Reminiscences of a Topologist* Ioan James

In my undergraduate days (1947-1951) the Oxford mathematics syllabus had not changed greatly from that of the thirties. G. H. Hardy, of course, had done much to break down the insularity of British mathe- matics, and as a result Hardy's former research stu- dents, of whom E. C. Titchmarsh was the foremost member, were influential. I was fortunate in that my tutor, Haslam-Jones, was one of these. He took pride in being able to teach the whole of the undergraduate course expertly and I owe him a great deal.

At the beginning of the third and last year of the course I fell seriously ill and had to take a year off. I had been on the point of approaching Titchmarsh to see if he would accept me as a graduate student. How- ever, while convalescing from my illness I had the op- portunity to read mathematics much more widely than the Oxford course provided for. Gradually I came to feel that rather than study eigenfunctions under Titch- marsh I might do better to s tudy topology under J. H. C. (Henry) Whitehead.

Whitehead had a rather unusual career. At Eton he had not distinguished himself academically and the College was quite surprised when he won a Balliol ex-

hibition, in mathematics. Although he did well at Ox- ford he decided, when he graduated, not to stay on but to begin a career in the City of London as a stock- broker's clerk. After a few years, however, he changed his mind, returned to Balliol briefly, and before long was on his way to Princeton, as one of the first Com- monwealth (now Harkness) fellows, to study under Oswald Veblen. The Cambridge Tract Foundations of Differential Geometry resulted from this. In those days the leading topologist in Princeton was James Alex- ander; Whitehead became a member of this circle,

* Text of an address given on 27 June 1988 at the fifth Oxford To- pology Symposium held at the Palazzone, CoRona, Italy. The two portraits of Henry Whi tehead were taken by Professor A. Kosinski and are reproduced wi th his kind permission.

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Hardy leading out his cricket team. From left to right: G. H. Hardy, H. O. Newboult (?), W. L. Ferrar, U. S. Haslam-Jones (?), E. C. Titchmarsh, ?, E. H. Linfoot, L. S. Bosanquet.

which included people like Samuel Eilenberg and So- lomon Lefschetz. While he left for America a geometer he returned a topologist, and set out to revitalise Ox- ford mathematics by promoting the study of topology.

Before World War II Whitehead had just one re- search student, Graham Higman. Whitehead's career was interrupted by the war during which he served, along with various other well-known mathematicians --notably Alan Turing--at the celebrated defence re- search centre at Bletchley. He was always proud of his work there although secrecy regulations prevented him saying much about it. One of his juniors at Bletchley was a young man named Peter Hilton, who became a founding member of the group of research students Whitehead gathered around him after the war.

I was fortunate to join this lively and distinguished group, which included people like Michael Barratt, Wilfred Cockroft, and Victor Gugenheim, as well as Hilton. Of these I was probably closest to Michael Bar- ratt. Being reluctant to trouble the great man himself about my struggles to understand what was going on, I troubled Michael instead and will always be grateful to him for the help he gave me.

It may be difficult for the present generation to ap- preciate how homotopy theory stood at that stage. Of course, research had been rather interrupted by the war, and there were many questions outstanding. For

example, it was not known that the homotopy groups of spheres were finitely generated. At the last Interna- tional Congress of Mathematicians before the war Lev Pontrjagin had announced that ~s(S 3) = 0. The argu- ment he had in mind was essentially what we now know as the Pontrjagin-Thom construction. This in- volves framed manifolds, of course, in this case 2- manifolds. Unfortunately, Pontrjagin had somehow overlooked the torus and so, as George Whitehead showed, his announcement was wrong and ~5(S 3) is non-trivial, in fact Z2.

The next real difficulty was ~6 ($3) �9 This group was known to be of order 12, either Z 3 + Z 4 or Z 3 + Z 2 + Z 2. Most of the Whitehead students had a go at trying to settle this question. In the end it was Barratt and George Paechter who got the right idea, essentially a special case of what we now refer to as Toda brackets. I well remember Barratt's lengthy arguments on the subject with a sceptical Henry Whitehead.

In those days there were few books relevant to grad- uate studies in homotopy theory. There was Alexan- droff and Hopf, also Seifert and Threlfall. There was Lefschetz's Introduction and his Colloquium volume. That was about all. The first new textbook to appear after the war was Steenrod's Topology of Fibre Bundles. That book was an enormous help to me and I practi- cally knew it by heart. Naturally a "first organization" of a new branch of mathematics stimulates a lot of fur-

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ther research and tends to make itself obsolete after a time, but Steenrod 's book remains recommended reading to this day.

