INTRODUCCIN. El material presentado es una recopilacin de trabajos no completa-sobre la conjetura de Poincar. Est ordenado de la siguiente manera: John Milnor reporte 2002 p 2 -6 John Milnor reporte 2003 p 7 14 John Milnor AMS Notices 2003 p 8 22 Allyn Jackson AMS Notices 2006 p 23 -27 HUAI-DONG-CAO XI-PING ZHU Asian 2006 p 28- 355 Morgan Tian 2007 p 356 -847 Gregory Perelman 2003 p 848- 869 Los trabajos de Milnor contienen notas histricas muy interesantes, despus de todo es uno de los grandes gemetras. Jackson nos presenta un veredicto favorable a los trabajos de Perelman. El artculo de Dong Zhu es polmico, pues ellos afirman que ellos resolvieron el problema. Morgan- Tian nos ofrecen un verdadero libro, en el que se reconoce plenamente a Perelman y sus mtodos como el matemtico que resuelve la conjetura de Poincar El artculo de Perelman se presenta para tener una idea de como escribe este extraordinario matemtico ruso. Afortunadamente casi todos los artculos de Perelman estn disponibles en la red. Vernor Arguedas Mayo 2007

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The Poincar Conjecture 99 Years Later: A Progress Report e The topology of 2-dimensional manifolds or surfaces was well understood in the 19th century. 1 In fact, there is a simple list of all possible smooth compact orientable surfaces. Any such surface has a well-dened genus g 0 , which can be described intuitively as the number of holes; two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have the same genus.

Figure 1. Sketches of smooth surfaces of genus 0, 1, and 2. The corresponding question in higher dimensions is much more dicult. Henri Poincar e was perhaps the rst to try to make a similar study of 3-dimensional manifolds. The most basic example of such a manifold is the 3-dimensional unit sphere, that is, the locus of all points (x , y , z , w) in 4-dimensional Euclidean space which have distance exactly 1 from the origin: x2 + y2 + z2 + w2 = 1 . He noted that a distinguishing feature of the 2-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere. In 1904, he asked a corresponding question in dimension three. 2 In more modern language, it can be phrased as follows: If a smooth compact 3-dimensional manifold M 3 has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M 3 is homeomorphic to the sphere S 3 ? He commented, with considerable foresight, Mais cette question nous entranerait trop loin. Since then, the hypothesis that every simply connected closed 3-manifold is homeomorphic to the 3-sphere has been known as the Poincar Conjecture. It has inspired topologists ever e since, and attempts to prove it have led to many advances in our understanding of the topology of manifolds. 3 Early Missteps. From the rst, the apparently simple nature of this statement has led mathematicians to overreach. Four years earlier, in 1900, Poincar himself had been the rst to err, stating e a false theorem that can be phrased as follows. 4 Every compact polyhedral manifold with the homology of an n-dimensional sphere is actually homeomorphic to the n-dimensional sphere.1 For denitions and other background material, see for example Massey, or Munkres 1975, as well as

Thurston 1997. (Names in small caps refer to the list of references at the end.) 2, 4 See Poincare, pages 486, 498; and also 370. 3 For a representative collection of attacks on the Poincar Conjecture, see Papakyriakopoulos, Birman, e Jakobsche, Thickstun, Gillman and Rolfsen, Gabai 1995, Rourke, and Poenaru.

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With further study, Poincar found a beautiful counterexample to his own claim. This e example can be described geometrically as follows. Consider all possible regular icosahedra inscribed in the 2-dimensional unit sphere. In order to specify one particular icosahedron in this family, we must provide three parameters. For example, two parameters are needed to specify a single vertex on the sphere, and then another parameter to specify the direction to a neighboring vertex. Thus each such icosahedron can be considered as a single point in the 3dimensional manifold M 3 consisting of all such icosahedra. 5 This manifold meets Poincars e preliminary criterion: By the methods of homology theory, it cannot be distinguished from the 3-dimensional sphere. However, he could prove that it is not a sphere by constructing a simple closed curve that cannot be deformed to a point within M 3 . The construction is not dicult: Choose some representative icosahedron and consider its images under rotation about one vertex through angles 0 2/5 . This denes a simple closed curve in M 3 that cannot be deformed to a point.

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Figure 2. The Whitehead link. The next important false theorem was by Henry Whitehead in 1934. As part of a purported proof of the Poincar Conjecture, he claimed that every open 3-dimensional manifold e that is contractible (that is, can be continuously deformed to a point) is homeomorphic to Euclidean space. Following in Poincars footsteps, he then substantially increased our une derstanding of the topology of manifolds by discovering a counterexample to his own theorem. (See Whitehead, pp. 2150.) His counterexample can be briey described as follows. Start with two disjoint solid tori T0 and T1 in the 3-sphere that are embedded as shown in Figure 2, so that each one individually is unknotted, but so that the two are linked together with linking number zero. Since T1 is unknotted, its complement T1 = S 3 interior(T1 ) is another unknotted solid torus that contains T0 . Choose a homeomorphism h of the 3sphere that maps T0 onto this larger solid torus T1 . Then we can inductively construct an increasing sequence of unknotted solid tori T0 T 1 T 2 in S 3 by setting Tj+1 = h(Tj ) . The union M 3 = Tj of this increasing sequence is the required Whitehead counterexample, a contractible manifold that is not homeomorphic to Euclidean space. To see that 1 (M 3 ) = 0 , note that every closed loop in T0 can be shrunk to a point (after perhaps crossing through itself) within the larger solid torus T 1 . But every closed loop in M 3 must be contained in some Tj , and hence can be shrunk to a point within5 In more technical language, this M 3 can be dened as the coset space SO(3)/I 60 where SO(3) is the

group of all rotations of Euclidean 3-space and where I60 is the subgroup consisting of the 60 rotations which carry a standard icosahedron to itself. The fundamental group 1 (M 3 ) , consisting of all homotopy classes of loops from a point to itself within M 3 , is a perfect group of order 120. 2

Tj+1 M 3 . On the other hand, M 3 is not homeomorphic to Euclidean 3-space since, if K M 3 is any compact subset which is large enough to contain T0 , one can check that the dierence set M 3 K is not simply connected. Since this time, many false proofs of the Poincar Conjecture have been proposed, some e of them relying on errors that are rather subtle and dicult to detect. For a delightful presentation of some of the pitfalls of 3-dimensional topology, see Bing. Higher Dimensions. The late 1950s and early 1960s saw an avalanche of progress with the discovery that higher dimensional-manifolds are actually easier to work with than 3-dimensional ones. One reason for this is the following: The fundamental group plays an important role in all dimensions even when it is trivial, and relations between generators of the fundamental group correspond to 2-dimensional disks, mapped into the manifold. In dimension 5 or greater, such disks can be put into general position so that they are disjoint from each other, with no self-intersections, but in dimension 3 or 4 it may not be possible to avoid intersections, leading to serious diculties. Stephen Smale announced a proof of the Poincar Conjecture in high dimensions in e 1960. He was quickly followed by John Stallings, who used a completely dierent method, and by Andrew Wallace, who had been working along lines quite similar to those of Smale. Let me rst describe the Stallings result, which has a weaker hypothesis and easier proof, but also a weaker conclusion. He assumed that the dimension is 7 or greater, but Christopher Zeeman later extended his argument to dimensions 5 and 6. Stallings-Zeeman Theorem. If M n is a nite simplicial complex of dimension n 5 which has the homotopy type 6 of the sphere S n and is locally piecewise linearly homeomorphic to the Euclidean space Rn , then M n is homeomorphic to S n under a homeomorphism which is piecewise linear except at a single point. In other words, the complement M n (point) is piecewise linearly homeomorphic to Rn . (The method of proof consists of pushing all of the diculties o toward a single point; hence there can be no control near that point.) The Smale proof, and the closely related proof given shortly afterward by Wallace, depended rather on dierentiable methods, building a manifold up inductively, starting with an n-dimensional ball, by successively adding handles. Here a k -handle can be added to a manifold M n with boundary by rst attaching a k -dimensional cell, using an attaching homeomorphism from the (k 1)-dimensional boundary sphere into the boundary of M n , and then thickening and smoothing corners to obtain a larger manifold with boundary. The proof is carried out by rearranging and canceling such handles. (Compare the presentation in Milnor, Siebenmann and Sondow.)6 In order to check that a manifold M n has the same homotopy type as the sphere S n , we must check

not only that it is simply connected, 1 (M n ) = 0 , but also that it has the same homology as the sphere. The example of the product S 2 S 2 shows that it is not enough to assume that 1 (M n ) = 0 when n > 3 . 3

