by Rob Kirby
Cameron McAllan Gordon was born on March 2, 1945, in the small village of Lumsden in Aberdeenshire, Scotland. His ancestors are Scots as far back as Cameron knows. His father’s full name, Cameron Grant Gordon, and his own are distinguished by the middle names of McAllan and Grant, which are the maiden names of Cameron Jr.’s mother and grandmother, respectively, for that was a custom in Scotland — preserving the mother’s maiden name by making it the middle name of a son or daughter.
Until she got married, Cameron’s mother, Agnes Emma McAllan, was a schoolteacher. His father was a baker, as was his father before him. After an eight-year sojourn in New York, Cameron Grant Gordon returned to Scotland, and, in 1942, bought a Lumsden bakery, which he made a success. Besides the shop in the village, he had several vans which would deliver baked goods to the surrounding farms. As the business matured, much of the day-to-day routine passed into the hands of the workers, but the wedding and birthday cakes remained Cameron’s father’s domain. His “butteries”, a northeastern Scottish specialty — a heavier version of the croissant — were well known in the region. They’re really good; a buttery with marmalade in the morning, man…
Despite his father’s success, and the comparative wealth that resulted from it, the family of four (which now included Cameron’s younger brother, Kenneth) lived in tiny quarters above the bakery. His parents’ marriage was a “disaster”, and he had “a miserable childhood”.
Cameron attended the local primary school until the age of eight, when he was sent to Blairmore, a nearby boarding school situated in an old country house with extensive grounds. Cameron recounts that in the wake of World War II, there were many ex-military officers who ended up teaching at such schools. Blairmore itself was founded in 1947 by one Colonel Ainslie, and among a faculty that included many excellent teachers, there were Major Davidson, Colonel Collard, and Squadron Leader Hurndall. The ethos of the school was thus rather military. With only 50 students, it was not a large institution, and but for two girls (the daughters of Colonel Ainslie and one of his friends), it was all boys.
Cameron attended Blairmore from the ages of eight to twelve when he shifted to Fettes College, a “public” (in the British sense) boarding school in Edinburgh. It, too, was tinged with militarism: students had to take a cold shower every morning, and skipping one meant getting caned if caught. In some sense it was fairly rigorous. But by my day, in 1957, it was beginning to mellow. And it did have a strong academic tradition, with a good record of getting pupils into Oxford and Cambridge. A contemporary of mine at Fettes was Angus Deaton, who won the Nobel Prize in Economics in 2015. Tony Blair also went there, a bit later. I had very good teachers, especially in science and math. The teaching was elitist in the British tradition: the bright kids got the good teachers, and the rest got the poorer, often ex-military, teachers.
Cameron started playing the guitar at Fettes, at around age 14. One of the big regrets of my life was when I was living in Aberlour on the Spey river, near a town called Elgin. There was a dance hall in Elgin called the Two Red Shoes, and one day, around Christmas 1962, I was passing by and saw a poster outside advertising “The Beatles, The ’Love Me Do’ Boys from Liverpool”, and I never went because who had ever heard of them? When school was not in session, he lived in Aberlour with his mother, who had by then decided to separate from his father, and had resumed her teaching career by accepting a post at a well-known orphanage in town.
At Blairmore I did Latin, and eventually Greek. When I went to Fettes, the first day in class we were asked whether we wanted to do classics or science. I had never had science so, along with one other boy who had been at Blairmore, I chose classics. We had this rather sadistic Greek teacher whose method of imparting knowledge was to grab hold of one of our sideburns and say “Decline ‘strategos’ ”, or whatever, increasingly twisting the sideburn at each error. Quite Dickensian! So needless to say we weren’t enjoying classics very much. However, 1957–58 was the year of the Asian flu pandemic, which sickened a lot of people at the school, and necessitated a temporary reorganization of a lot of the classes. My friend and I were moved into the science classes, and we thought these novel subjects physics and chemistry were such fun that we asked if we could make the change permanent. To the school’s credit, they said yes. So I started doing science and loved it — got really hooked.
At age 16, Cameron sat the entrance exam in Natural Sciences at Cambridge University. One applied not to the university as a whole, but to a specific college, and Cameron chose Trinity, because it had a strong tradition in science and math. He was offered a “place” at Trinity but because of his age he was encouraged to go back to Fettes for a year. He did so, and during that time became increasingly interested in math.
