Differential Topology Forty-six Years Later John Milnor n the 1965 Hedrick Lectures,1 I described its uniqueness up to a PL-isomorphism that is the state of differential topology, a field that topologically isotopic to the identity. In particular, was then young but growing very rapidly. they proved the following. During the intervening years, many problems Theorem 1. If a topological manifold Mn without in differential and geometric topology that boundary satisfies Ihad seemed totally impossible have been solved, 3 n 4 n often using drastically new tools. The following is H (M ; Z/2) = H (M ; Z/2) = 0 with n ≥ 5 , a brief survey, describing some of the highlights then it possesses a PL-manifold structure that is of these many developments. unique up to PL-isomorphism. Major Developments (For manifolds with boundary one needs n > 5.) The corresponding theorem for all manifolds of The first big breakthrough, by Kirby and Sieben- dimension n ≤ 3 had been proved much earlier mann [1969, 1969a, 1977], was an obstruction by Moise [1952]. However, we will see that the theory for the problem of triangulating a given corresponding statement in dimension 4 is false. topological manifold as a PL (= piecewise-linear) An analogous obstruction theory for the prob- manifold. (This was a sharpening of earlier work lem of passing from a PL-structure to a smooth by Casson and Sullivan and by Lashof and Rothen- structure had previously been introduced by berg. See [Ranicki, 1996].) If B and B are the Top PL Munkres [1960, 1964a, 1964b] and Hirsch [1963]. stable classifying spaces (as described in the lec- (See also [Hirsch-Mazur, 1974].) Furthermore, Cerf tures), they showed that the relative homotopy had filled in a crucial step, proving by a difficult group π (B ,B ) is cyclic of order two for j = 4, j Top PL geometric argument that the space of orientation- and zero otherwise. Given an n-dimensional topo- preserving diffeomorphisms of the three-sphere logical manifold Mn, it follows that there is an is connected. (See the Cartan Seminar Lectures of obstruction o ∈ H4(Mn; Z/2) to triangulating Mn 1962/63, as well as [Cerf, 1968].) Combined with ≥ as a PL-manifold. In dimensions n 5 this is other known results, this led to the following. the only obstruction. Given such a triangulation, there is a similar obstruction in H3(Mn; Z/2) to Theorem 2. Every PL-manifold of dimension n ≤ 7 possesses a compatible differentiable structure, John Milnor is professor of mathematics at Stony Brook and this structure is unique up to diffeomorphism University. His email address is [email protected]. whenever n < 7. The author would like to thank Mike Hopkins, Rob Kirby, For more information see Further Details at the Mark Mahowald, John Morgan, Larry Siebenmann, and Jonathan Sondow for their help. He also wants to thank end of this article. the NSF for its support under grant DMSO757856. The next big breakthrough was the classifi- 1These lectures have recently been digitized by MSRI and cation of simply connected closed topological should be available soon. Thanks to Dusa McDuff for 4-manifolds by Freedman [1982]. He proved, us- unearthing the original tapes. (With regard to Wilder’s ing wildly nondifferentiable methods, that such a introduction, compare [Milnor, 1999].) manifold is uniquely determined by 804 Notices of the AMS Volume 58, Number 6 (1) the isomorphism class of The Poincaré Hypothesis: Three Versions the symmetric bilinear form First consider the purely topological version. H2 ⊗ H2 → H4 ≅ Z , where Theorem 3. The topological Poincaré hypothesis is Hk = Hk(M4; Z), together with true in all dimensions. (2) the Kirby-Siebenmann invariant o ∈ H4(M4; Z/2) ≅ Z/2. That is, every closed topological manifold with the homotopy type of an n-sphere is actually These can be prescribed arbitrarily, except for homeomorphic to the n-sphere. For n > 4 this was two restrictions: The bilinear form must have proved by Newman [1966] and by Connell [1967], determinant ±1; and in the “even case” where both making use of the “engulfing method” of x ∪ x ≡ 0 (mod 2 H4) for every x ∈ H2, the Stallings [1960]. For n = 4 it is of course due to Kirby-Siebenmann class must be congruent to Freedman. For n = 3 it is due to Perelman, using (1/8)th of the signature. As an example, the Moise [1952] to pass from the topological to the Poincaré hypothesis for 4-dimensional topological PL case, and then using the Munkres-Hirsch-Mazur manifolds is an immediate consequence. For if obstruction theory to pass from PL to smooth. M4 is a homotopy sphere, then both H2 and the obstruction class must be zero. Theorem 4. The piecewise-linear Poincaré hypoth- One year