Figure 3. A 3-dimensional ball with a 1-handle attached. Smale Theorem. If M n is a dierentiable homotopy sphere of dimension n 5 , then M n is homeomorphic to S n . In fact M n is dieomorphic to a manifold obtained by gluing together the boundaries of two closed n-balls under a suitable dieomorphism. This was also proved by Wallace, at least for n 6 . (It should be noted that the 5-dimensional case is particularly dicult.) The much more dicult 4-dimensional case had to wait twenty years, for the work of Michael Freedman. Here the dierentiable methods used by Smale and Wallace and the piecewise linear methods used by Stallings and Zeeman do not work at all. Freedman used wildly non-dierentiable methods, not only to prove the 4-dimensional Poincar Conjecture e for topological manifolds, but also to give a complete classication of all closed simply connected topological 4-manifolds. The integral cohomology group H 2 of such a manifold is free abelian. Freedman needed just two invariants: The cup product : H 2 H 2 H 4 Z = is a symmetric bilinear form with determinant 1 , while the Kirby-Siebenmann invariant is an integer mod 2 that vanishes if and only if the product manifold M 4 R can be given a dierentiable structure. Freedman Theorem. Two closed simply connected 4-manifolds are homeomorphic if and only if they have the same bilinear form and the same KirbySiebenmann invariant . Any can be realized by such a manifold. If (x x) is odd for some x H 2 , then either value of can be realized also. However, if (x x) is always even, then is determined by , being congruent to one eighth of the signature of . In particular, if M 4 is a homotopy sphere, then H 2 = 0 and = 0 , so M 4 is homeomorphic to S 4 . It should be noted that the piecewise linear or dierentiable theories in dimension 4 are much more dicult. It is not known whether every smooth homotopy 4-sphere is dieomorphic to S 4 ; it is not known which 4-manifolds with = 0 actually possess dierentiable structures; and it is not known when this structure is essentially unique. The major results on these questions are due to Simon Donaldson. As one indication of the complications, Freedman showed, using Donaldsons work, that R4 admits uncountably many inequivalent dierentiable structures. (Compare Gompf.) In dimension 3, the discrepancies between topological, piecewise linear, and dierentiable theories disappear (see Hirsch, Munkres 1960, and Moise). However, diculties with the fundamental group become severe. The Thurston Geometrization Program. In the 2-dimensional case, each smooth compact surface can be given a beautiful geometrical structure, as a round sphere in the genus 0 case, as a at torus in the genus 1 case,4

and as a surface of constant negative curvature when the genus is 2 or more. A far-reaching conjecture by William Thurston in 1983 claims that something similar is true in dimension 3. His conjecture asserts that every compact orientable 3-dimensional manifold can be cut up along 2-spheres and tori so as to decompose into essentially unique pieces, each of which has a simple geometrical structure. There are eight possible 3-dimensional geometries in Thurstons program. Six of these are now well understood, 7 and there has been a great deal of progress with the geometry of constant negative curvature. 8 However, the eighth geometry, corresponding to constant positive curvature, remains largely untouched. For this geometry, we have the following extension of the Poincar Conjecture. e Thurston Elliptization Conjecture. Every closed 3-manifold with nite fundamental group has a metric of constant positive curvature, and hence is homeomorphic to a quotient S 3 / , where SO(4) is a nite group of rotations that acts freely on S 3 . The Poincar Conjecture corresponds to the special case where the group 1 (M 3 ) e = is trivial. The possible subgroups SO(4) were classied long ago by Hopf, but this conjecture remains wide open. Approaches through Dierential Geometry and Dierential Equations. In recent years there have been several attacks on the geometrization problem (and hence on the Poincar Conjecture) based on a study of the geometry of the innite-dimensional e space consisting of all Riemannian metrics on a given smooth 3-dimensional manifold. By denition, the length of a path on a Riemannian manifold is computed, in terms gij dxi dxj . From the rst and second of the metric tensor gij , as the integral ds = derivatives of this metric tensor, one can compute the Ricci curvature tensor Rij , and the scalar curvature R . (As an example, for the at Euclidean space one gets Rij = R = 0 , while for a round 3-dimensional sphere of radius r , one gets Ricci curvature Rij = 2gij /r2 and scalar curvature R = 6/r 2 .) One approach, due to Michael Anderson and building on earlier work by Hidehiko Yamabe, studies the total scalar curvature S =M3

R dV

as a functional on the space of all smooth unit volume Riemannian metrics. The critical points of this functional are the metrics of constant curvature. Another approach, due to Richard Hamilton, studies the Ricci ow , that is, the solutions to the dierential equation dgij = 2Rij . dt In other words, the metric is required to change with time so that distances decrease in directions of positive curvature. This is essentially a parabolic dierential equation and7 See, for example, Gordon and Heil, Auslander and Johnson, Scott, Tukia, Gabai 1992, and

Casson and Jungreis. 8 See Sullivan, Morgan, Thurston 1986, McMullen, and Otal. The pioneering papers by Haken and Waldhausen provided the basis for much of this work.

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behaves much like the heat equation studied by physicists: If we heat one end of a cold rod, then the heat will gradually ow throughout the rod until it attains an even temperature. Similarly, the initial hope for a 3-manifold with nite fundamental group was that, under the Ricci ow, positive curvature would tend to spread out until, in the limit (after rescaling to constant size), the manifold would attain constant curvature. If we start with a 3-manifold of positive Ricci curvature, Hamilton was able to carry out this program and construct a metric of constant curvature, thus solving a very special case of the Elliptization Conjecture. In the general case there are serious diculties, since this ow may tend toward singularities. He conjectured that these singularities must have a very special form, however, so that the method could still be used to construct a constant curvature metric. Three months ago, Grisha Perelman in St. Petersburg posted a preprint describing a way to resolve some of the major stumbling blocks in the Hamilton program and suggesting a path toward a solution of the full Elliptization Conjecture. The initial response of experts to this claim has been carefully guarded optimism, although, in view of the long history of false proofs in this area, no one will be convinced until all of the details have been carefully explained and veried. Perelman is planning to visit the United States in April, at which time his arguments will no doubt be subjected to detailed scrutiny. I want to thank the many mathematicians who helped me with this report. John Milnor, Stony Brook University, February 2003 References. M. T. Anderson, Scalar curvature and geometrization conjectures for 3-manifolds, M.S.R.I Publ. 30, 1997. , On long-time evolution in general relativity and geometrization of 3-manifolds, Comm. Math. Phys. 222 (2001), no. 3, 533567. L. Auslander and F.E.A. Johnson, On a conjecture of C.T.C. Wall , J. Lond. Math. Soc. 14 (1976) 331332. R. H. Bing, Some aspects of the topology of 3-manifolds related to the Poincar conjecture, e in Lectures on Modern Mathematics II, edit. T. L. Saaty, Wiley, 1964. J. Birman, Poincars conjecture and the homeotopy group of a closed, orientable 2-manifold , e Collection of articles dedicated to the memory of Hanna Neumann, VI. J. Austral. Math. Soc. 17 (1974), 214221. A. Casson and D. Jungreis, Convergence groups and Seifert bered 3-manifolds, Invent. Math. 118 (1994) 441456. S. K. Donaldson, Self-dual connections and the topology of smooth 4-manifolds, Bull. Amer. Math. Soc. 8 (1983) 8183. M. H. Freedman, The topology of four-dimensional manifolds, J. Di. Geom. 17 (1982), 357453. D. Gabai, Convergence groups are Fuchsian groups, Annals of Math. 136 (1992) 447510. , Valentin Poenarus program for the Poincar conjecture, Geometry, topology, & e physics, 139166, Conf. Proc. Lecture Notes Geom. Topology, VI, Internat. Press, Cambridge, MA, 1995.6

D. Gillman and D. Rolfsen, The Zeeman conjecture for standard spines is equivalent to the Poincar conjecture, Topology 22 (1983), no. 3, 315323. e R. Gompf, An exotic menagerie, J. Dierential Geom. 37 (1993) 199223. C. Gordon and W. Heil, Cyclic normal subgroups of fundamental groups of 3-manifolds, Topology 14 (1975) 305309. W. Haken, Uber das Homomorphieproblem der 3-Mannigfaltigkeiten I , Math. Z. 80 (1962) o 89120. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Dierential Geom. 17 (1982), no. 2, 255306. , The formation of singularities in the Ricci ow , Surveys in dierential geometry, Vol. II (Cambridge, MA, 1993), 7136, Internat. Press, Cambridge, MA, 1995. , Non-singular solutions of the Ricci ow on three-manifolds Comm. Anal. Geom. 7 (1999), no. 4, 695729. M. Hirsch, Obstruction theories for smoothing manifolds and maps, Bull. Amer. Math. Soc. 69 (1963) 352356. H. Hopf, Zum Cliord-Kleinschen Raumproblem, Math. Annalen 95 (192526) 313319. W. Jakobsche, The Bing-Borsuk conjecture is stronger than the Poincar conjecture, Fund. e Math. 106 (1980) 127134. W. S. Massey, Algebraic Topology: An Introduction, Harcourt Brace, 1967; Springer, 1977; or A Basic Course in Algebraic Topology, Springer, 1991. C. McMullen, Riemann surfaces and geometrization of 3-manifolds, Bull. Amer. Math. Soc. 27 (1992) 207216. J. Milnor with L. Siebenmann and J. Sondow, Lectures on the h-Cobordism Theorem, Princeton Math. Notes, Princeton U. Press, 1965. E. E. Moise, Geometric Topology in Dimensions 2 and 3, Springer, 1977. J. Morgan, On Thurstons uniformization theorem for three-dimensional manifolds, pp. 37 125 of The Smith Conjecture, edit. Bass and Morgan, Pure and Appl. Math. 112, Academic Press, 1984. J. Munkres, Obstructions to the smoothing of piecewise-dierentiable homeomorphisms, Annals Math. 72 (1960) 521554. , Topology: A First Course, Prentice Hall, 1975. J.-P. Otal, The hyperbolization theorem for bered 3-manifolds, Translated from the 1996 French original by Leslie D. Kay. SMF/AMS Texts and Monographs, 7. American Mathematical Society, Providence, RI; Socit Mathmatique de France, Paris, 2001. C. Papakyriakopoulos, A reduction of the Poincar conjecture to group theoretic conjectures, e Annals Math. 77 (1963) 250305. G. Perelman, The entropy formula for the Ricci ow and its geometric applications, (preprint available form the Los Alamos National Laboratory archive: math.DG/0211159v1, 11 Nov 2002). V. Ponaru, A program for the Poincar conjecture and some of its ramications, Topics e e in low-dimensional topology (University Park, PA, 1996), 6588, World Sci. Publishing, River Edge, NJ, 1999. H. Poincar, Oeuvres, Tome VI, Paris, 1953. e7

