Dehn surgery
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224844733
2008-07-10T17:52:11Z
Rybu
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fix it up a bit...
A '''Dehn surgery''' is a specific 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'''.
In the drilling process one removes an open tubular neighbourhood of the link from the 3-manifold. The resulting manifold we call the [[link complement]].
Given a 3-manifold with torus boundary components, we may glue in a [[solid torus]] by a [[homeomorphism]] 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 <math>m</math> and <math>l</math> 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 <math>m</math> and <math>l</math> 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 <math>T</math> 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 <math>T</math> given by a [[Seifert surface]]. When the ratio p/q are all integers, the surgery is called an ''integral surgery'' or a [[Surgery theory|genuine surgery]], since such surgeries are closely related to handlebodies and cobordism.
==See also==
*[[Lickorish-Wallace theorem]]
*[[Kirby calculus]]
*[[Hyperbolic Dehn surgery]]
[[Category:3-manifolds]]
[[it:Chirurgia di Dehn]]