aa r X i v : . [ m a t h . G T ] J a n PC4 AT AGE 40
MICHAEL FREEDMAN
This article is not a proof of the Poincar´e conjecture but a discussion of the proof, its context,and some of the people who played a prominent role. It is a personal, anecdotal account. Theremay be omission or transpositions as these recollections are 40 years old and not supported bycontemporaneous notes, but memories feel surprisingly fresh. I have not looked up old papers tocheck details of statements; this article is merely a download from my current mental state. Poincar´e liked to argue in many ways: analytically, combinatorially, and topologically. Heseemed averse to even fixing a definition for the term “manifold,” so one should not impose mod-ern notions like TOP, PL, and DIFF on his famous conjecture. He himself seems to have thoughtlittle about his own conjecture, since it asked if simply connected manifolds were spheres; clearlyhe meant to also specify that the homology should also vanish (except in the highest and lowestdimensions). The modern statement, now known to be true in every dimension is: a closed topo-logical manifold M that is homotopy equivalent to a sphere is homeomorphic to that sphere. Onecannot replace homeomorphism with diffeomorphism throughout the statement because of exam-ples like Milnor’s exotic 7-spheres. In all dimensions you still experience most of the excitementif you presume from the start that the manifold M n has a smooth structure and the goal is to proveit homeomorphic to S n . We will take this perspective. When the dimension n is 0 or 1 there is not much to prove. The n = n , n ≥
5. I willneed to say something about his proof since the case of PC4 starts the same way but encounters aspecial problem when n =
4. Solving this problem brings in the topological category in a way notpresent in Smale’s work. The reader might think Smale must also have labored in the topologicalcategory since the conclusion is only homeomorphism not diffeomorphism. But 99% of Smale’swork is in a smooth setting; his great achievement, the h -cobordism, theorem is smooth category.The step from the h -cobordism theorem to PC n is small and in some dimensions involves a gluingalong an S n − which might not be extended to a diffeomorphism over D n . So PC4 amounts toSmale’s outline with a topological twist. But here the tail wags the dog. When you delve into thisdetail the twist expands to fill your entire field of view. The final case (historically), dimension3, was proved by Perelman [Per03, Per03b] using Hamilton’s theory of Ricci flow. It is entirelydifferent in outline, more like Beethoven’s 9th than a conventional proof, and still stands as thegreatest accomplishment of 21st century mathematics. This is a written version for a CMSA lectured delivered at Harvard on September 28, 2020. In dimension > M and this can be usedto bridge to the strongest statement, but the cost is one must use a technically more difficult (proper) version of thework of Smale discussed below. In dimensions < I worked towards PC4 from 1974 to 1981, roughly half time. I had many mental pictures butno notes when it came to writing the proof. I realized I had no letters (in my mind) for any ofthe spaces, maps, or relations. Apparently as a youngster I had not learned yet to think in or withsymbols. When written, the proof was hard for others to understand; when I tried to explain it, Idid not know where to begin. To this day I regret that in my lectures at Berkeley with Smale in theaudience, I never even mentioned that it was his proof, with a twist, that I was presenting. Fromthe perspective of a youth in 1981, 1959 might as well have been 1859, I did not feel the historicalconnections. Now they seem obvious and I will try to capture them here.Three streams of thought enter the proof and I will personify them with the names of threemathematicians: Steve Smale, Andrew Casson, and Bob Edwards. Of course, each represents afield and there are other names as well, but in all three cases the power of their individual ideas isso strong and so determinative of the form of the final proof that little is lost by identifying themwith their fields of work. To put a name to their work (as relevant here): • Smale: Classical smooth topology and dynamics. • Casson: How to get started in 4D: “finger moves” simplify π , leading eventually to “Cas-son handles.” • Edwards: Manifold factors, decomposition spaces, shrinking.I will explain, in order, the input from each of these three streams and recall the occasionalanecdote.By the time (1959) of Smale’s paper Morse theory had already moved on to infinite dimen-sions. People knew the Morse inequalities and that seemed to be regarded as a satisfactory linkbetween critical points and homology of manifolds. Smale took a much closer look. Let us re-call his set up. Henceforth, all manifolds are assumed to be compact and smooth. Let W n + bean ( n + ) -dimensional manifold with two boundary components, M and M . Let us assume allthree fundamental groups are trivial, and the relative homology vanishes: H ∗ ( W , M ; Z ) =
