View source for 3-manifold

[[Image:3-Manifold 3-Torus.png|right|thumb|250px| An image from inside a [[Three-torus|3-torus]]. 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.]]

In [[mathematics]], a '''3-manifold''' is a [[space (mathematics)|space]] that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible [[shape of the universe]]. Just as a [[sphere]] looks like a [[plane (geometry)|plane]] to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

==Introduction==

===Definition===
A [[topological space]] ''X'' is a 3-manifold if it is a [[second-countable]] [[Hausdorff space]] and if every point in ''X'' has a [[neighbourhood (mathematics)|neighbourhood]] that is [[homeomorphic]] to [[Euclidean 3-space]].

===Mathematical theory of 3-manifolds===
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.

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]].

A key idea in the theory is to study a 3-manifold by considering special [[Surface (topology)|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.

[[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.

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.

== Important examples of 3-manifolds ==

===Euclidean 3-space===
{{main|Euclidean 3-space}}
Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional [[vector space]] over the real numbers.

===3-sphere===
{{Main|3-sphere}}
[[Image:Hypersphere coord.PNG|right|frame|[[Stereographic projection]] of the hypersphere's parallels (red), [[Meridian (perimetry, visual field)|meridians]] (blue) and hypermeridians (green).  Because this projection is [[conformal map|conformal]], the curves intersect each other orthogonally (in the yellow points) as in 4D.  All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line). ]]

A 3-sphere is a higher-dimensional analogue of a [[sphere]]. It consists of the set of points equidistant from a fixed central point in 4-dimensional [[Euclidean space]]. Just as an ordinary sphere (or 2-sphere) is a two-dimensional [[Surface (topology)|surface]] that forms the boundary of a [[ball (mathematics)|ball]] in three dimensions, a 3-sphere is an object with three [[dimension]]s that forms the boundary of a ball in four dimensions.

===Real projective 3-space===
{{main|Real projective space}}
Real projective 3-space, or '''RP'''<sup>''3''</sup>, is the [[topological space]] of lines passing through the origin 0 in '''R'''<sup>4</sup>. It is a [[compact space|compact]], [[smooth manifold]] of dimension ''3'', and is a special case '''Gr'''(1, '''R'''<sup>4</sup>) of a [[Grassmannian]] space.

'''RP'''<sup>3</sup> is ([[diffeomorphic]] to) [[SO(3)]], hence admits a group structure; the covering map ''S''<sup>3</sup> → '''RP'''<sup>3</sup> is a map of groups Spin(3) → SO(3), where [[Spin group|Spin(3)]] is a [[Lie group]] that is the [[universal cover]] of SO(3).

===3-torus===
{{main|Torus#n-dimensional torus}}
The 3-dimensional torus is the product of 3 circles. That is:

:<math>\mathbf{T}^3 = S^1 \times S^1 \times S^1.</math>

The 3-torus, '''T'''<sup>3</sup> can be described as a quotient of '''R'''<sup>3</sup> under integral shifts in any coordinate. That is, the 3-torus is '''R'''<sup>3</sup> modulo the [[Group action (mathematics)|action]] of the integer [[lattice (group)|lattice]] '''Z'''<sup>3</sup> (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional [[cube]] by gluing the opposite faces together.

A 3-torus in this sense is an example of a 3-dimensional [[Compact space|compact]] [[manifold]]. It is also an example of a compact [[abelian group|abelian]] [[Lie group]]. This follows from the fact that the [[unit circle]] is a compact abelian Lie group (when identified with the unit [[complex number]]s with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.