C. Rourke, Algorithms to disprove the Poincar conjecture, Turkish J. Math. 21 (1997) e 99110. P. Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980) 241277. , There are no fake Seifert bre spaces with innite 1 , Annals of Math. 117 (1983) 3570 , The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983) 401487. S. Smale, Generalized Poincars conjecture in dimensions greater than four , Annals Math. e 74 (1961) 391-406. (See also: Bull. Amer. Math. Soc. 66 (1960) 373375.) , The story of the higher dimensional Poincar conjecture (What actually happened e on the beaches of Rio), Math. Intelligencer 12 (1990) 4451. J. Stallings, Polyhedral homotopy spheres, Bull. Amer. Math. Soc. 66 (1960) 485488. D. Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et sur les varits hyperee boliques de dimension 3 bres sur le cercle, Sm. Bourbaki 554, Lecture Notes Math. e e 842, Springer, 1981. T. L. Thickstun, Open acyclic 3-manifolds, a loop theorem and the Poincar conjecture, e Bull. Amer. Math. Soc. (N.S.) 4 (1981) 192194. W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, in The Mathematical heritage of Henri Poincar, Proc. Symp. Pure Math. 39 (1983), e Part 1. (Also in Bull. Amer. Math. Soc. 6 (1982) 357381.) , Hyperbolic structures on 3-manifolds, I, deformation of acyclic manifolds , Annals of Math. 124 (1986) 203246 , Three-Dimensional Geometry and Topology, Vol. 1. edit. Silvio Levy. Princeton Mathematical Series 35. Princeton University Press, 1997. P. Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 154. F. Waldhausen, On irreducible 3-manifolds which are suciently large, Annals of Math. 87 (1968) 5688. A. Wallace, Modications and cobounding manifolds, II , J. Math. Mech 10 (1961) 773809. J. H. C. Whitehead, Mathematical Works, Volume II, Pergamon, 1962. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 2137. E. C. Zeeman, The Poincar conjecture for n 5 , in Topology of 3-Manifolds and Related e Topics Prentice Hall, 1962. (See also Bull. Amer. Math. Soc. 67 (1961) 270.)

8

Towards the Poincar Conjecture and the Classification of 3-ManifoldsJohn Milnor

T

he Poincar Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3-dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelmans work will be provided in a subsequent note by Michael Anderson.

space SO(3)/I60 . Here SO(3) is the group of rotations of Euclidean 3-space, and I60 is the subgroup consisting of those rotations which carry a regular icosahedron or dodecahedron onto itself (the unique simple group of order 60). This manifold has the homology of the 3-sphere, but its fundamental group 1 (SO(3)/I60 ) is a perfect group of order 120. He concluded the discussion by asking, again translated into modern language: If a closed 3-dimensional manifold has trivial fundamental group, must it be homeomorphic to the 3-sphere? The conjecture that this is indeed the case has come to be known as the Poincar Conjecture. It has turned out to be an extraordinarily difficult question, much harder than the corresponding question in dimension five or more,2 and is a key stumbling block in the effort to classify 3-dimensional manifolds. During the next fifty years the field of topology grew from a vague idea to a well-developed discipline. However, I will call attention only to a few developments that have played an important role in the classification problem for 3-manifolds. (For further information see: GORDON for a history of 3-manifold theory up to 1960; HEMPEL for a presentation of the theory up to 1976; BING for a description of some of the difficulties associated with 3-dimensional topology; JAMES for a general history of topology; W HITEHEAD for homotopy2 Compare SMALE 1960, STALLINGS, ZEEMAN, and WALLACE

Poincars QuestionAt the very beginning of the twentieth century, Henri Poincar (18541912) made an unwise claim, which can be stated in modern language as follows: If a closed 3-dimensional manifold has the homology of the sphere S 3 , then it is necessarily homeomorphic to S 3 . (See POINCAR 1900.1 ) However, the concept of fundamental group, which he had introduced already in 1895, provided the machinery needed to disprove this statement. In POINCAR 1904 he presented a counter-example that can be described as the cosetJohn Milnor is professor of mathematics at SUNY at Stony Brook. His email address is jack@math.sunysb.edu.1 Names in small caps refer to the list of references at the

end. Poincars terminology may confuse modern readers who use the phrase simply-connected to refer to a space with trivial fundamental group. In fact, he used simplyconnected to mean homeomorphic to the simplest possible model, that is, to the 3-sphere.

for dimension five or more, and FREEDMAN for dimension four.

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theory; and DEVLIN for the Poincar Conjecture as a Millennium Prize Problem.)

Results Based on Piecewise-Linear MethodsSince the problem of characterizing the 3-sphere seemed so difficult, Max DEHN (18781952) tried the simpler problem of characterizing the unknot within S 3 . Theorem claimed by Dehn (1910). A piecewiselinearly embedded circle K S 3 is unknotted if and only if the fundamental group 1 (S 3 K) is free cyclic. This is a true statement. However, Kneser, nineteen years later, pointed out a serious gap in Dehns proof. The question remained open for nearly fifty years, until the work of Papakyriakopoulos. Several basic steps were taken by James Waddel ALEXANDER (18881971). In 1919 he showed that the homology and fundamental group alone are not enough to characterize a 3-manifold. In fact, he described two lens spaces which can be distinguished only by their linking invariants. In 1924 he proved the following. Theorem of Alexander. A piecewise-linearly embedded 2-sphere in S 3 cuts the 3-sphere into two closed piecewise-linear 3-cells. Alexander also showed that a piecewise-linearly embedded torus must bound a solid torus on at least one of its two sides. Helmut KNESER (18981973) carried out a further step that has played a very important role in later developments.3 He called a closed piecewise-linear 3-manifold irreducible if every piecewise-linearly embedded 2-sphere bounds a 3-cell, and reducible otherwise. Suppose that we start with such a manifold M 3 which is connected and reducible. Then, cutting M 3 along an embedded 2-sphere which does not bound a 3-cell, we obtain a new manifold (not necessarily connected) with two boundary 2-spheres. We can again obtain a closed (possibly disconnected) 3-manifold by adjoining a cone over each of these boundary 2-spheres. Now either each component of the resulting manifold is irreducible or we can iterate this procedure. Theorem of Kneser (1929). This procedure always stops after a finite number of steps, yielding a manifold M 3 such that each connected component of M 3 is irreducible. In fact, in the orientable case, if we keep careful track of orientations and the number n of nonseparating cuts, then the original connected manifold M 3 can3 Parts of Knesers paper were based on Dehns work. In

be recovered as the connected sum of the components of M 3 , together with n copies of the handle S 1 S 2. (Compare SEIFERT 1931, MILNOR 1962.) In 1933 Herbert SEIFERT (19071966) introduced a class of fibrations which play an important role in subsequent developments. For our purposes, a Seifert fibration of a 3-manifold can be defined as a circle action which is free except on at most finitely many short fibers, as described below. Such an action is specified by a map (x, t) xt from M 3 (R/Z) to M 3 satisfying the usual conditions that x0 = x and xs+t = (xs )t . We require that each fiber xR/Z should be a circle and that the action of R/Z should be free except on at most finitely many of these fibers. Here is a canonical model for a Seifert fibration in a neighborhood of a short fiber: Let be a primitive n-th root of unity, and let D C be the open unit disk. Form the product D R and then identify each (z, t) with (z , t + 1/n) . The resulting quotient manifold is diffeomorphic to the product D (R/Z) , but the central fiber under the circle action (z, t)s = (z , t + s) is shorter than neighboring fibers, which wrap n times around it, since (0, t)1/n (0, t) . There were dramatic developments in 3-manifold theory, starting in the late 1950s with a paper by Christos PAPAKYRIAKOPOULOS (19141976). He was a quiet person who had worked by himself in Princeton for many years under the sponsorship of Ralph Fox. (I was also working with Fox at the time, but had no idea that Papakyriakopoulos was making progress on such an important project.) His proof of Dehns Lemma, which had stood as an unresolved problem since Kneser first pointed out the gap in Dehns argument, was a tour de force. Here is the statement: Dehns Lemma (Papakyriakopoulos 1957). Consider a piecewise-linear mapping f from a 2-dimensional disk into a 3-manifold, where the image may have many self-intersections in the interior but is not allowed to have any self-intersections near the boundary. Then there exists a nonsingular embedding of the disk which coincides with f throughout some neighborhood of the boundary. He proved this by constructing a tower of covering spaces, first simplifying the singularities of the disk lifted to the universal covering space of a neighborhood, then passing to the universal covering of a neighborhood of the simplified disk and iterating this construction, obtaining a nonsingular disk after finitely many steps. Using similar methods, he proved a result which was later sharpened as follows. Sphere Theorem. If the second homotopy group 2 (M 3 ) of an orientable 3-manifold is nontrivial, then there exists a piecewise-linearly embedded 2-sphere which represents a nontrivial element of this group.OF THE

a note added in proof, he pointed out that Dehns argument was wrong and hence that parts of his own paper were not proven. However, the result described here was not affected.