0. (Ofcourse by duality, the relative homology groups also vanish if we replace M with M .) With thesehypotheses W is called a (simply connected) h -cobordism. Theorem 1 (Smale’s h -cobordism theorem (hCT)) . A simply connected h-cobordism W n + is dif-feomorphic to a product M × I, provided n > . The idea of the proof is to relentlessly match the algebra of the Morse complex of a smoothMorse function, f : W → [ , ] to the geometry, or as Smale would say, the dynamics of the gradient( ∇ f ) flow. The Morse complex has critical points of index k as its chain group generators indimension k , and the differential of the complex records in matrix form the algebraic number i , j of times the boundary of the i th k -cell wraps over the j th ( k − ) -cell. The Morse cancelationlemma states that if the geometric count i , j = ±
1, the two critical points can be cancelled, and theMorse functions simplified. Smale realized that the in the simply connected case, the vanishing ofthe relative homology groups was the only obstruction to performing a series of deformations to f (called “handle slides,” or in the algebra, “row operations”) that lead to cancelation of all criticalpoints. Once there are no critical points left, the gradient flow lines define the desired productstructure. C4 AT AGE 40 3
Matching the geometry to the algebra consists of turning the algebraic information that thesigned sum of intersection points i , j = ∈ Z into the stronger geometric information that the i thdescending sphere truly meets the j th ascending sphere in one transverse point, not for examplethree points in a + , − , + pattern. To accomplish this he employed Whitney’s trick [Whi44] forremoving pairs of oppositely signed double points. These double points live in an n -dimensionalcross-section L n of W , often called the middle level, and Whitney’s method requires n >
4. Thedimension restriction arises because a 2 dimensional disk, the “Whitney disk,” guides the can-cellation of the extra (+ , − ) pair and we need that disk to be embedded; an easy thing whendim ( L ) = n >
4, and a difficult thing otherwise. The proof of PC4 follows Smale up to this point,from there forward it is all about how to locate the Whitney disks necessary to make { A } and { D } ,the ascending and descending 2-spheres in the middle level L , intersect each other geometrically in single transverse points, as the algebra would indicate. So Smale has gotten us started but weare left with the problem of simplifying, say, a canonical picture in the middle level L of twoembedded 2-spheres A and D which meet in three points + , − , + .It is here where Casson’s work begins. Before turning to Casson, this is a good place to explainhow the hCT implies various PCn. There are actually two overlapping techniques (both due toSmale). Suppose Σ n is a homotopy n sphere. We can cut out two disjoint balls from Σ n to obtainan n dimensional h -cobordism W n . If n is at least 6 the hCT recognizes W as diffeomorphicto S n − × I . Σ n is now obtained by gluing back the two balls, but the gluing diffeomorphismsreflect the complexity of the product structure output of the hCT and we don’t know that theresult is diffeomorphic to S n . We see from this picture that Σ is two balls glued together by ahomeomorphism of their boundaries. It is elementary (via Alexander’s coning trick) that the unionis homeomorphic to S n . To handle the lowest dimensional case treated by Smale, Σ , a differentstrategy is required. Some algebraic topology and a bit of surgery theory is used to construct a 6D h -cobordism W between Σ and S and the hCT is applied to W . Interestingly, this proves more,that Σ is diffeomorphic, not just homeomorphic, to S . This approach using surgery only succeedsin finding W in some dimensions but when it works the conclusion is stronger. In our case n = h -cobordism W from Σ to S . Thereason that our proof concludes only a homeomorphism at the end of the day is that there is a terrificstruggle involving lots of limits and point set topology to find the Whitney disks and they end upbeing merely topological category objects. The Whitney disks we find spoil the smooth categorycontext of the rest of the proof. It is an astonishing historical