1. Early work
The annulus conjecture (AC) as stated by Milnor is: Is the region bounded by two locally flat \( n \)-spheres in \( (n+1) \)-space necessarily homeomorphic to \( S^n \times [0,1] \)?
I was pleased to see problems that I could understand with my limited knowledge of mathematics. But I did know Morton Brown’s proof of the topological Schoenflies conjecture, and a canonical form of the Alexander isotopy used by Jim Kister in his proof that micro bundles contain bundles [e4]. I also knew that a bounded homeomorphism of \( R^n \) was canonically isotopic to the identity, from reading a paper of Ed Connell, and did not know until decades later that this fact was first proved by Kister [e1].
Occasionally I would have an idea for proving the annulus conjecture. One was embarrassing, for when I went to show it to my advisor, Eldon Dyer, we were joined by Saunders Mac Lane, and it was discovered that I had overlooked that the intersection of nested closed sets in a metric space might be empty if I did not know that the metric space was compact.
I did however prove that if one could nicely fit a “pillar” between the two spheres in the annulus conjecture, then there was indeed an annulus [1]. Dyer seemed mildly impressed, perhaps that this dubious student had managed to actually prove something. He was an editor of the Proceedings of the AMS and accepted the paper. Later I found out that this may have been folklore in the Bing topology community. If so, it’s not surprising.
I finished my PhD in 1965 with a thesis on a different topic and went to UCLA as an assistant professor. While there I read a paper of Jim Cantrell in which he showed that Morton Brown’s proof of the topological Schoenflies conjecture still held when the embedded codimension one sphere had a point not known to be locally flat, if its dimension was not equal to 2 (the 2-sphere boundary of the neighborhood of the Fox–Artin arc in \( S^3 \) is not locally flat at the wild point of the arc). The real point here was that a locally flat embedding could not fail to be flat at just one isolated point (except when \( n=2 \)). I managed to show that it could not fail to be flat at a Cantor set if the Cantor set was tame in the embedded sphere and in the ambient sphere. I submitted this to the Annals, and Milnor obtained an excellent referee in Dale Rolfsen who improved the proof substantially [3].
This paper led to a tenure track offer from the University of Wisconsin. I’d become a Westerner and did not wish to go back to the midwest, but I did spend the fall semester of 1967 at Wisconsin where I formed what became a lifelong friendship with Raymond Lickorish who was visiting for the year.
2. The torus trick
In August 1968 I had gone to the yearly topology conference in Athens, Georgia, and returned with a preprint of Černavskiĭ [e6] showing that the space of homeomorphisms of a topological manifold is locally contractible.
I was home, babysitting my 4 month old son, when something in the paper caused me to consider the lift of a homeomorphism \( h:T^n \to T^n \) of the torus to its universal cover \( H: R^n\to R^n \). Obviously \( H \) is periodic, and if \( h \) was homotopic to the identity, then \( H \) would be bounded by the same constant that held for a fundamental domain.
But if \( H \) is bounded, then it is isotopic to the identity (by a version of the Alexander isotopy [e1]) and therefore stable. Stable means that \( H \) is the composition of finitely many homeomorphisms, each of which is the identity on some open set. These had been carefully studied in [e3] and stable homeomorphism and the annulus conjecture were closely related. In particular, if \( H \) is stable, then the original \( h \) is stable, and I knew that was a very interesting fact (the requirement that \( h \) be homotopic to the identity was no problem, for one could arrange it to be so by composing with a diffeomorphism).
That last paragraph was a quick observation if one had the right tools and questions in hand. Now the obvious question was how to turn an arbitrary homeomorphism of \( R^n \) into a homeomorphism of \( T^n \) so as to show that those homeomorphisms were also stable.
I quickly thought of immersing a punctured \( T^n \) into \( R^n \) (nothing-else-to-do theory). Then the road forked, one fork leading to Černavskiĭ’s theorem on local contractibility and the other to PL structures.
The easier fork was to assume that \( h \) moved points less that some \( \epsilon \), for then \( h \) lifted to an embedding of \( T^n \) with a bigger (roughly by \( \epsilon \)) puncture into \( T^n \) minus a point. Then a canonical version of the topological Schoenflies theorem (see [e8]) allowed the extension of the embedding to a homeomorphism of \( T^n \), which was stable, but also agreed with the original \( h \) on an open set, so \( h \) was stable and isotopic (canonically) to the identity, thus proving local contractibility for \( R^n \). I think I knew this still in August.
