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| [[Image:3-Manifold_3-Torus.png|right|frame| An image from inside a 3-Torus, generated by [[Jeffrey_Weeks_(mathematician)|Jeff Weeks]]' CurvedSpaces software. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.]]
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| In [[mathematics]], a '''3-manifold''' is a 3-dimensional [[manifold]]. The topological, [[Piecewise linear manifold|piecewise-linear]], and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
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| Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as [[knot theory]], [[geometric group theory]], [[hyperbolic geometry]], [[number theory]], [[Teichmüller space|Teichmüller theory]], [[topological quantum field theory]], [[gauge theory]], [[Floer homology]], and [[partial differential equations]]. 3-manifold theory is considered a part of [[low-dimensional topology]] or [[geometric topology]].
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| A key idea in the theory is to study a 3-manifold by considering special [[surface]]s embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an [[incompressible surface]] and the theory of [[Haken manifold]]s, or one can choose the complementary pieces to be as nice as possible, leading to structures such as [[Heegaard splitting]]s, which are useful even in the non-Haken case.
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| [[William Thurston|Thurston's]] contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
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| The [[fundamental group]]s of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between [[group theory]] and topological methods.
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| == Important examples of 3-manifolds ==
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| * [[Euclidean 3-space]]
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| * [[3-sphere]]
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| * [[SO(3)]] (or 3-dimensional [[real projective space]])
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| * [[Torus|3-torus]]
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| * [[Hyperbolic 3-space]]
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| * [[Homology sphere#Poincaré homology sphere|Poincaré dodecahedral space]]
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| * [[Seifert–Weber space]]
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| * [[Gieseking manifold]]
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| ===Hyperbolic link complements===
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| The following examples are particularly well-known and studied.
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| * [[Figure-eight knot (mathematics)|Figure eight knot]]
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| * [[Whitehead link]]
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| * [[Borromean rings]]
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| == Some important classes of 3-manifolds ==
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| * [[Graph manifold]]
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| * [[Haken manifold]]
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| * [[Homology sphere]]s
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| * [[Hyperbolic 3-manifold]]
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| * [[I-bundle]]s
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| * [[Knot and link complements]]
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| * [[Lens space]]
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| * [[Seifert fiber spaces]], [[Circle bundle]]s
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| * [[Spherical 3-manifold]]
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| * [[Surface bundles over the circle]]
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| * [[Torus bundle]]
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| The classes are not necessarily mutually exclusive.
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| == Some important structures on 3-manifolds ==
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| * [[Contact geometry]]
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| * [[Essential lamination]]
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| * [[Haken manifold]]
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| * [[Heegaard splitting]]
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| * [[Taut foliation]]
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| * [[Trigenus]]
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| == Foundational results ==
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| Some results are named as conjectures as a result of historical artifacts.
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| We begin with the purely topological:
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| * [[Moise's theorem]] – Every 3-manifold has a triangulation, unique up to common subdivision
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| ** As corollary, every compact 3-manifold has a [[Heegaard splitting]].
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| * [[prime decomposition (3-manifold)|Prime decomposition theorem]]
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| * Kneser–Haken finiteness
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| * [[Loop theorem|Loop]] and [[Sphere theorem (3-manifolds)|sphere]] theorems
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| * [[Annulus theorem|Annulus]] and [[torus theorem]]
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| * [[JSJ decomposition]], also known as the toral decomposition
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| * [[Scott core theorem]]
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| * [[Lickorish-Wallace theorem]]
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| * [[Friedhelm Waldhausen|Waldhausen]]'s theorems on topological rigidity
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| * [[Waldhausen conjecture]] on Heegaard splittings
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| Theorems where geometry plays an important role in the proof:
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| * [[Smith conjecture]]
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| * [[Cyclic surgery theorem]]
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| Results explicitly linking geometry and topology:
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| * Thurston's [[hyperbolic Dehn surgery]] theorem
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| * The Jørgensen–Thurston theorem that the set of finite volumes of hyperbolic 3-manifolds has order type <math>\omega^\omega</math>.
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| * Thurston's [[hyperbolization theorem]] for Haken manifolds
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| * [[Tameness conjecture]], also called the Marden conjecture or tame ends conjecture
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| * [[Ending lamination conjecture]]
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| == Important conjectures ==
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| Some of these are thought to be solved, as of March 2007. Please see specific articles for more information.
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| * [[Poincaré conjecture]] — see also [[Solution of the Poincaré conjecture]]
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| * [[Thurston's geometrization conjecture]]
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| * [[Virtually fibered conjecture]]
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| * [[Virtually Haken conjecture]]
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| * [[Cabling conjecture]]
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| * [[Surface subgroup conjecture]]
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| * [[Simple loop conjecture]]
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| * The smallest hyperbolic 3-manifold is the [[Weeks manifold]].
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| * [[Lubotzy-Sarnak conjecture]] on [[property tau]]
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| == References ==
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| *{{citation |last=Hempel |first=John |title=3-manifolds |year=2004 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8218-3695-1 }}
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| *{{citation |last=Jaco |first=William H. |title=Lectures on three-manifold topology |year=1980 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8218-1693-4 }}
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| *{{citation |last=Rolfsen |first=Dale |title=Knots and Links |year=1976 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-914098-16-0 }}
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| *{{citation |last=Thurston |first=William P. |title=Three-dimensional geometry and topology |year=1997 |publisher=Princeton University Press |location=Princeton, NJ |isbn=0-691-08304-5 }}
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| *{{citation |last=Adams |first=Colin Conrad |title=The Knot Book |year=2002 |publisher=W. H. Freeman |location=New York |isbn=0-8050-7380-9|pages= |url= }}
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| *{{citation |last=Bing |first=R. H. |title=The Geometric Topology of 3-Manifolds |year=1983 |publisher=American Mathematical Society |series=Colloquium Publications |volume=40 |location=Providence, RI |isbn=0-8218-1040-5 |pages= |url= }}
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| ==External links==
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| *{{citation |last=Hatcher |first=Allen |title=Notes on basic 3-manifold topology |publisher=Cornell University |url=http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html }}
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| [[Category:Geometric topology]]
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| [[Category:3-manifolds]]
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