Commentary by M. Atiyah
Early papers on K-theory (1959–1962)
These papers, covering the years 1959–62, consist mainly of my joint papers with , my joint paper with , is in a sense the start of this development. Having been exposed, at the Arbeitstagung, to many hours of expounding his generalization of the Hirzebruch–Riemann–Roch theorem, I was playing about with his formulae for complex projective space. From (at that time a colleague in Cambridge) I had heard about the periodicity theorems and also about stunted projective spaces. I soon realized that Grothendieck’s formulae led to rather strong results for the James problems. Moreover the Bott periodicity theorems fitted in with the Grothendieck formalism, so that one could draw genuine topological conclusions. It was this which convinced me that a topological version of Grothendieck’s \( K \)-theory, based on the Bott periodicity theorem, would be a powerful tool in algebraic topology. In fact the purely algebraic problem which emerged from applying these ideas to the James problem on stunted projective spaces proved to be intriguingly hard. It involved determining how large \( n \) has to be in order for the power series \( \log ((1+t)/t)^n \) to have its first \( r \) coefficients integral. I showed this problem to several colleagues and in due course Todd came up with the complete solution, leading to the joint paper . Many years later Adams and Grant–Walker gratifyingly proved our results were best possible.on \( K \)-theory. However
Motivated by this problem I was in the process of developing my ideas on \( K \)-theory more systematically. Since most of the potential applications involved cohomological calculations with characteristic classes and homogenous spaces, in order to get integrability theorems of the type pioneered by Hirzebruch, it was natural that I should get Hirzebruch’s assistance. In this way, our extensive collaboration began and we soon formalized our ideas on \( K \)-theory as a generalized cohomology theory.  gave the first general applications of the theory.
I remember that introducing the odd-dimensional \( K \)-groups seemed at the time a daring generalization, following the vogue set by Grothendieck. Nevertheless all our early papers on \( K \)-theory were aimed at concrete applications. Extracting the optimal results from these new methods frequently involved some sophisticated algebra, as for example in . Originally I had noted that the differentiable Riemann–Roch theorem of  had an “unstable” version which could be used to prove strong non-embedding theorems for manifolds. Hirzebruch, using the elegant apparatus of “Hilbert polynomials” developed in  vastly extended my initial results. This paper also introduced (through their characters) the \( \Psi^k \)-operations which Adams was subsequently to exploit in brilliant fashion.
Hirzebruch and I spent the Fall of 1959 at the Institute for Advanced Study developing \( K \)-theory and giving various applications. The report of the Tucson conference , based on our work at the time, was the first major exposition of \( K \)-theory. It established the relation between the representation ring \( R(G) \) of a compact connected Lie group and the \( K \)-theory of the classifying space. The crucial fact that \( K(B_G)\to K(B_T) \) (where \( T\subset G \) is the maximal torus) is injective (unlike cohomology) depends on the existence of the direct image map \( K(B_T)\to K(B_G) \). I clearly remember how, in a brief walk round the Institute housing project, this fact suddenly dawned on me. I had been having lengthy discussions with Borel and Hirzebruch on the topic and on returning from the walk we were able to clinch the matter.
Paper  extended the results about \( R(G) \) to finite groups \( G \), and it depended heavily on help from the strong team that year of Serre, Tate, and Borel who were then introducing us into the mystery of Grothendieck’s schemes.
In the early fifties Hirzebruch had discovered some intriguing relations between Steenrod operations and the Todd polynomials. With the advent of \( K \)-theory it was possible to shed new light on these questions and this led to . The fact that it is in German clearly points to its authorship and reflects the fact that the ideas originated from Hirzebruch’s early work.
work on the Steenrod algebra was an important ingredient in our paper. This in turn is related to Quillen’s work on formal groups which led to an elegant new proof of Milnor’s theorem on the structure of the unitary cobordism ring.