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As an immediate corollary, it follows that 2 (S 3 K) = 0 for a completely arbitrary knot K S 3 . More generally, 2 (M 3 ) is trivial for any orientable 3-manifold which is irreducible in the sense of Kneser. Within a few years of Papakyriakopouloss breakthrough, Wolfgang HAKEN had made substantial progress in understanding quite general 3-manifolds. In 1961 Haken solved the triviality problem for knots; that is, he described an effective procedure for deciding whether a piecewiselinearly embedded circle in the 3-sphere is actually knotted. (See SCHUBERT 1961 for further results in this direction and a clearer exposition.) Friedhelm WALDHAUSEN made a great deal of further progress based on Hakens ideas. In 1967a he showed that there is a close relationship between Seifert fiber spaces and manifolds whose fundamental group has nontrivial center. In 1967b he introduced and analyzed the class of graph manifolds. By definition, these are manifolds that can be split by disjoint embedded tori into pieces, each of which is a circle bundle over a surface. Two key ideas in the Haken-Waldhausen approach seem rather innocuous but are actually extremely powerful: Definitions. For my purposes, a two-sided piecewise-linearly embedded closed surface F in a closed manifold M 3 will be called incompressible if the fundamental group 1 (F) is infinite and maps injectively into 1 (M 3 ) . The manifold M 3 is sufficiently large if it contains an incompressible surface. As an example of the power of these ideas, Waldhausen showed in 1968 that if two closed orientable 3-manifolds are irreducible and sufficiently large, with the same fundamental group, then they are actually homeomorphic. There is a similar statement for manifolds with boundary. These ideas were further developed in 1979 by JACO and SHALEN and by JOHANNSON, who emphasized the importance of decomposing a space by incompressible tori. Another important development during these years was the proof that every topological 3-manifold has an essentially unique piecewise-linear structure (see MOISE) and an essentially unique differentiable structure (see MUNKRES or HIRSCH, together with SMALE 1959). This is very different from the situation in higher dimensions, where it is essential to be clear as to whether one is dealing with4 The statement that a piecewise-linear manifold has an

differentiable manifolds, piecewise-linear manifolds, or topological manifolds.4

Manifolds of Constant CurvatureThe first interesting family of 3-manifolds to be classified were the flat Riemannian manifolds those which are locally isometric to Euclidean space. David Hilbert, in the eighteenth of his famous problems, asked whether there were only finitely many discrete groups of rigid motions of Euclidean nspace with compact fundamental domain. Ludwig BIEBERBACH (18861982) proved this statement in 1910 and in fact gave a complete classification of such groups. This had an immediate application to flat Riemannian manifolds. Here is a modern version of his result. Theorem (after Bieberbach). A compact flat Riemannian manifold M n is characterized, up to affine diffeomorphism, by its fundamental group. A given group occurs if and only if it is finitely generated, torsionfree, and contains an abelian subgroup of finite index. Any such contains a unique maximal abelian subgroup of finite index. It follows easily that this maximal abelian subgroup N is normal and that the quotient group = /N acts faithfully on N by conjugation. Furthermore, N Zn where n is the dimension. Thus the finite group embeds naturally into the group GL(n, Z) of automorphisms of N . In particular, it follows that any such manifold M n can be described as a quotient manifold T n / , where T n is a flat torus, is a finite group of isometries which acts freely on T n , and the fundamental group 1 (T n ) can be identified with the maximal abelian subgroup N 1 (M n ). In the 3-dimensional orientable case, there are just six such manifolds. The group SL(3, Z) is either cyclic of order 1, 2, 3, 4, or 6 or is isomorphic to Z/2 Z/2 . (For further information see CHARLAP, as well as ZASSENHAUS, MILNOR 1976a, or THURSTON 1997.) Compact 3-manifolds of constant positive curvature were classified in 1925 by Heinz H OPF (18941971). (Compare SEIFERT 1933, MILNOR 1957.) These included, for example, the Poincar icosahedral manifold which was mentioned earlier. Twenty-five years later, Georges DE RHAM (1903 1990) showed that Hopfs classification, up to isometry, actually coincides with the classification up to diffeomorphism. The lens spaces, with finite cyclic fundamental group, constitute a subfamily of particular interest. The lens spaces with group of order 5 were introduced already by ALEXANDER in 1919 as examples of 3-manifolds which could not be distinguished by homology and fundamental group alone. Lens spaces were classified up to piecewise-linear homeomorphism in 1935 by Reidemeister, Franz, and de Rham, using an invariant which they called VOLUME 50, NUMBER 10

essentially unique differentiable structure remains true in dimensions up to six. (Compare CERF.) However, KIRBY AND SIEBENMANN showed that a topological manifold of dimension four or more may well have several incompatible piecewise-linear structures. The four-dimensional case is particularly perilous: Freedman, making use of work of Donaldson, showed that the topological manifold R4 admits uncountably many inequivalent differentiable or piecewise-linear structures. (See GOMPF.)

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torsion. (See MILNOR 1966 as well as MILNOR AND BURLET 1970 for expositions of these ideas.) The topological invariance of torsion for an arbitrary simplicial complex was proved much later by CHAPMAN. One surprising byproduct of this classification was Horst SCHUBERTS 1956 classification of knots with two bridges, that is, knots that can be placed in R3 so that the height function has just two maxima and two minima. He showed that such a knot is uniquely determined by its associated 2-fold branched covering, which is a lens space. Although 3-manifolds of constant negative curvature actually exist in great variety, few examples were known until Thurstons work in the late 1970s. One interesting example was discovered in 1912 by H. GIESEKING. Starting with a regular 3-simplex of infinite edge length in hyperbolic 3-space, he identified the faces in pairs to obtain a nonorientable complete hyperbolic manifold of finite volume. SEIFERT AND WEBER described a compact example in 1933: starting with a regular dodecahedron of carefully chosen size in hyperbolic space, they identified opposite faces by a translation followed by a rotation through 3/10 of a full turn to obtain a compact orientable hyperbolic manifold. (An analogous construction using 1/10 of a full turn yields Poincars 3-manifold, with the 3-sphere as a 120-fold covering space.) One important property of manifolds of negative curvature was obtained by Alexandre PREISSMANN (19161990). (Preissmann, a student of Heinz Hopf, later changed fields and became an expert on the hydrodynamics of river flow.) Theorem of Preissmann (1942). If M n is a closed Riemannian manifold of strictly negative curvature, then any nontrivial abelian subgroup of 1 (M n ) is free cyclic. The theory received a dramatic impetus in 1975, when Robert RILEY (19352000) made a study of representations of a knot group 1 (S 3 K) into PSL2 (C) . Note that PSL2 (C) can be thought of either as the group of orientation-preserving isometries of hyperbolic 3-space or as the group of conformal automorphisms of its sphere-at-infinity. Using such representations, Riley was able to produce a number of examples of knots whose complement can be given the structure of a complete hyperbolic manifold of finite volume. Inspired by these examples, Thurston developed a rich theory of hyperbolic manifolds. See the discussion in the following section, together with KAPOVICH 2001 or MILNOR 1982. NOVEMBER 2003

The Thurston Geometrization ConjectureThe definitive conjectural picture of 3-dimensional manifolds was provided by William THURSTON in 1982. It asserts that: The interior of any compact 3-manifold can be split in an essentially unique way by disjoint embedded 2-spheres and tori into pieces which have a geometric structure. Here a geometric structure can be defined most easily 5 as a complete Riemannian metric which is locally isometric to one of the eight model structures listed below. For simplicity, I will deal only with closed 3-manifolds. Then we can first express the manifold as a connected sum of manifolds which are prime (that is, not further decomposable as nontrivial connected sums). It is claimed that each prime manifold either can be given such a geometric structure or else can be separated by incompressible tori into open pieces, each of which can be given such a structure. The eight allowed geometric structures are represented by the following examples: the sphere S 3 , with constant curvature +1; the Euclidean space R3 , with constant curvature 0 ; the hyperbolic space H 3 , with constant curvature 1 ; the product S 2 S 1; the product H 2 S 1 of hyperbolic plane and circle; a left invariant 6 Riemannian metric on the special linear group SL(2, R) ; a left invariant Riemannian metric on the solvable Poincar-Lorentz group E(1, 1) , which consists of rigid motions of a 1+1-dimensional spacetime provided with the flat metric dt 2 dx2 ; a left invariant metric on the nilpotent Heisenberg group, consisting of 3 3 matrices of the form 1 0 1 . 0 0 1 In each case, the universal covering of the indicated manifold provides a canonical model for the5 More formally, the canonical model for such a geometric

structure is one of the eight possible pairs (X, G) where X is a simply-connected 3-manifold and G is a transitive group of diffeomorphisms such that G admits a left and right invariant volume form such that the subgroup fixing any point of X is compact and such that G is maximal as a group of diffeomorphisms with this last property.6 See MILNOR 1976b 4 for the list of left invariant metrics

in dimension 3.

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corresponding geometry. Examples of the first three geometries were discussed in the section on constant curvature. A closed orientable manifold locally isometric to S 2 S 1 is necessarily diffeomorphic (but not necessarily isometric) to the manifold S 2 S 1 itself, while any product of a surface of genus two or more with a circle represents the H 2 S 1 geometry. The unit tangent bundle of a surface of genus two or more represents the SL(2, R) geometry. A torus bundle over the circle represents the Poincar-Lorentz solvegeometry, provided that its monodromy is represented by a transformation of the torus with a matrix such as 21 which has an eigenvalue greater than one. 11 Finally, any nontrivial circle bundle over a torus represents the nilgeometry. Six of these eight geometries, all but the hyperbolic and solvegeometry cases, correspond to manifolds with a Seifert fiber space structure. Two special cases are of particular interest. The conjecture would imply that: A closed 3-manifold has finite fundamental group if and only if it has a metric of constant positive curvature. In particular, any M 3 with trivial fundamental group must be homeomorphic to S 3 . This is a very sharp version of the Poincar Conjecture. Another consequence would be the following: A closed manifold M 3 is hyperbolic if and only if it is prime, with an infinite fundamental group which contains no Z Z. In the special case of a manifold which is sufficiently large, Thurston himself proved this last statement and in fact proved the full geometrization conjecture. (See MORGAN, THURSTON 1986, and MCMULLEN 1992.) Another important result by Thurston is that a surface bundle over the circle is hyperbolic if and only if (1) its monodromy is pseudo-Anosov up to isotopy and (2) its fiber has negative Euler characteristic. See SULLIVAN, MCMULLEN 1996, or OTAL. The spherical and hyperbolic cases of the Thurston Geometrization Conjecture are extremely difficult. However, the remaining six geometries are well understood. Many authors have contributed to this understanding (see, for example, GORDON AND HEIL, AUSLANDER AND JOHNSON, SCOTT, TUKIA, GABAI, and CASSON AND JUNGREIS). See THURSTON 1997 and SCOTT 1983b for excellent expositions.