My first independent piece of research arose from that book. Steenrod gives a classification of sphere- bundles over spheres, as fibre bundles, and I tried to classify them by homotopy type instead. Classification by fibre homotopy type is quite easy, of course, but the classification by ordinary homotopy type was just the sort of project I needed in order to get started. The outcome was two joint papers by Whitehead and my- self. The first, on the case where there exists a cross- section, was largely wri t ten by me, while Henry Whitehead largely wrote the second. In fact there was a difficulty for (n-1)-sphere bundles over S n (n = 4 or 8), which we were unable to resolve. Later I was able to settle n = 4, with some hints from Barratt, who has also settled the case n = 8 in work which, alas, re- mains unpublished.

Topology at Oxford has always benefited greatly from academic visitors. Some of these come to lecture at the seminar Whitehead established in 1946 and which always met, in term time, on Mondays at 5 (ex- cept in the cricket season when it met after dinner). This seminar is still running and I often regret that we didn't maintain a "visitors book" in which to record the many notable lectures given in the seminar and the people who gave them. One I particularly re- member is Norman Steenrod's lecture on the coho- mology operations now called the Steenrod squares; this led to the wel l -known note by Steenrod and Whitehead about vector fields on spheres.

In most years there have generally also been one or two topologists visiting Oxford, and they are often asked if they would like to give a course. I particularly remember Everett Pitcher's course on Morse theory. In this course Pitcher talked about Morse's theorem con- cerning the space of loops on S n+l . Essentially he showed that this could be approximated, from the ho- motopy theoretic point of view, by spaces of geodesic segments; thus fl(S n+x) has the homotopy type of a CW-complex S~ of the form S n U e 2n U e 3n U . . . .

Pitcher himself had shown that the 2n-cell in this com- plex was attached by the Whitehead square of the gen- erator of ~rn(S n) and had obtained some information about the attachment of the 3n-cell. I tried to improve on this and somehow got the idea that the complex was what I came to call the reduced product space- - essentially the free monoid on S n. I found out later that the same idea had occurred independently to the young Japanese topologist Hiroshi Toda--al though it was not suggested to him by Morse theory as far as I know. About that time George Whitehead published his famous A n n a l s paper concerning the exactness of the sequence

�9 . . ~ "lTr S n ~ T f r + l S n + l ~ "lTr+l $ 2 n + 1 ~ " l rr_ l S n ~ . . .

in homotopy groups of spheres. Whitehead showed that exactness holds as far as r = 2n, approximately, and I began to wonder if this could be extended fur- ther. Because the reduced product is such a simple gadget I began to play around with it and eventually found a family of maps S~ ~ S~ that seemed to have

52 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990

interesting properties. However, to exploit their full potential I needed some new machinery and fortu- nately this arrived in the form of Jean-Pierre Serre's thesis Homologie singuli~re des espaces fibrds.

The machinery Serre made such effective use of was, of course, the spectral sequence. This was origi- nally developed by Leray, for use in sheaf theory. However, Serre realized that a singular version of the spectral sequence would find many applications in ho- motopy theory. He forged this new weapon and re- sults came cascading out. For example, he showed the homotopy groups of spheres to be finitely generated.

In those years I always went skiing after Christmas, usually to Klosters, where I stayed at a picturesque inn called the Chesa Grischuna. After a hard day on the slopes I found it quite refreshing and certainly stimu- lating to study Serre's beautiful work. And with it I was able to establish that the sequence of George Whitehead had exactness properties far beyond what I had ever conjectured. In fact, the sequence is exact without limit for odd values of n and the same is true for even n if odd torsion is disregarded.

Although Serre immediately leapt to fame amongst us homotopy theorists, he was not at all well known in the outside world. About this time Whitehead per- suaded the Maison Franchise, the French cultural out- post in Oxford, to invite Serre over for a few days. The Director's doubts about inviting anyone so young were dispelled when, just before his visit, Serre was elected to membership of the Coll~ge de France, a most unusual honour for anyone of his age. The lec- ture Serre gave at the Monday seminar was full of ideas. He started by talking about %-theory--the sub- ject of his second Annals paper - -bu t he also told us about other ideas he had, for example the expression for ~~(S p + I ~/ S q+l) as a product of loop-spaces ~SP +1 x f~Sq +1 x f~Sv+q+l x . . . . This was taken up by Hilton who developed it in his well-known paper in the Proceedings of the Cambridge Philosophical Society. Chatting after dinner, I remember, Serre made a con-

