View source for Prime decomposition (3-manifold)

In [[Mathematics|mathematics]], the '''prime decomposition theorem for 3-manifolds''' states that every [[Compact space|compact]], [[Orientability|orientable]] [[3-manifold|3-manifold]] is the [[Connected sum|connected sum]] of a unique ([[Up to|up to]] [[Homeomorphism|homeomorphism]]) finite collection of [[Prime manifold|prime 3-manifold]]s.

A manifold is ''prime'' if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension. This condition is necessary since for any manifold M of dimension <math>n</math> it is true that

:<math>M=M\#S^n.</math>

(where ''M#S<sup>n</sup>'' means the connected sum of ''M'' and ''S<sup>n</sup>''). If ''P'' is a prime 3-manifold then either it is ''S''<sup>2</sup>&nbsp;&times;&nbsp;''S''<sup>1</sup> or the non-orientable ''S''<sup>2</sup> bundle over ''S''<sup>1</sup>,
or it is irreducible, which means that any embedded 2-sphere bounds a ball.  So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of ''S''<sup>2</sup> over ''S''<sup>1</sup>.

The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable ''S''<sup>2</sup> bundles over ''S''<sup>1</sup>.  This sum is unique as long as we specify that each summand is either irreducible or a non-orientable&nbsp;''S''<sup>2</sup> bundle over&nbsp;''S''<sup>1</sup>.

The proof is based on [[Normal surface|normal surface]] techniques originated by [[Biography:Hellmuth Kneser|Hellmuth Kneser]].  Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by [[Biography:John Milnor|John Milnor]].

==References==

* {{cite journal|first=John|last= Milnor|title= A unique decomposition theorem for 3-manifolds|journal=[[American Journal of Mathematics]]|volume= 84 |year=1962|pages= 1–7|doi=10.2307/2372800|mr=0142125}}

[[Category:3-manifolds]]

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