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3-manifold

- [Contact structure]
- [Trigenus]

Foundational results
Some results are named as conjectures as a result of historical artifacts.

We begin with the purely topological:

- [Moise's theorem] - Every 3-manifold has a triangulation, unique up to common subdivision
-- As corollary, every compact 3-manifold has a [Heegaard splitting] .
- [Prime decomposition theorem]
- Kneser-Haken finiteness
- [Loop] and [sphere] theorems
- [Annulus] and [Torus theorem]
- [JSJ decomposition] , also known as the toral decomposition
- [Scott core theorem]
- [Lickorish-Wallace theorem]
- [Waldhausen] 's theorems on topological rigidity
- [Waldhausen conjecture] on Heegaard splittings

Theorems where geometry plays an important role in the proof:

- [Smith conjecture]
- [Cyclic surgery theorem]

Results explicitly linking geometry and topology:

- Thurston's [hyperbolic Dehn surgery] theorem
- Jorgensen-Thurston theorem that the set of finite volumes of hyperbolic 3-manifolds has order type \omega^\omega.
- Thurston's [geometrization theorem] for Haken manifolds
- [Tameness conjecture] , also called the Marden conjecture or tame ends conjecture
- [Ending lamination conjecture]

Important conjectures
Some of these are thought to be solved, as of March 2007. Please see specific articles for more information.

- [Poincaré conjecture] — see also [Solution of the Poincaré conjecture]
- [Thurston's geometrization conjecture]
- [Virtually fibered conjecture]
- [Virtually Haken conjecture]
- [Cabling conjecture]
- [Surface subgroup conjecture]
- [Simple loop conjecture]
- The smallest hyperbolic 3-manifold is the [Weeks manifold] .
- [Lubotzy-Sarnak conjecture] on [property tau]

References
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External links
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