Academic "family tree" of Henry Whitehead, showing re- search students he supervised, grandstudents, and so on (in some cases more than one supervisor was involved, but for simplicity only one is shown).

jecture about symmetric products which rather stuck in my mind. It had long been known that the r th sym- metric product of S 1 had the same homotopy type as S I, and that the r th symmetric product of S 2 was ho- meomorphic to the r-dimensional complex projective space. Serre observed that in the limit, as r ~ % this indicated that the infinite symmetric product of S n was the Eilenberg-Mac Lane space K(z,n) for n = 1,2 and conjectured that this might be true generally.

After I got my doctorate I applied, with Henry Whitehead's encouragement, for one of the Common- wealth Fellowships that he had himself held when vis- iting the States in Prohibition days. I was fortunate enough to be selected and in 1954 found myself on the Queen Mary bound for New York along with a dozen or so other new Commonwealth Fellows. The Fellow- ship could be held anywhere in the States. I chose Princeton so as to be able to work with Steenrod along with Emery Thomas, whom I had gotten to know when he was a Rhodes Scholar at Oxford and who was just completing his Ph.D. at Princeton under Steenrod. Princeton was still the leading place in the States for topology, and probably in the whole world. Although Alexander had retired prematurely, Lef- schetz, Ralph Fox, Deane Montgomery, and other lu- minaries were in their prime, as well as Steenrod. I received a lot of encouragement and continued to de- velop some of the ideas in my thesis.

I am particularly proud of one result I obtained at that time. Hilton had obtained, arising out of his work on the loop-space on a product of spheres, some pow- erful results about the composition law for homotopy groups. I tried to prove similar results by my own methods and found that there appeared to be a dis- crepancy between Hilton's results and my own. At first I put it down to a difference in sign conventions - - these had gotten into rather a mess. However, after laboriously working through these it transpired that the apparent discrepancy was genuine and this led to the conclusion that ~rr(Sn), for n odd and all r, contains no elements of order 2 ". This is best possible since ~r6(S 3) contains an element of order 4, as mentioned before. The following year Toda, who had also been working on the homotopy groups of spheres, proved a similar result for odd primes. Recently there has

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Henry Whitehead at his desk in the old Mathematical Insti- Norman Steenrod. tute at Oxford.

been a revival of interest in these results and substan- tial improvements have been obtained.

When I got to Princeton I was surprised to learn that Steenrod was going to spend the second half of the year at Berkeley. Accordingly, early in February, Steenrod, Emery Thomas, and I got into Steenrod's Buick and drove off with him to California via Florida, Louisiana, and Texas. In those days the mathematics department at Berkeley was quite small, although very distinguished. You went to Dwinelle Hall, the home of the Arts Departments, and in one corner of the fourth floor, I believe, you would find people like Kelley, Lehmer, and Tarski. Emery and I settled in happily.

While I was at Berkeley my thoughts returned to Serre's conjecture about infinite symmetric products. If this was correct, then might it not be true, more gen- erally, that the homotopy groups of the infinite sym- metric product of a space were just the homology groups of the original space? This suggested trying to prove that the homotopy groups of the infinite sym- metric p roduc t sat isf ied the E i lenberg-S teenrod axioms for homology. And for this one needed to show that the infinite symmetric product transforms cofibrations into fibrations. Although this is true, in a sense, the definition of fibration needs to be modified somewhat. I spent a lot of time that year trying to con- vince Steenrod that the modification I thought worked actually did so. When I returned to Princeton at the end of that summer I felt sufficiently sure of the truth of the result to announce it at Montgomery's seminar in the Institute. However, at the end of my lecture John Moore told me that Albrecht Dold and Ren6 Thorn had just published an announcement of the same result in the Comptes Rendus. I must say that I felt very d i s appo in t ed . The infinite symmetr ic product, essentially the free abelian monoid, seemed the perfect follow-on for the infinite reduced product,

essentially the free non-abelian monoid. Perhaps I should have taken a chance and announced the result before I had worked out all the details, but I have never liked to do that sort of thing.

Back at Princeton I began, with Montgomery's en- couragment, to investigate H-spaces. Here the great conjecture, the proof of which e luded me, was to show that of the spheres only S 1, S 3, and S 7 can be H-spaces. However, amongst other results I was able to show that S" is only a homotopy-commutative H- space for n = 1, and a homotopy-associative H-space for n = 1 and n = 3. Later Frank Adams established the great conjecture, as we all know.

