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Frank Adams J Reminiscences on the Life and Mathematics of J. Frank Adams J. Peter May Frank Adams was both my closest personal friend and Hopf-invariant-one maps for both n and 2n. However, my closest mathematical friend. I will say a little about since the paper introduced what is now called the his mathematical work here, but a fuller appreciation Adams spectral sequence, it can't be written off as a is being prepared for publication elsewhere. I will try total loss. In fact, the Adams spectral sequence is the to convey something of his style and of his feelings most important theoretical tool in stable homotopy about mathematics, again letting him speak in his own theory, and its introduction marked the real starting words. point of this fundamental branch of algebraic to- Adams was knowledgeable about many other fields, pology. but topology was his love. While all of his work was at The Adams spectral sequence converges from a very high level, two groups of early papers stand out E2 = ExtA(H*(X; Zp), Zp) particularly: to the p-primary component of the stable homotopy On the structure and applications of the Steenrod groups of the space X, where A denotes the Steenrod algebra (June 1957) algebra of stable operations in mod p cohomology. The On the non-existence of elements of Hopf invariant connection with the Hopf-invariant-one problem is one (April 1958) that the mod 2 cup square is a Steenrod operation, and Vector fields on spheres (October 1961) this allows a translation of the problem into a stable On the groups J(X)--I (May, 1963) one. Certain differentials in the Adams spectral se- II (September 1963) quence give decompositions of Steenrod operations III (November 1963) into composites of secondary operations. In a two-cell IV (July 1965) complex X, there are no intermediate dimensions in the rood 2 cohomology, hence such a decomposition The dates given are the dates of submission; actu- of the relevant Steenrod operation implies that the cup ally, according to J(X)--IV, much of the material square of the integral generator in dimension n is zero in the J(X) papers dates from the years 1960-1961. mod 2. The first two papers above were concerned with the In the second paper above, the Hopf-invariant-one Hopf-invariant-one problem. One way of motivating problem was solved by means of an explicit decompo- the problem is to ask the possible dimensions n of a sition of all of the relevant mod 2 Steenrod operations real division algebra D. Given D, we obtain a map f in terms of secondary operations. from the unit spher e S 2n- 1 C D x D to the one-point My own 1964 doctoral thesis was motivated by the compactification S n of D by sending (x,y) to x- ly if x Adams spectral sequence, specifically by the following 0 and to the point at infinity if x = 0. If we form the passage from Adams's 1960 Berkeley lecture notes: two-cell complex X = S n Ufe 2", we find that its coho- The groups E2 are recursively computable up to any given mology is Z in dimensions n and 2n and that the cup dimension; what is left to one's intelligence is finding the square of the generator in dimension n is a generator differentials in the spectral sequence, and the group ex- tensions at the end of it. in dimension 2n. We say that f has Hopf invariant one. This account would be perfectly satisfying to a mathe- The homotopical problem asks what dimensions n matical logician: an algorithm is given for computing E2; support a map of spheres f: S 2n-1 ~ S n of Hopf in- none is given for computing dr. The practical mathemati- variant one. In view of the real, complex, quaternion, cian, however, is forced to admit that the intelligence of and Cayley numbers, n = 1, 2, 4, and 8 are possible. mathematicians is an asset at least as reliable as their will- ingness to do large amounts of tedious mechanical work. Adams proved that these are the only possibilities. The history of the subject shows, in fact, that whenever a The first paper I mentioned can be viewed as a chance has arisen to show that a differential dr is non-zero, failed attempt to prove this result. All it obtained on the experts have fallen on it with shouts of joy--"Here is the problem was that, if n > 4, one couldn't have an interesting phenomenon! Here is a chance to do some THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1 91990 Springer-Verlag New York 45 nice, clean research!"--and they have solved the problem Frank was the most competitive man I have ever in short order. met. Let me give one example. In the spring of 1971 On the other hand, the calculation of Ext groups is nec- essary not only for this spectral sequence, but also for the my younger son was 21/2 years old, the age of language study of cohomology operations of the nth kind: each such acquisition and thus of most accurate memory. One group can be calculated by a large amount of tedious me- day Frank and he were playing the card game Con- chanical work: but the process finds few people willing to centration on our living room floor. My wife said take it on. something to Frank, and he snapped back "Be quiet, That was what I took on in my thesis. But my calcu- I'm concentrating!" lations in fact forced some calculations of differentials, In fact, by then he had mellowed. He was far more and those calculations did not all agree with the ones intense in earlier years. In his Spring 1960 Berkeley tabulated by Adams in his cited lecture notes. I wrote notes, he described some work in progress on the him on February 23, 1964, pointing out his mistakes. I vector-fields-on-spheres problem, which asks for the hasten to add that mistakes of any sort were most un- maximum number of linearly independent vector usual in Frank's work. That marked the beginning of fields on S" for each n. Hirosi Toda, in Japan, was also our friendship and the start of a correspondence working on the problem and had some partial results. which averaged one or two letters a month in each di- With this spur, Adams had polished off the problem rection over the last twenty-five years, interrupted completely by October 1960. Moreover, his methods only by his frequent visits to Chicago and my visits to were totally different from those he had been working Cambridge. on in the spring. Then, he was thinking in terms of 46 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 ordinary cohomology, higher order cohomology oper- K(X) specified in terms of Adams operations were ations, and differentials in the Adams spectral se- always in the kernel of the natural homomorphism quence. As he wrote in the published account, "The K(X) ~ J(X). The remaining J(X) papers made clear author's work on this topic may be left in decent ob- that the Adams conjecture was of fundamental impor- scurity, like the bottom nine-tenths of an iceberg." In tance in algebraic topology. fact, his solution of the problem was obtained by the The Adams conjecture was later proven by Sullivan introduction and exploitation of what are now called and Quillen, and their proofs led to a cornucopia of the Adams operations in topological K-theory K(X). new mathematics. Sullivan's proof led him to the now Recall that K(X) is the Grothendieck ring determined ubiquitously used theory of localization and comple- by the semi-ring of isomorphism classes of vector tion of topological spaces. QuiUen's proof led him in- bundles over X. The vector-fields problem is closely exorably to the now standard definition of the higher related to the study of the groups J(X). These are quo- algebraic K-groups of rings. tients of the groups K(X) obtained by classifying vector Rather than say more about Adams's mathematics, I bundles in terms of fiber homotopy equivalence rather will let him give an example of his style of exposition. than bundle equivalence. The first of the J(X) papers In going over his papers in England, I found his lec- contained a remarkable conjecture--now called the ture notes on the definitive proof, using the Adams Adams conjecture--and proved it in special cases. It operations in K-theory, of the non-existence of ele- gave an upper bound for J(X) in terms of the Adams ments of Hopf invariant one. This proof is due to operations. That is, it asserted that certain elements of Adams and Atiyah. The lecture notes assume a little THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 47 knowledge of the relationship between ordinary coho- ceding Memorial Address. His letters were always a mology and K-theory, as given by the Atiyah-Hirze- delight, although his handwriting required careful de- bruch spectral sequence, but the lecture was clearly ciphering. Imagine the pleasure of receiving the fol- intended to be accessible to graduate students. lowing piece of doggerel in the mail. It concerns an- It may be objected that the algebra at the end of the other aspect of Frank's role in policing the topological proof was left to the reader 9That reflects Adams's con- literature. sidered position on the relation between topology and algebra in his work.
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