STABLE ALGEBRAIC TOPOLOGY, 1945-1966 J. P. MAY Contents 1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 5. Steenrod operations, K(ss; n)'s, and characteristic classes * * 8 6. The introduction of cobordism 10 7. The route from cobordism towards K-theory 12 8. Bott periodicity and K-theory 14 9. The Adams spectral sequence and Hopf invariant one 15 10. S-duality and the introduction of spectra 18 11. Oriented cobordism and complex cobordism 21 12. K-theory, cohomology, and characteristic classes 23 13. Generalized homology and cohomology theories 25 14. Vector fields on spheres and J(X) 28 15. Further applications and refinements of K-theory 31 16. Bordism and cobordism theories 34 17. Further work on cobordism and its relation to K-theory 37 18. High dimensional geometric topology 40 19. Iterated loop space theory 42 20. Algebraic K-theory and homotopical algebra 43 21. The stable homotopy category 45 References 50 Stable algebraic topology is one of the most theoretically deep and calculati* *on- ally powerful branches of mathematics. It is very largely a creation of the sec* *ond half of the twentieth century. Roughly speaking, a phenomenon in algebraic topo* *l- ogy is said to be "stable" if it occurs, at least for large dimensions, in a ma* *nner independent of dimension. While there are important precursors of the understan* *d- ing of stable phenomena, for example in Hopf's introduction of the Hopf invaria* *nt [Hopf35, FS], Hurewicz's introduction of homotopy groups [Hur35], and Borsuk's introduction of cohomotopy groups [Bor36], the first manifestation of stability* * in algebraic topology appeared in Freudenthal's extraordinarily prescient 1937 pap* *er [Fr37, Est], in which he proved that the homotopy groups of spheres are stable * *in a range of dimensions. Probably more should be said about precursors, but I will skip ahead and begin with the foundational work that started during World War II but first reached p* *rint 1 2 J. P. MAY in 1945. Aside from the gradual development of homology theory, which of course dates back at least to Poincare, some of the fundamental precursors are treated elsewhere in this volume [Ma , BG , Mc , We ]. However, another reason for not attempting such background is that I am not a historian of mathematics, not even as a hobby. I am a working mathematician who is bemused by the extraordinarily rapid, and perhaps therefore jagged, development of my branch of the subject. I am less interested in who did what when than in how that correlated with the progression of ideas. My theme is the transition from classical algebraic topology to stable algebr* *aic topology, with emphasis on the emergence of cobordism, K-theory, generalized ho- mology and cohomology, the stable homotopy category, and modern calculational techniques. The history is surprising, not at all as I imagined it. For one exa* *mple, we shall see that the introduction of spectra was quite independent of the intr* *o- duction of generalized cohomology theories. While some key strands developed in isolation, we shall see that there was a sudden coalescence around 1960: this w* *as when the subject as we know it today began to take shape, although in far from its final form: I doubt that we are there yet even now. Younger readers are urged to remember the difficulty of communication in those days. Even in 1964, when I wrote my thesis, the only way to make copies was to type using carbon paper: mimeographing was inconvenient and the xerox machine had not been invented, let alone fax or e-mail. Moreover, English had not yet become the lingua franca. Many relevant papers are in French or German (which I read) and some are in Russian, Spanish, or Japanese (which I do not read); furt* *her, the Iron Curtain hindered communication, and translation from the Russian was spotty. On the other hand, the number of people working in topology was quite small: most of them knew each other from conferences, and correspondence was regular. Moreover, the time between submission and publication of papers was shorter than it is today, usually no more than a year. I have profited from a perusal of all of Steenrod's very helpful compendium [StMR ] of Mathematical Reviews in algebraic and differential topology published between 1940 and 1967. Relatively few papers before the mid 1950's concern stab* *le algebraic topology, whereas an extraordinary stream of fundamental papers was published in the succeeding decade. That stream has since become a torrent. I will focus on the period covered in [StMR ], especially the years 1950 through * *1966, which is an arbitrary but convenient cut-off date. For the later part of that p* *eriod, I have switched focus a little, trying to give a fairly complete indication of * *the actual mathematical content of all of the most important relevant papers of the period. I shall also point out various more recent directions that can be seen* * in embryonic form during the period covered, but I shall not give references to the modern literature except in cases of direct follow up and completion of earlier* * work. I plan to try to bring the story up to date in a later paper, but lack of time * *has prevented me from attempting that now. References to mathematical contributions give the year of publication, the on* *ly exception being that books based on lecture notes are dated by the year the lec* *tures were given. References to historical writings are given without dates. STABLE ALGEBRAIC TOPOLOGY, 1945-1966 3 1.Setting up the foundations A great deal of modern mathematics, by no means just algebraic topology, would quite literally be unthinkable without the language of categories, functors, and natural transformations introduced by Eilenberg and MacLane in their 1945 paper [EM45b ]. It was perhaps inevitable that some such language would have appeared eventually. It was certainly not inevitable that such an early systematization * *would have proven so remarkably durable and appropriate; it is hard to imagine that t* *his language will ever be supplanted. With this language at hand, Eilenberg and Steenrod were able to formulate their axiomatization of ordinary homology and cohomology theory. The axioms were announced in 1945 [ES45 ], but their celebrated book "The foundations of algebraic topology" did not appear until 1952 [ES52 ], by which time its essent* *ial ideas were well-known to workers in the field. It should be recalled that Eilen* *berg had set the stage with his fundamentally important 1940 paper [Eil40], in which* * he defined singular homology and cohomology as we know them today. I will say a little about the axioms shortly, but another aspect of their work deserves immediate comment. They clearly and unambiguously separated the alge- bra from the topology. This was part of the separation of homological algebra f* *rom algebraic topology as distinct subjects. As discussed by Weibel [We ], the subj* *ect of homological algebra was set on firm foundations in the comparably fundamental book "Homological algebra" of Cartan and Eilenberg [CE56 ]. Two things are conspicuously missing from Eilenberg-Steenrod. We think of it today as an axiomatization of the homology and cohomology of finite CW com- plexes, but in fact CW complexes are nowhere mentioned. The definitive treatment of CW complexes had been published