# Peter John Hilton

### Quick Info

London, England

Binghampton, New York, USA

**Peter Hilton**was an English mathematician who made important contributions to homotopy theory. He is best known for his text-books and for his work as a code-breaker in World War II.

### Biography

**Peter Hilton**'s parents were Elizabeth Freedman and Mortimer Hilton. Peter was born and brought up in London, England. His interest in mathematics was reinforced in a strange way in 1933 [9]:-

At the age of 10, I was run over by a Rolls Royce, no less. It was an extraordinary incident. A boy in school snatched the cap off my head and ran across the road. I was angry and ran after him. I didn't notice the on-coming Rolls Royce. ... what happened was that I had a long period of recuperation, much of which was spent in a hospital bed with plaster of Paris on my left leg, all the way up, in fact, to my navel. So I had this sort of white board permanently available to me sitting on my stomach. It simply turned out that I spent a lot of my leisure time doing mathematical problems, writing them on the plaster of Paris, and erasing them each morning. It gave me the opportunity to realize that I had this intensive love of mathematics. I realized earlier that I had a certain proficiency, but I hadn't realized before that it was the sort of thing I would do when I had the choice of doing other things. So then it came to me that I really loved mathematics and thoroughly enjoyed doing it. I recall even having unkind thoughts about visitors who came to see if I was all right, as they would interrupt me when I was really enjoying what I was doing.He attended St Paul's School, Hammersmith, London, completing his studies in 1940. In his final year at the school he studied German on his own, a decision which soon had an unexpected significance. He won a scholarship to The Queen's College, Oxford, in his final year at school and matriculated in 1940. He was only just beginning his second year of study in November 1941 when an interviewing board arrived in Oxford looking to recruit "a mathematician with knowledge of modern European languages" [5]:-

My tutor recommended me to attend the interview although I was not a mathematician - merely an undergraduate of mathematics - and my knowledge of German was rudimentary, since I had merely been teaching myself for a year.Hilton was offered a position in the Foreign Office and told he had to start in January 1942. Since he was not due to be conscripted into the Royal Artillery for war service until August 1942, he was a little reluctant to give up his university studies earlier than necessary. However he had already undergone compulsory military training in the Royal Artillery which he had found extremely boring so it did not take him long to decide to start work at Bletchley Park on 12 January 1942. Although he had only completed four terms at Oxford, this entitled him to a wartime B.A. Only when he arrived at Bletchley Park did he learn the nature of the decoding work which he was to undertake. Hilton made a substantial contribution to the vitally important work which was undertaken at Bletchley Park, but the most significant aspect as far as his future career was concerned was the mathematicians he came to know through working there [9]:-

... two of my colleagues [at Bletchley Park] were Henry Whitehead and Max Newman. Due to the peculiar circumstances of the war, I was on an equal footing with them. In fact, I was simply a young man who had taken a wartime degree and they were eminent mathematicians. I became, in particular, very, very friendly with Henry Whitehead, on first-name, beer-drinking terms. After the war, Henry Whitehead invited me to come back to Oxford and be his first post-war research student. I said, "But I don't know anything about topology." And he said, "Oh, don't worry, Peter, you'll like it." So in fact, I didn't even know what the subject was. I went back to Oxford, and I studied topology. Whatever Henry Whitehead had specialized in, I would have studied. His personality was so attractive, that it was clear that it was going to be great fun to work with him. It turned out to be not only fun but extremely exacting and demanding - it was a marvellous experience. I really took up topology because it was Whitehead's field.Hilton was awarded an M.A. by Oxford in 1948 and, two years later, a doctorate for his thesis

*Calculation of the Homotopy Groups of*$A_{n}^{2}$

*-polyhedra*. He published the main results of his thesis in two papers in the

*Quarterly Journal of Mathematics*, the first appearing in 1950 and the second in the following year. We will not give the technical definition of an $A_{n}^{2}$-polyhedron, let us just say that it is a certain type of finite connected polytope. The main tool used in this work is a "suspension" theorem of Henry Whitehead which Hilton also generalised and applied in his major 31-page paper

*Suspension theorems and the generalized Hopf invariant*(1951). During this period Hilton was employed as a Lecturer at the University of Manchester (1948-1952). He wrote to Max Newman 28 March 1949:-