My work with Todd on complex Stiefel manifolds had of course a more famous real counterpart, essentially the vector field problem on spheres. In retrospect it is perhaps no surprise that, using real \( K \)-theory,  how to apply similar ideas to the problem of embedding real projective spaces \( P_n(R) \) in \( R^N \), i.e., getting lower bounds for \( N \) in terns of \( n \). The analogous complex problem was one of the early applications Hirzebruch and I had made in .was eventually able to solve the vector-field problem. Of course the real case is technically more subtle because one cannot use real cohomology and Adams had to use \( K \)-theory operations. Stimulated by Adams’ work I showed in
Papers  and  are not directly concerned with \( K \)-theory but involve related ideas. I was struck by the results in Wall’s thesis on the oriented cobordism ring and I saw what they meant in terms of the appropriate generalized cohomology theory. I should say that general ideas about cohomology and homotopy at this time were very much in the air and I was certainly influenced by the expositions of and Eckman, which were elegant and clear. I also learnt much from in Bonn about cobordism theory and this is reflected in , which examines more systematically the James problem of stunted projective spaces, the original stimulus for \( K \)-theory.
Throughout this period I also had extensive correspondence with. I needed his help, in the first instance, to clarify certain aspects of his periodicity theorems. Once the formalism of \( K \)-theory had been developed it seemed clear that the Bott periodicity maps should be induced by tensor products. For the complex case this could be deduced by cohomological arguments, but for the real case a more direct verification was necessary, and we appealed to both to help out. Bott’s greater experience with Lie groups proved very helpful in a number of directions.
Although \( K \)-theory was proving to be a powerful new tool in algebraic topology, Hirzebruch and I were still interested in its algebro-geometric origin in the Grothendieck–Riemann–Roch theorem. In  we extended Grothendieck’s theorem to the case of complex analytic embeddings, showing that one could define compatible direct image maps in \( K \)-theory both for analytic and topological bundles. The same ideas were involved in , although the emphasis there was on singular varieties. We had noticed that the cohomology class of an algebraic subvariety was annihilated by all differentials in the spectral sequence relating cohomology to \( K \)-theory. Using a construction of Serre, producing algebraic varieties from finite groups, it followed that not all the torsion in the (even-dimensional) cohomology could be algebraic, disposing of the integral Hodge conjectures.
My talk  at the Stockholm Congress summarized the applications of \( K \)-theory and concluded with the \( K \)-theory interpretation of the symbol of an elliptic operator. From this paper onwards the index of elliptic operators becomes a dominating theme and the interaction with \( K \)-theory is a two-way process. The later papers on \( K \)-theory therefore appeared interspersed chronologically with papers on index theory. In some cases a paper is simultaneously devoted to both topics, and I have categorized it according to its main component, and although this is sometimes rather arbitrary.
Later papers on K-theory (1964–1970)
Paper  dealing with Clifford algebras and their relation to real \( K \)-theory originated with and . After Shapiro’s untimely death I joined forces with Bott and we eventually produced a rather careful treatment of the Thom isomorphism in real \( K \)-theory, based on spinors and Clifford algebras. My collaboration with Bott continued with , our elementary proof of the complex periodicity theorem. This was strongly motivated by the question of well-posed boundary conditions for the elliptic systems, the topic of [e1] and [e2]. Originally the proof was modeled on some ideas of and formulated in terms of sheaf theory, but Bott persuaded me to eliminate all high-brow machinery and so the “elementary” proof emerged.
Just before the advent of \( K \)-theory .had applied secondary cohomology operations to solve the long-standing problem of elements of Hopf invariant one. His paper was long and difficult and it was an indication of the power of \( K \)-theory that it led to new and much shorter proofs of the Hopf invariant problem. I was pleased to find a proof which could be written “on a post-card”. Actually this involved the \( \Psi^k \) operations which Adams had introduced in his subsequent work on the vector-field problem. I wrote to Adams explaining the “post-card” proof and he immediately saw how to generalize it to obtain new results for the case of odd primes. This led to our joint paper
In attempting to understand reality questions of elliptic operators Singer and I were, for a long time, held up by the fact that real \( K \)-theory behaves differently from complex \( K \)-theory. Eventually Singer pointed out that a different notion of reality was needed. The essential point is that the Fourier transform of a real-valued function is not real but instead satisfies the relation \( f(-x)=f(x) \). It was therefore necessary to develop a new version of real \( K \)-theory for spaces with involution. This was carried out in . In fact \( KR \)-theory, as I called it, turned out to provide a better approach even for purely topological purposes.
