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 hadI 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 2 222 6 992 1 3 2 28128 2 28 28

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. In fact this only up to dimension 64. However, it seems possi- group is: ble that enough is known to decide this question • trivial when n is even, one way or another.) • either trivial or cyclic of order two when Denote the stable homotopy groups of spheres n = 4k − 3, and by • cyclic of order 22k−2(22k−1 −1) numerator Sq 4Bk = − n = πn+q( ) for q > n + 1 , k when n 4k 1 > 3. Π and let Jn ⊂ n be the image of the stable (This last number depends on the computation Whitehead homomorphism J : πn(SO) → n. (See Π of |J4k−1| as described above.) In the odd cases, [Whitehead, 1942].) This subgroup Jn is cyclicΠ of 3 setting n = 2q − 1, an explicit generator for the order bp S2q−1 can be constructed using one basic building |Jn| = block, namely the tangent disk-bundle of the q-sphere, together with one of the following two denominator Bk for n = 4k − 1 ,  4k diagrams.  2 for n ≡ 0, 1 (mod 8), and 1 for n ≡ 2, 4, 5, 6(mod 8),  where the Bk are Bernoulli numbers, for example, 1 1 1 B = ,B = ,B = , 1 6 2 30 3 42 1 5 691 B = ,B = ,B = , Here each circle represents one of our 4 30 5 66 6 2730 2q-dimensional building blocks, which is a Bk and where the fraction 4k must be reduced to 2q-dimensional parallelizable manifold with lowest terms. (Compare [Milnor-Stasheff, 1974, boundary, and each dot represents a plumbing Appendix B].) construction in which two of these manifolds are According to Pontrjagin and Thom, the stable pasted across each other so that their central n-stem n can also be described as the group q-spheres intersect transversally with intersection of all framedΠ cobordism classes of framed man- number +1. The result will be a smooth paral- ifolds. (Here one considers manifolds smoothly lelizable manifold with corners. After smoothing embedded in a high-dimensional Euclidean space, these corners we obtain a smooth manifold X2q and a framing means a choice of trivialization with smooth boundary. for the normal bundle.) Every homotopy sphere For q odd, use the left diagram, and for q is stably parallelizable, and hence possesses such even use the right diagram. In either case, if a framing. If we change the framing, then the q = 2, the resulting smooth boundary ∂X2q will corresponding class in n will be changed by an be a homotopy sphere representing the required element of the subgroup Jn. Thus there is an exact bp Π generator of S2q−1. (The case q = 2 is exceptional sequence since ∂X4 has only the homology of the 3-sphere. In bp bp (1) 0 → Sn → Sn → n/Jn , all other cases where S2q−1 is trivial, the boundary Π will be diffeomorphic to the standard (2q − 1)- 3 This computation of |J4k−1| is a special case of the sphere.) Adams conjecture [Adams, 1963, 1965]. The proof was The exact sequence (1) can be complemented completed by Mahowald [1970], and the full Adams con- by the following information. jecture was proved by Quillen [1971], Sullivan [1974], and by Becker-Gottlieb [1975]. Adams also showed that Theorem 6. For n ≡ 2 (mod 4), every element Jn is always a direct summand of n. of n can be represented by a topological sphere. Π Π