coincidence that within 6 monthsof PC4, Simon Donaldson had figured out enough about which four dimensional homotopy typesdo not contain smooth manifolds to know for sure that the Whitney disks I built cannot generallybe smoothed. These Whitney disks are all that stand in the way of realizing the quadratic form E ⊕ E as the intersection form of a close smooth 4-manifold, which is excluded by Donaldson’s“Theorem C” [Don86]. So the excursion into the topological category is necessary if one speaksgenerally in terms of 4-manifold constructions. Of course there may be some as yet unimaginedworkaround—or entirely new method—satisfactory to the study of a smooth homotopy sphere Σ .I think it safe to say that the greatest open problem in topology (“The last man standing”) is thesmooth category PC4: Is every homotopy 4-sphere diffeomorphic to S ? MICHAEL FREEDMAN
Now to Casson. Casson’s first great insight was that there was some hope for Whitney disksin 4D. He realized that the problem of finding an embedded Whitney disk is qualitatively differentfrom finding a slice disk for a classical knot in S . That prototypical 2-disk embedding problemhad been studied starting in the 50s by Fox and Milnor [FM66]: Given a knot K in S does itbound a smoothly embedded disk in the 4-ball B ? This problem is infinitely rich, and generallyhas a negative answer. But Casson realized that Whitney disks in all applications would havehomological duals, so by standing farther back, and taking more of the manifold topology intoaccount, the embedding problem might turn out to be less obstructed than in knot theory. Thishope bore fruit. It is easy to show that these three spaces are simply connected: The middle level L , and the complements L \ A , and L \ D . To look for a Whitney disk D in L meeting A ∪ D onlywhere it is supposed to, long the boundary of D , we would also like L \ ( A ∪ D ) to be simplyconnected. The duality Casson observed implies that π ( L \ ( A ∪ D )) is perfect. He realized thatit might help to make things worse before trying to make them better. If one takes ones fingerand pushes a generic patch of A along an arc and through a generic point on D this “finger move”will create two new + , − points of intersection. Each such point will microscopically look like theintersection of two complex lines in C and linking these complex lines is a “Clifford torus.” Thealgebraic point is that the top cell of these Clifford tori are relations saying the two loops linkingthe two sheets (complex lines in the model) commute in π of their joint complement. If youtake a perfect group and start forcing pairs of elements to commute pretty soon you have a trivialgroup. In this way Casson got off the ground, by doing enough finger moves between A and D tokill π ( L \ ( A ∪ D )) . This enabled him to at least immerse the Whitney disk W (with the requirednormal framing data) he was looking for [Cas86].From here the finger moves simply explode in number. Fearlessly, Casson decides to cure theproblem of the Whitney disk double points by capping them off with subsidiary Whitney disks ina hierarchy later called a Casson tower. When taken to its infinite limit, the open, tapered, regularneighborhood is called a “Casson handle.” I draw a dimensionally reduced schematic of a Cassonhandle in Figure 1.Casson (1974) showed that his Casson handles C had the correct proper homotopy type to serveas neighborhoods of embedded Whitney disks, C P ≃ ( D × R , ∂ D × R ) . My contribution sevenyears later was to replace proper homotopy equivalence—which was not strong enough for mostgeometric applications—with homeomorphism. As I mentioned, I spent those seven years staringat his handles about half time. I think this was the correct fraction. I was young and needed todo some projects likely to work in order to get tenure, raise a family, and preserve sanity. Thoseyears I was either climbing rocks, writing pleasant papers, or struggling with Casson handles. Notmuch else. My mother had taught me that everything must be fun, “fun first, pleasure second” asshe put it. My father who was always at work on something (and always enjoyed it) told me thatthere is “your work and, then, the world’s work.” The idea of course was to do both. My paperswere the “world’s work.” I enjoyed them greatly and would surely have lost confidence in myselfif I was not publishing regularly. Casson handles were “my work.” The climbing sheer pleasure. Ihad a reputation among my friends for “not thinking on lead,” I just would tear ahead and see whathappened. That is the way I liked it: When I did math I tried to think (a bit); when I climbed I justlet it rip. Many of my friends asked me variations of, “Since Casson is obviously so much smarter C4 AT AGE 40 5 . . . attaching region ... ... ... ... ...unbranched branchedWhitney disksF
