Geometrization conjecture

Thurston's geometrization conjecture states that compact 3manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3manifolds of the uniformization theorem for surfaces. It was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds 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. There are now four 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.
Contents
The conjecture
A 3manifold is called closed if it is compact and has no boundary.
Every closed 3manifold has a prime decomposition: this means it is the connected sum of prime threemanifolds (this decomposition is essentially unique except for a small problem in the case of nonorientable manifolds). This reduces much of the study of 3manifolds to the case of prime 3manifolds: those that cannot be written as a nontrivial connected sum.
Here is a statement of Thurston's conjecture:
 Every oriented prime closed 3manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an Anosov map of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)
For nonoriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover. It is also possible to work directly with nonorientable manifolds, but this gives some extra complications: it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure so this gives a minor extra complication.
In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.
The eight Thurston geometries
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold M is a diffeomorphism from M to X/Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.)
There is some connection with the Bianchi groups: the 3dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However S^{2}×R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable nonunimodular Bianchi groups, most of which give model geometries with no compact representatives.
Spherical geometry S^{3}
The point stabilizer is O_{3}(R), and the group G is the 6dimensional Lie group O_{4}(R), with 2 components. The corresponding manifolds are exactly the closed 3manifolds with finite fundamental group. Examples include the 3sphere, the Poincaré homology sphere, Lens spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type IX. Manifolds with this geometry are all compact, orientable, and have the structure of a Seifert fiber space (often in several ways). The complete list of such manifolds is given in the article on Spherical 3manifolds. Under Ricci flow manifolds with this geometry collapse to a point in finite time.
Euclidean geometry E^{3}
The point stabilizer is O_{3}(R), and the group G is the 6dimensional Lie group R^{3}.O_{3}(R), with 2 components. Examples are the 3torus, and more generally the mapping torus of a finite order automorphism of the 2torus; see torus bundle. There are exactly 10 finite closed 3manifolds with this geometry, 6 orientable and 4 nonorientable. This geometry can be modeled as a left invariant metric on the Bianchi groups of type I or VII_{0}. Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways). The complete list of such manifolds is given in the article on Seifert fiber spaces. Under Ricci flow manifolds with Euclidean geometry remain invariant.
Hyperbolic geometry H^{3}
The point stabilizer is O_{3}(R), and the group G is the 6dimensional Lie group O_{1,3}(R)^{+}, with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the Weeks manifold. Other examples are given by the Seifert–Weber space, or "sufficiently complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the Bianchi group of type V. Under Ricci flow manifolds with hyperbolic geometry expand.
The geometry of S^{2}×R
The point stabilizer is O_{2}(R)×Z/2Z, and the group G is O_{3}(R)×R.Z/2Z, with 4 components. The four finite volume manifolds with this geometry are: S^{2}×S^{1}, the mapping torus of the antipode map of S^{2}, the connected sum of two copies of 3 dimensional projective space, and the product of S^{1} with twodimensional projective space. The first two are mapping tori of the identity map and antipode map of the 2sphere, and are the only examples of 3manifolds that are prime but not irreducible. The third is the only example of a nontrivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a Seifert fiber space (often in several ways). Under normalized Ricci flow manifolds with this geometry converge to a 1dimensional manifold.
The geometry of H^{2}×R
The point stabilizer is O_{2}(R) × Z/2Z, and the group G is O_{1,2}(R)^{+} × R.Z/2Z, with 4 components. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.^{[1]}) The classification of such (oriented) manifolds is given in the article on Seifert fiber spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type III. Under normalized Ricci flow manifolds with this geometry converge to a 2dimensional manifold.
The geometry of the universal cover of SL_{2}(R)
is the universal cover of SL_{2}(R), which fibers over . The point stabilizer is O_{2}(R). The group G has 2 components. Its identity component has the structure . Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the Brieskorn homology spheres (excepting the 3sphere and the Poincare dodecahedral space). This geometry can be modeled as a left invariant metric on the Bianchi group of type VIII. Finite volume manifolds with this geometry are orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow manifolds with this geometry converge to a 2dimensional manifold.