The other fork was more complicated, for it involved using the immersion of a punctured \( T^n \) into \( R^n \) to pull back a PL structure from the standard one on \( R^n \) to one on the punctured torus. This could be extended to a PL structure on \( T^n \) when \( n\geq 6 \), giving a homeomorphism from \( T^n \) with a possibly exotic PL structure \( \Sigma \) to the standard \( T^n \). Whether the identity homeomorphism was isotopic to a PL homemorphism was a topic in nonsimply connected surgery theory, which had just been worked out by Terry Wall, although the calculations for fundamental group equal to a free Abelian group were not entirely worked out or known by all.
I had already arranged to teach in the summer of 1968 so as to have a free fall quarter to spend at the Institute for Advanced Study, a very propitious decision.
Shortly after arriving at the IAS, I generalized the torus trick to a handle version, using \( B^k \times R^n \), rather than just the zero-handle case when \( k=0 \). I circulated a one-page document [2], that spawned a few papers about my conjectures (see, for example, [e7]). In October I found myself sitting next to Larry Siebenmann at a colloquium dinner. He started asking questions about the torus trick and by the time dessert was well over, I thought he had drained every bit of useful information out of my brain. That began our collaboration. Larry worked late and I remember many mornings finding a sheaf of notes inside my screen door. We progressed quickly, for Larry knew all the PL to DIFF smoothing theory, and much else, that I would have needed months to absorb.
I listened in on discussions about nonsimply connected surgery; Bill Browder explained a lot to us and then Larry had the idea of first taking a \( 2^n \)-fold cover of the homeomorphism \( T^n_{\Sigma} \to T^n \), which would then kill Wall’s obstruction, and then take the lift to \( T^n \). This was an added-in-proof paragraph to my Annals paper, and settled the annulus conjecture except in dimensions \( 4,5 \).
Looking back, the moment I considered the universal lift of a homeomorphism of \( T^n \) and observed it was bounded, an observation known to many starting in the 19th century or earlier, I was on a downhill slope in the sense that there was never a difficulty that was a serious long-term problem. I thought of immersing the punctured torus quickly and it was easy to generalize to the handle version. So, a marvelous torus trick just fell in my lap, an amazing piece of luck. When you work on hard problems, it is often true that there is no nice solution to be found. Think of the Poincaré conjecture which so many people devoted years of their lives to, and a topological proof still hasn’t been found; Perelman was needed. It could have been the same with the annulus conjecture.
And then I was lucky again to be at the IAS and meet the perfect collaborator. Our work would not have been done by me in isolation.
And I was lucky in another way, in that I didn’t think of the torus trick four years earlier when I just as well could have. Then I’d have written the Annals paper, but that would have only been the next-to-last piece of the puzzle, and Wall would have done his surgery theory during those four years and put in the last piece of the puzzle, gaining most of the glory. So, I was lucky to be smart but not too smart.
Larry and I had our key results and wrote them up in late December, 1968, as a Bulletin of the AMS article. I went back to UCLA to teach, but met Larry again (with a few others) to ski at Heavenly Valley at the south end of Lake Tahoe in February, 1969. We skiers didn’t see much of Larry, for it turned out he was finishing up his argument that \( \pi_3 (\mathrm{TOP}/\mathrm{PL}) = Z/2 \). I, in a desultory sort of way, had been trying to prove the opposite, but for Larry his result was crucial, for otherwise there would have been no pathology, no obstruction to triangulation, and the subject would have boiled down to just the torus trick.
Back at UCLA, I received a letter from Larry explaining his proof. I remember well sitting at the kitchen table of my apartment beginning to read his proof when lightening struck and I saw a simple proof, the one in the literature, and stopped reading. Recently, over 40 years later, I looked for the letter and was surprised I couldn’t find it. Bob Edwards was in Paris so I asked him to ask Larry if he had a copy. Larry’s response was: “There’s no letter. I came to UCLA in spring 1969 and gave a lecture on the proof.”
Perhaps Larry is wrong, but I suspect that something that was very clear in my memory was wrong, an example of the fallibility of memory. That then raised the question of whether my memory is wrong in claiming that the simple proof is mine, but here I think I will rely on my memory. For one thing, the proof which is sketched in [4] is not what I would have written (it was submitted March 3, 1969, soon after our ski trip).