To supplement my income next year, in the hope, eventually, of being able to put down the deposit for a house, I have offered to give a course of extra-mural lectures on "The Development of Mathematics". Messrs Styler and Pedley seemed interested and asked me to give a syllabus, which I based on the course I gave in Oxford in 1947-48. They want me to give an extension course (once a week) starting in October, but I thought before agreeing I should make sure that you have no objections. I think I can promise that neither the preparation time nor the lecturing time would eat into my research time!The reason he was keen to buy a house was that he was soon to marry. He married Margaret Mostyn on 14 September 1949; they had two sons Nicholas and Timothy. Peter and Margaret (Meg) shared a love of the theatre and of acting; Peter enjoyed amateur acting, Meg was a professional. He spent the three years 1952-55 at Cambridge, receiving a second doctorate from Cambridge in 1952. Then he returned to Manchester where he was promoted to Senior Lecturer in 1956.

This part of Hilton's career was, like his doctoral studies, highly influenced by the friends he had made among the mathematicians at Bletchley. When Hilton first arrived at Bletchley Park he had worked in Hut 8 where Hugh Alexander and Shaun Wylie introduced him to the cryptanalysis methods used in decoding Naval Enigma messages. Later both Hilton and Wylie moved to the Newmanry run by Max Newman, where early computers (the Colossus machines) were used in decoding, and Hilton and Wylie remained close friends. When Hilton was appointed to the University of Manchester in 1948 it was to work in Max Newman's Department. When he went to Cambridge University in 1952 he became a colleague of his friend Shaun Wylie. Together they wrote the classic textbook

*Homology theory: An introduction to algebraic topology*published in 1960. A Heller writes in a review:-

This admirable book is designed to make the transition between the elementary textbook and the research-level treatise in algebraic topology. It begins with concrete geometrical fundamentals and manages nevertheless to introduce its reader to enough of the heavyweight machinery of modern topology so that he may hope to attack the contemporary literature of the field. The plan of the book is bipartite. The first portion is concerned principally with the homology theory of simplicial polyhedra, the second with singular homology. ... [An] account of the contents does not convey the richness of the discussion and the plenitude of examples which indeed are calculated not only to illuminate the systematic discussion but also to introduce the reader to the atmosphere of algebraic topology. This aim is even more evident in the very extensive collection of problems, which range from straightforward exercises to items out of the recent literature ... This is a badly needed book. It does an excellent job of carrying the serious beginning student of algebraic topology to a genuine acquaintance with the field, and seems to this reviewer likely to become a standard work in a domain where indeed it is essentially without a rival.However, this was not the first book that Hilton had published. He wrote

*An introduction to homotopy theory*which was published in 1953. He wrote in the Preface:-

At the moment, no textbook of homotopy theory exists at any level, with the result that the newcomer to this branch of mathematics is obliged to plunge straight into the study of original papers, often of very considerable complexity. This monograph is designed to fill the gap. It does not claim to be a comprehensive treatment of its subject (the recent work of the French school, for example, is not included, except for a brief introduction to it in the last section of Chapter V); but it is hoped that the reader familiar with Lefschetz's 'Introduction to Topology' will obtain an understanding of the fundamental ideas of homotopy theory from the first six chapters of this book.Shaun Wylie writes in a review [10]:-

This book occupies an important place in the literature of topology. ... It is a book for the serious student. For such a person, it is excellent, both in the clarity and discipline of the presentation and for the range of results covered.Hilton had also published

*Differential calculus*, a 56-page text in the

*Library of Mathematics*series. Hilton explain his aim in the Introduction:-

This book is intended to provide the university student in the physical sciences with information about the differential calculus which he is likely to need ... The emphasis is on detailed discussion of the ideas ... rather than on complete proofs.By the time that

*Homology theory: An introduction to algebraic topology*was published, Hilton was Mason Professor of Pure Mathematics at Birmingham University. He had been appointed to this Chair in 1958 and remained at Birmingham for four years. He spoke in [9] of his decision to move to the United States in 1962:-