Nil geometry
See also: NilmanifoldThis fibers over E^{2}, and is the geometry of the Heisenberg group. The point stabilizer is O_{2}(R). The group G has 2 components, and is a semidirect product of the 3dimensional Heisenberg group by the group O_{2}(R) of isometries of a circle. Compact manifolds with this geometry include the mapping torus of a Dehn twist of a 2torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the Bianchi group of type II. Finite volume manifolds with this geometry are compact and orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow compact manifolds with this geometry converge to R^{2} with the flat metric.
Sol geometry
See also: SolvmanifoldThis geometry fibers over the line with fiber the plane, and is the geometry of the identity component of the group G. The point stabilizer is the dihedral group of order 8. The group G has 8 components, and is the group of maps from 2dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component has a normal subgroup R^{2} with quotient R, where R acts on R^{2} with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the Bianchi group of type VI_{0} and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with sol geometry are compact. The compact manifolds with sol geometry are either the mapping torus of an Anosov map of the 2torus (an automorphism of the 2torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as ), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the sol manifolds could in principle be classified in terms of the units and ideal classes of this order, though the details do not seem to be written down anywhere. Under normalized Ricci flow compact manifolds with this geometry converge (rather slowly) to R^{1}.
Uniqueness
A closed 3manifold has a geometric structure of at most one of the 8 types above, but finite volume noncompact 3manifolds can occasionally have more than one type of geometric structure. (However a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π_{1}(M):
 If π_{1}(M) is finite then the geometric structure on M is spherical, and M is compact.
 If π_{1}(M) is virtually cyclic but not finite then the geometric structure on M is S^{2}×R, and M is compact.
 If π_{1}(M) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
 If π_{1}(M) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
 If π_{1}(M) is virtually solvable but not virtually nilpotent then the geometric structure on M is sol geometry, and M is compact.
 If π_{1}(M) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H^{2}×R or the universal cover of SL_{2}(R). The manifold M may be either compact or noncompact. If it is compact, then the 2 geometries can be distinguished by whether or not π_{1}(M) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is noncompact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
 If π_{1}(M) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or noncompact.
Infinite volume manifolds can have many different types of geometric structure: for example, R^{3} can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any nonunimodular 3dimensional Lie group.
There can be more than one way to decompose a closed 3manifold into pieces with geometric structures. For example:
 Taking connected sums with several copies of S^{3} does not change a manifold.
 The connected sum of two projective 3spaces has a S^{2}×R geometry, and is also the connected sum of two pieces with S^{3} geometry.
 The product of a surface negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.
It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
History
The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.
The case of 3manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by Ricci flow with surgery. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries S^{3} and S^{2}×R, while what is left at large times should have a thickthin decomposition into a "thick" piece with hyperbolic geometry and a "thin" graph manifold.
In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above. The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7.4 in the preprint 'Ricci Flow with surgery on threemanifolds'. This theorem was stated by Perelman without proof. There are now several different proofs of Perelman's Theorem 7.4, or variants of it which are sufficient to prove geometrization. There is the paper of Shioya and Yamaguchi ^{[2]} that uses Perelman's stability theorem ^{[3]} and a fibration theorem for Alexandrov spaces.^{[4]} This method, with full details leading to the proof of Geometrization, can be found in the exposition by B. Kleiner and J. Lott in 'Notes on Perelman's papers' in the journal Geometry & Topology.^{[5]}
A second route to Geometrization is the method of Bessières et al.,^{[6]}^{[7]} which uses Thurston's hyperbolization theorem for Haken manifolds ^{[8]} and Gromov's norm for 3manifolds.^{[9]} A book by the same authors with complete details of their version of the proof has been published by the European Mathematical Society.^{[10]}
Also containing proofs of Perelman's Theorem 7.4, there is a paper of Morgan and Tian,^{[11]} another paper of Kleiner and Lott,^{[12]} and a paper by Cao and Ge.^{[13]}
Notes
 ^ Ronald Fintushel, Local S^{1} actions on 3manifolds, Pacific J. o. M. 66 No1 (1976) 111118, http://projecteuclid.org/...
 ^ T. Shioya and T. Yamaguchi, 'Volume collapsed threemanifolds with a lower curvature bound,' Math. Ann. 333 (2005), no. 1, 131155.