===Hyperbolic 3-space===
{{main|hyperbolic 3-space}}
[[Image:Hyperbolic orthogonal dodecahedral honeycomb.png|thumb|A perspective projection of a [[Hyperbolic small dodecahedral honeycomb|dodecahedral tessellation]] in '''[[Hyperbolic 3-manifold|H<sup>3</sup>]]'''.<br>Four [[dodecahedron|dodecahedra]] meet at each edge, and eight meet at each vertex, like the cubes of a [[Cubic honeycomb|cubic tessellation]] in ''[[Euclidean space|E<sup>3</sup>]]'']]

Hyperbolic space is a [[homogeneous space]] that can be characterized by a [[constant function|constant]] negative [[Curvature of Riemannian manifolds|curvature]]. It is the model of [[hyperbolic geometry]]. It is distinguished from [[Euclidean space]]s with [[zero]] curvature that define the [[Euclidean geometry]], and models of [[elliptic geometry]] (like the [[3-sphere]]) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a [[saddle point]]. Another distinctive property is the [[Riemannian volume form|amount of space]] covered by the [[3-ball]] in hyperbolic 3-space: it increases [[exponential growth|exponentially]] with respect to the radius of the ball, rather than polynomially.

===Poincaré dodecahedral space===
{{main|Homology sphere#Poincaré homology sphere}}
The [[Henri Poincaré|Poincaré]] homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere.  Being a [[spherical 3-manifold]], it is the only  homology 3-sphere (besides the [[3-sphere]] itself) with a finite [[fundamental group]].  Its fundamental group is known as the [[binary icosahedral group]] and has order 120.  This shows the [[Poincaré conjecture]] cannot be stated in homology terms alone.

In 2003, lack of structure on the largest scales (above 60 degrees) in the [[cosmic microwave background]] as observed for one year by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] spacecraft led to the suggestion, by [[Jean-Pierre Luminet]] of the [[Observatoire de Paris]] and colleagues, that the [[shape of the universe]] is a Poincaré sphere.<ref name="physwebLum03">[http://physicsworld.com/cws/article/news/18368 "Is the universe a dodecahedron?"], article at PhysicsWorld.</ref><ref name="Nat03">{{cite journal
  | last = Luminet
  | first = Jean-Pierre
  | authorlink = Jean-Pierre Luminet
  | author2 = [[Jeffrey Weeks (mathematician)|Jeff Weeks]]|author3=Alain Riazuelo|author4=Roland Lehoucq|author5= Jean-Phillipe Uzan
  | title = Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
  | volume = 425
  | issue = 6958
  | pages = 593–595
  |journal = [[Nature (journal)|Nature]]
  | location = 
  | date = 2003-10-09
  | arxiv = astro-ph/0310253
  | issn = 
  | doi = 10.1038/nature01944
  | jstor = 
  | pmid = 14534579
  | bibcode=2003Natur.425..593L}}</ref> In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.<ref name="RBSG08">{{cite journal
  | last =Roukema
  | first =Boudewijn
  | authorlink =
  |author2=Zbigniew Buliński |author3=Agnieszka Szaniewska |author4=Nicolas E. Gaudin
  | title =A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
  | journal = Astronomy and Astrophysics
  | volume =482
  | issue =3
  | pages =747–753
  | publisher =
  | year = 2008
  | arxiv =0801.0006
  | doi =10.1051/0004-6361:20078777
  | jstor =
| bibcode=2008A&A...482..747L}}</ref>
However, there is no strong support for the correctness of the model, as yet.

===Seifert–Weber space===
{{main|Seifert–Weber space}}
In [[mathematics]], '''Seifert–Weber space''' (introduced by [[Herbert Seifert]] and Constantin Weber) is a [[Closed manifold|closed]] [[hyperbolic 3-manifold]].  It is also known as '''Seifert–Weber dodecahedral space''' and '''hyperbolic dodecahedral space'''.  It is one of the first discovered examples of closed hyperbolic 3-manifolds.

It is constructed by gluing each face of a [[dodecahedron]] to its opposite in a way that produces a closed 3-manifold.  There are three ways to do this gluing consistently.  Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the [[Poincaré homology sphere]], and rotation by 5/10 gives 3-dimensional [[real projective space]].