I’m rambling a bit, but there is a good example here of the educational philosophy, more prevalent then than now, which believed in criticism of failure rather than praise of success. Before I took the Cambridge entrance exam the housemaster of my boarding house at Fettes had told me and another boy to apply for Trevelyan Scholarships; these were scholarships awarded by a private trust that could be held at Oxford or Cambridge. We did, but were unsuccessful, and the housemaster made no secret of his displeasure. Later, when I got the letter from Trinity offering me a place, I was quite excited. But when I told the housemaster, who was an ex-Naval officer, he just grunted and said: “Hmmph. At least we managed to salvage something from the wreck”.
The following year Cameron went back to Cambridge and took the entrance exam in math. He was offered an “exhibition” at Trinity, one notch below a “scholarship”. (After his first year at Cambridge he was upgraded from exhibitioner to scholar.)
The exam was given in a large hall with desks arranged alphabetically. There was a three-hour session in the morning, another in the afternoon, and this went on for several days. Most exams had eight questions, and because they were difficult, getting five or so right was considered good. Of course everyone discussed the problems afterwards, and how they thought they’d done. Cameron remembers being somewhat intimidated by one fellow sitting next to him who would typically report that he’d been able to do seven of the questions, and maybe most of the eighth. It turned out to be Peter Goddard! (Goddard is a physicist and professor and ex-director of the Institute of Advanced Study.)
Cameron recalls: I had a rather idiosyncratic career at Cambridge. I was very irresponsible and extremely lucky that the system didn’t give up on me. My train of thought was this. Getting into Trinity was great, and you’re surrounded by all these clever people, so in my first year I worked really hard, went to all the lectures, etc. I took Part I of the Tripos at the end of that year, and I did alright, got a First Class and saw a lot of friends drop out, so I thought this is not so bad. Now at the end of the second year you take a Preliminary Exam to Part II, which doesn’t really count: the important exam, Part II (the equivalent of the BS in the US, as Part III is a year of graduate work) is at the end of the third year. I thought, well, the second year doesn’t count for much, and I can always work hard in the third year, and so I literally didn’t go to a single math lecture in my second year. Predictably, even though I borrowed Peter Goddard’s lecture notes, I did disastrously on the Preliminary Exam. Of course the (rather obvious!) flaw in my plan was that you had to know a lot of the second year material in order to understand the third year material. Consequently, I was told that on my Part II exam, on each paper I either got a reasonably high score or zero! Despite this very mixed performance the authorities were nice enough to let me stay on for Part III, where I did well enough to get into the PhD program.
I started working with John Hudson, who had been a student of Christopher Zeeman. In 1968, Hudson got a chair at the University of Durham, and Sue and I (we had just got married) decided to accompany him to Durham. I finished after two years at Durham in 1970.
I had met Sue (née Watson) a few years earlier when she was at a women’s teacher training college, Balls Park, in Hertford, about 30 miles from Cambridge. There were only two women’s colleges at Cambridge, Newnham and Girton, so there was a huge gender imbalance. On weekends some of the women from Balls Park would come to Cambridge for parties, sometimes taking a bus laid on by the college which Sue maintained was called “the passion wagon”. I met Sue at such a party, and eventually we got together.
Sue was from Chinley, a little town in Derbyshire, in a beautiful part of England known as the Peak District. There’s a bit of a math connection here, for L. J. Mordell, the well-known number theorist, was a professor at Manchester University from 1922–45, and had a country cottage in Chinley which he and his wife would visit at the weekends.
Hudson was a powerful PL topologist, and got a professorship unusually early in his career. But Cameron suspects he regretted his move to Durham for he missed the social life in Cambridge and left Durham after two years — and left topology not long after.
