Celebratio Mathematica — Atiyah

Papers on K-theory (1959–1970)

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 Hirzebruch on \( K \)-theory. However [3], my joint paper with Todd, is in a sense the start of this development. Having been exposed, at the Arbeitstagung, to many hours of Grothendieck expounding his generalization of the Hirzebruch–Riemann–Roch theorem, I was playing about with his formulae for complex projective space. From Ioan James (at that time a colleague in Cambridge) I had heard about the Bott 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 [3]. Many years later Adams and Grant–Walker gratifyingly proved our results were best possible.

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. [1] 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 [2]. Originally I had noted that the differentiable Riemann–Roch theorem of [1] 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 [2] 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 [5], 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 [6] 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 [8]. 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.

Milnor’s 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, Adams 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 [15] 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 [2].

Papers [9] and [7] 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 Hilton and Eckman, which were elegant and clear. I also learnt much from Milnor in Bonn about cobordism theory and this is reflected in [7], 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 Bott. 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 [12] 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 [14], 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 [16] 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 [17] dealing with Clifford algebras and their relation to real \( K \)-theory originated with Bott and Shapiro. 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 [18], 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 Grothendieck 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 Adams 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 [21].

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 [20]. 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. Hirzebruch and I had used the result in [8] 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 [22] where I went back to first principles in studying power operations and the interaction with the symmetric group.

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 [23], 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 [25] and [26] 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 Bott and I had put into [18] 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 Serre who had come across them in the context of algebraic geometry or number theory. In [6] 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 [28] with my student David Tall. This involved a formal treatment of Grothendieck’s \( \lambda \)-rings and was essentially an algebraic version of Adams’ work.

The main theorem of [6] for finite groups \( G \) had also been established earlier in [5] 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 [6] 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 [27] with Segal. In fact, Segal, who had been my student, wrote his thesis on equivariant \( K \)-theory and then rapidly became my collaborator in this field. [32] 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 [29] and [31]. A much more extensive treatment is given in my joint paper [33] with Dupont, 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 [34] 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 [35] 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.

References

[e1]M. F. Atiyah and R. Bott: “The index problem for manifolds with boundary,” pp. 175–186 in Differential Analysis: Papers presented at the international colloquium (Bombay, 7–14 January 1964). Tata Institute of Fundamental Research Studies in Mathematics 2. Oxford University Press (London), 1964. MR 0185606 Zbl 0163.34603 incollection

[e2]M. F. Atiyah: “The index theorem for manifolds with boundary,” pp. 337–351 in Seminar on the Atiyah–Singer index theorem. Edited by R. S. Palais. Annals of Mathematics Studies 57. Princeton University Press, 1965. Appendix I. Atiyah’s sole contribution to Seminar on the Atiyah–Singer index theorem (1965). Republished in Atiyah’s Collected works, vol. 3. See also similarly-titled article in Differential analysis (1965). incollection

Works

[1] M. F. Atiyah and F. Hirzebruch: “Riemann–Roch theorems for differentiable manifolds,” Bull. Am. Math. Soc. 65 : 4 (1959), pp. 276–281. MR 110106 Zbl 0142.40901 article

The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \). It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958] and [Milnor 1960].

More recently Grothendieck has obtained [Borel and Serre 1958] a more general Riemann–Roch Theorem for a map \( f: Y\to X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1960] on the stable \( J \)-homomorphism.

Another differentiable Riemann–Roch Theorem, with applications to embeddability problems of differentiable manifolds, will be found in [Atiyah and Hirzebruch 1959].

The proofs of our theorems rely heavily on the Bott periodicity of the classical groups [Bott 1957, 1958, 1959], and are altogether different from the earlier methods of [Borel and Hirzebruch 1958] and [Milnor 1960], which were based on Thom’s cobordism theory and Adams’ spectral sequence.