I had been appointed head of department at Birmingham University in 1958, and it looked as if I would be there for the rest of my career. That is to say that I would have approximately another 35 years as head of department. ... I was doing a lot of committee work. I was very conscientious and I don't believe unusually proficient, and I realized that it was having a very deleterious effect on my research. I saw no escape, because I certainly wasn't a big enough person to downgrade myself again, say to the rank of senior lecturer, to halve my salary and lose my position. And yet on the other hand I could not see the possibility of my aspiring to one of the very few positions in the country, at that time, where you could have professorial status without departmental and university-wide responsibilities. ... I had visited Cornell and thoroughly enjoyed it. When an offer came from Cornell, my wife Meg and I talked it over seriously. Then we realized that this was really the unique way of escaping from the situation. So we decided to try it experimentally, and it worked very well.During the 1950s Hilton had been an invited speaker at the British Mathematical Colloquium on three occasions (he is the only person to have had three invitations in a 10 year period). He gave the lectures:

*The Hilbert problem for three dimensional groups*at Oxford in 1950,

*Fibre spaces*at Exeter in 1955, and

*Lusternic-Schnitelmann category in homotopy theory*at Cardiff in 1959.

Hilton was appointed Professor at Cornell University, Ithaca, New York, in 1962, remaining there for nine years. Then, in 1971 he moved to the University of Washington. After two years there, he was appointed Louis D Beaumont Professor at Case Western Reserve University in Cleveland, Ohio. In the interview [9], when asked if there is anything he wished he had done differently, Hilton replied:-

I would not have accepted the position at Case Western Reserve University.After holding this position for nine years, Hilton became Distinguished Professor at the State University of New York (SUNY), Binghamton. The title Distinguished Professor, granted only by SUNY trustees, is highest academic rank only conferred on those who have achieved national or international prominence. He remained at SUNY until he retired in 1993 when the university conferred on him the title Professor Emeritus. Also in 1993 he became Distinguished Professor at the University of Central Florida in Orlando.

Hilton has made a remarkable contribution both as a research mathematician and as an author of textbooks. In November 2009, MathSciNet records an incredible 338 publications under his name. Let us indicate his research contribution by quoting his own words [9]:-

I think I have made more of a contribution as an expositor in algebraic topology than as a researcher, in bringing ideas into good order so that they would be accessible to students of algebraic topology and homological algebra. In my own research I think the best paper I ever wrote is a paper I wrote under the influence of Jean-Pierre Serre on the homotopy groups of the union of spheres [On the homotopy groups of the union of spheres (1955)]. This was, I think, the first time that Lie algebras were used in homotopy theory in an effective way. It was basic and it is a paper very frequently cited. I think then I would have to jump and say that a series of papers I did with Joseph Roitberg and later also with Guido Mislin on questions relating to failures of cancellation in homotopy theory are good papers. There are two sorts of cancellation you can ask about in homotopy theory. You can take the union of two spaces with a single common point - it's like addition - so you're asking about failure of cancellation under addition. We were able to show systematically how to construct examples where one could take two different spaces, and add on the same space to each, so that the two unions had the same homotopy type; and then, what turned out to be a more difficult problem, we showed how to construct examples with the topological product replacing the union. In the course of that work Joe Roitberg and I were able to construct the first new example of a Hopf manifold. And that, I think, began a whole new industry in mathematics. So I was very happy about that paper. Then I would say the work that the three of us did on localization theory - the new results and the systematization of known results - was a very significant contribution to the whole structure of the subject. I think also that Eckmann and I did significant work in applying categorical notions. Both of us felt these notions were appropriate for looking at concepts and problems in algebraic topology and homological algebra. We were neither of us pure category- theorists and I think that these series of papers we wrote on group-like structures in general categories, and on general homotopy theory and duality, were two very significant contributions to the applications of these categorical notions that suggested ideas and problems. I always feel enormously grateful to Norman Steenrod, that he not only encouraged us, but also when he compiled his list of papers in algebraic topology, he gave Eckmann-Hilton duality a special heading for the papers that had been written in that area. I do think that Eckmann and I did in that way systematize some ideas and we showed how certain ideas are naturally related. Some of these things were intuitively clear to certain people but had not been systematized. By systematizing them, we made them more readily accessible to students, but also we broadened and extended them substantially.Hilton has written many outstanding books - from first year undergraduate level, to graduate text level, to research monograph. We cannot do justice to them in this biography but let us at least mention most of them. Following on from those already mentioned,

*Partial derivatives*(1960) "is intended primarily to serve the needs of the university student in the physical sciences". Then there are three texts

