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{{Expert-subject|Mathematics|date=February 2009}} | |||
In mathematics, a '''handle decomposition of a 3-manifold''' allows simplification of the original [[3-manifold]] into pieces which are easier to study. An important method used to decompose into [[handlebody|handlebodies]] is the [[Heegaard splitting]], which gives us a decomposition in two handlebodies of equal genus.<ref>{{cite book|title=[[Quantum invariant|Quantum Invariants]] of Knots and 3-manifolds|first=Vladimir G. |last=Turaev|publisher=Walter de Gruyter|year= 1994|isbn=3-11-013704-6}}</ref> | |||
As an example: [[lens space]]s are orientable 3-spaces, and allow decomposition into two '''solid-tori''' which are genus-one-handlebodies. The genus one non-orientable space is a space which is the union of two [[solid Klein bottle]]s and corresponds to the twisted product of the 2-sphere and the 1-sphere: <math>\scriptstyle S^2\tilde{\times}S^1</math>. | |||
Each orientable 3-manifold is the union of exactly two orientable handlebodies; meanwhile, each non-orientable one needs three orientable handlebodies. | |||
The minimal genus of the glueing boundary determines what is known as the [[Heegaard genus]]. For non-orientable spaces an interesting invariant is the [[trigenus|tri-genus]]. | |||
==References== | |||
{{reflist}} | |||
*J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. ''Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies'', Topology Appl. 60 (1994), 267-280. | |||
*J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. ''Stiefel-Whitney surfaces and the trigenus of non-orientable 3-manifolds'', Manuscripta Math. 100 (1999), 405-422. | |||
[[Category:3-manifolds]] | |||
[[Category:Topology]] | |||
{{topology-stub}} | |||
Revision as of 03:43, 15 January 2014
Template:Expert-subject In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study. An important method used to decompose into handlebodies is the Heegaard splitting, which gives us a decomposition in two handlebodies of equal genus.[1]
As an example: lens spaces are orientable 3-spaces, and allow decomposition into two solid-tori which are genus-one-handlebodies. The genus one non-orientable space is a space which is the union of two solid Klein bottles and corresponds to the twisted product of the 2-sphere and the 1-sphere: .
Each orientable 3-manifold is the union of exactly two orientable handlebodies; meanwhile, each non-orientable one needs three orientable handlebodies.
The minimal genus of the glueing boundary determines what is known as the Heegaard genus. For non-orientable spaces an interesting invariant is the tri-genus.
References
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- J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies, Topology Appl. 60 (1994), 267-280.
- J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel-Whitney surfaces and the trigenus of non-orientable 3-manifolds, Manuscripta Math. 100 (1999), 405-422.
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