With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the [[dihedral angle]] between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in [[hyperbolic space]] there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold.
It is a [[Quotient space (topology)|quotient space]] of the [[order-5 dodecahedral honeycomb]], a [[regular polytope|regular]] [[tessellation]] of [[hyperbolic 3-space]] by dodecahedra with this dihedral angle.

===Gieseking manifold===
{{main|Gieseking manifold}}
In [[mathematics]], the '''Gieseking manifold''' is a [[hyperbolic 3-manifold|cusped hyperbolic]] 3-manifold of finite volume.  It is [[orientability|non-orientable]] and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161.  It was discovered by {{harvs|txt|last=Gieseking|first=Hugo|year=1912}}.

The Gieseking manifold can be constructed by removing the vertices from a [[tetrahedron]], then gluing the faces together in pairs using affine-linear maps.  Label the vertices 0, 1, 2, 3.  Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order.  Glue the face 0,2,3 to the face 3,2,1 in that order.   In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of [[David B. A. Epstein]] and Robert C. Penner.<ref>{{cite journal|first1=David B.A. |last1=Epstein |
authorlink1= David B. A. Epstein| first2=Robert C. |last2=Penner|title=Euclidean decompositions of noncompact hyperbolic manifolds| journal=  [[Journal of Differential Geometry]] |volume= 27 |year=1988|issue= 1|pages= 67–80|mr=0918457 |doi=10.4310/jdg/1214441650}}</ref>  Moreover, the angle made by the faces is <math>\pi/3</math>.  The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.

== Some important classes of 3-manifolds ==

* [[Graph manifold]]
* [[Haken manifold]]
* [[Homology sphere]]s
* [[Hyperbolic 3-manifold]]
* [[I-bundle]]s
* [[Knot and link complements]]
* [[Lens space]]
* [[Seifert fiber spaces]], [[Circle bundle]]s
* [[Spherical 3-manifold]]
* [[Surface bundles over the circle]]
* [[Torus bundle]]

===Hyperbolic link complements===
[[File:BorromeanRings.svg|thumb|[[Borromean rings]] are a hyperbolic link.]]

A '''hyperbolic link''' is a [[link (knot theory)|link]] in the [[3-sphere]] with [[knot complement|complement]] that has a complete [[Riemannian metric]] of constant negative [[curvature]], i.e. has a [[hyperbolic geometry]]. A '''hyperbolic knot''' is a hyperbolic link with one [[connected space|component]].

The following examples are particularly well-known and studied.

* [[Figure-eight knot (mathematics)|Figure eight knot]]
* [[Whitehead link]]
* [[Borromean rings]]

The classes are not necessarily mutually exclusive.

== Some important structures on 3-manifolds ==

=== Contact geometry ===
{{main|Contact geometry}}
'''Contact geometry''' is the study of a geometric structure on [[smooth manifold]]s given by a hyperplane [[distribution (differential geometry)|distribution]] in the [[tangent bundle]] and specified by a [[differential form|one-form]], both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the [[Frobenius theorem (differential topology)|Frobenius theorem]], one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one [[foliation]] on the manifold ('complete integrability').

Contact geometry is in many ways an odd-dimensional counterpart of [[symplectic geometry]], which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of [[classical mechanics]], where one can consider either the even-dimensional [[phase space]] of a mechanical system or the odd-dimensional [[extended phase space]] that includes the time variable.

=== Haken manifold ===
{{main|Haken manifold}}
A '''Haken manifold''' is a [[compact space|compact]], [[P²-irreducible]] 3-manifold that is '''sufficiently large''', meaning that it contains a properly embedded [[2-sided|two-sided]] [[incompressible surface]]. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.

A 3-manifold finitely covered by a Haken manifold is said to be '''virtually Haken'''.  The [[Virtually Haken conjecture]] asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.

Haken manifolds were introduced by  Wolfgang Haken.  Haken proved that Haken manifolds have a '''hierarchy''', where they can be split up into 3-balls along incompressible surfaces.  Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. Jaco and Oertel gave an algorithm to determine if a 3-manifold was Haken.