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1. Casson handles.than you, why was it that he didn’t analyze his own handles.” The answer, I think, is “opportunitycosts”; he was so full of brilliant ideas: secondary obstructions to knot cobordism (with Gordon),the Casson invariant, what became “weak Heegaard reduction” (also with Gordon), that he wassimply too busy.Let us turn to the third stream, Bob Edwards and the world of decomposition space theory, theBing shrinking criterion (BSC), and manifold factors. In 1980, Bob was the undisputed heavy-weight champ in what might variously be called Texas-topology, Bing-topology, the R.L.Moore-R.H.Bing school. It is an area at the wild end of manifold topology, with roots and emphasis in lowdimensions. The Alexander horned sphere is what you should picture. It drew on point set topologybut was a very pictorial, concrete, and vivid undertaking. It also had an enjoyably sportive feel thatallows me to call Bob “Champ” whereas it would never occur to me to call Hiranaka or Delignethe “Champ” of algebraic geometry. During those seven years, I had discovered something, nowcalled re-imbedding. If you have a Casson tower of some critical minimum height, then 7, laterreduced to 5 (see [CP16, GS84]), you can leave the lowest stage alone and build a new extensioninside a neighborhood of the original stage construction which has any, arbitrarily large, numberof stages. This quickly allows geometric control to be added to Casson handles. The original con-struction just kind of flops around, the higher stages are in no sense smaller than the lower ones.But with re-imbedding it is only a small step to build Casson towers in which the higher stages con-verge beautifully to a Cantor set. Now if one takes a tapered regular neighborhood and completesit with this limiting Cantor set one obtains a compactified C in any application where formerly aCasson handle C was present. A handle has its boundary divided into the attaching region andthe co-attaching or “belt” region. See Figure 2. In early fall of 1980 I used the Kirby calculus todraw an exact picture of the belt region for the standard unbranching C . For the everyday closed2-handle ( D × D , S × D ) , the belt region is D × S , a solid torus. What I drew belt ( C ) waswhat I later learned was a decomposition space D × S / Wh, the solid torus with a compactumcalled the Whitehead continuum crushed to a point (and endowed with the quotient topology). A
MICHAEL FREEDMAN few weeks later I was sitting at a pizza restaurant after a meeting of the recurring Southern Cali-fornia Topology meeting with Bob Edwards. Apropos of nothing in particular, he was putting pento napkin to explain the “Andrews-Rubin” [AR65] shrinking argument which proves that the space D × S / Wh is a manifold factor. Even the concept to me was astonishing: D × S / Wh is mostcertainly not a manifold but it turns out that its product with the real line ( D × S / Wh ) × R ishomeomorphic to D × S × R . Edwards understood this phenomenon well as he had just com-pleted his proof that the double suspension of the homology sphere bounding the Mazur manifoldwas S . The single suspension is of course not a manifold. My excitement was overwhelming.The frontier of my controlled Casson handle C , while not a manifold, was close—if it turned outto have a product collar neighborhood that would, by Andrews-Rubin, be a manifold and wouldharbor the Whitney disk I had been seeking for six years. In the end, the proof proceeded some-what differently but the fact that the belt region of “frontier” of the controlled Casson handle wasa manifold factor convinced me that PC4 was likely true.... belt region = D × S / Wh = attaching region Wh = ∩ ∞ i = ( D × S ) i ,each nested the sameway as its predecessor. ( D × S ) ( D × S ) WhF