one-parameter family of Riemannian metrics gij = gij (t) satisfying the differential equation

gij /t = 2 Rij ,where Rij = Rij ({ghk }) is the associated Ricci tensor. This particular differential equation was chosen by Hamilton for much the same reason that Einstein introduced the Ricci tensor into his theory of gravitation7 he needed a symmetric 2-index tensor which arises naturally from the metric tensor and its derivatives gij /uh and 2 gij /uh uk . The Ricci tensor is essentially the only possibility. The factor of 2 in this equation is more or less arbitrary, but the negative sign is essential. The reason for this is that the Ricci flow equation is a kind of nonlinear generalization of the heat equation

/t = of mathematical physics. For example, as gij varies under the Ricci flow, the associated scalar curvature R = g ij Rij varies according to a nonlinear version R/t = R + 2 R ij Rij of the heat equation. Like the heat equation, the Ricci flow equation is well behaved in forward time and acts as a kind of smoothing operator but is usually impossible to solve in backward time. If some parts of a solid object are hot and others are cold, then, under the heat equation, heat will flow from hot to cold, so that the object gradually attains a uniform temperature. To some extent the Ricci flow behaves similarly, so that the curvature tries to become more uniform, but there are many complications which have no easy resolution. To give a very simple example of the Ricci flow, consider a round sphere of radius r in Euclidean (n + 1) -space. Then the metric tensor takes the form gij = r 2 gij where gij is the metric for a unit sphere, while the Ricci tensor

Rij = (n 1)gijis independent of r . The Ricci flow equation reduces to dr 2 /dt = 2(n 1) with solution

r 2 (t) = r 2 (0) 2(n 1) t.Thus the sphere collapses to a point in finite time. More generally, Hamilton was able to prove the following. Theorem of Hamilton. Suppose that we start with a compact 3-dimensional manifold whose Ricci tensor is everywhere positive definite. Then, as the7 For relations between the geometrization problem and

The Ricci FlowA quite different method was introduced by Richard HAMILTON 1982. Consider a Riemannian manifold with local coordinates u1 , . . . , un and with metric ds 2 = gij dui duj . The associated Ricci flow is a 1230 NOTICESOF THE

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manifold shrinks to a point under the Ricci flow, it becomes rounder and rounder. If we rescale the metric so that the volume remains constant, then it converges towards a manifold of constant positive curvature. Hamilton tried to apply this technique to more general 3-manifolds, analyzing the singularities which may arise, but was able to prove the geometrization conjecture only under very strong supplementary hypotheses. (For a survey of such results, see CAO AND CHOW.) In a remarkable trio of preprints, Grigory PERELMAN has announced a resolution of these difficulties and promised a proof of the full Thurston conjecture based on Hamiltons ideas, with further details to be provided in a fourth preprint. One way in which singularities may arise during the Ricci flow is that a 2-sphere in M 3 may collapse to a point in finite time. Perelman shows that such collapses can be eliminated by performing a kind of surgery on the manifold, analogous to Knesers construction described earlier. After a finite number of such surgeries, he asserts that each component either: 1. converges towards a manifold of constant positive curvature which shrinks to a point in finite time, or possibly 2. converges towards an S 2 S 1 which shrinks to a circle in finite time, or 3. admits a Thurston thick-thin decomposition, where the thick parts correspond to hyperbolic manifolds and the thin parts correspond to the other Thurston geometries. I will not attempt to comment on the details of Perelmans arguments, which are ingenious and highly technical. However, it is clear that he has introduced new methods that are both powerful and beautiful and made a substantial contribution to our understanding. References[Alexander, 1919] J. W. ALEXANDER, Note on two threedimensional manifolds with the same group, Trans. Amer. Math. Soc. 20 (1919), 339342. [Alexander, 1924] , On the subdivision of 3-space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 68. [Anderson, 1997] M. T. ANDERSON, Scalar Curvature and Geometrization Conjectures for 3-Manifolds, Math. Sci. Res. Inst. Publ., vol. 30, 1997. [Anderson, 2001] , On long-time evolution in gen eral relativity and geometrization of 3-manifolds, Comm. Math. Phys. 222 (2001), 533567. [Auslander and Johnson, 1976] L. AUSLANDER and F. E. A. JOHNSON, On a conjecture of C. T. C. Wall, J. London. Math. Soc. 14 (1976), 331332. [Bieberbach, 1910] L. BIEBERBACH, ber die Bewegungsgruppen des n -dimensionalen euklidischen Raumes mit einem endlichen Fundamentalbereich, Gtt. Nachr. (1910), 7584.

[Bieberbach, 1911/12] , ber die Bewegungsgrup pen der Euklidischen Rume I, II, Math. Ann. 70 (1911), 297336; 72 (1912), 400412. [Bing, 1964] R. H. BING, Some aspects of the topology of 3-manifolds related to the Poincar conjecture, in Lectures on Modern Mathematics II (T. L. Saaty, ed.), Wiley, 1964. [Cao and Chow, 1999] H.-D. CAO and B. CHOW, Recent developments on the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 5974. [Casson and Jungreis, 1994] A. CASSON and D. JUNGREIS, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994), 441456. [Cerf, 1968] J. CERF, Sur les diffomorphismes de la sphre de dimension trois (4 = 0) , Lecture Notes in Math., vol. 53, Springer-Verlag, 1968. [Chapman, 1974] T. A. CHAPMAN, Topological invariance of Whitehead torsion, Amer. J. Math. 96 (1974), 488497. [Charlap, 1986] L. CHARLAP, Bieberbach Groups and Flat Manifolds, Springer, 1986. (See also: Compact flat Riemannian manifolds. I, Ann. of Math. 81 (1965), 1530.) [Dehn, 1910] M. DEHN, ber die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), 137168. [Devlin, 2002] K. DEVLIN, The Millennium Problems, Basic Books, 2002. [Freedman, 1982] M. H. FREEDMAN, The topology of fourdimensional manifolds, J. Differential Geom. 17 (1982), 357453. [Gabai, 1992] D. GABAI, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992), 447510. [Gieseking, 1912] H. GIESEKING, Analytische Untersuchungen ueber topologische Gruppen, Thesis, Muenster, 1912. [Gompf, 1993] R. GOMPF, An exotic menagerie, J. Differential Geom. 37 (1993), 199223. [Gordon, 1999] C. MCA. GORDON, 3-dimensional topology up to 1960, pp. 449490 in [James, 1999]. [Gordon and Heil, 1975] C. MCA. GORDON and W. HEIL, Cyclic normal subgroups of fundamental groups of 3-manifolds, Topology 14 (1975), 305309. [Haken, 1961a] W. HAKEN, Ein Verfahren zur Aufspaltung einer 3 -Mannigfaltigkeit in irreduzible 3 -Mannigfaltigkeiten, Math. Z. 76 (1961), 427467. [Haken, 1961b] , Theorie der Normalflchen, Acta Math. 105 (1961), 245375. [Haken, 1962] , ber das Homomorphieproblem der 3-Mannigfaltigkeiten I, Math. Z. 80 (1962), 89120. [Hamilton, 1982] R. S. HAMILTON, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255306. [Hamilton, 1995] , The formation of singularities in the Ricci flow, Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), International Press, Cambridge, MA, 1995, pp. 7136. [Hamilton, 1999] , Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), 695729. [Hempel, 1976] J. HEMPEL, 3-Manifolds, Ann. of Math. Studies, vol. 86, Princeton Univ. Press, 1976. [Hirsch, 1963] M. HIRSCH, Obstruction theories for smoothing manifolds and maps, Bull. Amer. Math. Soc. 69 (1963), 352356. [Hopf, 1925] H. HOPF, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95 (1925), 313319.

NOVEMBER 2003

NOTICES

OF THE

AMS

1231

[Jaco and Shalen, 1979] W. JACO and P. SHALEN, Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21, no. 220 (1979). [James, 1999] I. M. JAMES (ed.), History of Topology, NorthHolland, Amsterdam, 1999. [Johannson, 1979] K. JOHANNSON, Homotopy Equivalences of 3-Manifolds with Boundaries, Lecture Notes in Math., vol. 761, Springer, 1979. [Kapovich, 2001] M. KAPOVICH, Hyperbolic Manifolds and Discrete Groups, Progress Math., vol. 183, Birkhuser, 2001. [Kirby and Siebenmann, 1969] R. KIRBY and L. SIEBENMANN, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742749. [Kneser, 1929] H. KNESER, Geschlossene Flchen in dreidimensionalen Mannigfaltigkeiten, Jahresber. Deutsch. Math. Verein. 38 (1929), 248260. [Milnor, 1957] J. MILNOR, Groups which act on S n without fixed points, Amer. J. Math. 79 (1957), 623630; reprinted in [Milnor, 1995]. [Milnor, 1962] , A unique decomposition theorem for 3 -manifolds, Amer. J. Math. 84 (1962), 17; reprinted in [Milnor, 1995]. [Milnor, 1966] , Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358426; reprinted in [Milnor, 1995]. [Milnor, 1976a] , Hilberts problem 18: on crystal lographic groups, fundamental domains, and on sphere packing, Mathematical Developments Arising from Hilbert Problems, Proc. Sympos. Pure Math., vol. 28, Part 2, Amer. Math. Soc., 1976, pp. 491506; reprinted in [Milnor, 1994]. [Milnor, 1976b] , Curvatures of left invariant met rics on Lie groups, Adv. Math. 21 (1976), 293329; reprinted in [Milnor, 1994]. [Milnor, 1982] , Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 924; also in Proc. Sympos. Pure Math., vol. 39, Amer. Math. Soc., 1983; and in [Milnor, 1995]. (See also: How to compute volume in hyperbolic space in this last collection.) [Milnor, 1994] , Collected Papers 1, Geometry, Publish or Perish, 1994. [Milnor, 1995] , Collected Papers 2, The Fundamen tal Group, Publish or Perish, 1995. [Milnor and Burlet, 1970] J. MILNOR and O. BURLET, Torsion et type simple dhomotopie, Essays on Topology and Related Topics, Springer, 1970, pp. 1217; reprinted in [Milnor, 1995]. [McMullen, 1992] C. MCMULLEN, Riemann surfaces and geometrization of 3-manifolds, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 207216. [McMullen, 1996] , Renormalization and 3-Manifolds Which Fiber over the Circle, Ann. of Math. Studies, vol. 142, Princeton Univ. Press, 1996. [Moise, 1977] E. E. MOISE, Geometric Topology in Dimensions 2 and 3, Springer, 1977. [Morgan, 1984] J. MORGAN, On Thurstons uniformization theorem for three-dimensional manifolds, The Smith Conjecture (Bass and Morgan, eds.), Pure Appl. Math., vol. 112, Academic Press, 1984, pp. 37125. [Munkres, 1960] J. MUNKRES, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. 72 (1960), 521554. (See also: Concordance is equivalent to smoothability, Topology 5 (1966), 371389.)

