ADAMS, JOHN FRANK

(b. Woolwich, United Kingdom, 5 November 1930; d. near Brampton, United Kingdom, 7 January 1989)

mathematics, algebraic topology, stable homotopy theory.

Adams was a leading figure in algebraic topology. His work was the first to systematize the branch of algebraic topology that became known as stable homotopy theory. He solved several of the outstanding problems in algebraic topology by stable methods, which he pioneered. His name is attached to the Adams spectral sequence, the Adams operations, and the Adams conjecture. Adams was a problem solver, and each of these named contributions came from Adams’s work on a particularly important specific problem, namely the Hopf invariant one problem, the vector fields on spheres problem, and the problem of computing certain groups J (X). His work on each of these problems led to substantial advances in mathematics that reached far beyond the specific problem at hand. The development of stable homotopy theory as a subject, the development of topological K theory as an effective calculational tool, and the development of guiding principles and goals in the study of stable homotopy theory derive from his work on these problems.

Life and Career. Adams’s mother was Jean Mary Baines, a biologist; his father, William Frank Adams, was a civil engineer. His early education was somewhat disrupted by the events of World War II, but occurred mainly at Bedford School. Adams served in the Royal Engineers during 1948 and 1949 before beginning his university education. He entered Trinity College, Cambridge, in 1949, taking Part II of the Mathematical Tripos in 1951 and Part III in 1952. He married Grace Rhoda Cathy in 1953; their family eventually included one son and three daughters (one adopted).

Adams obtained his PhD at Cambridge in 1955. His thesis advisor was Shaun Wylie and his thesis examiners were Peter Hilton and Henry (J. H. C.) Whitehead, who was at the time the dominant figure in algebraic topology in England. It is surely Whitehead’s influence that led Adams to this subject. Roughly speaking, algebraic topology assigns discrete algebraic invariants to continuous phenomena, giving an algebraic picture of topological shapes. At the time, the subject was in its infancy, and Adams was to become one of its leading pioneers.

From 1955 to 1958, Adams was a Research Fellow at Trinity College, and he succeeded Wylie as a Fellow and College Lecturer at Trinity Hall, Cambridge, in 1958. He moved to Manchester University in 1962 and returned to Cambridge as a Fellow of Trinity College and Lowndean Professor of Astronomy and Geometry in 1970. He spent the rest of his career there. He made many trips to the United States, visiting Princeton in 1957–1958 and 1961 and making frequent visits to the University of Chicago from 1957 to 1985. He was killed in a car crash on 7 January 1989 while driving from London to his home in Hemingford Gray.

Stable Homotopy Theory In algebraic topology, one obtains conclusions about spaces by studying algebraic invariants. The simplest such invariants are the homotopy groups π_n(X) of a space with base point. These are obtained by dividing the base-point preserving continuous maps Sⁿ→X, where Sⁿ is the n-sphere, into equivalence classes, where two maps are equivalent if one can be continuously deformed into the other. There is a way of suspending a space, creating a new space X. For example, Sⁿ is Sⁿ+1. There is a homomorphism π_n (X)→π_n+1(X), and Hans Freudenthal proved in 1937 that if n is large relative to the dimension of X, then this homomorphism is an isomorphism. For example π_n+q+1 ((Sⁿ⁺¹)) if q< n-1. The common value is called the q th stable homotopy group of spheres. Algebraic invariants of spaces that are independent of dimension, in the sense that this illustrates, are said to be stable. Adams was among the first to articulate the notion of stable phenomena and to articulate the principle that translation of a seemingly unstable problem into a stable one can be the first step towards its solution.

For example, one can ask for which n the sphere Sⁿ admits a multiplication with a two-sided identity element. This is an unstable problem, called the Hopf invariant one problem. Adams solved it, showing that only n =0, 1, 3, and 7 are possible, by first translating it into a stable problem. In the course of his work on this problem, he introduced an algebraic tool, later called the Adams spectral sequence, for the calculation of stable homotopy groups. This tool gave a powerful method for tackling stable problems calculationally. His work led to a changed point of view on algebraic topology as a whole, and it is now understood that stable homotopy theory, although typically dealing with infinite dimensional objects, is by far the most calculationally accessible part of the subject as a whole. Adams is rightly viewed as the creator of the subfield of stable homotopy theory.

K Theory. Stable homotopy theory is the natural home of homology and cohomology theories, which are pairs of sequences k_q and k^q of abelian groups satisfying certain axioms, among them the stability axiom that k^q(X) is isomorphic to k^q+1(X). The earliest homology theories to be developed were ordinary ones, which are characterized by the dimension axiom k^q S⁰ = 0 if q 0, where S⁰ is the zero sphere. The first “extraordinary theory” to be introduced was K theory, which was defined around 1960 by Michael Atiyah and Friedrich Hirzebruch. A theorem of Raoul Bott states that K theory is periodic, so that K^q(X) is isomorphic to K^q+2(X) for all X. This theory starts from a group K⁰(X) defined in terms of complex vector bundles over the space X. There is a variant, KO*(X), defined in terms of real vector bundles over X, that is periodic of period eight rather than two and carries more information. A second main stream of Adams’s work was the use of K theory to solve problems in algebraic topology

For example, one can ask how many linearly independent vector fields there are on an n-sphere. This is again an unstable problem that can be translated into a stable problem. However, while Adams solved the Hopf invariant one problem by use of secondary cohomology operations in ordinary cohomology, he solved the vector fields on spheres problem by use of “primary cohomology operations” in real K theory. Adams introduced the appropriate operations, and they are now called the Adams operations. Adams and Atiyah later showed that these primary operations in complex K theory can be used to obtain a simplified solution of the Hopf invariant one problem.

