# John Frank Adams

### Quick Info

Woolwich, London, England

Near Brampton, Huntingdonshire, England

**Frank Adams**was an English algebraic topologist who pioneered methods for calculating the homotopy of spheres

### Biography

**Frank Adams**'s mother was Jean Mary Baines, a biologist, and his father was William Frank Adams, a civil engineer. He was the eldest of his parents two children, having one younger brother. The family was evacuated from London during World War II which somewhat disrupted his early education. As a consequence he attended school in a number of places but his education was mainly at Bedford School. By the time his schooling was completed World War II had ended but Britain still had national service and all young men were required to serve for two years. Adams served in the Royal Engineers during 1948 and 1949 before beginning his university education.

Adams entered Trinity College, Cambridge, in 1949 to study mathematics. He took Part II of the Mathematical Tripos in 1951, and Part III in the following year. After taking his first degree he started graduate work at Cambridge with Besicovitch on geometric measure theory. He married Grace Rhoda Carty in 1953. James writes in [1]:-

Soon after their marriage she became a minister in the Congregational church. They had a son and three daughters (one adopted). Family life was extremely important to Adams, though he preferred to keep it separate from his professional life. The family used to do many things together, especially fell-walking in the Lake District.He changed supervisors and began working on algebraic topology with Shaun Wylie. However, he was most strongly influenced by Henry Whitehead, who led the foremost British school of algebraic topology. This happened during the year 1954 which Adams spent as a junior lecturer at the University of Oxford. He won a Fellowship at Trinity College, Cambridge, with his doctoral thesis on spectral sequences

*On Special Sequences of Self-Obstruction Invariants*which he submitted in 1955. He returned to Cambridge in 1956 to take up the Fellowship and during this period he developed the spectral sequence which today is called the "Adams' spectral sequence".

Adams won a Commonwealth Scholarship which enabled him to visit Chicago as a research associate in 1957-58. While in the United States he also visited Princeton. Adams said:-

... I regard the progress of my researches in America as most successful. ... By good luck, moreover, my new methods were sufficiently powerful to answer one of the classical problems of my subject, that proposed by H Hopf in 1935.The conjecture that Adams solved was the famous conjecture about the existence of $H$-structures on spheres.

On his return from the United States he became a College Lecturer at Trinity Hall Cambridge. His work turned towards $K$-theory, the generalised cohomology theory on vector bundles. Using this theory he solved another important conjecture, this one being about vector fields on spheres.

After spending further time in Princeton, Adams took up a post at Manchester as a Reader in 1962, being appointed to Newman's chair when he retired in 1964. At this time he became Fielden professor. He continued to produce work of outstanding depth and originality, and during his first few years at Manchester he wrote a series of papers

*On the groups J(X*) which were highly influential in homotopy theory.

In 1964 Adams was elected a Fellow of the Royal Society. In [3] James says:-

It was in 1965, however, that he suffered the first attack of a psychiatric illness, as a result of which he was on sick leave for some months. It was apparently brought on by the worry caused by his responsibilities as head of department ...In 1970 Adams succeeded Hodge as Lowndean Professor of Astronomy and Geometry at Cambridge, and at this time he returned to Trinity College. His research continued to be of fundamental importance in the homotopy theory of classifying spaces of topological groups, finite $H$-spaces and equivariant homotopy theory. Around this time in addition to his research papers, he began to publish expository work, some resulting from lecture courses. These books are of major importance, and include

*Stable homotopy theory*(1964),

*Lectures on Lie groups*(1969),

*Algebraic topology: a student's guide*(1972),

*Stable homotopy theory and generalized homology*(1974),

*Localisation and completion*(1975), and

*Infinite loop spaces*(1978). Let us say a little about these works.

*Stable homotopy theory*(1964) is a short 74 page book which is based on six lectures Adams gave at the University of California at Berkeley in 1961.