Hudson gave Cameron a thesis problem that stemmed from an embedding theorem of M. C. Irwin [e3]. Using engulfing methods pioneered by Christopher Zeeman, Irwin had shown that a map \( f:M^m \to Q^q \) was homotopic to a PL embedding if \( q \geq m+3 \), \( M \) is \( (2m{-}q) \)-connected and \( Q \) is \( (2m{-}q{+}1) \)-connected. Via a theorem of Stallings (in an unpublished manuscript [e1]) together with an early version of the Hauptvermutung proved by Casson and Sullivan (see [e15]), Irwin’s theorem can be improved by putting connectivity conditions on the map rather than on the manifolds: \( f \) is homotopic to a PL embedding if \( f \) is \( (2m{-}q{+}1) \)-connected, i.e., \( f_*:\pi_i(M) \to \pi_i(Q) \) is an isomorphism for \( i\leq 2m-q \) and is an epimorphism for \( i=2m-q+1 \). (Hudson had proved an earlier version of this which had the additional hypothesis that \( M \) and \( Q \) are \( (3m{-}2q{+}2) \)-connected [9].) Hudson had also proved a relative version of Irwin’s Theorem for manifolds with boundary [e7]: a map \( f: (M^m, \partial M) \to (Q^q, \partial Q) \) is homotopic, as a map of pairs, to a PL embedding if \( q \ge m+3 \), \( (M, \partial M) \) is \( (2m{-}q) \)-connected and \( (Q, \partial Q) \) is \( (2m{-}q{+}1) \)-connected.
Hudson’s suggestion to Cameron was to prove the obvious missing fourth theorem: the relative case with the connectivity conditions on \( (M, \partial M) \) and \( (Q, \partial Q) \) replaced by the corresponding connectivity condition on the map — i.e., that \( (f, \partial f) \) be \( (2m{-}q{+}1) \)-connected — and he gave hints for the expected proof. Nothing seemed to work, and Hudson repeated his hints and encouragement, but still nothing.
Then one day I read Fox’s “A quick trip through knot theory” and I found it fascinating. I particularly remember being intrigued by a table of the homology of the cyclic branched covers of the figure eight knot. So I got interested in knot theory.
And then I read Zeeman’s paper
[e4]
on twisted spun knots.
It’s one of the most beautiful papers I’ve read, a model for good
mathematical writing. He starts off with an example, the 5-twist spun
trefoil, and, using the fibration of the complement of the trefoil,
shows that the resulting 2-knot is fibered with fiber the
Poincaré homology sphere. Then Zeeman gets more general, points
out that the fact that the trefoil is fibered is a red herring, and
proves his main theorem: if \( X^n \) is a knotted sphere in \( S^{n+2} \),
then for any \( k > 0 \) there is a codimension one knot (the \( k \)-twist
spin of \( K \)) whose fiber is the \( k \)-fold cyclic branched cover of
\( S^{n+2} \) branched over \( X \).
At the end of his paper Zeeman asked some questions, and I don’t think he had thought very hard about them, for I was able to answer one of them. So I started writing papers on knots, and was having great fun. After working on knots for a year or so, I went back to the PL stuff, and, with a more mature perspective perhaps, saw that the thing I’d been trying to prove was actually false. I wrote up the counterexample in [1]. I might add that another benefit of reading Zeeman’s paper was that later I was able to use twist-spinning to show that smooth 2-knots in \( S^4 \) are not determined by their complements [2]. An important part of the proof was showing that the Gluck construction on a twist-spun 2-knot always gives \( S^4 \) (incidentally answering another of Zeeman’s questions). By the way, this experience has made me appreciate the value of asking questions in papers. I think we are sometimes reluctant to do this, feeling perhaps that if a question is too easily answered it makes us look foolish for asking it. But we shouldn’t be.
How did Cameron end up going to Florida State University in Tallahassee? Well, he’d read the “little red book”, the conference proceedings from the 1961 workshop in Georgia (which included Fox’s “Quick Trip Through Knot Theory” and other gems), and noted all the topologists in the southeastern United States, especially at Florida State, which in 1970 had De Witt Sumners, John Bryant, Chris Lacher, Jim Andrews (of the Andrews–Curtis Conjecture [e2]), Perrin Wright, Wolfgang Heil, and Orville Harrold (who was chairman). Cameron applied elsewhere in the US and UK, but guesses that getting an offer from Florida State was not unrelated to the fact that De Witt Sumners was Hudson’s only other Ph.D. student; he had been a Marshall Scholar at Cambridge and got his Ph.D. in 1967. Cameron and Sue had a great time in Tallahassee with that young group of geometric topologists.
After two years at Florida State, and despite the fact that he had a tenure-track job, Cameron figured he’d go back to the UK. The government had instituted a scheme to try and reverse “brain drain”, a prominent issue in those days when the US had many jobs which lured “brains” from all over Europe. So Cameron got a three-year fellowship to return to Cambridge. While there, he applied and got a Fellowship at Gonville and Caius College (Caius, pronounced “Keys”, for short). Zeeman had been at Caius before leaving to found the Maths Institute at Warwick, and wrote a letter for Cameron which the latter suspects helped greatly with his fellowship application.