[2] M. F. Atiyah and F. Hirzebruch: “Quelques théorèmes de non-plongement pour les variétés différentiables” [Some non-immersion theorems for differentiable manifolds], Bull. Soc. Math. France 87 (1959), pp. 383–396. MR 114231 Zbl 0196.55903 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

[3]M. F. Atiyah and J. A. Todd: “On complex Stiefel manifolds,” Proc. Cambridge Philos. Soc. 56 : 4 (1960), pp. 342–353. MR 0132552 Zbl 0109.16102 article

[4]M. F. Atiyah and F. Hirzebruch: “Quelques théoremes de non-plongement pour les variétés différentiables” [Some non-embedding theorems for differentiable manifolds], Colloques Int. Centre Nat. Rech. Sci. 89 (1960), pp. 383–396. See also Bull. Soc. Math. France 87 (1959). Zbl 0108.18202 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch, 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958; 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espance euclidien (ou ce qui est équivalent, une spère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui ’\etaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott ([Borel and Hirzebruch, 1958/59] et [Bott 1958]) qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

[5] M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 7–38 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. Republished in Algebraic topology: A student’s guide (1972). MR 139181 Zbl 0108.17705 incollection

F. Hirzebruch	Related	Volume
Carl Barnett Allendoerfer	Related

[6]M. F. Atiyah: “Characters and cohomology of finite groups,” Inst. Hautes Études Sci. Publ. Math. 9 : 1 (1961), pp. 23–64. MR 0148722 Zbl 0107.02303 article

[7]M. F. Atiyah: “Thom complexes,” Proc. London Math. Soc. (3) 11 : 1 (1961), pp. 291–310. MR 0131880 Zbl 0124.16301 article

The spaces which form the title of this paper were introduced by Thom in [1954] as a tool in his study of differentiable manifolds. In addition certain special Thom complexes have been studied by James [1959] in connexion with Stiefel manifolds (cf. [James 1958a; 1958b]). The purpose of this paper is to prove a number of general results on Thom complexes, and to deduce the main theorems of James [1958b; 1959] as immediate consequences. Our main result (3.3) is a duality theorem (in the Whitehead–Spanier \( S \)-theory) for Thom complexes over differentiable manifolds. Besides its application this is a result of some independent interest, since it provides a satisfactory place for manifolds in \( S \)-theory.

[8] M. F. Atiyah and F. Hirzebruch: “Cohomologie-Operationen und charakteristische Klassen” [Cohomology operations and characteristic classes], Math. Z. 77 : 1 (1961), pp. 149–187. Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday. MR 156361 Zbl 0109.16002 article

F. Hirzebruch	Related	Volume
Friedrich Karl Schmidt	Related

[9]M. F. Atiyah: “Bordism and cobordism,” Proc. Cambridge Philos. Soc. 57 (1961), pp. 200–208. MR 0126856 Zbl 0104.17405 article

In [1959], [1960] Wall determined the structure of the cobordism ring introduced by Thom in [1953]. Among Wall’s results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin [1953], [1958] and Dold [1959/60] which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are “cohomology-type” exact sequences arising in the well-known way from mappings into a universal space. The appropriate “cohomology” theory is constructed by taking as universal space the Thom complex \( \mathit{MSO}(n) \), for \( n \) large. This gives rise to (oriented) cobordism groups \( \mathit{MSO}^*(X) \) of a space \( X \).

[10] M. F. Atiyah and F. Hirzebruch: “Bott periodicity and the parallelizability of the spheres,” Proc. Camb. Philos. Soc. 57 : 2 (April 1961), pp. 223–226. MR 126282 Zbl 0108.35902 article

The theorems of Bott [1958, 1959a] on the stable homotopy of the classical groups imply that the sphere \( S^n \) is not parallelizable for \( n \neq 1 \), \( 3, 7 \). This was shown independently by Kervaire [1958] and Milnor [1958; Bott and Milnor 1958]. Another proof can be found in [Borel and Hirzebruch 1959, §26.11]. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that \( S^n \) with any (perhaps extraordinary) differentiable structure is not parallelizable if \( n \neq 1 \), \( 3, 7 \). Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in [Bott 1958, 1959a]. The purpose of this note is to show how the refined form of Bott’s results given in [Bott 1959b] leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the folowing theorem due to Milnor [1958] which implies the non-parallelizability.

