Prime decomposition of 3-manifolds

In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds.

A manifold is prime if it is not homeomorphic to any connected sum of manifolds, except for the trivial connected sum of the manifold with a sphere of the same dimension,  M ≅ M # S n {\textstyle M\cong M\\#S^{n}} . If  P {\displaystyle P} is a prime 3-manifold then either it is  S 2 × S 1 {\displaystyle S^{2}\times S^{1}} or the non-orientable  S 2 {\displaystyle S^{2}} bundle over  S 1 , {\displaystyle S^{1},} 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 2 {\displaystyle S^{2}} over  S 1 . {\displaystyle S^{1}.}

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 2 {\displaystyle S^{2}} bundles over  S 1 . {\displaystyle S^{1}.} This sum is unique as long as we specify that each summand is either irreducible or a non-orientable  S 2 {\displaystyle S^{2}} bundle over  S 1 . {\displaystyle S^{1}.}

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

## References

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* Hempel, John (1976). 3-Manifolds. Annals of Mathematics Studies. Vol. 86\. Princeton, NJ: Princeton University Press. doi:10.1090/chel/349. ISBN 0-8218-3695-1. MR 0415619. Zbl 0345.57001.
* Jaco, William (1980). Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics. Vol. 43\. Providence, RI: American Mathematical Society. doi:10.1090/cbms/043. ISBN 0-8218-1693-4. MR 0565450. Zbl 0433.57001.
* Kneser, Hellmuth (1929). "Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten". Jahresbericht der Deutschen Mathematiker-Vereinigung. 38: 248–259. doi:10.1515/9783110894516.147. JFM 55.0311.03.
* Milnor, J. (1962). "A unique decomposition theorem for 3-manifolds". American Journal of Mathematics. 84 (1): 1–7. doi:10.2307/2372800. MR 0142125. S2CID 122595895. Zbl 0108.36501.

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Manifolds (Glossary)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main results (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Closed
* (Almost) Complex
* (Almost) Contact
* Fibered
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* Flat
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* Hadamard
* Hermitian
* Hyperbolic
* Kähler
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* Lie group
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* Oriented
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* Poisson
* Prime
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* (Pseudo−, Sub−) Riemannian
* Rizza
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
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* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
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* bundle
* De Rham cohomology
* Differential form
* Vector-valued
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
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* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
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* Dual
* Fiber
* (Co) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
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Connections|

* Affine
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* Koszul
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* Principal
* Vector
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Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach manifold
* Diffeology
* Diffiety
* Fréchet manifold
* K-theory
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

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