Cameron recalls that when he visited Cambridge in 2017 to participate in a program at the Newton Institute, he was honored to be appointed by Caius as the G. C. Steward Visiting Fellow. Soon after arriving he was informed that one of the duties of the Steward Fellow was to attend the Venn Dinner, the annual dinner of the Venn Society, the college’s undergraduate mathematical society. (John Venn had been a Fellow of the college; he is commemorated with a stained glass window of his famous diagram in the college dining hall. The annual black-tie dinner is apparently the only function of the Venn Society.) But Cameron’s enthusiasm at accepting the invitation was dampened somewhat on being informed that another of his duties was to give the after-dinner speech! To make matters worse, he was told that Zeeman, as the Steward Fellow some years earlier, had begun his speech by climbing onto the banquet table, throwing a boomerang around the dining hall, catching it, and then doing it again to prove that the first time wasn’t a fluke. A hard act to follow!
At Cambridge Cameron started working with Andrew Casson and fairly soon they had what is known as the Casson–Gordon invariant [3]. A second version based on Casson’s lectures at IHES in 1974 appeared in the “lost topology” book [7].
Cameron visited Bob Edwards at UCLA in 1975, and went on to the Institute for Advanced Study in 1976–77 for a special year in topology. (I recall visiting for a few days, describing a handle description of K3, and soliciting questions for the problem list [e9] which had started at the Stanford conference in August 1976. Thurston suggested a number of problems most of which he promptly solved himself during the next few years.)
That year was a great year. I decided I was going to stay in the States and I applied for jobs, and went to Austin in late summer 1977.
I visited Cameron in September and brought with me a just-out copy of Bill Meeks’ lecture notes [e8], sketching a proof of an equivariant Dehn’s Lemma. Cameron immediately realized that this, together with work of Thurston, meant that P. A. Smith’s famous problem asking whether a finite cyclic group action on \( S^3 \) could ever have a knot as the fixed point set was settled in the negative; only the unknot could be the fixed point set. Moreover, the action is by the restriction to \( \mathbb{Z}_n \subset S^1 \) of the circle action on the unit 3-sphere in \( \mathbb{C}^2 \) given by \( (z,w) \mapsto (uz,w) \), \( u \in S^1 \). The whole story is well told in [e13]. Cameron had worked on the problem with Rick Litherland, in particular looking for a proof of an equivariant Dehn’s Lemma, but minimal surface theory was needed to reduce daunting technical difficulties. The history is described in the preface to [e13]. from which I quote:
During a conversation with Bass, Thurston learned of Bass’s result (Chapter VI). He saw, in the light of his own work (Chapter V), the relevance to the Smith conjecture. He also saw the need to treat the cases covered in Part C. What was needed to deal with these missing cases came clearly into focus during conversations between Thurston and Gordon (the latter being motivated by his earlier work with Litherland [5]; see Chapter VII). At about the same time, Meeks and Yau had established exactly the required result (Chapter VIII). However, there was a gap in communication between Thurston and Gordon, on the one hand, and Meeks and Yau, on the other. This gap was bridged by Gordon when he learned of the existence of the work of Meeks and Yau. With that, the proof was complete.
When I asked Cameron for his three favorite theorems or pieces of work, he began with the Casson–Gordon invariant [3] [7]. Well, the stuff with Andrew, it just worked out so well. In 1973 Frank Adams decided that in the topology seminar at Cambridge we would go through the Atiyah–Singer Index Theorem, and everyone was expected to give a talk. I was terrified, people were talking about Fredholm operators and stuff I knew nothing about. What was I going to do? But I found this paper by Hsiang and Szczarba [e6] where they used the Index Theorem to get results about surfaces in 4-manifolds. It was the 4-dimensional \( G \)-Signature Theorem where the fixed points were surfaces and isolated points, explicit enough that I could understand it and give a talk on it. And it came in handy when I started talking with Andrew.
(I will interject into Cameron’s story the fact that he later came up with a beautiful, elementary proof of the 4-dimensional \( G \)-Signature Theorem which appears in the lost topology book [6].)