Here is what I would have written: Start with \( Q \), an exotic, rel boundary, \( [-1,+1] \times T^n \), meaning that the standard PL structures on the boundary components do not extend to a PL homeomorphism from \( Q \) to the product structure. But the \( s \)-cobordism theorem tells us (\( n \geq 6 \)) that \( Q \) is PL homeomorphic to the product if we do not require that it be the identity on both ends, but just at \( -1 \). Measure the difference at the \( +1 \) end by a PL homeomorphism \( h \). Now a large odd cover leaves \( Q \) still exotic, but \( h \) is now arbitrarily close to the identity on \( +1 \times T^n \), and is thus topologically isotopic to the identity. But this cover of \( h \) cannot be PL isotopic to the identity. This is the difference between TOP and PL.
In general we do not pick apart contributions to joint papers, but in this case it seems to me in retrospect that after I met Larry, everything was really due to him, with the exception of two of his proofs which I substantially improved upon. One I have just mentioned, and the other is the proof of the product structure theorem using the “windowblind” lemma [5], p. 35.
Larry and I intended to get a full paper out fairly quickly, for I visited IAS again in fall 1969, and then we met in Cambridge, England, in summer 1971, but by then our goals had diverged. I would have settled for a shorter paper along the lines of Chapters 1 and 3 of our eventual book, but Larry had greater goals including parameterized versions of our theorems and more. So he eventually wrote the vast majority of our book.
3. The calculus
In early 1973, I went to a conference in Tokyo and while there overheard Takao Matumoto talking to someone and mentioning the Kummer surface. I was curious and luckily had been jogging with Arnold Kas who was a fairly recent student of Kodaira. It was natural to ask him about this interesting 4-manifold. He taught me many things about complex surfaces, and did so in a way that I could turn much of it into handlebody theory. Our AMS memoir [14] with John Harer contains much of this material. The algebraic geometers had a way of showing that various descriptions of the complex surface \( E(1) \) as a Lefschetz fibration with various singular fibers, were all diffeomorphic to \( CP^2 \) blown up nine times. The methods for doing so translated into connect summing with \( \pm CP^2 \) and sliding 2-handles over 2-handles.
At some point I began to wonder if different ways of adding 2-handles to \( B^4 \), which gave the same boundary, could be equivalent under the same moves that the complex geometers used. This worked nicely in some examples, in particular when the boundary was the Poincaré homology 3-sphere. By summer 1974, when I was visiting Warwick, the conjecture was firmly in my mind, and I explained it to Colin Rourke. That fall, the proof was boiled down to some Cerf theory. I queried Jack Wagoner at length for he was our local expert, having just sorted out the pseudo-isotopy problem with Allen Hatcher [e9] using advanced Cerf theory. He told me enough, and I lectured on the theorem in the fall of 1974. It took me a while to write it up, as usual, and then I sent it to the Annals. Browder then suggested it wouldn’t look proper if they accepted my paper after rejecting a related paper, for I was an associate editor of the Annals at that time. So it went to Inventiones, where the referee (Allen Hatcher I believe) found an error which took me a few weeks to sort out. The paper [6] appeared in 1978.
Meanwhile Rourke, with Roger Fenn, was working on a PL version of a proof, and got stuck with 3-handles. Cerf theory had been an easier path for me. But they did see a neat way to combine my two moves into one [e11].
This theorem was really a uniqueness theorem to go with the existence theorem (every oriented 3-manifold can be obtained by surgery on a framed link in \( S^3 \)) proved by Lickorish [e2] in the early 60s. Nowadays an existence statement automatically generates a uniqueness question, but back then I was not aware of anyone asking about uniqueness.
This theorem and the framed link pictures became known as the Kirby calculus (it rolls easily off the tongue) despite the fact that others, especially Akbulut, developed it far beyond what I had done. Oddly, it had no applications (other than giving moral support for trying to prove manifolds were the same with these methods) until Reshetikhin and Turaev [e17] used the moves to show that certain combinations of their invariants for framed links were unchanged under the calculus moves, with the result that they had 3-manifold quantum invariants, as they came to be called.
In the mid 70s John Harer joined Kas and me in working on handlebody descriptions of complex surfaces. This became an AMS memoir [14] with an expository first chapter written by Kas describing complex surfaces from a topological point of view, followed by handlebody/calculus pictures of \( E(1) \) and \( E(2) \) (we called them the half-Kummer and Kummer (K3) surfaces), and then a chapter drawing the 3-handle attaching maps for \( E(1)_{2,3} \), which is \( E(1) \) with two logarithmic transforms of degrees 2 and 3. We hoped to see that the 3-handles could not be canceled and thus that these logarithmic transforms were not diffeomorphic to \( E(1) \), but this had to wait for Donaldson and gauge theory. Our work was done by 1979, but didn’t appear in a timely fashion.