806 Notices of the AMS Volume 58, Number 6 Hence the exact sequence (1) takes the more pre- Further Details cise form Here is a brief outline of the current knowledge bp of . Since the direct summand J is known (2) 0 → Sn → Sn → n/Jn → 0 . n n precisely, we need only look at the quotient /J . However, for n = 4k − 2, itΠ rather extends to an Π n n exact sequence The mostdifficultpartis the 2-primarycomponent,Π which has been computed by Kochman [1990], with bp (3) 0 = S4k−2 → S4k−2 → 4k−2/J4k−2 corrections by Kochman and Mahowald [1995], in Π all cases with n ≤ 64. The 3 and 5 primary →k Z/2 → Sbp → 0 . Φ 4k−3 components have been computed in a much larger Brumfiel [1968, 1969, 1970] sharpened this range by Ravenel [1986]. The primary components result by showing that the exact sequence (2) is for p ≥ 7 are trivial for n < 82. (In fact, for any split exact, except possibly in the case where n has p, the p-primary component of n/Jn is trivial the form 2k − 3. (In fact it could fail to split only whenever n < 2p(p − 1) − 2 and isΠ cyclic of order in the cases n = 2k − 3 ≥ 125. See the discussion p when p = 2p(p − 1) − 2.) below.) Thus the stable stem n is precisely known for The Kervaire homomorphism k in (3) was in- n ≤ 64, and hence the groupΠ Sn is precisely known troduced in [Kervaire, 1960]. (TheΦ image k(θ) ∈ Z for n ≤ 64 , n = 4. In Table 2 the notation bk /2 is called the Kervaire invariant of theΦ homo- bp topy class θ.) Thus there are two possibilities: stands for the order of the subgroup S4k−1 ⊂ S4k−1, 3 bp Z whereas a notation such as 2 4 stands for the • If k = 0, then S4k−3 ≅ /2, generated by 4k−2 direct sum of three copies of Z/2 and one copy of the manifoldΦ ∂X described above, and every Z/4. The trivial group is indicated by a heavy dot. element of 4k−2 canberepresentedbyahomotopy bp sphere. Π All entries corresponding to the subgroups Sn bp Sbp • If k = 0, then S4k−3 = 0. This means that have been underlined. (Note that 4k−3 is cyclic of the boundaryΦ of X4k−2 is diffeomorphic to the order two, indicated by a 2 in the 1 or 5 column, standard S4k−3. We can glue a 4k − 2 ball onto this except in the cases k = 1, 2, 4, 8, 16.) Within this boundary to obtain a framed (4k − 2)-manifold range, the group Sn is trivial only in the cases that is not framed cobordant to any homotopy sphere. In this case the kernel of k forms a n = 1, 2, 3, 5, 6, 12, 61 (and possibly 4). subgroup of index two in 4k−2/J4k−2Φ consisting The corresponding values of b are not difficult of those framed cobordismΠ classes that can be k represented by homotopy spheres. to compute but grow very rapidly. See Table 3 The question as to just when k = 0 was the (with approximate values for k > 5). Note that last major unsolved problem in understandingΦ the those bk for which k has many divisors tend to be group of homotopy spheres. It has recently been somewhat larger. solved in all but one case by Hill, Hopkins, and In conclusion, here is an argument that was Ravenel: postponed above. Theorem 7. The Kervaire homomorphism k is nonzero for k = 1, 2, 4, 8, 16, and possibly for Φ Outline Proof of Theorem 2. It is not difficult k = 32, but is zero in all other cases. + to check that the group π Diff (Sn) consist- 0 In fact Browder [1969] showed that k can ing of all smooth isotopy classes of orienta- be nonzero only if n is a power of two,Φ and tion preserving diffeomorphisms of the unit Barratt, Jones, and Mahowald [1984] completed n-sphere is abelian. Define n to be the quotient the verification that k is indeed nonzero for k = + n−1 of π0 Diff (S ) by the subgroupΓ consisting of 1, 2, 4, 8, 16. Finally, Hill, Hopkins, and Ravenel Φ those isotopy classes that extend over the closed [2010] have shown that k = 0 whenever k > unit n-disk. There is a natural embedding n ⊂ Sn 32. (Their basic tool is aΦ carefully constructed generalized cohomology theory of period 256.) that sends each (f ) ∈ n to the “twisted n-sphere”Γ Thus only the case k = 32, with 4k − 2 = obtained by gluing theΓ boundaries of two n-disks 126, remains unsettled. In particular, for n = together under f . It followed from [Smale, 1962] 4, 125, 126, if the order | n| is known, then we that n = Sn for n ≥ 5, and from [Smale, 1959] can compute the number |SΠn| of exotic n-spheres that Γ3 = 0. Since it is easy to check that 1 = 0 precisely. In fact, if we exclude 4, 126, and numbers and Γ2 = 0, we have Γ k of the form 2 − 3 ≥ 125, then the group Sn can Γ be described completely whenever the structure n = Sn for every n = 4 . 4 of n is known. Γ

4 Π One also needs the fact that the kernel of k is always Mahowald tells me that this is true in dimension 62, and a direct summand of 4k−2 (at least for 4k −Φ 2 = 126). the remaining four cases are straightforward. Π

June/July 2011 Notices of the AMS 807 Table 2 n 0 1 2 3 4 5 6 7

0 + − • • • ? • • b2 2 8 + 2 2 2 6 b3 • 3 2 b4 2 3 2 2 16 + 2 2 2 2 8 b5 2 24 2 2 2 b6 2 24 2 24 + 2 2 2 2 6 b7 2 3 3 b8 2 3 4 3 2 4 32 + 2 2 2 2 4 b9 2 6 2 2 6 2 60 b10 2 6 4 4 2 3 3 3 40 + 2 12 2 2 2 24 b11 8 2 2 720 2 6 b12 2 12 3 2 2 2 4 48 + 2 4 2 6 2 6 b13 2 4 2 6 2 2 2 4 b14 3 3 2 2 3 56 + 2 2 2 2 b15 2 4 • 2 12 b16 2