There exists a real vector bundle \( \xi \) over the sphere \( S^n \) with \( w_n(\xi) \neq 0 \) only for \( n = 1 \), \( 2,4 \) or 8

[11] M. F. Atiyah and F. Hirzebruch: “Charakteristische Klassen und Anwendungen” [Characteristic classes and applications], Enseignement Math. (2) 7 : 1 (1961), pp. 188–213. MR 154294 Zbl 0104.39801 article

[12] M. F. Atiyah and F. Hirzebruch: “The Riemann–Roch theorem for analytic embeddings,” Topology 1 : 2 (April–June 1962), pp. 151–166. MR 148084 Zbl 0108.36402 article

In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y\to X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \)). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:

\( f \) is an embedding,
\( f \) is a projection \( X\times P_N \to X \), where \( P_N \) is a projective space,

and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.

[13]M. F. Atiyah: “Vector bundles and the Künneth formula,” Topology 1 : 3 (1962), pp. 245–248. MR 0150780 article

The purpose of this note is to establish a Künneth formula for the ring \( K^*(X) \) introduced in [Atiyah and Hirzebruch 1960]. We shall prove the following:

Let \( X \), \( Y \) be finite CW-complexes, then we have a natural exact sequence \[ 0 \rightarrow K^*(X)\otimes K^*(Y) \stackrel{\alpha}{\longrightarrow} K^*(X\times Y)\stackrel{\beta}{\longrightarrow} \operatorname{Tor}(K^*(X),K^*(Y)) \rightarrow 0. \]

Here \( \otimes \), \( \operatorname{Tor} \) are applied to abelian groups, and \( \alpha \) is the natural map induced by the product in \( K^* \). Moreover the sequence is \( \mathbb{Z}_2 \)-graded with \( \deg\alpha = 0 \), \( \deg\beta = 1 \), where \( K^p\otimes K^q \) and \( \operatorname{Tor}(K^p,K^q) \) are given degree \( p+q \) (\( p,q \in \mathbb{Z}_2 \)).

[14] M. F. Atiyah and F. Hirzebruch: “Analytic cycles on complex manifolds,” Topology 1 : 1 (January–March 1962), pp. 25–45. MR 145560 Zbl 0108.36401 article

Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \)-dimensional complex analytic subspace of \( X \). Then \( Y \) defines or “carries” a \( 2k \)-dimensional integral homology class \( y \) of \( X \), although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \). These conditions are that certain cohomology operations should vanish on \( u \), for example \( \mathrm{Sq}^3u = 0 \): they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

[15]M. F. Atiyah: “Immersions and embeddings of manifolds,” Topology 1 : 2 (April–June 1962), pp. 125–132. MR 0145549 Zbl 0109.41101 article

[16]M. F. Atiyah: “The Grothendieck ring in geometry and topology,” pp. 442–446 in Proceedings of the International Congress of Mathematicians 1962 (Stockholm, 15–22 August 1962), vol. 1. Inst. Mittag-Leffler (Djursholm), 1963. MR 0180975 Zbl 0121.39702 incollection

[17]M. F. Atiyah, R. Bott, and A. Shapiro: “Clifford modules,” Topology 3 : Supplement 1 (July 1964), pp. 3–38. MR 0167985 Zbl 0146.19001 article

Raoul H. Bott	Related	Volume
Arnold Samuel Shapiro	Related

[18]M. Atiyah and R. Bott: “On the periodicity theorem for complex vector bundles,” Acta Math. 112 : 1 (1964), pp. 229–247. MR 0178470 Zbl 0131.38201 article

The periodicity theorem for the infinite unitary group [Bott 1959] can be interpreted as a statement about complex vector bundles. As such it describes the relation between vector bundles over \( X \) and \( X\times S^2 \), where \( X \) is a compact space and \( S^2 \) is the 2-sphere. This relation is most succinctly expressed by the formula \[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \] where \( K(X) \) is the Grothendieck group of complex vector bundles over \( X \). The general theory of these \( K \)-groups, as developed in [Atiyah and Hirzebruch 1961], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof.