IGURE ( D × S / Wh ) × R is homeomorphic to D × S × R .To appreciate why D × S / Wh is a manifold factor we introduce the Bing Shrinking Criterion(BSC), as exposited by Edwards [Edw80].Let X and Y be compact metric spaces. A map f : X → Y is approximable by homeomorphisms(ABH) if, for every ε > ∃ a homeomorphism: g : X → Y with dist Y ( f ( x ) , g ( x )) < ε for all x ∈ X . Bing Shrinking Criterion (BSC) : f : X → Y is ABH iff for all ε > h : X → X so that(1) dist Y ( f ◦ h ( x ) , f ( x )) < ε for all x ∈ X and(2) diam X ( h ( f − ( y ))) < ε for all y ∈ Y .Andrew and Rubin found self-homeomorphisms h ε of ( D × S ) × R so that π ◦ h ε is ε -closeto π , and diam ( h ( π − ( r ))) < ε , where π : ( D × S ) × R ε −→ ( D × S / Wh ) × R is the projectioncrushing each Wh × r to a point.Their idea is to lift, at any deep stage i , the self clasp of the Whitehead double ( D × S ) i × r to an infinite chain of clasps and then apply a global twist making each chain of the link smalldiameter. Figure 3 gives a hint of how this works. C4 AT AGE 40 7 R lift i th stage ( i + ) th stage, ( D × S ) j × h twist i th stage solidtori appear smallafter lift/twistand contain Wh × r F IGURE D of the standard (open) 2-hanlde ( D × R , S × R ) , but usingthe re-imbedding lemma, and its concomitant geometric control is also a subset of the (generalCasson handle) C . Actually this topographic map varies in ways that are not terribly importantwith different choices of C , but in all cases the design D includes into both C and the standard(open) handle H . If this map was fully extensive and covered both C and H it would by itselfdescribe the desired homeomorphism, but unfortunately it has a countable number of bits missing,which we call “holes” in H , and “gaps” in C . The gaps are bits of C which remain terra incognitaafter the exploration that lead to the design. To sketch the rest of the proof, I need to tell you howthe design is built, why it has these gaps, and finally what to do about them.There will be a lot a branching going on in this paragraph, and I hope to keep two quite differenttypes straight as they are described. Let me call them “bulk” and “radial” branching. In Figure 1 wealready met bulk branching, this comes from the fact that every Whitney disk that Casson installsto kill a double point has itself many double points and requires that many Whitney disks at thenext level. Bulk branching, in the presence of geometric control, limits to a Cantor set (which playsthe role of the singular points in the Andrews-Rubin argument). This branching is really only adistraction and does not materially affect the proof. The reader would be served well by the (false)fantasy that all Casson handles are unbranched (even when obtained via the re-imbedding theorem)and just forget about bulk branching. The more interesting branching is what I’m calling radial.Recall a 7-stage tower contains a 14-stage tower (Casson handles are alive and can replicate!). Wecan think of 14 = + T and T ′ for the second tower with T ′ contained in T . This binary choice now occurs countablyoften as we may now identify 14 stage towers within both T ′ and T , and again create a binarychoice (move in or stay out) in attaining the final 7 stages in these 14-ers. Continuing in this waywe realize each of two choices “in or out” every 7th stage, always while maintaining geometriccontrol so that the limits are as we expect. We call this branching “radial” because when D isembedded in H , indeed at every such branch, the primed choice has a smaller ρ coordinate in the MICHAEL FREEDMAN polar coordinates of the transverse R . The design carries a singular foliation. At radial coordinateswhich are not in the standard Cantor set the leaf is a compact 3-maniold with boundary, at a Cantorset radius the leaf is an Andrew-Rubin-like decomposition space. Corresponding to the “middlethirds” are the holes or gaps where this procedure has not succeeded in corresponding points of H to points of C .There is now a technical point, all but one of the holes is homeomorphic to S × R (the outlieris homeomorphic to R ). At this point we don’t know what the topology of the gaps is-—but longafter this proof is finished we do learn enough to know that they all along had the topology of theircorresponding holes. Our plan for the { holes } and the corresponding { gaps } is eventually to crushthem to points and analyze the results. It is much better in applying the Bing/Edwards mathematicsthat we crush only closed “cell-like” [Edw80] sets. Cell-like means that when embedded in aEuclidean space (or Hilbert space) for any neighborhood U there is a smaller neighborhood V thatcontracts within U . Our holes/gaps are neither closed nor cell-like. The first deficiency is easilycorrected by taking the closures, the second by locating certain 2-disks to add to the holes or gaps.We use the notations hole+ and gap+ for holes or gaps that have been closed and augmented withan additional 2-disk to become cell-like. In a sense adding the + is taking a step back because weare giving up some of the “explored region” of C covered by the design, but sometimes to take twosteps forward one must take a step backwards. This is an example.Now comes the key diagram which will divide what remains of the proof into two separatetheorems: 1) the Edwards shrink and 2) the “Sphere to