[Otal, 1996] J.-P. OTAL, Le thorme dhyperbolisation pour les varits fibres de dimension 3, Astrisque 235 (1996); translated in SMF/AMS Texts and Monographs, vol. 7, Amer. Math. Soc. and Soc. Math. France, Paris, 2001. [Papakyriakopoulos, 1957] C. PAPAKYRIAKOPOULOS, On Dehns lemma and the asphericity of knots, Ann. of Math. 66 (1957), 126. (See also: Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 169172.) [Papakyriakopoulos, 1960] , The Theory of Three Dimensional Manifolds since 1950, Proc. Internat. Congr. Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 433440. [Perelman, 2002] G. PERELMAN, The entropy formula for the Ricci flow and its geometric applications (available on the Internet from: arXiv:math.DG/0211159 v1, 11 November 2002). [Perelman, 2003] , Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109 v1, 10 March 2003; Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245, 17 July 2003). [Poincar, 1895] H. POINCAR, Analysis situs, J. de lcole Polytechnique 1 (1895), 1121. (See Oeuvres, Tome VI, Paris, 1953, pp. 193288.) [Poincar, 1900] , Second complment lanalysis situs, Proc. London Math. Soc. 32 (1900), 277308. (See Oeuvres, Tome VI, Paris, 1953, p. 370.) [Poincar, 1904] , Cinquime complment lanalysis situs, Rend. Circ. Mat. Palermo 18 (1904), 45110. (See Oeuvres, Tome VI, Paris, 1953, p. 498.) [Preissmann, 1942] A. PREISSMANN, Quelques proprits globales des espaces de Riemann, Comment. Math. Helv. 15 (1942), 175216. [de Rham, 1950] G. DE RHAM, Complexes automorphismes et homomorphie diffrentiables, Ann. Inst. Fourier 2 (1950), 5167. [Riley, 1975] R. RILEY, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281288. [Riley, 1979] , An elliptical path from parabolic representations to hyperbolic structures, Topology of Low-Dimensional Manifolds (Proc. Second Sussex Conf., Chelwood Gate), Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 99133. [Riley, 1982] , Seven excellent knots, Low-Dimen sional Topology (Bangor), London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, CambridgeNew York, 1982, pp. 81151. [Schubert, 1956] H. SCHUBERT, Knoten mit zwei Brcken, Math. Z. 65 (1956), 133170. [Schubert, 1961] , Bestimmung der Primfaktorzer legung von Verkettungen, Math. Z. 76 (1961), 116148. [Scott, 1980] P. SCOTT, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980), 241277. [Scott, 1983a] , There are no fake Seifert fibre spaces with infinite 1, Ann. of Math. 117 (1983), 3570. [Scott, 1983b] , The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401487. [Seifert, 1931] H. SEIFERT, Konstruktion dreidimensionaler geschlossener Rume, Ber. Verh. Schs. Akad. Wiss. Leipzig 83 (1931), 2666. [Seifert, 1933] , Topologie dreidimensionales gefaserter Raum, Acta Math. 60 (1933), 147288; translated in Pure Appl. Math., vol. 89, Academic Press, 1980.

1232

NOTICES

OF THE

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[Seifert and Threlfall, 1934] H. SEIFERT and W. THRELFALL, Lehrbuch der Topologie, Teubner, 1934; translated in Pure Appl. Math., vol. 89, Academic Press, 1980. [Seifert and Weber, 1933] H. SEIFERT and C. WEBER, Die beiden Dodekaederrume, Math. Z. 37 (1933), 237253. [Smale, 1959] S. SMALE, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621626. [Smale, 1960] , The generalized Poincar conjecture in higher dimensions, Bull. Amer. Math. Soc. 66 (1960), 373375. (See also: Generalized Poincars conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391406; as well as: The story of the higher dimensional Poincar conjecture (What actually happened on the beaches of Rio), Math. Intelligencer 12 (1990), 4451.) [Stallings, 1960] J. STALLINGS, Polyhedral homotopy spheres, Bull. Amer. Math. Soc. 66 (1960), 485488. [Sullivan, 1981] D. SULLIVAN, Travaux de Thurston sur les groupes quasi-fuchsiens et sur les varits hyperboliques de dimension 3 fibres sur le cercle, Sm. Bourbaki 554, Lecture Notes in Math., vol. 842, Springer, 1981. [Thurston, 1982] W. P. THURSTON, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357381. (Also in The Mathematical Heritage of Henri Poincar, Proc. Sympos. Pure Math., vol. 39, Part 1, Amer. Math. Soc., 1983.) [Thurston, 1986] , Hyperbolic structures on 3-manifolds, I. Deformation of acyclic manifolds, Ann. of Math. 124 (1986), 203246. [Thurston, 1997] , Three-Dimensional Geometry and Topology, Vol. 1 (Silvio Levy, ed.), Princeton Math. Ser., vol. 35, Princeton Univ. Press, 1997. [Tukia, 1988] P. TUKIA, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 154. [Waldhausen, 1967a] F. WALDHAUSEN, Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505517. [Waldhausen, 1967b] , Eine Klasse von 3-dimen sionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308333; ibid. 4 (1967), 87117. [Waldhausen, 1968] , On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 5688. [Wallace, 1961] A. WALLACE, Modifications and cobounding manifolds. II, J. Math. Mech. 10 (1961), 773809. [Whitehead, 1983] G. W. WHITEHEAD, Fifty years of homotopy theory, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 129. [Zassenhaus, 1948] H. ZASSENHAUS, ber einen Algorithmus zur Bestimmung der Raumgruppen, Comment. Math. Helv. 21 (1948), 117141. [Zeeman, 1962] E. C. ZEEMAN, The Poincar conjecture for n 5 , Topology of 3-Manifolds and Related Topics, Prentice-Hall, 1962, pp. 198204. (See also: Bull. Amer. Math. Soc. 67 (1961), 270.)

About the Cover Regular Polytopes The recipe for producing all regular polyhedra in Euclidean space of arbitrary dimension does not seem to be as well known as it should be, although it is the principal result of the popular and well-beloved book Regular polytopes by the late H. S. M. Coxeter, memorials of whom appear in this issue. Start with a finite group generated by reflections in Euclidean space, which may be identified with a group of transformations of the unit sphere. The hyperplanes of all reflections in this group partition the unit sphere into spherical simplices, called chambers. If one chamber is fixed, then the reflections in its walls generate the entire group, and the chamber is a fundamental domain. If the walls of this fixed chamber are colored, then all the walls of all chambers may be colored consistently. The Coxeter graph of this configuration has as its nodes the generating reflections, and its links, which are labeled, record the generating relations of the products of the generators. There is implicit order 3 on unlabeled links, and no link for a commuting pair. The red and green reflections, for example, on the first cover image generate a group of order ten, thus giving rise to the link of the graph labeled by 5. If one node of the graph is chosen and the walls labeled by the complementary colors are deleted, there results a partition of the sphere by spherical polytopes. On the cover, deleting the red and green edges gives rise to a partition by spherical pentagons. This in turn will correspond to a polytope of one dimension larger inscribed in the sphere, a dodecahedron in this case. The principal result in the subject is that this polytope will be regular if and only if the Coxeter graph is connected and linear (thus excluding the graphs called Dn and En ), and the node is an end. On the cover the two choices of end node give rise to the dodecahedron and icosahedron, but the choice of middle node only to a semiregular solid. All regular polytopes arise in this way. Thus the connected, linear Coxeter graphs associated to finite reflection groups, together with a choice of end node, classify regular polytopes completely. This beautiful formulation of nineteenth-century results is due to Coxeter and explained in his book. It clarifies enormously the classical theory of Book XIII of the Elements. This result has also more recently been intriguingly generalized by Ichiro Satake in a paper (Ann. of Math. 71, 1960) well known to specialists in representation theory, in which he describes all the facets of the convex hull of the reflection group orbit of an arbitary point in Euclidean space. He uses this generalization in his description of compactifications of symmetric spaces. In view of Satakes results, it is not too surprising that these convex hulls also play an important role in modern work on automorphic forms, particularly that of James Arthur. Bill Casselman, Covers Editor (notices-covers@ams.org)

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Conjectures No More?Consensus Forming on the Proof of the Poincar and Geometrization ConjecturesAllyn Jackson

Have the Poincar Conjecture and the Thurston Geometrization Conjecture been proved? This question has been on the minds of mathematicians for more than three years, ever since Grigory Perelman posted his now-famous papers on the Web. In midsummer 2006, as the International Congress of Mathematicians in Madrid approaches and speculation about the Fields Medals is buzzing, some experts who had been making cautious statements for the past three years sound increasingly confident that the conjectures are finally yielding. In particular, many believe the Poincar Conjecture is now a bona fide theorem. The picture is slightly less clear for the Geometrization Conjecture, but there is much optimism that this result will soon be established as well.

Of the Dollars and the GloryFor mathematicians, the million dollars that the Clay Mathematics Institute (CMI) has offered for the solution of the Poincar Conjecture is mere icing on the cake. The real prize is the glory of settling a question that has tantalized mathematicians for more than a century. The statement dates back to 1904, when Henri Poincar conjectured that it is the property of being simply connected that topologically distinguishes the three-sphere from other compact three-manifolds. Since that time there have been many incorrect attempts to prove the Poincar Conjecture, some of them by such wellknown mathematicians as Edwin Moise, Christos Papakyriakopolous, Valentin Poenaru, and ColinAllyn Jackson is senior writer and deputy editor of the Notices. Her email address is axj@ams.org.