[19]M. F. Atiyah: “On the \( K \)-theory of compact Lie groups,” Topology 4 : 1 (1965), pp. 95–99. MR 0178092 Zbl 0136.21001 article

[20]M. F. Atiyah: “\( K \)-theory and reality,” Quart. J. Math. Oxford Ser. (2) 17 : 1 (1966), pp. 367–386. MR 0206940 Zbl 0146.19101 article

The \( K \)-theory of complex vector bundles [Atiyah 1965; Atiyah and Hirzebruch 1961] has many variants and refinements. Thus there are:

\( K \)-theory of real vector bundles, denoted by \( \mathit{KO} \),
\( K \)-theory of self-conjugate bundles, denoted by \( \mathit{KC} \) [Anderson 1964] or \( \mathit{KSC} \) [Green 1964],
\( K \)-theory of \( G \)-vector bundles over \( G \)-spaces [Atiyah and Segal 1965], denoted by \( K_G \).

In this paper we introduce a new \( K \)-theory denoted by \( \mathit{KR} \) which is, in a sense, a mixture of these three. Our definition is motivated partly by analogy with real algebraic geometry and partly by the theory of real elliptic operators. In fact, for a thorough treatment of the index problem for real elliptic operators, our \( \mathit{KR} \)-theory is essential. On the other hand, from the purely topological point of view, \( \mathit{KR} \)-theory has a number of advantages and there is a strong case for regarding it as the primary theory and obtaining all the others from it. One of the main purposes of this paper is in fact to show how \( \mathit{KR} \)-theory leads to an elegant proof of the periodicity theorem for \( \mathit{KO} \)-theory, starting essentially from the periodicity theorem for \( K \)-theory as proved in [Atiyah and Bott 1964].

[21]J. F. Adams and M. F. Atiyah: “\( K \)-theory and the Hopf invariant,” Quart. J. Math. Oxford Ser. (2) 17 (1966), pp. 31–38. MR 0198460 Zbl 0136.43903 article

[22]M. F. Atiyah: “Power operations in \( K \)-theory,” Quart. J. Math. Oxford Ser. (2) 17 : 1 (1966), pp. 165–193. Russian translation published in Mathematika 14:2 (1970). MR 0202130 Zbl 0144.44901 article

[23]M. F. Atiyah and D. W. Anderson: \( K \)-theory. Mathematics Lecture Notes 7. W. A. Benjamin (New York and Amsterdam), 1967. Lectures by Atiyah (Fall 1964), notes by Anderson. Russian translation published as Lekcii po \( K \)-teorii (1967). 2nd edition published in 1989. MR 0224083 book

[24]M. F. Atiyah and D. W. Anderson: Lekcii po \( K \)-teorii [Lectures on \( K \)-theory]. Mir (Moscow), 1967. Russian translation of \( K \)-theory (1967). Zbl 0159.53401 book

[25]M. F. Atiyah: “Bott periodicity and the index of elliptic operators,” Quart. J. Math. Oxford Ser. (2) 19 (1968), pp. 113–140. MR 0228000 Zbl 0159.53501 article

[26]M. F. Atiyah: “Algebraic topology and operators in Hilbert space,” pp. 101–121 in Lectures in modern analysis and applications, vol. I. Edited by C. T. Taam. Lecture Notes in Mathematics 103. Springer (Berlin), 1969. MR 0248803 Zbl 0177.51701 incollection

[27]M. F. Atiyah and G. B. Segal: “Equivariant \( K \)-theory and completion,” J. Differential Geometry 3 (1969), pp. 1–18. MR 0259946 Zbl 0215.24403 article

It was shown in [Atiyah 1961] that, for any finite group \( G \), the completed character ring \( R(G)^{\wedge} \) was isomorphic to \( K^*(B_G) \) where \( B_G \) denotes a classifying space for \( G \). The corresponding result for compact connected Lie groups was established in [Atiyah and Hirzebruch 1961], and a combination of the methods of [Atiyah and Hirzebruch 1961] and [Atiyah 1961] (together with certain basic properties of \( R(G) \) given in [Segal 1968b]) can be used to derive the theorem for general compact Lie groups. Such a proof however would be extremely lengthy, the worst part being in fact the treatment for finite groups where one climbs up via cyclic and Sylow subgroups.