by J.H.C. Whitehead in 1948 [Whi48 ], but they were not yet in regular use. Many later authors continued to refer to poly* *he- dra where we would refer to finite CW complexes, and I shall sometimes take the liberty of describing their results in terms of finite CW complexes. Even more surprisingly, Eilenberg-Mac Lane spaces are nowhere mentioned. These spaces had been introduced in 1943 [EM43 , EM45a ], and the relation (1.1) H"n(X; ss) ~=[X; K(ss; n)] was certainly known to Eilenberg and Steenrod. It seems that they did not belie* *ve it to be important. Nowadays, the proof of this relation is seen as one the most immediate and natural applications of the axiomatization. However, there was something missing for the derivation of this relation. Des* *pite their elementary nature, the theory of cofiber sequences and the dual theory of* * fiber sequences were surprisingly late to be formulated explicitly. They were implici* *t, at least, in Barratt's papers on "track groups" [Ba55 ], but they were not clearly* * ar- ticulated until the papers of Puppe [Pu58 ] and Nomura [Nom60 ]. The concomitant principle of Eckmann-Hilton duality also dates from the late 1950's [Eck57, EH5* *8 ] (see also [Hil65]). The language of fiber and cofiber sequences pervades modern homotopy theory, and its late development contrasts vividly with the earlier in* *tro- duction of categorical language. Probably not coincidentally, the key categori* *cal notion of an adjoint functor was also only introduced in the late 1950's, by Kan [Kan58 ]. Although a little peripheral to the present subject, a third fundamental text of the early 1950's, Steenrod's "The topology of fiber bundles" [St51] neverthe* *less 4 J. P. MAY must be mentioned. In the first flowering of stable algebraic topology, with t* *he introduction of cobordism and K-theory, the solidly established theory of fiber bundles was absolutely central to the translation of problems in geometric topo* *logy to problems in stable algebraic topology. 2.The Eilenberg-Steenrod axioms The functoriality, naturality of connecting homomorphism, exactness, and ho- motopy axioms need no comment now, although their economy and clarity would not have been predicted from earlier work in the subject. Remember that these a* *re axioms on the homology or cohomology of pairs of spaces. The crucial and subtle axiom is excision. A triad (X; A; B) is excisive if X is the union of the inter* *iors of A and B. In homology, the excision map H*(B; A \ B) -! H*(X; A) must be an isomorphism. One subtlety is that I have stated the axiom in the form that Eilenberg and Steenrod verify it for singular homology. With a view towards oth* *er theories, they state the axiom under the stronger hypothesis that B is closed i* *n X. Conveniently for later developments, the dimension axiom was stated last. The fundamental theorem is that homology and cohomology with a given coefficient group is unique on triangulable pairs or, more generally, on finite CW pairs. Several important extensions of the axioms came later. First, one wants an ax* *iom that characterizes ordinary homology and cohomology on general CW pairs. For that Milnor [Mil62] added the additivity axiom. It asserts that homology conver* *ts disjoint unions to direct sums and cohomology converts disjoint unions to direct products. It implies that the homology of a CW complex X is the colimit of the homologies of its skeleta Xn. In cohomology, it implies lim1exact sequences (2.1) 0 -! lim1Hq-1(Xn) -! Hq(X) -! limHq(Xn) -! 0: This allows the extension of the uniqueness theorem to infinite CW pairs. One next wants an axiom that distinguishes singular theories from other theo- ries on general pairs of spaces. I do not know who first formulated it; it appe* *ars in [Swi75] and may be due to Adams. This is the weak equivalence axiom. It asserts that a weak equivalence of pairs induces an isomorphism on homology and cohomology. Any space is weakly equivalent to a CW complex, any pair of spaces is weakly equivalent to a CW pair, and any excisive triad is weakly equivalent * *to a triad that consists of a CW complex X and a pair of subcomplexes A and B. Here B=A \ B ~=X=A as CW complexes, which neatly explains the excision axiom. The weak equivalence axiom reduces computation of the homology and cohomology of general pairs to their computation on CW pairs. Thus it implies the uniqueness theorem for homology and cohomology on general pairs. Finally, one wants an axiom system for the reduced homology and cohomology of based spaces. The earliest published account is in the 1958 paper [DT58 ] of* * Dold and Thom, who ascribe it to Puppe. They use it to prove that the homotopy groups of the infinite symmetric products SP 1X of based spaces X can be computed as the reduced integral homology groups of X. There are several slightly later papers [Ke59 , BP60 , Hu60] devoted to single space axioms for the homology and cohomology of both based spaces and, curiously, unbased spaces. For the reduced homology of nondegenerately based spaces, the axioms just require functors "kqtogether with natural suspension isomorphisms (2.2) * : "kq(X) ~="kq+1(X) STABLE ALGEBRAIC TOPOLOGY, 1945-1966 5 that satisfy the exactness, wedge, and weak equivalence axioms. Here the exactn* *ess axiom requires the sequences (2.3) "kq(X)-f*!"kq(Y ) -! "kq(Cf) to be exact for a map f : X -! Y with cofiber Cf = Y [f CX. The wedge axiom requires the functors "kqto carry wedges (1-point unions) to direct sums. The weak equivalence axiom requires a weak equivalence to induce isomorphisms on all homology groups. Given such a reduced homology theory, one obtains an unreduced homology theory by setting kq(X) = "kq(X+ ), where X+ is the union of X and a disjoint basepoint, and kq(X; A) = "k(Cf), where f : A -! X is the inclusion. For an unreduced homology theory k*, one obtains a reduced homology theory by setting "kq(X) = kq(X; *). Thus reduced and unreduced homology theories are equivalent notions. The same is true for cohomology theories. The summary in this paragraph makes no reference to the dimension axiom and applies in general. In view of (2.2), all of ordinary homology and cohomology theory is actually part of stable algebraic topology. As an informal rule of thumb, when thinking * *in terms of classical algebraic topology, one uses unreduced theories. When thinki* *ng in terms of stable algebraic topology, one wants the suspension axiom to hold with* *out qualification in all degrees and one therefore works with reduced theories. In * *fact, in a great deal of recent work, it is an accepted convention that k* means redu* *ced homology, and one writes k*(X+ ) for unreduced homology. I shall not take that point of view here, however. This summary of the axioms is skewed towards singular homology and coho- mology. The viewpoint of someone working in, say, algebraic geometry would be quite different. However, there are two footnotes to the axioms that are not we* *ll- known and may be worth mentioning. To characterize Cech cohomology on compact Hausdorff spaces, Eilenberg and Steenrod add the continuity axiom. Keesee [Kee5* *1] observed that this axiom implies the homotopy axiom. More substantively, let us go back to (1.1) above. If X has the homotopy type* * of a CW complex, then the square brackets denote homotopy classes of