later, Donaldson [1983] used gauge esis is true for n-dimensional manifolds except pos- theoretic methods to show that many of these sibly when n = 4. topological manifolds cannot possess any smooth That is, any closed PL manifold of dimension structure (and hence by Theorem 2 cannot be n = 4 with the homotopy type of an n-sphere is triangulated as PL-manifolds). More explicitly, if PL-homeomorphic to the n-sphere. For n > 4 this 4 M is smooth and simply connected with positive was proved by Smale [1962]; while for n = 3 it definite bilinear form, he showed that this form follows from Perelman’s work, together with the must be diagonalizable; hence M4 must be home- Munkres-Hirsch-Mazur obstruction theory. omorphic to a connected sum of copies of the The differentiable Poincaré hypothesis is more complex projective plane. There are many posi- complicated, being true in some dimensions tive definite bilinear forms with determinant 1, and false in others, while remaining totally and with signature divisible by 16 in the even mysterious in dimension 4. We can formulate case, which are not diagonalizable. (See, for ex- the question more precisely by noting that the ample, [Milnor-Husemoller, 1973].) Each of these set of all oriented diffeomorphism classes of corresponds to a topological manifold M4 with closed smooth homotopy n-spheres (= topological 4 no smooth structure but such that M × R does n-spheres) forms a commutative monoid Sn under have a smooth structure that is unique up to the connected sum operation. In fact this monoid diffeomorphism. is actually a finite abelian group except possibly The combination of Freedman’s topological re- when n = 4. Much of the following outline is sults and Donaldson’s analytic results quickly led based on [Kervaire-Milnor, 1963], which showed in to rather amazing consequences. For example, it principle how to compute these groups2 in terms followed that there are uncountably many noniso- of the stable homotopy groups of spheres for morphic smooth or PL structures on R4. (Compare n > 4. Unfortunately, many proofs were put off to [Gompf, 1993].) All other dimensions are better part 2 of this paper, which was never completed. behaved: For n > 4, Stallings [1962] showed that However, the missing arguments have been the topological space Rn has a unique PL-structure supplied elsewhere; see especially [Levine, 1985]. up to PL-isomorphism. Using the Moise result for Using Perelman’s result for n = 3, the group Sn n < 4 together with the Munkres-Hirsch-Mazur ob- can be described as follows for small n (Table 1). struction theory, it follows that the differentiable (Here, for example, 2 · 8 stands for the group Z Z structure of Rn is unique up to diffeomorphism /2 ⊕ /8, and 1 stands for the trivial group.) for all n = 4. Thus the differentiable Poincaré hypothesis is A satisfactory theory of three-dimensional man- true in dimensions 1, 2, 3, 5, 6, and 12, but ifolds took longer. The first milestone was the unknown in dimension 4. I had conjectured that it geometrization conjecture by Thurston [1982, would be false in all higher dimensions. However, 1986], which set the goal for what a theory Mahowald has pointed out that there is at least of three-manifolds should look like. This con- 2The Kervaire-Milnor paper worked rather with the jecture was finally verified by Perelman [2002, group n of homotopy spheres up to h-cobordism. This 2003a, 2003b], using a difficult argument based makesΘ a difference only for n = 4, since it is known, on the “Ricci flow” partial differential equation. using the h-cobordism theorem of Smale [1962], that (Compare the expositions of Morgan-Tian [2007] ≅ Sn → n for n = 4. However, the difference is important and Kleiner-Lott [2008].) The three-dimensional in theΘ four-dimensional case, since 4 is trivial, while the Poincaré hypothesis followed as a special case. structure of S4 is the great unsolvedΘ problem. June/July 2011 Notices of the AMS 805 Table 1 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Sn 1 1 1 ? 1 1 28 2 2·2·2 6 992 1 3 2 2·8128 2 2·8 2·8 bp one more exceptional case: The group S61 is also where Sn ⊂ Sn stands for the subgroup trivial. (Compare Further Details below.) represented by homotopy spheres that bound parallelizable manifolds. This subgroup is the Problem. Is the group S nontrivial for all n part of S that is best understood. It can be n > 6 , n = 12, 61? n partially described as follows. (Any precise computation for large n is impossi- Theorem 5. = Sbp ble at the present time, since the stable homotopy For n 4 the group n is finite cyclic groups of spheres have been computed completely with an explicitly known generator.