Rourke. A recent incorrect proof, by Martin Dunwoody of Southampton University, came in 2002, about six months before Perelman posted his first paper on the subject. Almost as soon as news stories started to appear about Dunwoodys proof (an April 2002 article in the New York Times carried the headline UK Math Wiz May Have Solved Problem), the proof fell apart. In fact, there have been so many wrong proofs of the Poincar Conjecture that John Stallings of the University of California, Berkeley, has posted on his webpage a paper he wrote in 1966 called How not to prove the Poincar Conjecture, which describes his own failed attack, as a warning to others who might hit upon the same idea. One characteristic that most of the failed attempts share is a reliance on topological arguments. But, noted John Morgan of Columbia University, It seems like this problem does not succumb to that type of argument. Rather, he said, one needs tools from outside topology, from geometry and analysis, to tackle this topological question. In contrast to the multiple failed attempts on Poincar, it appears that, before Perelman's work appeared, no one had seriously claimed to be able to prove the full Thurston Geometrization Conjecture. In fact, this is a much deeper and more farreaching statement than the Poincar Conjecture and includes Poincar as a special case. First proposed in the 1970s by William Thurston, who is now at Cornell University, the Geometrization Conjecture provides a way to classify all three-manifolds. Thurstons great insight was to see how geometry could be used to understand the topology ofOF THE

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three-manifolds. The Geometrization Conjecture states that any three-manifold can be split into pieces in an essentially unique way and that each of these pieces carries a geometric structure given by one of eight model geometries. The conjecture was not wide open before the work of Perelman; it had been established in many cases. Thurston himself proved the conjecture for manifolds that are sufficiently large. Several mathematicians contributed to establishing the full conjecture for six of the eight geometries. The two remaining geometries are the spherical and hyperbolic ones, where the metrics have constant positive and constant negative curvature, respectively. The Poincar Conjecture comes under the case of metrics of constant positive curvature. (An excellent historical account is [Milnor].) Against this background, mathematicians were naturally skeptical when Perelman posted his articles on the arXiv, the first in November 2002, the second in March 2003, and the third in July 2003 [Perelman13]. Nevertheless, his efforts were from the outset taken quite seriously. One reason is that Perelman is a well-regarded mathematician who had already made distinguished contributions to geometric analysis. He was an invited speaker at the 1994 ICM in Zurich, where he gave a lecture in the geometry section about spaces with curvature bounded below. In 1996 he was awarded one of the ten prizes given to outstanding young mathematicians every four years by the European Mathematical Society (Perelman refused to accept that prize). Another reason Perelmans work was taken seriously is that it fits into a well-known program to use the Ricci flow to prove the Geometrization Conjecture. The originator of this program is Richard Hamilton, now at Columbia University, who will be a plenary speaker at the 2006 ICM in Madrid. The abstract for Hamiltons talk says that the Ricci flow program was developed by him and Shing-Tung Yau of Harvard University. The idea, first described in a 1982 paper by Hamilton [Hamilton], is to use the Ricci flow, a partial differential equation that is a nonlinear version of the heat equation, to homogenize the geometry of threemanifolds to show that they fit into Thurstons classification. It was generally believed that, philosophically, Hamiltons approach ought to work. This belief strengthened as Hamilton and others worked out much of the analysis that was needed. The toughest obstacle was handling the singularities that could develop in the Ricci flow. It was this obstacle that Perelman, by introducing deep new ideas in geometric analysis, was able to overcome to such spectacular effect. (An excellent expository account about the Ricci flow is [Anderson].) 898 NOTICESOF THE

Poring over PerelmanIn the spring of 2003, after his first two papers had appeared on the Web, Perelman gave lectures at several universities in the U.S., including Columbia University, the Massachusetts Institute of Technology, and Princeton University, as well as a series of lectures at Stony Brook University. Soon thereafter he returned to his home base in St. Petersburg, and he has given only a very few lectures on the subject since then. He answered mathematical questions by email, but some mathematicians report that after a while he stopped even that form of communication. It is not clear what Perelman has made of the acclaim that has surrounded his achievements. Many articles about his work have come out in the popular press, though it appears that he never consented to be interviewed by reporters. As mathematicians began to read the papers carefully, they found them tough going. Perelmans articles are remarkably carefully written if one takes into account how much new ground he breaks in a relatively few number of pages, explained John Lott of the University of Michigan. However, they are not written in such a way that one can just sit down and quickly decide whether his arguments are complete. Morgan remarked that Perelman omits certain technicalities that turned out to be standard, but rather tricky, to work out in detail. And, Morgan said, sometimes arguments are justified by a statement that they are analogous to arguments presented earlier, but it is not always clear exactly how the earlier arguments can be adapted. On top of these difficulties, there are some outright mistakes in the paper, though none has proven serious. It appears that Perelman never submitted his articles to any journal. Had he done so, they probably would not have been accepted without substantial revisions. Soon after Perelmans papers appeared on the Web, mathematicians undertook efforts to understand and verify them. In June 2003 Lott, together with Bruce Kleiner, who is now at Yale University, started a webpage in which they presented notes about Perelmans work as they went carefully through his papers. In late 2003 the American Institute of Mathematics in Palo Alto and the Mathematical Sciences Research Institute in Berkeley jointly sponsored a workshop on Perelmans first article; another workshop, about Perelmans second article, was held in the summer of 2004 at Princeton University. The Clay Institute, which has an obvious interest in knowing whether Perelmans work is correct, provided funding for the Princeton workshop and also sponsored a month-long summer school held at MSRI in the summer of 2005. In addition Clay provided some support to Kleiner and Lott, who continued to add to and post their notes on the Web, as well as Morgan and VOLUME 53, NUMBER 8

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Gang Tian of Princeton University, who are collaborating on a book about Perelmans work on the Poincar Conjecture. In June 2005 Grard Besson of the University of Grenoble presented a Bourbaki lecture on the work of Perelman; the lecture will appear in the Astrisque series in September 2006. In the fall of 2005 Xi-Ping Zhu of Zhongshan University gave a six-month series of lectures at Harvard University, describing the content of a paper that he has written with HuaiDong Cao of Lehigh University and that appeared in the June 2006 issue of the Asian Journal of Mathematics. There have been other workshops and summer schools on the subject, not to mention the many lectures given in mathematics departments and at conferences. Study groups were formed to go through Perelmans papers in several countries, including China, France, Germany, and the United States. While it seems that Perelmans papers were never refereed in the traditional sense, they have been subjected to extraordinary scrutiny in the three and a half years since their posting on the Web. The simple passage of time without anyone finding a serious problem in his work has, at least for many nonexperts, led to a conviction that it must be correct. For example, Koji Fujiwara of Tohoku University is not an expert in this area, but he believes Perelmans work must be right, for two reasons. If there were something philosophically wrong, so that the approach could not work, after three years someone would have found the philosophical problem, he reasoned. And second, Fujiwara said, Perelman is a well-known expert on Ricci curvature, and his previous papers have been reliable and have not been found to contain mistakes. Of course, this kind of confidence is the privilege of the nonexpert. Experts have to work much harder.

Filling in the DetailsThey should give [Perelman] a Fields Medal for the Poincar Conjecture, declared John Morgan in an interview in May 2006. I believe the argument is correct, as do, I think, all who have looked at it seriously.This is clearly the most exciting thing that has happened in mathematics in the last four years, since the previous batch of Fields Medals were awarded. Morgan said that the book he is writing with Tian, which is to appear in early 2007, will provide a full exposition of the proof of the Poincar Conjecture la Perelman.1 Morgan said that he has no doubts that Perelman can also prove the Geometrization Conjecture, but Morgan has1 On July 25, 2006, Morgan and Tian posted on the arXiv

a 473-page manuscript Ricci Flow and the Poincar Conjecture, http://arXiv.org/abs/math/0607607.

not personally gone through that proof in detail, as he has done with Poincar. Indeed, many mathematicians express more confidence in the proof of Poincar than in the proof of Geometrization. Perelman himself provided a shortcut to proving Poincar, and there is a more extensive body of material that is needed for the proof of the full Geometrization Conjecture. Some believe that the best way to ensure that Poincar has really been proved is to verify the proof of Geometrization. So what is the status of the proof of the Geometrization Conjecture? In May 2006 Kleiner and Lott posted on the arXiv an article titled Notes on Perelmans papers. They say that their article, along with a 2005 paper by T. Shioya and T. Yamaguchi, provides details for Perelmans arguments for the Geometrization Conjecture. Lott cautioned that Perelmans work has to be further examined by the mathematical community before there can be any universally accepted verdict. The Kleiner-Lott paper is based on the set of notes they began posting on the Web in the summer of 2003. In the three years over which they developed the notes and made them public, Kleiner and Lott received corrections and comments from many mathematicians. They plan to submit their paper to a journal. In late April 2006 the Asian Journal of Mathematics announced on its website the upcoming publication of the paper by Cao and Zhu, A complete proof of the Poincar and Geometrization ConjecturesApplication of the Hamilton-Perelman theory of the Ricci flow. The announcement included the papers abstract, which states in full: In this paper, we give a complete proof of the Poincar and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. The 330-page paper appeared in print in the June 2006 issue of the Asian Journal. The issue has not been made available electronically on the journals website and is available only as a printed paper publication. The Cao-Zhu article did not circulate as a preprint, but the work presented there was described in Zhus lectures at Harvard during the 20052006 academic year. Some have noted the short amount of time between the submission date for the Cao-Zhu paper, December 12, 2005, and the date when it was accepted for publication, April 16, 2006, and wondered whether such an important paper of over 300 pages could have been refereed in a serious way. In a May 2006 interview, Yau, who is one of the editors-in-chief of the Asian Journal, said that the manuscript had been around for a year, but we have been very careful not to distribute it, to make sure everything is right before it is in print. AskedOF THE