*Lectures on Lie groups*(1969) is described by N R Wallach as follows:-

This book covers in a concise manner the fine structure and representation theory of compact Lie groups, with emphasis on the classical groups. The exposition of the book is aimed at the reader who has some understanding of algebraic topology and would like to understand the aspects of the theory of compact Lie groups that are relevant to algebraic topology. The book fulfils its aims admirably and should be a useful reference for any mathematician who would like to learn the basic results on compact Lie groups.

*Algebraic topology: a student's guide*(1972) is rather unusual. It is in two parts, the first contains a description of the topics that Adams thought essential for any young mathematician interested in algebraic topology. It links to a wide variety of textbooks with Adams indicating the one which treats the topic in the way he considers best. The second part contains excerpts from some famous papers on algebraic topology together with surveys of generalized cohomology theories and complex cobordism written by Adams.

*Stable homotopy theory and generalized homology*(1974) comprises of three lecture courses, one on the algebra of stable operations in complex cobordism delivered in 1967, the second on complex cobordism theory delivered in 1970, and the third on stable homotopy and generalized homology theories delivered in 1971.

Stewart B Priddy, reviewing

*Infinite loop spaces*(1978), writes:-

Over the past few years, various topologists have been heard to complain about the lengthy and technical nature of infinite loop space theory. Even if one suspected that some of these detractors had not come to grips with the problems involved, there was still an undeniable need for a compact and moderately elementary introduction to the subject and to the current literature. Adams' book fills this need nicely and it can be recommended to anyone seeking a substantial overview of the main topics.As is evident from the lecture courses which Adams published, his lectures were well prepared but usually hard. He once received a letter from a second year undergraduate class saying:-

The class wishes to inform Professor Adams that it has been left behind.He replied:-

At any rate I have done exterior algebra, even if the second year haven't.In [1] James describes his attitude to research students and to research:-

Adams was an awe-inspiring teacher who expected a great deal of his research students and whose criticism of work which did not impress him could be withering. For those who were stimulated rather than intimidated by this treatment, he was generous with his help. The competitive instinct in Adams was highly developed, for example in his attitude to research. Priority of discovery mattered a great deal to him and he was known to argue such questions not just as to the day but as to the time of day. In a subject where 'show and tell' is customary he was extraordinarily secretive about research in progress.Adams received many awards for his work. Among these was the Sylvester Medal of the Royal Society of London which was awarded to him in 1982:-

... in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of that subject.The London Mathematical Society awarded him their junior Berwick Prize in 1963, and their senior Whitehead Prize in 1974. He was elected to the National Academy of Sciences (United States) in 1985 and the Royal Danish Academy of Sciences in 1988.

His health continued to cause him problems with another psychiatric illness in 1986. Perhaps his health contributed to his death since he decided to go to London, despite feeling unwell, to a celebration for the retirement of a friend. He was killed in a car crash only a few miles from his home on the return journey. He had apparently always had a reputation as a car driver. According to [3]:-

He drove cars with remarkable skill but in a style that left a lasting impression on his passengers.Finally we note that seven years after Adams died another book was published based on his lecture courses. This is

*Lectures on exceptional Lie groups*published in 1996. The book is based on lectures which Adams gave at Cambridge which he considered to be sequel to his book

*Lectures on Lie groups*(1969).

### References (show)

- Biography by I M James, in
*Dictionary of National Biography*(Oxford, 2004). See THIS LINK. - J P May and C B Thomas (eds),
*The selected works of J Frank Adams*(2 Vols) (Cambridge, 1992). - I M James, John Frank Adams,
*Biographical Memoirs of Fellows of the Royal Society of London***36**(1990), 3-16. - I M James and C T C Wall, Obituary : John Frank Adams,
*Bull. London Math. Soc.***29**(4) (1997), 489-501. - J P May, Memorial address for J Frank Adams,
*The Mathematical Intelligencer***12**(1) (1990), 40-44. - J P May, The work of J F Adams, in
*Adams Memorial Symposium on Algebraic Topology***1,**Manchester 1990 (Cambridge, 1992), 1-27. - J P May, Reminiscences on the life and mathematics of J Frank Adams,
*The Mathematical Intelligencer***12**(1) (1990), 45-48.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update February 2005

Last Update February 2005