In the latter 70s, Selman and I wrote a paper [9] on Mazur manifolds, as we called them, which is noteworthy because these examples turned into Akbulut’s “corks” which he used to give the smallest exotic 4-manifold [e15], [e16]. Selman also introduced the notation for a 1-handle, namely an unknotted circle with a dot on it. If you want to trade in a 1-handle for a 2-handle, just replace the dot with a zero.
Cappell and Shaneson constructed smooth 4-manifolds homotopy equivalent to \( RP^4 \) but not diffeomorphic [e10]. This raised the question of whether the double cover was diffeomorphic to \( S^4 \) and that these were exotic involutions or, less likely, that the double cover was a counterexample to the smooth 4-dimensional Poincaré conjecture. Selman and I drew handlebody pictures of the double cover and showed that it was indeed \( S^4 \) [8]. Meanwhile, Fintushel and Stern had elegantly constructed exotic involutions on \( S^4 \) by different methods [e12]. However, Iain Aitchison, then a Master’s student in Melbourne, read our paper carefully and pointed out that a key framing was 1 rather than 0.
So, the double covers of the Cappell–Shaneson exotic \( RP^4 \)s were still only homotopy spheres. We, meaning Selman with me cheering from the sideline, learned a lot more about the first interesting case, showing that it was also the Gluck construction on the knotted 2-sphere made from the two distinct slices of the \( 8_9 \) knot, that there was a natural presentation of the trivial fundamental group given by \( \{x,y\mid xyx=yxy, x^4 = y^5\} \), and there was a comparatively simple description with two 1-handles and two 2-handles. We published this [13]. A few years later, Bob Gompf found an elegant way to add a \( 2-3 \)-handle pair and show that this homotopy 4-sphere was indeed \( S^4 \) [e13].
In the summer of 1976 a group of us met in Cambridge, imposing on the hospitality of Lickorish. While there, Paul Melvin and I concocted a very simple argument (a three-page paper) showing that if 0-surgery on a knot \( K \) gave \( S^1 \times S^2 \), then \( K \) had to be a slice knot. In fact, \( K \) had to be the intersection of the equatorial 3-sphere in \( S^4 \) with an unknotted \( S^2 \) in \( S^4 \). This proves Property R for almost all knots, but the result was soon forgotten, not because Dave Gabai proved Property R for the remaining cases, but because his methods in [e14] were so far reaching.
From Berkeley, I sent the three-page paper to Frank Adams, the topology editor at Inventiones. Three days later I received an acceptance letter. How’s that for turn-around time?! I surmise that Frank received the submission, opened it around tea-time in Cambridge, saw Lickorish, who vouched for the result, and replied by return mail.
In the mid-70s Siebenmann was visiting Berkeley and he gave a few lectures on descriptions of the Poincaré homology 3-sphere. Marty Scharlemann and I added other descriptions to produce [10]; the contents are nicely described in a math review by Anatoly Libgober.
In 1978 during another summer visit to Cambridge, I had a few conversations with Raymond Lickorish and suddenly there was a preprint of a nice theorem: every knot is concordant to a prime knot — with my name on it. I’m very pleased to have a joint paper [11] with Raymond, but I could have done more to earn it.
Akbulut and I saw a nice way to see the double branched cover of a link \( L \) in \( S^3 \) bounding a Seifert surface \( F \) pushed into \( B^4 \), using the calculus [12]. This worked well for complex curves in \( CP^2 \), and led to my drawing the quintic with its 53 2-handles, using plastic forms. The senior editor at Math Annalen wrote back, “Are all these cartoons really necessary?”
4. A book and low-dimensional bordisms
By 1987 I had taught courses on 4-manifolds several times and in particular had talked with Iain Aitchison about proofs of the classical 4-manifold theorems using geometric techniques in dimension four. When Chern invited me to visit Nankai, his Institute in Tianjin, China, for the month of May, 1987, I agreed and offered to teach a course on 4-manifolds, with the idea that the students would help with notes.