based maps. Huber [Hu61 ] proved that if X is a paracompact Hausdorff space, then the Cech cohomology group Hn (X; ss) is isomorphic to the set of homotopy classes of maps X -! K(ss; n). In contrast, for the general representation of singular cohomolo* *gy in the form (1.1), we must understand [X; K(ss; n)] to be the set of maps in the category that is obtained from the homotopy category of based spaces by adjoini* *ng formal inverses to the weak equivalences; equivalently, we must replace X by a * *CW complex weakly equivalent to it before taking homotopy classes of maps. 3. Stable and unstable homotopy groups Another important precursor of stable algebraic topology was a substantial in- crease in the understanding of the relationship between stable and unstable hom* *o- topy groups and of certain fundamental exact sequences relating homotopy groups in different dimensions. I am here thinking of what was achieved by bare hands work, in the early to mid 1950's, using CW complexes and homotopical methods rather than the contemporaneous and overlapping progress that came with the introduction of spectral sequences. We have seen that the critical axiom for homology is excision. In the early 1950's, Blakers and Massey [BM51 , BM52 , BM53 ] made a systematic study of 6 J. P. MAY excision in homotopy theory, proving that homotopy groups satisfy the excision axiom in a range of dimensions. This gave a new proof of the Freudenthal suspen* *sion theorem and considerably clarified the conceptual relationship between homology and homotopy. The proofs were quite difficult, and it soon became fashionable to prove versions of their results using homology and spectral sequences. However, Boardman later came up with a quite accessible direct homotopical proof, which is presented in [Swi75], for example. It is worth emphasizing that the homotopi* *cal proof gives a stronger result than can be obtained by homological methods. The Freudenthal suspension theorem establishes the stable range for homotopy groups, roughly twice the connectivity of a space. It was shown by G.W. Whitehe* *ad [Wh53 ] that there is a metastable range for the homotopy groups of spheres. The suspension homomorphism E fits into the EHP exact sequence . .-.! ssq(Sn)-E!ssq+1(Sn+1)-H!ssq+1(S2n+1)-P!ssq-1(Sn)-E!ssq(Sn+1) -! . . . when q 3n - 2. Here H is a (generalized) Hopf invariant that Whitehead had introduced earlier [Wh50 ] and P is the (J.H.C.) Whitehead product. There were many extensions and refinements of these results. For example, Hilton [Hil51] g* *ave a definition of the Hopf invariant in the next range of dimensions, q < 4n, in * *the sequence above. The extrapolation of calculations and understanding in stable homotopy theory to calculations and understanding in the metastable range, and further, has been a major theme ever since. James [Ja55, Ja56a, Ja56b, Ja57] and Toda [To62a] went much further with this. James proved that, on 2-primary components, there is an EHP exact sequence that is valid for all values of q, and Toda proved an appropriate analogue for * *odd primes. James introduced the James construction JX for the purpose. Here JX is the free topological monoid generated by a based space X. For a connected CW complex X, James proved that JX is homotopy equivalent to MX. The space JX comes with a natural filtration, and its simple combinatorial structure allows direct construction of suitable Hopf invariant maps. Milnor [Mil56b] pro* *ved that JX splits up to homotopy as the wedge of the suspensions of its filtration quotients. These arguments were the prototypes for a great deal of later work in which combinatorial approximations to the n-fold loop spaces MnnX have been used to obtain stable decompositions of such spaces, leading to a great deal of* * new calculational information in stable homotopy theory. However, this goes beyond the present story. The power and limitations of such direct homotopical methods of calculation a* *re well illustrated in Toda's series of papers [To58a, To58b, To58c, To59] and mon* *o- graph [To62b]; while cohomology operations, spectral sequences, and the method of killing homotopy groups are used extensively, most of the work in these calc* *u- lations of the groups ssn+k(Sn) for small k consists of direct elementwise indu* *ctive arguments in the EHP sequence. Later work of this sort gave these groups for a * *few more values of k, but it was apparent that this was not the route towards major progress in the determination of the homotopy groups of spheres. 4.Spectral sequences and calculations in homology and homotopy Although the credit for the invention of spectral sequences belongs to Leray [Le49, Mc], for algebraic topology the decisive introduction of spectral sequen* *ces is due to Serre [Se51]. For a fibration p : E -! B with connected base space B and STABLE ALGEBRAIC TOPOLOGY, 1945-1966 7 fiber F , the Serre spectral sequence in homology has E2p;q= Hp(B; Hq(F ; ss)),* * where local coefficients are understood, and it converges in total degree p + q to H** *(E; ss). The analogous cohomology spectral sequence with coefficients in a commutative r* *ing ss is a spectral sequence of differential algebras, and it converges to the ass* *ociated graded algebra of H*(E; ss) with respect to a suitable filtration. With the Serre spectral sequence, algebraic topology emerged as a subject in which substantial calculations could be made. While its applications go far bey* *ond our purview, many of the calculations that it made possible and ideas to which * *it led were essential prerequisites to the emergence of stable algebraic topology. Work of Borel [Bo53a, Bo53b] and others gave a systematic understanding of the homology and cohomology of the classical Lie groups and of their classifying sp* *aces and homogeneous spaces. The basic characteristic classes had all been defined earlier, but the precise detailed analysis of the various cohomology algebras a* *nd their induced maps was vital to future progress. Serre's introduction of class theory [Se53a], and his use of the spectral seq* *uence to prove the finiteness of the homotopy groups of spheres, save for ssn(Sn) and ss4n-1(S2n), were to change the way people thought about algebraic topology. Ea* *r- lier calculations had generally had as their goal the understanding of homology and cohomology with integer or with real coefficients. In the years since, calc* *ula- tions have largely focused on mod p homology and cohomology, especially in stab* *le algebraic topology where the rational theory is essentially trivial. Moreover, * *this change in point of view led ultimately to the study of all of homotopy theory in terms of localized and completed spaces. The method of killing homotopy groups introduced by Cartan and Serre [CS52a , CS52b] was also profoundly influential. It provided the first systematic route* * to the computations of homotopy groups. The idea is easy enough. Let X be a simple space. Inductively, by killing homotopy groups and passing to homotopy fibers, * *one can construct a sequence of fibrations pn : X[n + 1; 1) -! X[n; 1) with fibre K(ssn(X); n - 1), where X[n; 1) is (n - 1)-connected and its higher homotopy groups are those of X. The initial map p1 is just the universal coveri* *ng of X. Assuming that one knows the first n homotopy groups of X, one should