The purpose of this paper is to give a new and much simpler proof of the theorem about \( K^*(B_G) \) which applies directly to all compact Lie groups \( G \). The main feature of our new proof is that we generalize the whole problem in a rather natural way by working with the equivariant \( K \)-theory developed in [Segal 1968a]. We shall formulate and prove a general theorem about the completion \( K_G^*(X)^{\wedge} \) for any compact \( G \)-space \( X \). The theorem about \( R(G) \) then follows by taking \( X \) to be a point.

[28]M. F. Atiyah and D. O. Tall: “Group representations, \( \lambda \)-rings and the \( J \)-homomorphism,” Topology 8 : 3 (July 1969), pp. 253–297. MR 0244387 Zbl 0159.53301 article

[29]M. F. Atiyah: Vector fields on manifolds. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen 200. Westdeutscher Verlag (Cologne), 1970. MR 0263102 Zbl 0193.52303 book

[30]M. F. Atiyah: “Power operations in \( K \)-theory,” Matematika 14 : 2 (1970), pp. 35–65. Russian translation of article in Q. J. Math., Oxf. 17:1 (1966). Zbl 0208.51503 article

[31]M. F. Atiyah: “Elliptic operators and singularities of vector fields,” pp. 207–209 in Actes du Congrès International des Mathématiciens (Nice, 1–10 September 1970), vol. 2. Gauthier-Villars (Paris), 1971. MR 0415688 Zbl 0222.58004 incollection

[32]M. F. Atiyah and G. B. Segal: “Exponential isomorphisms for \( \lambda \)-rings,” Quart. J. Math. Oxford Ser. (2) 22 : 3 (1971), pp. 371–378. MR 0291250 Zbl 0226.13008 article

[33]M. F. Atiyah and J. L. Dupont: “Vector fields with finite singularities,” Acta Math. 128 : 1 (1972), pp. 1–40. MR 0451256 Zbl 0233.57010 article

In this paper we give some generalizations of the famous theorem of H. Hopf which states that the number of singularities of a tangent vector field on a compact smooth manifold is equal to the Euler characteristic. Instead of a single vector field we consider \( r \) vector fields \( u_1,\dots,u_r \) and we are interested in their “singularities”, that is, the set \( \Sigma \) of points on the manifold at which they become linearly dependent. In general \( \Sigma \) will have dimension \( r-1 \), it is a cycle and its homology class is the \( (n-r+1) \)-th Stiefel–Whitney class of the manifold. This is the standard primary obstruction theory and it provides one way of generalizing the classical Hopf Theorem. However, this theory says nothing about \( \Sigma \) if \( \dim\Sigma < r-1 \). In this paper following E. Thomas [1967] we shall generalize the Hopf theorem by considering the other extreme case in which \( \Sigma \) is finite, so that \( \dim\Sigma = 0 \).

[34]M. F. Atiyah and L. Smith: “Compact Lie groups and the stable homotopy of spheres,” Topology 13 : 2 (1974), pp. 135–142. MR 0343269 Zbl 0282.55008 article

[35]M. F. Atiyah: “A survey of \( K \)-theory,” pp. 1–9 in \( K \)-theory and operator algebras (University of Georgia, Athens, GA, 21–25 April 1975). Edited by B. B. Morrel and I. M. Singer. Springer (Berlin), 1977. MR 0474299 Zbl 0345.55005 incollection

Isadore M. Singer	Related	Volume
Bernard B. Morrel	Related

[36]M. Atiyah: Collected works, vol. 2: \( K \)-theory. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951893 Zbl 0724.55001 book

[37]M. F. Atiyah and D. W. Anderson: \( K \)-theory, 2nd edition. Advanced Book Classics. Addison-Wesley (Redwood City, CA), 1989. Original edition published by W. A. Benjamin in 1967. MR 1043170 book