Towards the Poincaré Conjecture and the Classification of 3-Manifolds John Milnor he Poincaré Conjecture was posed ninety- space SO(3)/I60 . Here SO(3) is the group of rota- nine years ago and may possibly have tions of Euclidean 3-space, and I60 is the subgroup been proved in the last few months. This consisting of those rotations which carry a regu- note will be an account of some of the lar icosahedron or dodecahedron onto itself (the Tmajor results over the past hundred unique simple group of order 60). This manifold years which have paved the way towards a proof has the homology of the 3-sphere, but its funda- and towards the even more ambitious project of mental group π1(SO(3)/I60) is a perfect group of classifying all compact 3-dimensional manifolds. order 120. He concluded the discussion by asking, The final paragraph provides a brief description of again translated into modern language: the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work If a closed 3-dimensional manifold has will be provided in a subsequent note by Michael trivial fundamental group, must it be Anderson. homeomorphic to the 3-sphere? The conjecture that this is indeed the case has Poincaré’s Question come to be known as the Poincaré Conjecture. It At the very beginning of the twentieth century, has turned out to be an extraordinarily difficult Henri Poincaré (1854–1912) made an unwise claim, question, much harder than the corresponding which can be stated in modern language as follows: question in dimension five or more,2 and is a If a closed 3-dimensional manifold has key stumbling block in the effort to classify the homology of the sphere S3 , then it 3-dimensional manifolds. is necessarily homeomorphic to S3 . During the next fifty years the field of topology 1 grew from a vague idea to a well-developed disci- (See POINCARÉ 1900. ) However, the concept of “fun- pline. However, I will call attention only to a few damental group”, which he had introduced already developments that have played an important role in 1895, provided the machinery needed to disprove in the classification problem for 3-manifolds. (For this statement. In POINCARÉ 1904 he presented a further information see: GORDON for a history of counter-example that can be described as the coset 3-manifold theory up to 1960; HEMPEL for a pre- John Milnor is professor of mathematics at SUNY at Stony sentation of the theory up to 1976; BING for a Brook. His email address is [email protected]. description of some of the difficulties associated 1Names in small caps refer to the list of references at the with 3-dimensional topology; JAMES for a general end. Poincaré’s terminology may confuse modern readers history of topology; WHITEHEAD for homotopy who use the phrase “simply-connected” to refer to a space 2 with trivial fundamental group. In fact, he used “simply- Compare SMALE 1960, STALLINGS, ZEEMAN, and WALLACE connected” to mean homeomorphic to the simplest possi- for dimension five or more, and FREEDMAN for dimension ble model, that is, to the 3-sphere. four. 1226 NOTICES OF THE AMS VOLUME 50, NUMBER 10 theory; and DEVLIN for the Poincaré Conjecture as be recovered as the connected sum of the compo- a Millennium Prize Problem.) nents of M3, together with n copies of the “handle” 1 2 S × S . (Compare SEIFERT 1931, MILNOR 1962.) Results Based on Piecewise-Linear Methods In 1933 Herbert SEIFERT (1907–1966) introduced Since the problem of characterizing the 3-sphere a class of fibrations which play an important role seemed so difficult, Max DEHN (1878–1952) tried the in subsequent developments. For our purposes, a simpler problem of characterizing the unknot Seifert fibration of a 3-manifold can be defined as within S3 . a circle action which is free except on at most fi- nitely many “short” fibers, as described below. Theorem claimed by Dehn (1910). A piecewise- Such an action is specified by a map (x, t) → xt linearly embedded circle K ⊂ S3 is unknotted if and from M3 × (R/Z) to M3 satisfying the usual only if the fundamental group π (S3 K) is free 1 conditions that x0 = x and xs+t =(xs )t. We require cyclic. that each fiber xR/Z should be a circle and that the This is a true statement. However, Kneser, nineteen action of R/Z should be free except on at most years later, pointed out a serious gap in Dehn’s finitely many of these fibers. Here is a canonical proof. The question remained open for nearly fifty model for a Seifert fibration in a neighborhood years, until the work of Papakyriakopoulos. of a short fiber: Let α be a primitive n-th root of Several basic steps were taken by James Wad- unity, and let D ⊂ C be the open unit