===Essential lamination===
{{main|Essential lamination}}

An '''essential lamination''' is a [[lamination (topology)|lamination]] where every leaf is incompressible and end incompressible, if the complementary regions of the lamination are irreducible, and if there are no spherical leaves.

Essential laminations generalize the incompressible surfaces found in Haken manifolds.

=== Heegaard splitting ===
{{main|Heegaard splitting}}
A '''Heegaard splitting''' {{IPAc-en|audio=En-heegaard.ogg|}} is a decomposition of a compact oriented 3-manifold that results from dividing it into two [[handlebody|handlebodies]].

Every closed, [[orientable]] three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to [[Edwin E. Moise|Moise]].  This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of [[Smale]] about handle decompositions from Morse theory.

=== Taut foliation ===
{{main|Taut foliation}}
A '''taut foliation''' is a [[codimension]] 1 [[foliation]] of a 3-manifold with the property that there is a single transverse circle intersecting every leaf.  By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation.  Equivalently, by a result of [[Dennis Sullivan]], a codimension 1 foliation is taut if there exists a [[Riemannian metric]] that makes each leaf a [[minimal surface]].

Taut foliations were brought to prominence by the work of [[William Thurston]] and [[David Gabai]].

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

We begin with the purely topological:

===Moise's theorem===
{{main|Moise's theorem}}
In [[geometric topology]], '''Moise's theorem''', proved by [[Edwin E. Moise]] in {{harvtxt|Moise|1952}}, states that any topological 3-manifold has an essentially unique [[piecewise-linear structure]] and [[smooth structure]].

As corollary, every compact 3-manifold has a [[Heegaard splitting]].

===Prime decomposition theorem===
{{main|prime decomposition (3-manifold)}}
The '''prime decomposition theorem for 3-manifolds''' states that every [[compact space|compact]], [[orientability|orientable]] 3-manifold is the [[connected sum]] of a unique ([[up to]] [[homeomorphism]]) 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.

=== Kneser–Haken finiteness===
Kneser-Haken finiteness says that for each 3-manifold, there is a constant '''C''' such that any collection of surfaces of cardinality greater than '''C''' must contain parallel elements.

===Loop and Sphere theorems===
{{main|Loop theorem|Sphere theorem (3-manifolds)}}
The '''loop theorem''' is a generalization of [[Dehn's lemma]] and should more properly be called the "disk theorem".  It was first proven by [[Christos Papakyriakopoulos]] in 1956, along with Dehn's lemma and the [[Sphere theorem (3-manifolds)|Sphere theorem]].

A simple and useful version of the loop theorem states that if there is a map

:<math>f\colon (D^2,\partial D^2)\to (M,\partial M) \, </math>

with <math>f|\partial D^2</math> not nullhomotopic in <math>\partial M</math>, then there is an embedding with the same property.

The '''sphere theorem''' of {{harvs|txt|last=Papakyriakopoulos|authorlink=Christos Papakyriakopoulos|year=1957}} gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.

One example is the following:

Let <math>M</math> be an [[orientable]] 3-manifold such that <math>\pi_2(M)</math> is not the trivial group.  Then there exists a non-zero element of
<math>\pi_2(M)</math> having a representative that is an [[embedding]] <math>S^2\to M</math>.

===Annulus and Torus theorems===
The '''annulus theorem''' states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus.  This should not be confused with the high dimensional theorem of the same name.