have enough inductive control on the space X[n; 1) to use the Serre spectral sequence to compute Hn+1(X[n + 1; 1)), which by the Hurewicz isomorphism is ssn+1(X). This is closely related to Postnikov systems [Pos51a, Pos51b, Pos51c], which we* *re not yet available to Cartan and Serre and so were implicitly reinvented by them. If in : X -! Xn is the nth term of the Postnikov tower of X, then in induces an isomorphism on ssq for q n and the higher homotopy groups of Xn are zero; X[n + 1; 1) is the homotopy fiber of in. An interesting companion to this method was given in Moore's study [Mo54 ] of the homotopy groups of spaces with a single non-vanishing homology group, which are now called Moore spaces. This work led later to the introduction of the mod* * p homotopy groups of spaces. Cohomotopy groups with coefficients were introduced and studied earlier, by Peterson [Pe56a, Pe56b]. Moore also gave a functorial, * *semi- simplicial, construction of Postnikov systems, in [Ca54-55] and [Mo58 ], which * *are sometimes called Moore-Postnikov systems as a result. This and related work of Moore in [Ca54-55], Heller in [He55], and especially Kan in [Kan55 ] and many l* *ater 8 J. P. MAY papers (see [May67 ]), began the modern systematic use of simplicial methods in algebraic topology. 5. Steenrod operations, K(ss; n)'s, and characteristic classes For the method of killing homotopy groups to be useful, one must know some- thing about the cohomology of Eilenberg-Mac Lane spaces. The problem of calcula* *t- ing these cohomology groups was intensively studied by Eilenberg and Mac Lane, notably in [EM50 ], and was solved a few years later by Cartan [Ca54-55], using methods of homological algebra. However, Cartan's original answer was not in the form we know it today. In fact, in mod p cohomology for odd primes p, it is sti* *ll not obvious how to correlate Cartan's calculations with the definitive calculat* *ions in terms of Steenrod operations. I will not say anything about the invention and development of the basic prop* *er- ties of the Steenrod operations [St47, St52, St53a, St53b, St57, ST57] since th* *at is interestingly discussed in [Ma ] and [Wh1 ]. Steenrod and Epstein [SE62 ] publi* *shed a systematic account of the results. Epstein [Ep66 ] later showed how to constr* *uct Steenrod operations in a general context of homological algebra. In fact, simp* *ly by separating the algebra from the topology, Steenrod's original definition can* * be adapted to a variety of situations in both topology and algebra [May70 ]. An essential point is that the Steenrod operations are stable, in the sense t* *hat the following diagrams commute, where Z2 is the field Z=2Z. i H"q(X; Z2)___Sq___//"Hq+i(X; Z2) (5.1) * || *|| fflffl| fflffl| H"q+1(X; Z2) _____//"Hq+1+i(X; Z2): Sqi The analogous diagram commutes for odd primes, where P ihas degree 2i(p - 1). Serre [Se53b] computed H*(K(ss2; n); Z2), where ss2 is cyclic of order 2, in * *mod- ern terms: it is the free commutative algebra on suitable composites of Steenrod operations acting on the fundamental class n 2 Hn(K(ss2; n); Z2). The analogue for odd primes was worked out by Cartan in [Ca54-55], in later exposes that are* * in fact independent of his original calculations published in the same place. Formulas for the iteration of the Steenrod operations were first proven by Ad* *em [Adem52 ] at the prime 2 and by Adem and Cartan [Adem53 , Adem57 , Ca55], in- dependently, at odd primes. However, it was Cartan who first defined the Steenr* *od algebra Ap and determined its basis of admissible monomials. In the paper [Se53b], Serre also formulated the modern viewpoint on cohomol- ogy operations. A cohomology operation OE of degree i is a natural transforma- tion kq -! `q+i for some fixed q, where k* and `* are any cohomology theo- ries. When k* is ordinary cohomology with coefficients in ss and `* is ordinary cohomology with coefficients in ae, OE is determined by naturality by the eleme* *nt OE(q) 2 Hq+i(K(ss; q); ae). Observe that, by (1.1), this element may be viewed * *as a homotopy class of maps K(ss; q) -! K(ae; q + i). A crucial point quickly understood was the calculation of the Steenrod operat* *ions in the cohomologies of Lie groups and their classifying spaces and homogeneous spaces. In particular, already in 1950 [Wu50a , Wu53 ], Wu proved his basic for* *mula STABLE ALGEBRAIC TOPOLOGY, 1945-1966 9 for the calculation of the Steenrod operations on the Stiefel-Whitney classes: X s - r + t - 1 (5.2) Sqr(ws) = wr-tws+t for s > r 0: t t Borel and Serre made a systematic study shortly afterwards [BS51 , BS53]. Also in 1950 [Wu50b ], Wu proved his formula giving an algorithm for the cal- culation of the Stiefel-Whitney classes of the tangent bundle of a manifold di- rectly in terms of its cup products; see Section 12. Wu was a close collaborato* *r of Thom, and his work was dependent on work of Thom, announced in part in 1950 [Thom50a , Thom50b ] and published in 1952 [Thom52 ]. In that paper, Thom proved the Thom isomorphism theorem and used it to give the now familiar description of Stiefel-Whitney classes in terms of Steenrod operations. Since [Thom52 ] was la* *ter overshadowed by Thom's great work on cobordism, it is well worth describing some of its original contributions. Thom considered locally trivial fiber bundles p : E -! B with fiber Sk-1, with no assumptions about the group of the bundle. Working sheaf theoretically and resolutely avoiding the use of spectral sequences, which were available to him,* * Thom proved the Thom isomorphism (5.3) OE : Hq(B) -! Hq+k(Mp; E); where Mp is the mapping cylinder of p. He worked with twisted integer coefficie* *nts, thus allowing for non-oriented fibrations, before studying the mod 2 case. Obse* *rve that, in the motivating example of the unit sphere bundle E = S(E0) of a k- dimensional vector bundle p0: E0 -! B with a Riemannian metric, the quotient space Mp=E is homeomorphic to the quotient space D(E0)=S(E0), where D(E0) is the unit disk bundle of E0. This quotient space is called the Thom space of p0a* *nd now usually denoted T p0or T (E0). Using mod 2 coefficients in the Thom isomorphism, Thom defined the Stiefel- Whitney classes of E by (5.4) wi= OE-1SqiOE(1); and he proved that, in the case of vector bundles, these are the classical Stie* *fel- Whitney classes of E. He rederived the properties of Stiefel-Whitney classes, * *in particular the Whitney duality theorem, from the new definition. This gave an elegant new proof of Whitney's result [Whit41 ] that the Stiefel-Whitney classe* *s of the normal bundle of an immersion f are invariants of the induced map f* on mod 2 cohomology. In particular, they are independent of the choice of the differen* *tiable structures on the manifolds in question. It is worth emphasizing that Whitney's foundational work in [Whit41 ] and other papers, for example on embeddings and immersions of smooth manifolds, was an essential prerequisite to virtually all * *of the later applications of algebraic topology to geometric topology. Thom then generalized to obtain results of this form for purely topological i* *m- mersions, with no hypothesis of differentiability. It should be remembered that* * this paper appeared four years before Milnor's discovery of exotic differential stru* *ctures on spheres [Mil56a]. For an embedding f, he went further and showed that the homotopy type of a tubular neighborhood of f is independent of the differentiab* *le structure on the ambient manifold. He then introduced the notion of fiber homo- topy equivalence and proved that the fiber homotopy type of the tangent bundle of a manifold is independent of its differentiable structure. He observed that * *the 10 J. P. MAY Stiefel-Whitney classes are invariant under fiber homotopy equivalence, and ask* *ed what other such classes there might be. The determination of all characteristic classes for spherical fibrations evolved over the following two decades. That * *is a long story, intertwined with the theory of iterated loop spaces, and is well be* *yond our present scope. 