I was pleased with the proof I’d worked out for Rohlin’s theorem. In it’s generalized form it states that the extra \( Z/2 \) factor comes from the spin bordism class (\( \Omega^{\mathrm{Spin}}_2 = Z/2 \)) of a characteristic surface \( \Sigma \) in \( X^4 \); \( \Sigma \) is dual to \( w_2 \) and the spin structure on \( X-\Sigma \) descends to the (codimension one) normal circle bundle to \( \Sigma \), and this descends to a spin structure on a section of \( \Sigma \) into the circle bundle; this is independent of the choice of section because \( \Sigma \) is characteristic [15], pp. 64–71.
I also included chapters sketching an outline of Freedman’s great work on topological 4-manifolds and included some history of how the existence of exotic smooth structures on \( R^4 \) came about. More history is given in Freedman’s volume at Celebratio Mathematica,1 but I can add yet one more bit.
In fall 1982 it became clear that there would be no paper announcing one of the more remarkable results in topology, the existence of exotic structures on \( R^4 \) (in all other dimensions exotic smooth structures did not exist). To remedy this, I wrote to Atiyah suggesting that a paper be written announcing this result authored by Casson, Donaldson, Freedman, Taubes and Uhlenbeck. The authors could have been just Donaldson and Freedman, but I felt that Casson deserved credit for his Casson handles, and that Taubes and Uhlenbeck also deserved credit for the foundations that Donaldson built upon. But Atiyah was the opposite of enthusiastic, probably not wanting to dilute the credit due his PhD student, and the idea died. But two Fields Medals (Donaldson and Freedman), and later an Abel Prize to Uhlenbeck gave each a good dose of glory.
In the late 1980s, Larry Taylor spent time in Berkeley during his vacations and we frequently met for lunch at the Musical Offering café. We began discussing low-dimensional bordism groups in the Spin and Pin\( _\pm \) cases. The Arf invariant and the Brown invariant with values in \( Z/8 \), used in the nonorientable cases, were key ingredients. We would work out a case, in a very geometric fashion, and the next day Larry would turn up with a well-written version typeset in \( \mathrm{\TeX} \). This happened lunch after lunch and magically (in my eyes) our paper [16] was done. One cannot ask for a better collaborator than Larry.
This is probably a good point at which to admit than in all of my collaborative publications, it is my coauthor who has done the lion’s share of the writing, and perhaps the lion’s share of the thinking as well. Yet another way I’ve been lucky.
5. Quantum invariants
In March 1989 Paul Melvin was visiting Stanford, and we had begun to think about projects when Kolya Reshetikhin came to talk there about his new work with Vladimir Turaev on the subject now known as quantum invariants of 3-manifolds. Paul and I were curious and wondered whether these invariants were a repackaging of known invariants or if not, what their topological nature might be.
We did show [17], [18] that they were known at the roots of unity, \( 2\pi i/q \) for \( q = 2,3,4,6 \) and we developed some properties of the invariants. The fact that the paper is so often cited is due to Paul’s great care with accuracy, for the subject was bedeviled by easy-to-get-wrong signs. Paul is more algebraically minded than I am, and he taught me more (elementary) representation theory than I ever expected to know.
This work lead to our being invited to the opening semester of the Newton Institute in Cambridge in the fall of 1992 for a program involving outgrowths of the Reshetikhin–Turaev methods. Besides the math, the visit included some British tradition.
Prince Philip (the Queen’s husband) came by for an hour to “open” the Institute. He made the rounds of some of the offices, and told Paul and me of an old monastery in Greece (his grandfather was George I of Greece) which had preserved bits of mathematics during a dark time. He made a good impression.
I was the Rothschild Visiting Professor and at an appropriate function at Trinity College (where Atiyah, Director of the Institute, was Master), I was briefed on how to thank Lady Rothschild and to be aware that she would let me know when our conversation was to end. She did, gracefully, after a few sentences, and awkwardness on both our parts was avoided.
And I was a Fellow at Emmanuel College which has a fantastic Oriental plane tree (google it!) in a back garden.
6. Gökova
In 1992 Selman Akbulut organized the first Gökova geometry/topology conference during the week of Memorial Day. It was held at the Hotel Yucelen in Akyaka at the east end of the Bay of Gökova in southwest Turkey. It was a beautiful site and an excellent group of mathematicians came, so it became a yearly conference which continues to this day. Besides Selman, the local organizer is Turgut Önder, an Emery Thomas PhD from Berkeley, now at Middle East Technical University, METU. Funding came from TUBITAK, the Turkish equivalent of the US National Science Foundation, and (eventually) from the NSF itself. Initially, the rooms were heavily subsidized by the hotel owner, but as outside support for the conference has declined, the hotel now is the biggest supporter of the conference.