*Homotopy theory and duality*(1965);

*Suites spectrales et théories de cohomologie générales*(1970); and

*Tópicos de álgebra homológica*(1970) each based on courses given by Hilton. Also in 1970, in collaboration with Brian Griffiths, he published

*A comprehensive textbook of classical mathematics*. This book, reprinted in 1978, is a classic. Keith Hirst, reviewing the book, writes:-

This book is written by two men who have each contributed much to the development of mathematical education in universities, as well as to mathematics itself. Both of these activities have aided in the writing of the book, whose starting point was an in-service course of mathematics for teachers at Birmingham University in 1961-62. There is a long introduction, which should be read with care, for it reveals as few other textbooks the existence of opinions about the presentation of mathematical ideas, their communication, and their importance. This leads to a personal commitment to sharing mathematics which pervades the whole body of the book.For further details of this book see the biography of Brian Griffiths.

Continuing in our run through of some of Hilton's books, there is:

*General cohomology theory and K-theory*(1971);

*Lectures in homological algebra*(1971); (with Urs Stammbach)

*A course in homological algebra*(1971); (with Yel Chiang Wu)

*A course in modern algebra*(1974); (with Joe Roitberg and Guido Mislin)

*Localization of nilpotent groups and spaces*(1975);

*Nilpotente Gruppen und nilpotente Räum*(1984); (with Dereck Holton and Jean Pedersen)

*Mathematical reflections : In a room with many mirrors*(1997); and (with Dereck Holton and Jean Pedersen)

*Mathematical vistas : In a room with many mirrors*(2002). Of these last two books, the first "is suitable for keen high school and college students and would serve as an admirable text for teacher-training." The second of the two is a companion book in which the authors write:-

The two books are dedicated to the same principal purpose - to stimulate the interest of bright people in mathematics.Finally let us give the following quote from a review that Hilton wrote in 1998 [4]:-

Just as any sensitive human being can be brought to appreciate beauty in art, music or literature, so that person can be educated to recognize the beauty in a piece of mathematics. The rarity of that recognition is not due to the "fact" that most people are not mathematically gifted but to the crassly utilitarian manner of teaching mathematics and of deciding syllabi and curricula, in which tedious, routine calculations, learned as a skill, are emphasized at the expense of genuinely mathematical ideas, and in which students spend almost all their time answering someone else's questions rather than asking their own.

### References (show)

- M F Atiyah, Review: Homology Theory by P J Hilton and S Wylie,
*The Mathematical Gazette***46**(357) (1962), 257-258. - L K Barrett, Review: Homology Theory, An Introduction to Algebraic Topology by P J Hilton and S Wylie,
*Amer. Math. Monthly***69**(5) (1962), 448-449. - R L Goodstein, Review: Classical Mathematics by H B Griffiths and P J Hilton,
*The Mathematical Gazette***55**(393) (1971), 341. - P Hilton, Review: The Pleasures of Counting by T W Körner,
*Amer. Math. Monthly***105**(5) (1998), 481-485. - P Hilton, Reminiscences and Reflections of a Codebreaker, in
*W D Joyner (ed.), Coding Theory and Cryptography: From Enigma and Geheimschreiber to Quantum Theory*(Springer-Verlag, 2000), 1-8. - P Hilton, Reminiscences of Bletchley Park, 1942-45, in
*P Duren (ed.), A century of mathematics in America*(American Mathematical Society, Providence, Rhode Island, 1988), 291-302. - T W Körner, 'Mathematical Reflections' and 'Mathematical Vistas',
*Notices Amer. Math. Soc.***50**(2) (2003), 224-227. - C Schochet, Review: A Course in Homological Algebra by P J Hilton and U Stammbach,
*Amer. Math. Monthly***80**(10) (1973), 1153-1154. - L A Steen and G L Alexanderson, Peter J Hilton, in D J Albers, G L Alexanderson (eds.),
*Mathematical People*(Birkhäuser, Boston-Basel-Stuttgart, 1985), 133-150. - S Wylie, Review: An Introduction to Homotopy Theory by P J Hilton,
*The Mathematical Gazette***39**(327) (1955), 82-83.

### Additional Resources (show)

Other pages about Peter Hilton:

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Honours awarded to Peter Hilton

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update February 2010

Last Update February 2010