6.The introduction of cobordism In the last chapter of [Thom52 ], Thom set up the modern theory of Poincare duality for manifolds with boundary and explained the now familiar necessary Eu* *ler characteristic and index conditions for a differentiable manifold to be the bou* *ndary of a compact differentiable manifold. The emphasis he placed on the index was a precursor of things to come. He also recalled Pontryagin's fundamental observat* *ion [Pon42, Pon47] that, for M to be such a boundary, it is necessary that all of its characteristic numbers be zero. He went on to observe that the vanishing of Stiefel-Whitney numbers is still a necessary condition when M is not assumed to be differentiable. He observed that "la recherche de conditions suffisantes est* * un probleme beaucoup plus difficile". Two years later, as announced in [Thom53a , Thom53b , Thom53c ] and published in his wonderful 1954 paper [Thom54 ], he had solved this problem for smooth co* *m- pact manifolds. The importance to modern topology, both geometric and algebraic, of his introduction and calculation of cobordism cannot be exaggerated. For ex- ample, Milnor's construction of exotic differentiable structures on S7 begins w* *ith Thom's theory and in particular with Thom's result that every smooth compact 7-manifold is a boundary. Cobordism theory was not wholly unprecedented. In 1950, Pontryagin [Pon50] showed that the stable homotopy groups of spheres, in low dimension at least, a* *re isomorphic to the framed cobordism groups of smooth manifolds. His motivation was to obtain methods for the computation of stable homotopy groups, and he used this technique to prove that ssn+2(Sn) ~=Z=2Z, thus correcting an earlier mista* *ke of his. While that motivation seems misguided in retrospect, it was an imaginat* *ive attack on the problem. Pontryagin's paper was in Russian, never translated, and it is not quoted by Thom. However, Thom does quote earlier papers of Pontryagin [Pon42, Pon47] in which the idea of pulling back the zero-section in Grassmanni* *ans along a smooth approximation to a classifying map plays a prominent role. Thom's paper [Thom54 ] reads a little surprisingly today. Its main focus is n* *ot cobordism, which does not appear until the last chapter, but rather the realiza* *tion of homology classes of manifolds by submanifolds. It seems that it was this that first motivated Thom to a detailed analysis of the cohomology and homotopy of Thom complexes, not just in the stable range relevant to corbordism but also in the unstable range. Moreover, the existence of a stable range for the homotopy groups of T SO(k) and T O(k) is proven by direct methods of algebraic topology, rather than as a consequence of the isomorphism between homotopy groups and cobordism groups. For a closed subgroup G of O(k), Thom lets T (G) be the Thom space of the universal bundle EG - ! BG with fiber Sk-1. He considers a compact oriented manifold V n and asks when a homology class x 2 Hn-k(V ) is realizable as the image of the fundamental class of submanifold W n-kof codimension k. He dualizes the question as follows. For any space X, say that a class y 2 Hk(X) is G-reali* *zable STABLE ALGEBRAIC TOPOLOGY, 1945-1966 11 if there is a map f : X -! T (G) such that f*() = y, where 2 Hk(T (G)) is the Thom class. Let y 2 Hk(V ) be Poincare dual to x. Then "le theoreme fondamental" asserts that x is realizable by a submanifold W such that the structure group of the normal bundle of W in V can be reduced to G if and only if y is G-realizabl* *e. Of course, the analogue with mod 2 coefficients does not need orientability. As* * we shall see in Section 16, Atiyah explained this result conceptually almost a dec* *ade later. Taking G to be the trivial group, it follows from a result of Serre [Se53a] t* *hat x is realizable if k is odd or if n < 2k and that Nx is realizable for some int* *eger N that depends only on k and n. However, the main focus is on G = O(k) and G = SO(k). Here Thom shows directly that ssk+i(T O(k)) is independent of k when i < k, and similarly for T SO(k). Moreover, crucially, he proves that T O(* *k) has the same 2k-type as a precisely specified product of Eilenberg-Mac Lane spa* *ces K(Z2; k + i). The Wu formula (5.2) is the key to the calculation. He goes on to study H*(T O(k); Z2) in low dimensions beyond the stable range for k 3. For the realizability problem, he deduces that x 2 Hi(V n; Z2) is realizable for i < [n* *=2], with further information in low codegrees n - i. The problem for T SO(k) is much harder, and ssk+i(T SO(k)) is only determined completely for i 7; more detailed information is obtained for k 4. However, Thom shows that T SO(k) has the rational cohomology type of an explicitly speci- fied product of Eilenberg-Mac Lane spaces K(Z; k+i). For the realizability prob* *lem, he deduces that some integer multiple of any x 2 Hi(V n; Z) is realizable, and * *that any x is realizable if i 5 or n 8. Before turning to cobordism, Thom studies the problem posed by Steenrod of determining which homology classes x 2 Hr(K) of a finite polyhedron K are real- izable as f*(z), where z is the fundamental class of a compact manifold Mr and f : Mr -! K is a map. By embedding K as a retract of a manifold with boundary M and taking the double V of M to obtain a manifold without boundary, Thom re- duces this question to the realizability question already studied. He thereby p* *roves that, in mod 2 homology, every class x is realizable. In retrospect, of course,* * this presages unoriented bordism and its relationship to ordinary mod 2 homology. Si* *m- ilarly, he proves that, in integral homology, some integer multiple of every cl* *ass x is realizable. Remarkably, he then proves that every class x is realizable if r* * 6, but that there are unrealizable classes in all dimensions r 7. Only after all of this does he prove the cobordism theorems. Let Nn be the set of cobordism classes of smooth compact n-manifolds, where two n-manifolds are cobordant if their disjoint union is the boundary of a smooth compact (n + 1)- manifold with boundary. Define megan similarly for oriented n-manifolds. Under disjoint union, Nn is a Z2-vector space and Mn is an Abelian group; any boundar* *y is the zero element. Under cartesian product, N* and M* are graded rings. Moreover, the index defines a ring homomorphism I : M* -! Z. The fundamental geometric theorem is the Thom isomorphism: Nn is isomorphic to the stable homotopy group ssk+n(T O(k)) and Mn is isomorphic to the stable homotopy group ssk+n(T SO(k)). While modern proofs are easier reading than Thom's, the