I’ve been to the conference about a dozen times, partly for the math and camaraderie, and also for the Wednesday afternoon hikes to beautiful spots, the Saturday boat trips in the Bay, and for an exceptional swim or kayak down a one mile river formed by underground springs. There are some photos in the gallery.
In 1998, Selman convinced the METU math department to invite an outside committee (Avner Friedman, Robert Langlands, Ron Stern and myself) to review the department, perhaps the one and only time this has been done. It turned out that Langlands had visited Turkey for a year in the late 1960s, learned the language, and was curious to meet old friends and see what had changed.
7. Problem lists
The AMS held its 24th Summer Research Institute at Stanford, August 2–21, 1976, and the topic was algebraic and geometric topology. Besides myself, the speakers included Raoul Bott, Greg Brumfiel, Sylvain Cappell, Julius Shaneson, Bob Edwards, Cameron Gordon, Allen Hatcher, Jack Wagoner, Wu-Chung Hsiang, William Jaco, Max Karoubi, Dick Lashof and Mel Rothenberg, Ronnie Lee, James Lin, Ib Madsen, Jim Milgram, John Morgan, Bob Oliver, Ted Petrie, Dan Quillen, Larry Siebenmann, Emery Thomas, Friedhelm Waldhausen, Terry Wall, and James West.
This conference may have been the last of the all-inclusive topology conferences with an all-star line-up of lectures, for within six to eight years low-dimensional topology had exploded in size with hyperbolic 3-manifolds (Thurston) in one constellation, topological 4-manifolds (Freedman) in another, exotic smooth 4-manifolds (Donaldson) in a third, and new knot invariants connected to operator algebras and physics (Jones) in a fourth.
It was still possible to have a problem session followed by a problem list of manageable size. This I did by 1978, with much help from many people [7].
This is the review by Louis Kauffman:
This is an excellent survey of problems in knot theory, surfaces, three-manifolds and four-manifolds. For specific topics and questions we can do no better than to refer the reader directly to the article. Here are a few up-dates and comments: Problem 1.1 is true. It follows from a remark and results of Thurston. The top part of the diagram for Problem 1.37 should contain left-half-twists. Problem 2.3 is false, by methods of Johnson and Johanson. In Problem 2.6, the conjectured mapping exists for a finite subgroup via results of Steve Kerkhoff. The answer to Dale Rolfsen’s question in Problem 3.13 is: exactly the commutators (by John Harer). The answer to Problem 3.33 (B) is “yes” (Thurston). The answer to Problem 3.38 (the Smith conjecture) is “yes” (solution by Thurston, Meeks–Yau, Gordon, Litherland, Bass, Shalen and Otto Schmink). Problem 3.43 has been proved for various cases where the manifold has a “geometric structure”. Problem 4.8 has been answered affirmatively by M. Freedman. In general the would-be problem solver should beware (be aware) of the presence of Thurston, Jaco, Shalen and Johannson.
The reader may never have heard of Otto Schmink, who exists solely as an inside joke. In 1954 a Canadian group satirized Senator Joe McCarthy by postulating that he died and went to heaven and then began to investigate the process by which dubious (in his mind) figures such as John Stuart Mill, John Milton, etc., had been admitted. Soon they were removed from “up here” and sent to “down there”. The distinguished names began to dwindle, and eventually even one Otto Schmink was also demoted, just as McCarthy went mad while investigating God herself. The satire is well done and can be found at Nick Voss’ blog. Recommended.
In 1993 I agreed to an update of the “old” problem list, not realizing that such a project would end up at 438 pages [19]. Of course I had even more help this time, especially from Geoff Mess who essentially wrote the updates for most of the old 3-manifold and knot theory problems.
The National Academy of Sciences has a yearly Prize for Scientific Reviewing. This refers to the practice in most other disciplines of writing reviews on a particular topic which brings researchers up to date in that topic. Mathematicians don’t really write such reviews, but the NAS did not wish to leave mathematicians out of competition, so they decided that my problem list(s) were the next best thing to a “review”. The Prize was awarded in April 1995, before the second list was published although there were drafts circulating. Perhaps the Prize was for past and future problem lists?
The same problem arose the next time it was math’s turn, and it was solved by awarding the Prize to Bruce Kleiner and John Lott for their opus on Perelman’s solution to the 3-dimensional Poincaré conjecture and Thurston’s geometrization conjecture [e18].