During my two years in the States I made many friends among contemporary and more senior mathe- maticians. This was particularly true in my second year, at the Institute for Advanced Study, where Mar- ston Morse and Oswald Veblen, alas no longer with us, were very kind, also Deane Montgomery, Hassler Whitney, and many others. Unfortunately, Einstein and von Neumann had passed away by that time, but the legendary Alexander was still in Princeton, living rather in seclusion, and one day Emery Thomas and I went to call on him and hear stories about Princeton in pre-war days.

Lefschetz figured prominently in these stories, of course. When I first knew him he was spending part of each year in Mexico, helping to build up the mathe- matics department at the University of Mexico. As part of this programme, in 1956 he was organising a major topology conference in Mexico City; I was de- lighted to be invited to attend and give a short talk. I travelled down to Mexico with Ralph, Cynthia, and Robin Fox, spending about a week on the drive which proceeded rather slowly becasue of Ralph's butterfly collecting and Cynthia's bird watching. Mexico was delightful--I spent the whole summer there, partly in Mexico City and partly in Acapulco--and I have re-

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Emery Thomas, Michael Barratt, Henry Whitehead, the au- thor, and Peter Hilton at the International Topology Sympo- sium in Mexico City, 1956.

tained the warmest of feelings towards that country and its people. The conference itself was a great occa- sion. Everybody was there and much good work was done. Alas it was marred, towards the end, by the news of the accidental death of Witold Hurewicz, who fell from the top of a Mayan pyramid when on a visit to Yucatan.

On my return to England I was elected to a Junior Research Fellowship at Caius College, Cambridge. This was before the great Cambridge diaspora. Adams was away in the States but Michael Atiyah and Chris- topher Zeeman were there working as college tutors while David Epstein and Terry Wall had gotten to the part III stage. All of us attended Hodge's seminar and I r emember a lively discussion taking place over Milnor's proof of the existence of exotic differential structures on the 7-sphere, one of the sensations of the Mexico conference. I was just getting accustomed to Cambridge life when I heard that a new Readership in Mathematics had been established at Oxford and was invited to apply. When I was offered it I accepted with alacrity, particularly since it would enable me to rejoin Whitehead and work in partnership with him. By now I knew Henry and Barbara Whitehead very well, and they kindly invited me to lodge at Manor Farm, Noke, where they had moved from Charlbury Road. This was a delightful place to live, and I greatly enjoyed life on the farm. There was never a dull moment where the Whiteheads were concerned.

One of the great challenges of homotopy theory in those days was the vector-field problem, which is dis- cussed in Steenrod's book on fibre bundles. When I first read about it as a research student I tried to see what could be done in the positive direction. My ef- forts led me to the 1920s work of Hurwitz and Radon; and I was delighted to find that this work could be adapted to give what I thought were new results about the problem. I took these along to Whitehead-- i t was

Henry Whitehead at Manor Farm, Noke.

the first thing I had to show him as a research student - - a n d was dashed to hear that Beno Eckmann had been there before me. Returning to the problem some ten years later I started to become interested in Stiefel manifolds, which are central to questions about vector fields. The work I did at that time is collected in my book on the subject. Of course, it was Adams who eventually solved the vector-field problem itself, but I enjoyed finding out about the beautiful properties of Stiefel manifolds and made a number of conjectures, some of which have recently been established by Haynes Miller.

Adams's solution to the vector-field problem used K-theory, particularly the family of operations he in- vented for the purpose. Many other applications have since been found for these operations. The immersion problem for real projective spaces has attracted the at- tention of almost every homotopy theorist at one time or another. My own contribution was to prove a non- immers ion result for d imens ions one less than a power of two, which according to Gitler and Maho- wald is best possible. It is a good illustration of the power of K-theory.

Perhaps that is a good point at which to stop. Looking back, I have the impression that it was much easier to make an impact on homotopy theory up to 1970, say, than it is nowadays. However, the powerful methods now available enable results to be proved that seemed quite out of reach not so very long ago. The architecture of homotopy theory, at any rate, of stable homotopy theory, is beginning to become more apparent, and in this I am particularly proud that one of my most recent research students, Mike Hopkins, has played a leading role.

Mathematics Institute Oxford University Oxford, England OX1 3LB

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