 ^ V. Kapovitch, 'Perelman's Stability Theorem', in Surveys of Differential Geometry, Metric and Comparison Geometry, vol. XI, International press, (2007), 103136. There is a preprint on http://arxiv.org/abs/math/0703002v3
 ^ T. Yamaguchi. A convergence theorem in the geometry of Alexandrov spaces. In Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), volume 1 of Semin. Congr., pages 601642. Soc. math. France, Paris, 1996.
 ^ B. Kleiner and J. Lott, 'Notes on Perelman's papers', Geometry & Topology, volume 12, pp. 25872855, 2008. There is a preprint at http://arxiv.org/abs/math/0605667v3
 ^ Bessieres, L.; Besson, G.; Boileau, M.; Maillot, S.; Porti, J. (2007). "Weak collapsing and geometrization of aspherical 3manifolds". arXiv:0706.2065 [math.GT].
 ^ Bessieres, L.; Besson, G.; Boileau, M.; Maillot, S.; Porti, J. (2010). "Collapsing irreducible 3manifolds with nontrivial fundamental group". Invent. Math. 179 (2): 435–460. doi:10.1007/s0022200902226.
 ^ J.P. Otal, 'Thurston's hyperbolization of Haken manifolds,'Surveys in differential geometry, Vol. III Cambridge, MA, 77194, Int. Press, Boston, MA, 1998.
 ^ M. Gromov. Volume and bounded cohomology. Inst. Hautes Etudes Sci. Publ. Math., (56):599 (1983), 1982.
 ^ L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. Available at http://wwwfourier.ujfgrenoble.fr/~lbessier/book.pdf
 ^ J. Morgan and G. Tian, 'Completion of the Proof of the Geometrization Conjecture', arXiv:math/0809.4040, 2008, http://arxiv.org/abs/0809.4040v1
 ^ B. Kleiner and J. Lott, 'Locally collapsed 3manifolds, arXiv:1005.5106, 2010
 ^ J. Cao and J. Ge, 'A proof of Perelman's collapsing theorem for 3manifolds', arXiv:0908.3229
References
 L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. [1]
 M. Boileau Geometrization of 3manifolds with symmetries
 F. Bonahon Geometric structures on 3manifolds Handbook of Geometric Topology (2002) Elsevier.
 Allen Hatcher: Notes on Basic 3Manifold Topology 2000
 J. Isenberg, M. Jackson, Ricci flow of locally homogeneous geometries on a Riemannian manifold, J. Diff. Geom. 35 (1992) no. 3 723741.
 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002
 G. Perelman,Ricci flow with surgery on threemanifolds, 2003
 G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, 2003
 Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006) (fills in the details of Perelman's proof of the geometrization conjecture).
 Cao, HuaiDong; Zhu, XiPing (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the HamiltonPerelman theory of the Ricci flow" (PDF). Asian Journal of Mathematics 10 (2): 165–498. http://www.intlpress.com/AJM/p/2006/10_2/AJM102165492Abstract.php. Retrieved 20060731. Revised version (December 2006): HamiltonPerelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture
 John W. Morgan. Recent progress on the Poincaré conjecture and the classification of 3manifolds. Bulletin Amer. Math. Soc. 42 (2005) no. 1, 5778 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture)
 Morgan, John W.; Fong, Frederick TszHo (2010). Ricci Flow and Geometrization of 3Manifolds. University Lecture Series. ISBN 9780821849637. http://www.ams.org/bookstoregetitem/item=ulect53. Retrieved 20100926.
 Scott, Peter The geometries of 3manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401487.
 Thurston, William P. (1982). "Threedimensional manifolds, Kleinian groups and hyperbolic geometry". American Mathematical Society. Bulletin. New Series 6 (3): 357–381. doi:10.1090/S027309791982150030. ISSN 00029904. MR648524 This gives the original statement of the conjecture.
 William Thurston. Threedimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0691083045 (in depth explanation of the eight geometries and the proof that there are only eight)
 William Thurston. The Geometry and Topology of ThreeManifolds, 1980 Princeton lecture notes on geometric structures on 3manifolds.
External links
 The Geometry of 3Manifolds(video) A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.
Categories: Geometric topology
 Riemannian geometry
 3manifolds
 Conjectures
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