Andrew told me he had shown that, apart from the stevedore’s knot, none of the infinitely many algebraically slice twist knots are ribbon, and we wondered if we could show that they weren’t slice either. Well we eventually did. Our first arguments were sort of ham-fisted, but very explicit, with branched covers and surgery pictures [3]. A more elegant account was written up by Andrew in [7].
Having mentioned Frank Adams let me say that I always had a soft spot for him. He was very encouraging to me, although he could be quite intimidating and had a reputation for being harsh on his students.
Then I asked Cameron about another paper with Andrew [9] that turned out to be quite important although published in a modest journal.
One day (when we were both at Austin) I went to Andrew’s office and said: “I know that if you have a Heegaard splitting and there are two disks on opposite sides that intersect in one point, then the splitting can be reduced, but what happens if the two disks are disjoint?” Well, we started talking and about three hours later we had a theorem. I wrote it up, probably badly, and we sent it to Topology. They rejected it, saying that the applications were already known, and Larry Siebenmann, who was editor of both Topology and Topology and its Applications, remarked that if Andrew and I had not been such famous authors(!) he would have rejected it out of hand, but instead he suggested that it be published in the lesser journal, so it was. It turned out to be quite influential, but only later.
Getting back to your question, Rob, about my three favorite pieces of work, I’ve already mentioned the Casson–Gordon invariant. The other two obvious choices are the Cyclic Surgery Theorem [8], with Marc Culler, John Luecke, and Peter Shalen, and the Knot Complement Theorem [10], with John Luecke. The former was a nice example of two completely different sets of techniques perfectly complementing each other. The theorem is that if \( M \) is a compact, connected, irreducible, orientable 3-manifold with torus boundary, that is not a Seifert fibered space, and if \( r \) and \( s \) are distinct slopes on \( \partial M \) such that the corresponding Dehn fillings \( M(r) \) and \( M(s) \) have cyclic fundamental groups, then the distance (minimal intersection number) \( \Delta(r,s) \) between \( r \) and \( s \) is 1. Hence there are at most three such slopes. In the case where the manifold \( M \) is hyperbolic, Marc and Peter, using a beautiful set of ideas involving the \( \operatorname{SL}_{2}(\mathbb{C}) \)character variety \( X(M) \) of \( \pi_1(M) \), showed that the ideal points of the projective completion \( \tilde{X}(M) \) of \( X(M) \) correspond to actions of \( \pi_1(M) \) on a simplicial tree, which in turn give rise to essential surfaces in \( M \). Further, they showed that for almost all slopes \( r \), either there is a point of \( X(M) \) corresponding to a noncyclic representation of \( \pi_1(M(r)) \), or there is an ideal point of \( \tilde{X}(M) \) whose associated surface is either closed and remains incompressible in \( M(r) \) or has boundary slope \( r \) and is not a fiber in any fibration of \( M \) over the circle (i.e., \( r \) is a strict boundary slope). The conclusion is that if \( \pi_1(M(r)) \) and \( \pi_1(M(s)) \) are cyclic then either \( \Delta(r,s) = 1 \) or one of \( r \) and \( s \) is a strict boundary slope. Using various combinatorial-topological methods John and I were able to describe in enough detail what happens when you do Dehn filling along a boundary slope to complete the proof in the hyperbolic case, and also to do the case where \( M \) contains an essential torus. One of the techniques we used was the combinatorial analysis of the intersection of a pair of essential punctured surfaces in \( M \) with different boundary slopes, a technique introduced by Rick Litherland [e11] in his study of Dehn surgery on satellite knots. By using Dave Gabai’s powerful idea of thin position used in his proof of the Property R Conjecture [e14] one can apply surface-intersection techniques to punctured spheres in knot complements that are not necessarily essential. By a considerable elaboration of the machinery developed in the proof of the Cyclic Surgery Theorem to study such intersections, John and I were eventually able to prove the Knot Complement Theorem.
In addition to the collaborations I have already mentioned I have more recently done a lot of joint work with Steve Boyer, some of which is ongoing. Given that mathematical research consists mostly of banging one’s head against a brick wall, one pleasant feature of joint work is that you know (OK, maybe not 100%…) that there is at least one other person who is interested in what you’re doing.
Cameron has two children: Catherine born in September, 1976, and Andrew in August, 1980, both in the same hospital in Cambridge.
Sue passed away in January 2018 of cancer.
