> of the original 3-manifold.  There are many inequivalent ways of doing this, in general. This process is called '''Dehn filling'''.
 
* '''Dehn surgery''' on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with '''Dehn filling''' on all the components of the boundary corresponding to the link.
 
We can pick two oriented simple closed [[curve]]s ''m'' and ''ℓ'' on the boundary torus of the 3-manifold that generate the [[fundamental group]] of the torus. This gives any simple closed curve <math>\gamma</math> on that torus two coordinates ''p'' and ''q'', each coordinate corresponding to the algebraic intersection of the curve with ''m'' and ''ℓ'' respectively.  These coordinates only depend on the [[homotopy class]] of <math>\gamma</math>.
 
We can specify a homeomorphism of the boundary of a solid torus to ''T'' by having the meridian curve of the solid torus map to a curve homotopic to <math>\gamma</math>. As long as the meridian maps to the '''surgery slope''' <math>[\gamma]</math>, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio ''p''/''q'' is called the '''surgery coefficient'''. 
 
In the case of links in the 3-sphere or more generally an oriented homology sphere, there is a canonical choice of the meridians and longitudes of ''T''.  The longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a [[Seifert surface]].  The meridian is the curve that bounds a disc in the tubular neighbourhood of the link.  When the ratios ''p''/''q'' are all integers, the surgery is called an ''integral surgery''. Such surgeries are closely related to [[handle decomposition|handlebodies]], [[cobordism]] and [[Morse theory|Morse functions]].
 
==Results==
 
Every [[closed manifold|closed]], [[orientable]], [[connected space|connected]] [[3-manifold]] is obtained by performing Dehn surgery on a link in the [[3-sphere]].  This result, the [[Lickorish-Wallace theorem]], was first proven by Wallace in 1960 and independently by Lickorish in a stronger form in 1962.  Via the now well-known relation between [[Surgery theory|genuine surgery]] and [[cobordism]], this result is equivalent to the theorem that the [[cobordism|oriented cobordism group]] of 3-manifolds is trivial, originally due to [[Vladimir Abramovich Rokhlin]] in 1951.
 
Since orientable 3-manifolds can all be generated by suitably decorated links, one might ask how distinct surgery presentations of a given 3-manifold might be related. The answer is called the [[Kirby calculus]].
 
==See also==
*[[Hyperbolic Dehn surgery]]
*[[Tubular neighborhood]]
*[[Surgery theory|Surgery]] on manifolds, in the general sense, also called spherical modification.
 
==References==
 
*{{citation|first=M.|last=Dehn|authorlink=Max Dehn|title=Die Gruppe der Abbildungsklassen|journal=[[Acta Mathematica]]|volume=69|issue=1|year=1938|pages=135–206|doi=10.1007/BF02547712}}.
*{{citation|first=René|last=Thom|authorlink=René Thom|mr=0061823| title=Quelques propriétés globales des variétés différentiables|journal= [[Commentarii Mathematici Helvetici]] |volume= 28|year=  1954|pages= 17–86 |url=http://retro.seals.ch/digbib/view?did=c1:389597&sdid=c1:389960| doi=10.1007/BF02566923}}
*{{citation|first=Rob|last=Kirby|authorlink=Robion Kirby|title=A calculus for framed links in ''S''<sup>3</sup>|journal=[[Inventiones Mathematicae]]|volume=45|issue=1|year=1978|pages=35–56|doi=10.1007/BF01406222}}.
*{{citation|first1=R. P.|last1=Fenn|first2=C. P.|last2=Rourke|title=On Kirby's calculus of links|journal=Topology|volume=18|issue=1|year=1979|pages=1–15|doi=10.1016/0040-9383(79)90010-7}}.
*{{citation|first1=Robert|last1=Gompf|first2=András|last2=Stipsicz|title=4-Manifolds and Kirby Calculus|year=1999|volume=20|series=Graduate Studies in Mathematics|publisher=American Mathematical Society|location=Providence, RI|isbn=0-8218-0994-6}}.
 
[[Category:3-manifolds]]
[[Category:Surgery theory]]

Latest revision as of 07:41, 24 May 2014

Once a association struggle begins, you will see The most important particular War Map, a map of this war area area association conflicts booty place. Friendly territories will consistently be more on the left, by having the adversary association on the inside the right. boondocks anteroom on the war map represents some sort or other of war base.

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