basic ideas are the same. In slightly modernized terms, an isomorphism OE : Nn -! ssk+n(T O(k)) is constructed as follows. Embed a given n-manifold M in Rk+n for k large, let be the normal bundle of the embedding, and construct a tubular neighborhood V of M in Rk+n. Define a map f from Sk+n to the Thom space T () by identifying V with the total space of and mapping points not in V to the basepoint. This is * *the 12 J. P. MAY Pontryagin-Thom construction. Classify and compose f with the induced map of Thom spaces T () -! T O(k) to obtain OE(M), checking that the homotopy class of the composite is independent of the choice of M in its cobordism class and of the embedding. To construct an inverse isomorphism to OE, view the classifying space BO(k) as a Grassmannian manifold of sufficiently high dimension. Up to homotopy, any map g : Sk+n -! T O(k) can be smoothly approximated by a map that is transverse to the zero section. Define (g) to be the cobordism class * *of the inverse image of the zero section, checking that this class is independent * *of the homotopy class of g. Transversality is the crux of the proof, and Thom was the first to develop this notion. From here, the earlier calculations in the paper immediately identify the gro* *ups Nn. Using this identification, Thom proves that two manifolds are cobordant if * *and only if they have the same Stiefel-Whitney numbers. By calculating the Stiefel- Whitney numbers of products, this allows him to determine the ring structure of N*: it is a polynomial algebra on one generator of dimension n for each n 2 not of the form 2j - 1. The even dimensional generators can be chosen to be the cobordism classes of the real projective spaces RP2n. Similarly, the groups Mn are identified modulo torsion by the earlier calcu- lations. Using this, Thom proves that if all Pontryagin numbers of an oriented manifold are zero, then the disjoint union of some number of copies of that man* *i- fold is a boundary. This allows determination of the ring M*Q: it is a polynomi* *al algebra on generators of dimension 4n for n 1. The generators can be chosen to be the cobordism classes of the complex projective spaces CP 2n. Dold [Dold56a] soon after identified odd dimensional generators of N*. The Wu formula for the computation of Stiefel-Whitney classes of manifolds give restri* *c- tions on which collections of Steifel-Whitney numbers actually correspond to the cobordism class of a manifold, and Dold [Dold56b] proved that these relations a* *re complete: a collection of Stiefel-Whitney numbers that satisfies the Wu relatio* *ns corresponds to a manifold. In modern invariant terms, the Stiefel-Whitney num- bers of manifolds define a monomomorphism N* -! Hom (H*(BO; Z2); Z2), and its image consists of those homomorphisms that annihilate the subgroup generated by the Wu relations. 7.The route from cobordism towards K-theory Hirzebruch [Hirz53] had already introduced multiplicative sequences of charac- teristic classes before Thom's paper. However, cobordism theory provided exactly the right framework for their study and allowed him to prove the index theorem [Hirz56]: the index of a smooth oriented 4n-manifold M is the characteristic nu* *m- ber , where L is the L-genus and o is the tangent bundle of M. Here L(o) is a polynomial in the Pontryagin classes of M determined in Hirzebruch's formalism by the power series L(x) = x=tanh(x). Using Thom's observation that the index defines a ring homomorphism M* -! Z, Hirzebruch's formalism shows that the index formula must hold for some power series L, and L(x) is the only power series that gives the correct answer on complex projective spaces. The purpose of Hirzebruch's monograph [Hirz56] was to prove the Riemann- Roch theorem for algebraic varieties of arbitrary dimension. It would take us t* *oo far afield to say much about this, and a quite detailed summary may be found in Dieudonne [Dieu, pp. 580-595]. Suffice it to say that Hirzebruch's essential st* *rategy STABLE ALGEBRAIC TOPOLOGY, 1945-1966 13 was to reduce the Riemann-Roch theorem to the index theorem. One key ingredient in the reduction should be mentioned, namely a method for splitting vector bund* *les that led later to the splitting principle in K-theory. Another nice discussion of [Hirz56] may be found in Bott's review [Bott61] of* * the second part of Borel and Hirzebruch's deeply influential work [BH58 , BH59 , BH* *60 ]. The Riemann-Roch theorem showed that the characteristic number of any projective non-singular variety M is an integer, namely the arithmetic genus of M; here oc is the complex tangent bundle of M and T is the Todd genus, which is determined by the power series T (x) = x=1 - e-x. Borel and Hirzebruch sought and proved an analogous integrality theorem for arbitrary differentiable manifo* *lds. The ^A-genus is related to the Todd genus by the formula T (x) = ex=2^A(x), and* * it satisfies ^A(x) = ^A(-x). As Bott explains clearly, this makes it plausible tha* *t the ^A genus should satisfy a similar integrality relation on arbitrary compact manifo* *lds, as Borel and Hirzebruch prove. More precisely, they prove it up to a factor of * *2 that was later eliminated by Milnor's proof (implicit in [Mil60]) that the Todd genu* *s of an almost complex manifold is an integer. Milnor and Kervaire [Mil58b, KM60 ] gave an important application of the in- tegrality of the A^-genus. In 1943, G.W. Whitehead introduced the stable J- homomorphism J : ssq(SO(n)) -! ssq+n(Sn), n large. Writing sssq= ssq+n(Sn) for the qth stable homotopy group of spheres and letting n go to infinity, this* * can be written J : ssq(SO) -! sssq. Milnor and Kervaire used the integrality theorem to prove that, when q = 4k - 1, the order jn of the image of J is divisible by * *the denominator of Bk=4k, where Bk is the kth Bernoulli number. This result gave the first sign of regularity in the stable homotopy groups of spheres, and their pr* *oof showed that the J-homomorphism is of considerable relevance to geometric topol- ogy. In fact, although this is a result in stable homotopy theory, they derive* * it from a generalization of a theorem of Rohlin in differential topology. Rohlin's* * the- orem [Ro51 , Ro52] states that the Pontrjagin number p1(M) of a compact oriented smooth 4-manifold M with w2(M) = 0 is divisible by 48. Milnor and Kervaire mimic his arguments to prove that the Pontrjagin number pn(M) of an almost par- allelizable smooth 4n-manifold is divisible by (2n - 1)!jnan, where an is 2 if * *n is even and 1 if n is odd, with equality for at least one such manifold M. For the historical story, one striking feature of the work of Borel and Hirze* *bruch is its systematic use of multiplicative functions FC(X) -! H**(X; R) and FR(X) -! H**(X; R), where FR(X) and FC(X) are the semi-groups of equivalence classes of complex and real vector bundles over X and H**(X; R) is the direct product of the real cohomology groups of X. A multiplicative function is one that converts sums to products. The authors are tantalizingly close to K-theory. Two things a* *re missing: the Grothendieck construction and Bott periodicity. The first was introduced by Grothendieck [BS58 ], who needed it to formulate * *his generalized, relative, version of the Riemann-Roch theorem in algebraic geometr* *y. Grothendieck is the inventor of the general subject of K-theory, and his ideas * *played a centrally important role in the introduction of topological K-theory. As to the second, as Bott notes in his review, the work of Borel and Hirzebru* *ch led them to an exact sequence (7.1) 0 -! Zn!