Adams had in 1961 proved a result relating Chern characters and Steenrod powers.  in our work on the Todd genus, but I was still mystified by the real significance of Adams’ result. Searching for the answer eventually led to  where I went back to first principles in studying power operations and the interaction with the symmetric group.and I had used the result in
My collaboration with Hirzebruch on \( K \)-theory was intended to culminate in a book on the subject. We had many meetings drawing up plans but alas we never seemed to have the necessary time. However, during my sabbatical term at Harvard in the Fall of 1964 I gave a course of lectures on \( K \)-theory. The notes of this course were eventually published , and this has had to act as a substitute for the projected joint book. In fact this course was given at a propitious time since I was able to incorporate the elementary proof of the periodicity theorem and to introduce the Adams operations. As a result the book is pure \( K \)-theory without any use or mention of operations. As an appendix I also explained the role of the space of Fredholm operators as a classifying space for \( K \)-theory, a result which had been independently found by Jänich and made a natural bridge to elliptic operator theory.
The interplay between index theory and \( K \)-theory at this time was very extensive and in  and  I gave what I considered definitive treatments of this relationship. In the course of developing this I was extremely surprised to find the proof of the periodicity theorem disintegrating into formalities. All the work which and I had put into  appeared to be unnecessary! It was in this respect somewhat reminiscent of Quillen’s miraculous treatment of complex cobordism.
The intimate relation between group representations and \( K \)-theory, leading eventually to the hybrid of equivariant \( K \)-theory, was another recurrent theme. I was first introduced to such ideas by  I had established the theorem \( R(G)=K(B_G) \) identifying the completed representation ring of a finite group with the \( K \)-group of its classifying space. After Adams’ series of papers on the \( J \)-homomorphism, determining the fibre homotopy classification of sphere bundles, it was natural to look at the corresponding question for group representations. This was carried out in my joint paper  with my student . This involved a formal treatment of Grothendieck’s \( \lambda \)-rings and was essentially an algebraic version of Adams’ work.who had come across them in the context of algebraic geometry or number theory. In
The main theorem of  for finite groups \( G \) had also been established earlier in  for compact connected Lie groups, and it was natural to conjecture that there should be a uniform treatment applying to all compact Lie groups. The proof in  I had long regarded with misgiving: it was lengthy, highly technical, and depended on many apparent pieces of good fortune. I was unhappy with it and constantly searched for something more natural. Eventually, working systematically with equivariant \( K \)-theory gave the right approach, and this was worked out in my joint paper  with . In fact, Segal, who had been my student, wrote his thesis on equivariant \( K \)-theory and then rapidly became my collaborator in this field.  was another joint paper establishing some rather peculiar algebraic properties of \( \lambda \)-rings.
The classical theorem of Hopf identifying the number of zeros of a vector field on a manifold with the Euler characteristic has interesting generalizations to several vector fields. These are discussed briefly in  and . A much more extensive treatment is given in my joint paper  with , who was then my assistant at the Institute for Advanced Study. These papers straddle the \( K \)-theory/index theory frontier in a fundamental way. In particular, they use rather refined index theorems for real operators where the indices are integers modulo high powers of 2. I found such links between analysis and homotopy theory particularly appealing. Most of the time analysis is linked with real cohomology, via differential forms, and it is a surprising novelty to connect analysis with subtle torsion effects.
My interest in homotopy theory was always unorthodox. I found the conventional approach rather ugly and cumbersome and I was always on the look-out for some indirect geometric or analytical approach. This was the spirit in which  was written. My hope was that the use of natural framed manifolds, e.g., Lie groups, to represent the stable homotopy of spheres might lead to a break-through. Although there has been some progress on this point, my expectations have not yet been realized.
The final \( K \)-theory paper  is a survey of the told of \( K \)-theory in various branches of algebra, geometry, and analysis. It gives a brief but perhaps helpful overview.