8. Journal publishing and MSP
In 1996 Colin Rourke wrote to me proposing a new, very low cost journal provisionally titled Geometry & Topology, to be edited by Brian Sanderson and himself. They had designed an excellent editorial system in which an editor would handle a paper and obtain referee reports and then make a recommendation; if to accept, two seconders were needed, and the paper would be under consideration for four weeks. A journal tradition thus began: handlers gave serious arguments for their recommendations, and were rarely overturned.
My only task was to help Colin recruit a distinguished board of editors (which we did) in order to send a message that G&T was at the level of the best specialized journals. Topologists were unhappy with the subscription price of Elsevier’s Topology, and we wanted to give authors an excellent alternative. We succeeded a decade later when Topology came to an end after the Oxford stalwarts resigned and Elsevier failed to recruit other editors.
G&T thrived and in 2000 we established a new journal, Algebraic & Geometric Topology with Bob Oliver and Marty Scharlemann as chief editors. By 2003 it became clear that the journals needed a legal home and subscriptions. Up until then, prices were hard to set because we didn’t know how many pages we would publish in a given year. Moreover, we printed and bound the journals only at the end of the year, and this meant that libraries tended to purchase them out of their book budget rather than renew them yearly from their journal budget. At a propitious moment, Paulo Ney de Souza (my PhD student and the systems administrator for the math department) mentioned that he was interested in publishing, and agreed to run the servers for the electronic version of the journals. He persuaded Silvio Levy, already well known as an editor of mathematics, to take on a role as production editor with us. Ron Stern, president of the Pacific Journal of Mathematics, let us know that he was ready to move production of PJM and was willing to take a chance on our team. With servers, high-level copy editing, and a source of money, we were ready to establish Mathematical Sciences Publishers as a nonprofit California company.
MSP began a period of growth in which its “core” mathematics journals were founded:
- Communication in Applied Mathematics and Computational Science (CAMCoS) in 2006 by Alexander Chorin and John Bell;
- Algebra & Number Theory in 2007 by David Eisenbud and Bjorn Poonen;
- Analysis & PDE in 2008 by Maciej Zworski.
In 2006, the editors of an Elsevier journal in mechanical engineering, led by chief editor Charles Steele (of Stanford University), were ready to resign en masse; they heard of our work as an independent nonprofit and decided to bring their journal to us. They founded the Journal of Mechanics of Materials and Structures, which MSP is still publishing today. And in 2008 Ken Berenhaut founded our journal of undergraduate research papers, Involve.
In the next decade MSP started or adopted nearly ten more journals. It continues to operate on a principal of lean efficiency, its aim being to set an example of sustainable publishing practices in a climate dominated by corporate giants. MSP is now very ably managed by Alex Scorpan (of The Wild World of 4-Manifolds fame).
Another endeavor integral to MSP’s success has been EditFlow®, our editorial software for journal management. It is fair to say that it is easily the best software for math journals, having been designed (originally by de Souza) for and by mathematicians. Early on it was licensed to the AMS and then to the London Math Society, and it is now used by well over 100 other journals.
Last but not least, MSP is the home of Celebratio Mathematica in which this article appears.
9. Morse 2-functions and trisections
My collaboration with Dave Gay began with a visit to Tucson (Arizona State) in 2003. It had recently been shown by Cliff Taubes and Ko Honda that a smooth 4-manifold \( X \) with \( \beta_2(X) \geq 1 \) has a symplectic structure on the complement of a 1-manifold. We thought it would be interesting to construct such almost symplectic forms, perhaps on the complements of the cores of round 1-handles [20].
This effort led to a second paper [21] in which we constructed broken Lefschetz fibrations on all smooth, closed, oriented 4-manifolds; at the time we thought we needed both Lefschetz and anti-Lefschetz singularities, but Yanki Lekili soon noticed that the anti-Lefschetz singularities were not in fact needed. The circles where the fibration “broke”, that is, where the genus of the fibers changed by one, were roughly the same as the circles on which an almost symplectic form vanished.
This work gradually morphed over time into our paper on Morse 2-functions [22], an analog of Morse functions in which the reals \( R \) are replaced by the plane \( R^2 \). Then Dave noticed a clever way to trisect a Morse 2-function, and thus trisections of smooth 4-manifolds were born [23].