The '''torus theorem''' is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus<ref>{{cite journal | last1 = Feustel | first1 = Charles D | year = 1976 | title = On the torus theorem and its applications | url = | journal = Transactions of the American Mathematical Society | volume = 217 | issue = | pages = 1–43 | doi=10.2307/1997556}}</ref>

===JSJ decomposition===
{{main|JSJ decomposition}}
The '''JSJ decomposition''', also known as the '''toral decomposition''', is a [[topological]] construct given by the following theorem:

:[[Irreducible (mathematics)|Irreducible]] [[Orientability|orientable]]  closed (i.e., compact and without boundary) 3-manifolds have a unique (up to [[homotopy|isotopy]]) minimal collection of disjointly [[Embedding|embedded]] [[incompressible surface|incompressible]] [[torus|tori]] such that each component of the 3-manifold obtained by cutting along the tori is either [[atoroidal]] or [[Seifert-fibered]].

The acronym JSJ is for [[William Jaco]], [[Peter Shalen]], and [[Klaus Johannson]].  The first two worked together, and the third worked independently.<ref>Jaco, William; Shalen, Peter B. ''A new decomposition theorem for irreducible sufficiently-large 3-manifolds.'' Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71&ndash;84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.</ref><ref>Johannson, Klaus, ''Homotopy equivalences of 3-manifolds with boundaries.'' Lecture Notes in Mathematics, 761. Springer, Berlin, 1979.  {{ISBN|3-540-09714-7}}
</ref>

===Scott core theorem===
{{main|Scott core theorem}}
The '''Scott core theorem''' is a theorem about the finite presentability of [[fundamental group]]s of 3-manifolds due to [[G. Peter Scott]].<ref>{{Citation | last=Scott | first=G. Peter | author-link=G. Peter Scott| title=Compact submanifolds of 3-manifolds | doi=10.1112/jlms/s2-7.2.246 |mr=0326737 | year=1973 | journal=[[Journal of the London Mathematical Society]] |series=Second Series | volume=7 | pages=246–250 | issue=2}}</ref>  The precise statement is as follows:

Given a 3-manifold (not necessarily [[compact manifold|compact]]) with [[finitely generated group|finitely generated]] fundamental group, there is a compact three-dimensional [[submanifold]], called the '''compact core''' or '''Scott core''', such that its [[inclusion map]] induces an [[isomorphism]] on fundamental groups.  In particular, this means a finitely generated 3-manifold group is [[presentation of a group|finitely presentable]].

A simplified proof is given in,<ref>{{Citation | last1=Rubinstein | first1=J. Hyam | authorlink1=J. Hyam Rubinstein|last2=Swarup | first2=Gadde A. | title=On Scott's core theorem | doi=10.1112/blms/22.5.495 |mr=1082023 | year=1990 | journal= [[Bulletin of the London Mathematical Society]] | volume=22 | issue=5 | pages=495–498}}</ref> and a stronger uniqueness statement is proven in.<ref>{{Citation | last1=Harris | first1=Luke | last2=Scott | first2=G. Peter |author2-link=G. Peter Scott| title=The uniqueness of compact cores for 3-manifolds | url=http://projecteuclid.org/getRecord?id=euclid.pjm/1102366188 |mr=1379290 | year=1996 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=172 | issue=1 | pages=139–150 | doi = 10.2140/pjm.1996.172.139 }}</ref>

===Lickorish–Wallace theorem===
{{main|Lickorish–Wallace theorem}}
The '''Lickorish–Wallace theorem''' states that any [[closed manifold|closed]], [[orientable]], connected 3-manifold may be obtained by performing [[Dehn surgery]] on a [[framed link]] in the [[3-sphere]] with <math>\pm 1</math> surgery coefficients.  Furthermore, each component of the link can be assumed to be unknotted.

===Waldhausen's theorems on topological rigidity===
[[Friedhelm Waldhausen]]'s theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.

===Waldhausen conjecture on Heegaard splittings===
Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.

===Smith conjecture===
{{main|Smith conjecture}}
The '''Smith conjecture'''  (now proven) states that if ''f'' is a [[diffeomorphism]]  of the [[3-sphere]] of [[Order (group theory)|finite order]], then the [[fixed point set]] of ''f'' cannot be a nontrivial [[knot (mathematics)|knot]].