The J-homomorphism. There is a connection between K theory and stable homotopy theory. It is given by the J-homomorphism, which is a general construction that starts from the groups KO_*(X) and constructs quotient groups J_*(X). When X is a point, the J-homomorphism connects the K theory of a point to the stable homotopy groups of spheres. Adams studied the groups J_*(X) in a fundamentally important series of papers. He arrived at a conjectural relationship between the Adams operations and the question of when vector bundles become equivalent under a weaker equivalence relation than that used to defined K theory, namely fiber homotopy equivalence, and he showed how to compute J(X) assuming the truth of this conjecture, which was later called the Adams conjecture. Although he himself did not prove the conjecture, he proved special cases to which the general case was reduced by later work of Daniel G. Quillen and, independently, Dennis Sullivan. Their solutions opened up vast new areas of algebraic topology, leading Quillen to introduce algebraic K theory and Sullivan to introduce localizations and completions of topological spaces.

The original importance of the J-homomorphism was its role in differential topology. One can ask how many different differential structures there are on an n- sphere, for example. The answer, in dimensions above four, is directly related to the J-homomorphism, as John Milnor and Michel A. Kervaire showed. However, Adams’s work led to a quite different perspective. The J-homomorphism gives a way of seeing a small part of stable homotopy theory, which is very far from being periodic, in terms of periodic K theory. Adams was convinced that this was only a starting point, and that there should be higher periodicities that reveal more of the structure of stable homotopy theory. That point of view has guided the development of stable homotopy theory ever since his original J(X) papers.

Later Work These contributions all date from the first decade of Adams’s mathematical work. His later work, although less spectacular, was also of considerable influence. His early work included a substantial amount of homological algebra related to the understanding and calculation of the Adams spectral sequence, and later homo-logical work played a key role in the proof of the Segal conjecture, a problem that greatly interested Adams in later years. A series of papers on maps between classifying spaces turned out to be prescient precursors of substantial later work in that direction. His many other contributions pale only in comparison with the extraordinary level of his early work.

Adams was the leading figure in the algebraic topology community for close to thirty years. He had voluminous correspondence, took the refereeing of papers very seriously, and wrote a great many reviews of papers for publication in Mathematical Reviews. His vibrant personality and great erudition, together with his intolerance of slovenly work and his competitiveness, made him a somewhat feared but wholly revered person. He was awarded the Berwick Prize of the London Mathematical Society in 1963 and the Sylvester Medal of the Royal Society of London in 1982. He was elected a Fellow of the Royal Society in 1964.

BIBLIOGRAPHY

A complete bibliography of Adams’s work may be found in N. Ray and G. Walker, eds. Adams Memorial Symposium on Algebraic Topology, London Mathematical Society lecture note series 175–176, 2 vols., Cambridge, U.K., and New York: Cambridge University Press, 1992.

WORKS BY ADAMS

“On the Structure and Applications of the Steenrod Algebra.” Commentarii Mathematici Helvetici 32 (1958): 180–214.

“On the Non-Existence of Elements of Hopf Invariant One.” Annals of Mathematics Second Series 72, no. 1 (1960): 20–104.

“Vector Fields on Spheres.” Annals of Mathematics Second Series 75, no. 3 (1962): 603–632.

“On the Groups J(X).” I–IV. Topology 2 (1963): 181–195; 3 (1965): 137–171; 3 (1965): 193–222; 5 (1966): 21–71.

The Selected Works of J. Frank Adams. Edited by J. P. May and Charles B. Thomas. 2 vols. Cambridge, U.K.; New York: Cambridge University Press, 1992. Most of Adams’s published work has been collected here.

OTHER SOURCES

James, I. M. “John Frank Adams.” Biographical Memoirs of Fellows of the Royal Society of London 36 (1990), 1–16.

May, J. P. “Memorial Address for J. Frank Adams.” Mathematical Intelligencer 12, no. 1 (1990): 40–44.

——. “Reminiscences on the Life and Mathematics oxsf J. Frank Adams.” Mathematical Intelligencer 12, no. 1 (1990): 45–48.

Ray, Nigel, and Grant Walker, eds. Adams Memorial Symposium on Algebraic Topology. London Mathematical Society lecture note series 175–176. 2 vols. Cambridge, U.K.; New York: Cambridge University Press, 1992. Includes a paper describing Adams’s work.

J. P. May