disk. Form del ALEXANDER (1888–1971). In 1919 he showed that the product D × R and then identify each (z,t) with the homology and fundamental group alone are not (αz , t +1/n). The resulting quotient manifold enough to characterize a 3-manifold. In fact, he de- is diffeomorphic to the product D × (R/Z), but scribed two lens spaces which can be distinguished the central fiber under the circle action (z,t)s = only by their linking invariants. In 1924 he proved (z,t+ s) is shorter than neighboring fibers, the following. which wrap n times around it, since (0,t)1/n ≡ (0,t). There were dramatic developments in 3-manifold Theorem of Alexander. A piecewise-linearly em- theory, starting in the late 1950s with a paper by bedded 2-sphere in S3 cuts the 3-sphere into two Christos PAPAKYRIAKOPOULOS (1914–1976). He was a closed piecewise-linear 3-cells. quiet person who had worked by himself in Alexander also showed that a piecewise-linearly Princeton for many years under the sponsorship embedded torus must bound a solid torus on at of Ralph Fox. (I was also working with Fox at least one of its two sides. the time, but had no idea that Papakyriakopoulos Helmut KNESER (1898–1973) carried out a further was making progress on such an important project.) step that has played a very important role in later His proof of “Dehn’s Lemma”, which had stood as developments.3 He called a closed piecewise-linear an unresolved problem since Kneser first pointed 3-manifold irreducible if every piecewise-linearly out the gap in Dehn’s argument, was a tour de embedded 2-sphere bounds a 3-cell, and reducible force. Here is the statement: otherwise. Suppose that we start with such a man- Dehn’s Lemma (Papakyriakopoulos 1957). ifold M3 which is connected and reducible. Then, 3 Consider a piecewise-linear mapping f from a cutting M along an embedded 2-sphere which 2-dimensional disk into a 3-manifold, where the does not bound a 3-cell, we obtain a new manifold image may have many self-intersections in the in- (not necessarily connected) with two boundary terior but is not allowed to have any self-intersections 2-spheres. We can again obtain a closed (possibly near the boundary. Then there exists a nonsingular disconnected) 3-manifold by adjoining a cone over embedding of the disk which coincides with f each of these boundary 2-spheres. Now either each throughout some neighborhood of the boundary. component of the resulting manifold is irreducible or we can iterate this procedure. He proved this by constructing a tower of cover- ing spaces, first simplifying the singularities of Theorem of Kneser (1929). This procedure al- the disk lifted to the universal covering space of a ways stops after a finite number of steps, yielding neighborhood, then passing to the universal cov- 3 a manifold M such that each connected compo- ering of a neighborhood of the simplified disk and 3 nent of M is irreducible. iterating this construction, obtaining a nonsingu- In fact, in the orientable case, if we keep careful track lar disk after finitely many steps. Using similar of orientations and the number n of nonseparating methods, he proved a result which was later sharp- cuts, then the original connected manifold M3 can ened as follows. 3Parts of Kneser’s paper were based on Dehn’s work. In Sphere Theorem. If the second homotopy group 3 a note added in proof, he pointed out that Dehn’s argu- π2(M ) of an orientable 3-manifold is nontrivial, ment was wrong and hence that parts of his own paper then there exists a piecewise-linearly embedded were not proven. However, the result described here was 2-sphere which represents a nontrivial element of not affected. this group. NOVEMBER 2003 NOTICES OF THE AMS 1227 As an immediate corollary, it follows that differentiable manifolds, piecewise-linear mani- 3 4 π2(S K)=0 for a completely arbitrary knot folds, or topological manifolds. 3 3 K ⊂ S . More generally, π2(M ) is trivial for any orientable 3-manifold which is irreducible in the Manifolds of Constant Curvature sense of Kneser. The first interesting family of 3-manifolds to be Within a few years of Papakyriakopoulos’s classified were the flat Riemannian manifolds— breakthrough, Wolfgang HAKEN had made sub- those which are locally isometric to Euclidean space. stantial progress in understanding quite general David Hilbert, in the eighteenth of his famous prob- 3-manifolds. In 1961 Haken solved the triviality lems, asked whether there were only finitely many problem for knots; that is, he described an effec- discrete groups of rigid motions of Euclidean n- tive procedure for deciding whether a piecewise- space with compact fundamental domain.