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3manifold
 [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 3manifold has a triangulation, unique up to common subdivision
 As corollary, every compact 3manifold has a [Heegaard splitting] .
 [Prime decomposition theorem]
 KneserHaken finiteness
 [Loop] and [sphere] theorems
 [Annulus] and [Torus theorem]
 [JSJ decomposition] , also known as the toral decomposition
 [Scott core theorem]
 [LickorishWallace 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
 JorgensenThurston theorem that the set of finite volumes of hyperbolic 3manifolds 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.
 [PoincarĂ© conjecture] â€” see also [Solution of the PoincarĂ© conjecture]
 [Thurston's geometrization conjecture]
 [Virtually fibered conjecture]
 [Virtually Haken conjecture]
 [Cabling conjecture]
 [Surface subgroup conjecture]
 [Simple loop conjecture]
 The smallest hyperbolic 3manifold is the [Weeks manifold] .
 [LubotzySarnak conjecture] on [property tau]
References






External links

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