Table 3 k 2 3 4 5 6 7 8 9 9 7 12 14 bk 28 992 8128 261632 1.45 × 10 6.71 × 10 1.94 × 10 7.54 × 10

k 10 11 12 13 14 15 16 16 17 21 20 25 29 31 bk 2.4×10 3.4×10 8.3×10 7.4×10 3.1×10 5.×10 1.8×10

+ On the other hand, Cerf proved5 that π Diff (S3) [Becker-Gottlieb, 1975] J. C. Becker and D. H. Gottlieb, 0 = 0 and hence that 4 = 0 (although S4 is com- The transfer map and fiber bundles, Topology 14 (1975), 1–12. pletely unknown). UsingΓ results about Sn as [Browder, 1969] W. Browder, The Kervaire invariant of described above, it follows that n = 0 for n < 7, framed manifolds and its generalization, Ann. of and that is finite abelian for all n. n Γ Math. 90 (1969), 157–186. The Munkres-Hirsch-MazurΓ obstructions to [Brumfiel, 1968] G. Brumfiel, On the homotopy groups the existence of a smooth structure on a given of BPL and PL/O, Ann. of Math. 88 (1968), 291–311. n k n PL-manifold M lie in the groups H (M ; k−1), [Brumfiel, 1969] , On the homotopy groups of BPL whereas obstructions to its uniqueness lieΓ in and PL/O. II, Topology 8 (1969), 305–311. k n H (M ; k). (Unlike most of the constructions [Brumfiel, 1970] , On the homotopy groups of BPL discussedΓ above, this works even in dimension 4.) and PL/O. III, Michigan Math. J. 17 (1970), 217–224. Evidently Theorem 2 follows.  [Cerf, 1968] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ( 4 = 0), Lecture Notes in For further historical discussion see Milnor Mathematics, 53, Springer-Verlag,Γ Berlin-New York, [1999, 2007, 2009]. 1968, xii+133 pp. [Connell, 1967] E. H. Connell, A topological H- cobordism theorem for n ≥ 5, Illinois J. Math. References 11 (1967), 300–309. [Adams, 1963] J. F. Adams, On the groups J(X) I, [Donaldson, 1983] S. K. Donaldson, An application Topology 2 (1963), 181–195. of gauge theory to four-dimensional topology, J. [Adams, 1966] , On the groups J(X) II, Topology Differential Geom. 18 (1983), 279–315. 3 (1965), 137–171. [Freedman, 1982] M. H. Freedman, The topology of [Antonelli, Burghelea and Kahn, 1972] P. L. Antonelli, four-dimensional manifolds, J. Differential Geom. D. Burghelea, and P. J. Kahn, The nonfinite 17 (1982), 357–453. homotopy type of some diffeomorphism groups, [Gompf, 1993] R. Gompf, An exotic menagerie, J. Topology 11 (1972), 1–49. Differential Geom. 37 (1993), 199–223. [Barratt-Jones-Mahowald, 1984] M. G. Barratt, J. D. S. [Hatcher, 1983] A. E. Hatcher, A proof of the Smale Jones, and M. E. Mahowald, Relations amongst conjecture, Diff(S3) ≃ O(4), Ann. of Math. 117 Toda brackets and the Kervaire invariant in di- (1983), 553–607. mension 62, J. London Math. Soc. 30 (1984), [Hill-Hopkins-Ravenel, 2010] M. A. Hill, M. J. Hopkins, 533–550. and D. C. Ravenel, On the nonexistence of ele- ments of Kervaire invariant one, arXiv 0908.3724v2 5Hatcher [1983] later proved the sharper result that the [math.AT] (November, 2010). inclusion SO(4) → Diff+(S3) is a homotopy equivalence. [Hirsch, 1963] M. W. Hirsch, Obstruction theories for On the other hand, for n ≥ 7, Antonelli, Burghelea, and smoothing manifolds and maps, Bull. Amer. Math. Kahn [1972] showed that Diff+(Sn) does not have the ho- Soc. 69 (1963), 352–356. motopy type of any finite complex. (For earlier results, see [Hirsch-Mazur, 1974] M. W. Hirsch and B. [Novikov, 1963].) For n = 6 the group Diff+(Sn) is not Mazur, Smoothings of Piecewise Linear connected since 7 = 0; but I am not aware of any results Manifolds, Annals of Mathematics Studies, for n = 4, 5. Γ No. 80. Princeton University Press,

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June/July 2011 Notices of the AMS 809