===Cyclic surgery theorem===
{{main|Cyclic surgery theorem}}
The '''cyclic surgery theorem''' states that, for a [[Compact space|compact]], [[Connected space|connected]], [[Orientability|orientable]], [[Irreducibility (mathematics)|irreducible]] three-manifold ''M'' whose boundary is a [[torus]] ''T'', if ''M'' is not a [[Seifert-fibered space]] and ''r,s'' are slopes on ''T'' such that their [[Dehn surgery|Dehn filling]]s have cyclic fundamental group, then the distance between ''r'' and ''s'' (the minimal number of times that two simple closed curves in ''T'' representing ''r'' and ''s'' must intersect) is at most 1. Consequently, there are at most three Dehn fillings of ''M'' with cyclic fundamental group.

===Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem===
{{main|Hyperbolic Dehn surgery}}
'''Thurston's hyperbolic Dehn surgery theorem''' states:  <math>M(u_1, u_2, \dots, u_n)</math> is hyperbolic as long as a finite set of ''exceptional slopes'' <math>E_i</math> is avoided for the ''i''-th cusp for each ''i''.  In addition, <math>M(u_1, u_2, \dots, u_n)</math> converges to ''M'' in ''H'' as all <math>p_i^2+q_i^2 \rightarrow \infty</math> for all <math>p_i/q_i</math> corresponding to non-empty Dehn fillings <math>u_i</math>.

This theorem is due to [[William Thurston]] and fundamental to the theory of hyperbolic 3-manifolds.  It shows that nontrivial limits exist in ''H''.  Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.

Another important result by Thurston is that volume decreases under hyperbolic Dehn filling.  In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic.  The proof relies on basic properties of the [[Gromov norm]].

Jørgensen also showed that the volume function on this space is a [[continuous function|continuous]], [[proper map|proper]] function.  Thus by the previous results, nontrivial limits in ''H'' are taken to nontrivial limits in the set of volumes.  In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has [[ordinal number|ordinal type]] <math>\omega^\omega</math>.  This result is known as the '''Thurston-Jørgensen theorem'''.  Further work characterizing this set was done by [[Mikhail Gromov (mathematician)|Gromov]].

Also, Gabai, Meyerhoff & Milley showed that the [[Weeks manifold]] has the smallest volume of any closed orientable hyperbolic 3-manifold.

===Thurston's hyperbolization theorem for Haken manifolds===
{{main|Hyperbolization theorem}}
One form of Thurston's geometrization theorem states:
If ''M'' is an  compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of ''M'' has a complete hyperbolic structure of finite volume.

The [[Mostow rigidity theorem]] implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.

The conditions that the manifold ''M'' should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a  closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.

===Tameness conjecture, also called the Marden conjecture or tame ends conjecture===
{{main|Tameness conjecture}}
The '''tameness theorem''' states that every complete [[hyperbolic 3-manifold]] with finitely generated [[fundamental group]] is [[topologically tame]], in other words [[homeomorphism|homeomorphic]] to the interior of a [[compact space|compact]] 3-manifold.

The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by [[Danny Calegari]] and [[David Gabai]]. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the [[density theorem for Kleinian groups]] and the [[ending lamination theorem]]. It also implies the [[Ahlfors measure conjecture]].

===Ending lamination conjecture===
{{main|Ending lamination conjecture}}
The '''ending lamination theorem''', originally conjectured by William Thurston and later proven by [[Jeffrey Brock]], [[Richard Canary]], and Yair Minsky, states that [[hyperbolic 3-manifold]]s with [[finitely-generated group|finitely generated]] [[fundamental group]]s are determined by their topology together with certain "end invariants", which are geodesic [[Lamination (topology)|lamination]]s on some surfaces in the boundary of the manifold.