-! ss2n(U(n)) -! ss2n(U(n + 1)) -! 0: 14 J. P. MAY More precisely, they proved the sequence to be exact modulo 2-torsion. As Bott writes: "The exact sequence conflicted, at the time of its discovery, with comp* *u- tations of homotopy theorists and led to a spirited controversy. At present it* * is known the sequence is exact even with regard to the prime 2." What he neglects to say is that the sequence also follows from Bott periodicity, and the conflic* *t for some time held up publication of that result. 8.Bott periodicity and K-theory One version of the Bott periodicity theorem asserts that there is a homotopy equivalence BU -! MSU. The periodicity is clearer in the equivalent reformula- tion BU x Z ' M2(BU x Z). The real analogue gives BO x Z ' M8(BO x Z). Bott's original proof of these beautiful results is based on the use of Morse t* *heory. Before proving the periodicity theorem, Bott had clearly demonstrated the power* * of Morse theory by using it to prove that there is no torsion in the integral homo* *logy of MG for any simply connected compact Lie group G [Bott56]. Bott announced the periodicity theorem in [Bott57], and he gave two somewhat different proofs, both based on Morse theory, in [Bott58, Bott59a]. It immediately became a challenge to reprove the periodicity theorems using the standard methods of algebraic topology. In the complex case, a proof was given by Toda [To62b], together with a rederivation of the Borel-Hirzebruch exa* *ct sequence (7.1), but his proof did not show that BU and MSU have the same homotopy type. The space BU is an H-space under Whitney sum, and Bott's proofs led to simple and explicit H-maps that give the equivalences. In the re* *al case, there are actually six maps that must be proven to be equivalences. These explicit maps were exploited by Dyer and Lashof [DL61 ] and Moore (written up by Cartan [Ca54-55]) to give direct calculational proofs. Actually, there is a cur* *ious simplification to be made: comparison of the proofs in [DL61 ] and [Ca54-55] sh* *ows that each finds particular difficulty in proving one of the required equivalenc* *es, but they find difficulty with different maps: combining the best of both proofs giv* *es a quite tractable argument. Finally, in their announcement [AH59 ], submitted in May, 1959, Atiyah and Hirzebruch introduce the functor K(X) for a finite CW complex X: it is the Grothendieck construction on the semi-group FC(X), and it is a ring with mul- tiplication induced by the tensor product of vector bundles. They define KO(X) similarly. They noticed a striking reinterpretation of Bott periodicity: tensor* * prod- uct of bundles induces a natural isomorphism fi that fits into the commutative diagram fi 2 K(X) K(S0) ___________//_K(X x S ) ch || |ch| fflffl| fflffl| H**(X; Q) H**(S2); Q)_ff__//H**(X x S2; Q); where ch is the Chern character and ff is the cup product isomorphism. They observe that, for connected X, the kernel K"(X) of the dimension map " : K(X) - ! Z can be identified with the set of homotopy classes of maps X -! BU. In principle, modulo a lim1 argument not yet available, this leads to a homotopy equivalence from BU to the basepoint component of M2BU. How- ever, their reinterpretation of Bott periodicity was by no means an obvious one* *. In STABLE ALGEBRAIC TOPOLOGY, 1945-1966 15 [Bott58], Bott related his explicit maps to tensor products of bundles and so p* *roved that his original version of the periodicity theorem really did imply the versi* *on no- ticed by Atiyah and Hirzebruch. Moreover, he gave the analogous reinterpretation in the real case, where a direct proof of the new version was less simple. Jumping ahead to 1963 for a moment, Atiyah and Bott together [AB64 ] then found a direct and elementary analytic proof of the complex case of the periodi* *city isomorphism in its tensor product formulation, using clutching functions to des* *cribe bundles over X x S2 explicitly. Their proof actually gives a more general resul* *t, namely a Thom isomorphism, and important refinements and generalizations are given in their lecture notes [At64] and [Bott63]. The analytic proof is relevan* *t to the Atiyah-Singer index theorem, which was already announced in 1963 [AS63 ] and which generalizes Hirzebruch's index theorem. The first published proof appeared in 1965 [Pa65], based on seminars in 1963-64. In their 1959 announcement [AH59 ] and also in [Hirz59], Atiyah and Hirzebruch give a Riemann-Roch theorem relative to a suitable map f : M -! N of differenti* *al manifolds; see Section 12 for the statement. They observe that their theorem can be rewritten for holomorphic maps between complex manifolds in the same form as Grothendieck's version of the Reimann-Roch theorem. Their results imply a new proof of the integrality of the ^A-genus, together with a sharpening in the* * case of Spin-manifolds of dimension congruent to 4 mod 8 that had been conjectured by Borel and Hirzebruch. They also rederive and give a conceptual sharpening of Milnor's result on the J-homomorphism. In [AH59 ], nothing is said about K(X) being part of a generalized cohomology theory. Moreover, it is clear that the authors as yet have no hint of K-homology and Poincare duality: their statement of the Riemann-Roch theorem involves a pushforward map f!, as it must, but that map was not well understood. They re- mark that "It is probable that f!is actually induced by a functorial homomorphi* *sm K(Y ) -! K(X)". Rather than proceed directly to 1960 and the first published account of K-the* *ory as a generalized cohomology theory, I shall interpolate a discussion of several* * quite different lines of work that were going on in the late 1950's. As preamble, Milnor [BM58 , Mil58c] saw immediately, in February, 1958, that Bott's results led to the solution of two longstanding problems; [BM58 ] is a p* *air of letters between Milnor and Bott on this subject, and [Mil58c] fills in the d* *e- tails. The relevant result of Bott is that the image of the Hurewicz homomorphi* *sm ss2n(BU) -! H2n(BU) is divisible by exactly (n - 1)!. This is closely related to the exact sequence (7.1). What Milnor deduces from this is: 1. The vector space Rn possesses a bilinear product without zero divisors only for n equal to 1, 2, 4, or 8. 2. The sphere Sn-1 is parallelizable only for n - 1 equal to 1, 3, or 7. The latter result was also proven at about the same time by Kervaire [Ker58]. 