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whether the paper had been refereed in the usual way, Yau said that it had and remarked that the Asian Journal has very high standards. Although not enough time has yet passed for the Cao-Zhu paper to have been subjected to much scrutiny by the mathematical community, the paper became widely known because of coverage about it in the Chinese press during June 2006. Chinese Mathematicians Solve Global Puzzle read the headline of an article that appeared on the Xinhua news service on June 3, 2006. The articles first sentence stated: Two Chinese mathematicians have put the final pieces together in the solution to a puzzle that has perplexed scientists around the globe for more than a century. Cao characterized the barrage of media attention to his work with Zhu as overwhelming. Some of the news articles were translated into English and posted on the Web. In those articles, the achievements of Cao and Zhu, both of whom are Chinese, are emphasized, while the achievements of Perelman are mentioned in a less prominent way. In one story from the Xinhua news agency, which appeared on June 21, 2006, the name of Perelman does not even appear. The coverage began after Yau held a news conference in Beijing on June 3, 2006, in which he announced the work of Cao and Zhu. Yau said that he was misquoted in some of the media accounts and does not endorse what is said there. On June 20, 2006, he presented a public lecture on the subject at the Morningside Center of Mathematics at the Chinese Academy of Sciences in Beijing, the slides of which are available on the centers website at http://www.mcm.ac.cn/Active/yau_new.pdf.

Doling Out the PrizesWith so many players, who will get credit for the proof of these monumental results? This is not a simple question. Often in mathematics credit for a result goes to the person who came up with the decisive ideas that really made the proof work, even if that person never wrote up a complete proof. As a historical example, Robion Kirby of the University of California, Berkeley, pointed to Thurstons orbifold theorem. Thurston described this result in a 1982 article in the Bulletin of the AMS [Thurston], using an argument that Kirby characterized as definitely sketchy. The orbifold theorem covers the Geometrization Conjecture when there is a discrete group acting on the threemanifold with fixed points, and this covers a lot of cases, although not the Poincar Conjecture. After more than a dozen years had passed without a complete proof, Kirby added the orbifold theorem to his well-known problem list in topology and declared it to be an open question. Two different groups of mathematicians independently produced complete proofs of the theorem (one group was Daryl Cooper, Craig Hodgson, and Steven Kerckhoff, 900 NOTICESOF THE

and the other was Michel Boileau, Bernhard Leeb, and Joan Porti). This was a lot of work, some pieces of Thurstons sketch were improved, and the community honors their work, Kirby said. But it is acknowledged that this is Thurstons theorem. The mathematical world is waiting to find out whether Perelman will receive a Fields Medal for his work. The traditional rule followed by the Fields Medal committees is that a recipient must not be over forty in the year in which the medal is given. Perelman turned forty in June 2006. Some believe that, even disregarding the Poincar and Geometrization Conjectures, Perelman may have done enough to deserve a Fields Medal. What Perelmans work says about singularity development in Ricci flow is an enormous advance that in itself would make him a serious candidate for a Fields Medal, Morgan said. The Poincar Conjecture is one of the CMIs seven Millennium Prize Problems, which were announced in 2000. Until Perelmans work, there were no serious solutions proposed to any of the problems, so no prizes have yet been given. The prize rules state that a proposed solution must be published in a refereed journal of worldwide repute and that this published solution must be out for two years before the CMI will consider awarding a prize. The rules are worded in such a way that the person considered for the prize need not be the author of the published solution, noted James Carlson, president of the Clay Mathematics Institute. The fact that Perelman pursued an unorthodox route and posted [his papers] on the arXiv and did not submit them to a journal is not itself an obstacle to him receiving the prize, Carlson said. At the appropriate time, he said, the Clay Institute will consider all the available materials and make a judgment about whether the proof of Poincar is correct. Only after that will it consider giving the prize. One question the Clay Institute faces is whether to give the prize solely to Perelman or to include others as joint recipientsperhaps Hamilton? Carlson said it would be premature for him to speculate on such possibilities. But no doubt the mathematical world will continue to speculate and to discuss the extraordinary saga of Perelmans work. One thing is clear: Perelman has made an enormous contribution to the field. Many of the things he didnot submitting his work to a journal, not lecturing much, completely shunning the limelightare not easy to understand. Perelman is a very talented and unusual individual, and this is the route that he has chosen, Carlson remarked. I think the most important thing is that he wrote those three papers and he posted them on the arXiv, and that has given mathematicians a great gift and lots of new ideas and things to think about. VOLUME 53, NUMBER 8

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References[Anderson] MICHAEL ANDERSON, Geometrization of 3-manifolds via the Ricci flow, Notices 2 (2004), 18493. [Besson] GRARD BESSON, Preuve de le conjecture de Poincar en dformant le mtrique par la courbure de Ricci, daprs G. Perelman, Astrisque 307, Socit Mathmatique de France (to appear September 2006). [Cao-Zhu] HUAI-DONG CAO and XI-PING ZHU, A complete proof of the Poincar and Geometrization ConjecturesApplication of the Hamilton-Perelman Theory of the Ricci flow, Asian J. Math. 10 (2006), 145492. [Kleiner-Lott] BRUCE KLEINER and JOHN LOTT, Notes on Perelmans papers, arXiv:math.DG/0605667, 25 May 2006. [Hamilton] R. S. HAMILTON, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 695729. [Milnor] JOHN MILNOR, Towards the Poincar Conjecture and the classification of 3-manifolds, Notices 10 (2003), 122633. [Perelman1] G. PERELMAN, The entropy formula for the Ricci flow and its geometric applications, http:// arXiv.org/abs/math.DG/0211159, 11 November 2002. [Perelman2] , Ricci flow with surgery on three-man ifolds, http://arxiv.org/abs/math.DG/0303109, 10 March 2003. [Perelman3] , Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, http://arxiv.org/abs/math.DG/0307245, 17 July 2003. [Shioya-Yamaguchi] T. SHIOYA and T. YAMAGUCHI, Volumecollapsed three-manifolds with a lower curvature bound, Math. Ann. 333 (2005), 13155. [Thurston] WILLIAM P. THURSTON, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N. S.) 6 (1982), 35781.

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c 2006 International Press001

A COMPLETE PROOF OF THE POINCARE AND GEOMETRIZATION CONJECTURES APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOWHUAI-DONG CAO AND XI-PING ZHU Abstract. In this paper, we give a complete proof of the Poincar and the geometrization e conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci ow. Key words. Ricci ow, Ricci ow with surgery, Hamilton-Perelman theory, Poincar Conjece ture, geometrization of 3-manifolds AMS subject classications. 53C21, 53C44

CONTENTS Introduction 1 Evolution Equations 1.1 The Ricci Flow . . . . . . . . . . . . . . . . . 1.2 Short-time Existence and Uniqueness . . . . . 1.3 Evolution of Curvatures . . . . . . . . . . . . 1.4 Derivative Estimates . . . . . . . . . . . . . . 1.5 Variational Structure and Dynamic Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 172 172 177 183 190 199 210 210 213 217 223 226 234 239 239 243 255 261 267 267 286 291 302

2 Maximum Principle and Li-Yau-Hamilton Inequalities 2.1 Preserving Positive Curvature . . . . . . . . . . . . . . . . 2.2 Strong Maximum Principle . . . . . . . . . . . . . . . . . 2.3 Advanced Maximum Principle for Tensors . . . . . . . . . 2.4 Hamilton-Ivey Curvature Pinching Estimate . . . . . . . . 2.5 Li-Yau-Hamilton Estimates . . . . . . . . . . . . . . . . . 2.6 Perelmans Estimate for Conjugate Heat Equations . . . . 3 Perelmans Reduced Volume 3.1 Riemannian Formalism in Potentially Innite Dimensions 3.2 Comparison Theorems for Perelmans Reduced Volume . . 3.3 No Local Collapsing Theorem I . . . . . . . . . . . . . . . 3.4 No Local Collapsing Theorem II . . . . . . . . . . . . . . 4 Formation of Singularities 4.1 Cheeger Type Compactness 4.2 Injectivity Radius Estimates 4.3 Limiting Singularity Models 4.4 Ricci Solitons . . . . . . . . Received Department

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December 12, 2005; accepted for publication April 16, 2006. of Mathematics, Lehigh University, Bethlehem, PA 18015, USA (huc2@lehigh.edu). Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China (stszxp@ zsu.edu.cn). 165

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5 Long Time Behaviors 307 5.1 The Ricci Flow on Two-manifolds . . . . . . . . . . . . . . . . . . . . 308 5.2 Dierentiable Sphere Theorems in 3-D and 4-D . . . . . . . . . . . . . 321 5.3 Nonsingular Solutions on Three-manifolds . . . . . . . . . . . . . . . . 336 6 Ancient -solutions 6.1 Preliminaries . . . . . . . . . . . . . . . . 6.2 Asymptotic Shrinking Solitons . . . . . . 6.3 Curvature Estimates via Volume Growth . 6.4 Ancient -solutions on Three-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 357 364 373 384 398 398 405 413 432 452 468 481 486 491

7 Ricci Flow on Three-manifolds 7.1 Canonical Neighborhood Structures . . . . . . . . . . . . 7.2 Curvature Estimates for Smooth Solutions . . . . . . . . . 7.3 Ricci Flow with Surgery . . . . . . . . . . . . . . . . . . . 7.4 Justication of the Canonical Neighborhood Assumptions 7.5 Curvature Estimates for Surgically Modied Solutions . . 7.6 Long Time Behavior . . . . . . . . . . . . . . . . . . . . . 7.7 Geometrization of Three-manifolds . . . . . . . . . . . . . References Index

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Introduction. In this paper, we shall present the