A striking, but easy to prove, application is this [24]: If we take the fundamental group of the various pieces of a trisection — the central fiber, the three slices, the three pieces of pie and the whole 4-manifold \( X \) — then we get a cube of groups with epimorphisms called a trisection of \( \pi_1(X) \). Then there is a bijection between smooth, orientable closed 4-manifolds and trisections of finitely presented groups, up to stabilization. Most striking is that the countably infinite exotic smooth structures on simply connected 4-manifolds correspond to different trisections of the trivial group!
10. Mathematical descendants
Over 53 years I’ve greatly enjoyed my 54 PhD students and their descendants. I knew early on that I would enjoy having students, from the first two at UCLA, Ted Turner and David Gauld, to the many more at Berkeley. I left UCLA for Berkeley primarily to escape the southern California smog and for the opportunity to have more grad students, and so I left behind many friends on the UCLA math faculty.
I’ve written elsewhere2 about my philosophy of advising students, but I’ll restate it here. Briefly, grad students should find their own problems, if possible; some problems won’t work out (this is also valuable knowledge), and others will. I’ve said to many students in their post-qual, thesis-writing phase that we are now embarking on the same adventure, proving our next theorem (never mind that I have already done so a few times).
When I was a student in the 1950s, I’d heard that 75% of new PhDs never write another paper after their thesis. This seemed to be a consequence of the grad student “three-year track”: that is, taking grad courses the first year, passing a qual in the second year, and then working on a problem assigned by an adviser — with hints as to how to solve it. The result was thus a thesis by spring of the third year. Naturally, many who have no experience of doing research independently are able to publish again.
In the 1970s problems were easy to find in low-dimensional topology, and it was not so hard to be familiar with most aspects of the subject. My first successful group of students at Berkeley — Scharlemann, Akbulut, Handel, Kaplan, Melvin and Nordstrom — could all work on different topics and yet still talk to each other. They were aided by many visitors, for in those days there was enough money for people like Edwards, Lickorish, Siebenmann, Larry Taylor, Hatcher, John Morgan and others to visit for up to a quarter.
There followed another excellent cohort of students, sometimes taught by their older “brothers”, including Harer, Giller, Livingston, Hass, Menasco, Cochran, Hughes, Ruberman, Gompf and Aitchison.
But then low-dimensional topology was dramatically changed, first by Thurston, then Freedman and Donaldson in 1981–2, and Jones in 1984. More advances followed, e.g. contact and symplectic topology, Floer theory, pseudoholomorphic curves, Heegaard Floer homology by Ozsváth and Szabó, geometric group theory and Perelman’s contributions. There was much more to be learned and better advice than I could give was needed. By now there must be at least a half dozen subfields of low-dimensional topology and geometry that are as lively as all of low-dimensional topology in the 1970s.
I tried teaching some of these subjects in graduate courses, rather poorly in some cases. That lead students into these newer subjects, but often I couldn’t help them much. Tom Mrowka bailed me out with several of my students in gauge theory as did other mathematicians in other areas.
I haven’t mentioned all of my graduate students, but I could include Kevin Walker whom I introduced to the Canyonlands and now he is the County Commissioner in Moab, Utah, looking after the environment, and also once hiked across Utah for 35 days without seeing another person (it was summer but he knew the geology well enough to find seeps, and most days there were thunderstorms near enough to get water in the water pockets); Joanna Kania-Bartoszynska who keeps NSF on their toes; William Chen who has won millions at poker and wrote a book which got Amazon ratings of zero (“hated the book, too much math”) or five (“you must buy this book if you wish to win at poker”); Stephen Bigelow who won the Blumenthal Prize for the best PhD thesis in five years, worldwide, for, “The Braid Group is Linear”; Rob Schneiderman who remains a world-class jazz pianist while writing fine papers in 4-manifolds, often with Peter Teichner; Eli Grigsby who deigned to jog with me and ran the Boston Marathon; and Andy Wand who was heading up a popular rock band in Istanbul before rediscovering topology in Gökova, and several more whose weddings I was privileged to attend.
So, I’ve been lucky in math (and in many other things); lucky that the torus trick existed and that I stumbled on it and then met Larry Siebenmann; lucky that I jogged with Arnold Kas which led to many discussions about complex surfaces and to the Kirby calculus; lucky to have had so many excellent collaborators who did more than their share of the work; and lucky to have had such an excellent bunch of mathematical sons and daughters, many of whom have remained good friends, as are the editors of this volume.