===Poincaré conjecture===
{{main|Poincaré conjecture}}
The 3-sphere is an especially important 3-manifold because of the now-proven [[Poincaré conjecture]]. Originally conjectured by [[Henri Poincaré]], the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a [[Closed manifold|closed]] 3-manifold).  The Poincaré conjecture claims that if such a space has the additional property that each [[path (topology)|loop]] in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.  An [[generalized Poincaré conjecture|analogous result]] has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, [[Grigori Perelman]] presented a proof of the conjecture in three papers made available in 2002 and 2003 on [[arXiv]]. The proof followed on from the program of [[Richard S. Hamilton]] to use the [[Ricci flow]] to attack the problem. Perelman introduced a modification of the standard Ricci flow, called ''Ricci flow with surgery'' to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.

===Thurston's geometrization conjecture===
{{main|Thurston's geometrization conjecture}}
'''Thurston's geometrization conjecture''' states that certain three-dimensional [[topological space]]s each have a unique geometric structure that can be associated with them. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surface (topology)|surface]]s, which states that every [[simply connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|hyperbolic]]).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.  The conjecture was proposed by {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}}, and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].

Thurston's  [[hyperbolization theorem]] implies that [[Haken manifold]]s satisfy the  geometrization conjecture.  Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.

[[Grigori Perelman]] sketched a proof of the full geometrization conjecture in 2003 using [[Ricci flow]] with [[Surgery theory|surgery]].
There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the [[spherical space form conjecture]] are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

===Virtually fibered conjecture and Virtually Haken conjecture===
{{main|Virtually fibered conjecture|Virtually Haken conjecture}}
The '''virtually fibered conjecture''', formulated by [[United States|American]] [[mathematician]] [[William Thurston]], states that every [[closed manifold|closed]], [[irreducible manifold|irreducible]], [[atoroidal]] 3-manifold with infinite [[fundamental group]] has a finite [[covering space|cover]] which is a [[surface bundle over the circle]].

The '''virtually Haken conjecture''' states that every [[compact manifold|compact]], [[orientable manifold|orientable]], [[irreducible manifold|irreducible]] three-dimensional manifold with infinite [[fundamental group]] is ''virtually Haken''. That is, it has a finite cover (a [[covering space]] with a finite-to-one covering map) that is a [[Haken manifold]].

In a posting on the ArXiv on 25 Aug 2009,<ref name="arxiv.org">Nicolas Bergeron and [[Daniel Wise (mathematician)|Daniel T. Wise]], ''A boundary criterion for cubulation'', https://arxiv.org/abs/0908.3609</ref> [[Daniel Wise (mathematician)|Daniel Wise]] implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.<ref>{{citation|authorlink=Daniel Wise (mathematician)|first=Daniel T.|last= Wise|title=Research announcement: The structure of groups with a quasiconvex hierarchy|journal= Electronic Research Announcements in Mathematical Sciences|volume= 16|pages=44–55|date=2009-10-29|url= http://www.aimsciences.org/journals/displayArticles.jsp?paperID=4703|doi= 10.3934/era.2009.16.44|mr=2558631}}</ref>
Several more preprints<ref>Haglund and Wise, ''A combination theorem for special cube complexes'', <br />
Hruska and Wise, ''Finiteness properties of cubulated groups'', <br /> Hsu and Wise, ''Cubulating malnormal amalgams'', <br />
http://comet.lehman.cuny.edu/behrstock/cbms/program.html
</ref> have followed, including the aforementioned longer manuscript by Wise.<ref name="docs.google.com">Daniel T. Wise, ''The structure of groups with a quasiconvex hierarchy'', https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1</ref> In March 2012, during a conference at [[Institut Henri Poincaré]] in Paris, [[Ian Agol]] announced he could prove the [[virtually Haken conjecture]] for closed hyperbolic 3-manifolds.<ref>Ian Agol, ''The virtual Haken conjecture'', https://arxiv.org/abs/1204.2810</ref> The proof built on results of Kahn and Markovic<ref>[[Jeremy Kahn]] and [[Vladimir Markovic]], ''Immersing almost geodesic surfaces in a closed hyperbolic manifold'' https://arxiv.org/abs/0910.5501, ''Counting essential surfaces in a closed hyperbolic 3-manifold'', https://arxiv.org/abs/1012.2828</ref> in their proof of the [[Surface subgroup conjecture]] and results of Wise in proving the Malnormal Special Quotient Theorem<ref name="docs.google.com"/> and results of Bergeron and Wise for the cubulation of groups.<ref name="arxiv.org"/> Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.