Sphere” theorem. Q CH α ABH β ABH F IGURE α crushes { holes+ } and β crushes { gaps+ } .Tautologically H and C have a common quotient Q , obtained by crushing the holes+, respec-tively gaps+, to points. This is the benefit of having found the design D on both sides. It turns outboth quotient maps can be shown to be approximable by homeomorphisms (ABH). This, of course,would finish the proof that Casson handles C are homeomorphic to the standard open handle H , bycomposing one approximating homeomorphism by the inverse of the other. And with this result inhand the Poincar´e Conjecture PC4, and much more within 4D topology follows.Edwards analysis of the first map α is a tour de force of Bing topology. Many of the bestideas in that field enter: It encompasses and generalizes the Andrews-Rubin shrinking technologyand also contains a compound use of the principle that “Countable, null, star-like equivalent”decompositions are shrinkable [Bea67]. It is a beautiful and pure application of the Bing shrinkingcriterion.The arrow β , might be called the “blindfold shrink.” Ric Ancel [Anc84], Jim Cannon, FrankQuinn, and Mike Starbird helped me revise this argument and express it as an extension of Brown’sproof of the Schoenflies theorem, a proof that I learned from them. The notation is a bit dauntingbut the idea is simple. After the Edwards shrink we know that both domain and range of β are C4 AT AGE 40 9 manifolds and ones that embed in R , in fact one readily reduces to the case that both domain andrange are standard 4-spheres. Then the idea is to look at the 8D graph of the function β and try toimprove it to a homeomorphism by systematically modifying it to remove its largest “flat spots.”The data (from the Edwards shrink) that the quotient space has nice local and global topology andallows you to insert a local “drawing” of the decomposition space structure into the target withina small neighborhood of the image of these largest decomposition elements. A natural procedure,see Figure 5, uses this inserted drawing to resolve these most singular points and make the functionone-to-one over them. There is a blending problem: the resolution creates countably many smallvertical spots, that is the resolution is only a relation not a function. But one should not worry, weare on the right track. One goes back and forth: sanding down first the largest horizontal spots,then flipping the workpiece over and sanding down the largest of these newly created vertical spots,flipping it back to sand down somewhat smaller horizontal spots, etc. One moves back and forthsanding and polishing until the limiting relation has neither horizontal nor vertical thickness, i.e.becomes a homeomorphism. XXR ⊇ Q ∼ = H flat spot ( s ) graph of β C = Casson handleF
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5. The neighborhood of the largest gap+, X , is drawn on the y -axis near β ( X ) to facilitate a resolution of β to a homeomorphism on X . This is good, butsmall vertical spots inevitably result from pulling points where β is already a home-omorphism over other smaller elements of { gaps+ } . This deficit is corrected later.I thank Arumina Ray and Mark Powell for including this lecture in their book; it is hopedthat these brief recollections may add context or at least amusement. There is much left to beunderstood in four dimensions—the best of luck! R EFERENCES [Anc84] Fredric Ancel,
Approximating cell-like maps of S by homeomorphisms , Four-manifold Theory (Durham,NH, 1982) (1984), 143–164.[AR65] J.J. Andrews and Leonard Rubin, Some spaces whose product with E is E , Bull. Amer. Math. Soc. (1965), no. 4, 675–677.[Bea67] Ralph Bean, Decompositions of E with a null sequence of starlike equivalent nondegenerate elements areE , Illinois J. Math. (1967), 21–23.[Cas86] Andrew Casson, Three lectures on new infinite constructions in 4-dimensional manifolds , A la Recherchede la Topologie Perdue (1986), 201–244.[CP16] Jae Choon Cha and Mark Powell,
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Connections, cohomology and the intersection forms of 4-manifolds , J. DifferentialGeom. (1986), no. 3, 275–341.[Edw80] Robert Edwards, The topology of manifolds and cell-like maps , Proceedings of the International Congressof Mathematicians (Helsinki, 1978), 1980, pp. 111–127.[FM66] Ralph Fox and John Milnor,
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Ricci flow with surgery on three-manifolds (2003), available at arXiv:math/0303109 .[Sma56] Stephen Smale,
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ICROSOFT R ESEARCH , S
TATION Q, AND D EPARTMENT OF M ATHEMATICS , U
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