===Simple loop conjecture===
If <math>f\colon S \rightarrow T</math> is a map of closed connected surfaces such that <math>f_\star \colon \pi_1(S) \rightarrow \pi_1(T)</math> is not injective, then there exists a non-contractible simple closed
curve <math>\alpha \subset S </math> such that <math>f|_a</math> is homotopically trivial. This conjecture was proven by [[David Gabai]].

===Surface subgroup conjecture===
{{main|Surface subgroup conjecture}}

The '''surface subgroup conjecture''' of [[Friedhelm Waldhausen]] states that the fundamental group of every closed, irreducible 3-manifold with infinite [[fundamental group]] has a surface subgroup.  By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in [[Robion Kirby]]'s problem list.<ref>[[Robion Kirby]], [http://math.berkeley.edu/~kirby/problems.ps.gz ''Problems in low-dimensional topology'']</ref>

Assuming the [[geometrization conjecture]], the only open case was that of closed [[hyperbolic 3-manifold]]s.  A proof of this case was announced in the Summer of 2009 by [[Jeremy Kahn]] and [[Vladimir Markovic]] and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009.<ref>'Immersing almost geodesic surfaces in a closed hyperbolic three manifold', [http://front.math.ucdavis.edu/0910.5501 arXiv:0910.5501]</ref> Their paper was published in the [[Annals of Mathematics]] in 2012.<ref>{{citation|first1=Jeremy |last1=Kahn|authorlink1=Jeremy Kahn| first2=Vladimir|last2= Markovic| authorlink2=Markovic| title=Immersing almost geodesic surfaces in a closed hyperbolic three manifold|url=http://annals.math.princeton.edu/2012/175-3/p04| doi = 10.4007/annals.2012.175.3.4 | arxiv = 0910.5501| journal = [[Annals of Mathematics]] | volume = 175| issue = 3| pages = 1127–1190| year = 2012}}</ref>  In June 2012, Kahn and Markovic were given the [[Clay Research Award]]s by the [[Clay Mathematics Institute]] at a ceremony in [[Oxford]].<ref>[http://www.claymath.org/research_conference/2012/ 2012 Clay Research Conference] {{webarchive|url=https://web.archive.org/web/20120604035509/http://claymath.org/research_conference/2012/ |date=2012-06-04 }}</ref>

== Important conjectures ==

===Cabling conjecture===
The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a <math>(p,q)</math>-cable on some other knot, and the surgery must have been performed using the slope <math>pq</math>.

===Lubotzky—Sarnak conjecture===
The fundamental group of any finite volume hyperbolic ''n''-manifold does
not have Property τ.

==References==
{{reflist|30em}}

== Further reading ==
*{{citation |last=Hempel |first=John |title=3-manifolds |year=2004 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8218-3695-1|mr=2098385 |doi=10.1090/chel/349}}
*{{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|mr=0565450 }}
*{{citation |last=Rolfsen |first=Dale |title=Knots and Links |year=1976 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-914098-16-0|mr=1277811 }}
*{{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|mr=1435975 }}
*{{citation |last=Adams |first=Colin Conrad |title=The Knot Book. An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original. |year=2004 |publisher=American Mathematical Society |location=Providence, RI |isbn=0-8050-7380-9|pages=xiv+307 |mr=2079925 }}
*{{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=x+238 |mr=0928227 }}

==External links==
*{{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 }}
{{Commonscat}}

[[Category:Geometric topology]]
[[Category:3-manifolds| ]]