9. The Adams spectral sequence and Hopf invariant one Milnor's results just cited are also among the many implications of Adams' ce* *l- ebrated theorem that ss2n-1(Sn) contains an element of Hopf invariant one if and only if n is 1, 2, 4, or 8 [Ad60 ]. This result was announced in [Ad58b ], whi* *ch was submitted in April, 1958. This work was a sequel to and completion of work 16 J. P. MAY begun in [Ad58a ], submitted in June, 1957, in which Adams first attacked the H* *opf invariant one problem and introduced the Adams spectral sequence. Fix a prime p, let A be the mod p Steenrod algebra, and let X be a space. In its original form in [Ad58a ], the Adams spectral sequence satisfies Es;t2= Exts;tA(H*(X); Zp); where s is the homological degree, t is the internal degree, and t - s is the t* *otal degree, so that Es;t2= 0 if s < 0 or t < s. The differentials are of the form dr : Es;tr-! Es+r;t+r-1r: There is a filtration of the stable homotopy groups sssn(X) such that Es;n+s1= F ssssn(X)=F s+1;n+s+1sssn(X): The intersection of the filtrations consists of the elements of sssn(X) that ar* *e of finite order prime to p. When X = S0, {E*;*r}is a spectral sequence of differen* *tial Zp-algebras and converges as an algebra to the associated graded algebra of the ring of stable homotopy groups of spheres under the composition product. The Adams spectral sequence can be thought of in several ways: it is a so- phisticated reformulation and generalization of the Cartan-Serre method of kill* *ing homotopy groups, and it is an extension and systematization of the method of studying homotopy groups by considering higher order cohomology operations. The idea of higher order operations first appeared with Steenrod's introducti* *on of functional cohomology operations [St49]. Let f : Y -! X be a map. Steenrod showed how to construct an element x [f x0in H*(Y ) from a pair of elements x; * *x0 in H*(X) such that x [ x0= 0 and f*(x0) = 0. He defined functional mod 2 Steen- rod operations similarly. These operations are defined on a subspace of H*(X), and they are well-defined up to indeterminacy. Adem [Adem56 ] made a system- atic study of functional cohomology operations associated to stable cohomology operations, and Peterson [Pe57] gave a presentation in terms of Postnikov syste* *ms with stable k-invariants. Although a few low dimensional examples had appeared earlier, Adem [Adem58 ] gave the first systematic study of secondary cohomology operations, building on his earlier proof of the Adem relations for the iterated Steenrod operations. He related secondary and functional cohomology operations in [Adem59 ]. Peterson and Stein [PS59 ] then gave a treatment of secondary and functional operations in terms of two-stage Postnikov systems. It was this kind of treatment that Adams had in mind. Secondary and higher operations come from relations between relations, and homological algebra is the natural tool for the study of relations between relations. The essential idea * *of the construction of the Adams spectral sequence is to construct a realization o* *f a free resolution of the A-module H*(X) (in a range of dimensions) by means of a resolution of the space X. This gives a kind of exact couple of spaces that lea* *ds to an exact couple giving the desired spectral sequence on passage to homotopy groups. Implicitly, as became much clearer with a later reformulation in terms * *of the homology of spectra rather than the cohomology of spaces, the fundamental points are the representation (1.1) of cohomology and the calculation of the cohomology of Eilenberg-Mac Lane spaces in terms of Steenrod operations. The relationship to the Hopf invariant one problem comes about as follows. There is an element of Hopf invariant one in ss2n-1(Sn) if and only if there is a (stable) two-cell complex such that the Steenrod operation Sqn connects the STABLE ALGEBRAIC TOPOLOGY, 1945-1966 17 bottom cell to the top cell in mod 2 cohomology. If n is not a power of two, then Sqn is decomposable as a linear combination of iterated Steenrod operation* *s, by the Adem relations, and no such two-cell complex is possible. Now, for any connected Zp-algebra A, Ext1;tA(Zp; Zp) is isomorphic to the dual of the vector space of degree t indecomposable elements of A. Take A to be the mod 2 Steenrod algebra and considerithe Adams spectral sequence for X = S0. Then we have elements hi 2 E1;22dual to the Steenrod operations Sq2i. It is direct from the construction ofithe spectral sequence that there is an element of Hopf invarian* *t one detected by Sq2 if and only if hi is a permanent cycle in the spectral sequence. The element h0 corresponds to the Bockstein Sq1 = fi, and multiplication by h0 in the spectral sequence detects multiplication by 2 in the stable homotopy gro* *ups of spheres. Adams computes enough of Es;*2, s = 2 and s = 3, to see that the elements h0h2iare non-zero in E2 for i 3. The only way that h0h2ican be a boundary is if d2(hi+1) = h0h2i. If i 3 and both hi and hi+1 are permanent cycles, we conclude that hi represents an odd dimensional homotopy class xi such that 2x2iis non-zero. This is impossible since sss*is a graded commutative ring* *. This implies the main theorem of [Ad58a ]: if both ss2n-1(Sn) and ss4n-1(S2n) contain elements of Hopf invariant one, then n 4, which was tantalizingly close to the expected answer. This line of argument doesn't work to solve the problem. However, the method of proof implies that Sq16, although indecomposable in A, admits a decomposition in terms of composites of primary and secondary operations, taking into account the relevant domains of definition and indeterminacy. In [Ad60 ], Adams constru* *cts such a decomposition of Sq2i for all i 4. While the argument makes no use of the Adams spectral sequence, it implies the differential d2(hi+1) = h0h2ifor i * * 3. The arguments in [Ad60 ] are very long, and I won't attempt a complete summar* *y. They require a more thorough exposition of the foundations of graded homological algebra than was needed in [Ad58a ], and this work has been used ever since. Th* *ey also require an axiomatization and construction of secondary cohomology opera- tions in terms of universal examples, together with a detailed study of how to * *relate the homological algebra to the analysis of the operations. Finally, particular * *oper- ations relevantnto the problem at hand are constructed, a putative decomposition formula for Sq2n is proven formally by means of the general theory, and the coe* *ffi- cient of Sq2 in the decomposition is proven to be non-zero by explicit calcula* *tion in a specific example. There are two crucially important ingredients in the work that must be singled out. First, the work of Milnor and Moore [MM65 ] on graded Hopf algebras plays a key role in the relevant homological algebra. Although [MM65 ] was not pub- lished until 1965, a mimeographed version was distributed much earlier and was an essential prerequisite to the higher level of algebraic sophistication that * *Adams introduced into algebraic topology. Second, Adams needed to make some calculations of E2 beyond those of [Ad58a ], and for this purpose he made substantial use of Milnor's remarkable analysis of* * the structure of the Steenrod algebra [Mil58a]. This analysis has played a central * *role in a great many later calculations in stable algebraic topology. The Steenrod alge* *bra A is a Hopf algebra. Its coproduct is determined by the Cartan formula and is cocommutative. Therefore the dual Hopf algebra, denoted A*, is commutative as an algebra. Milnor proved that it is a free commutative algebra in the graded s* *ense. 18 J. P. MAY Explicitly, for an odd prime p, it can be written as a tensor product (9.1) A* = E{oi|i 0} P {i|i 1} of an exterior algebra on odd degree generators oi and a polynomial algebra on even degree generators i. Moreover, the coproduct on the generators admits a simple explicit formula, in principle equivalent to the Adem relations but far * *more algebraically tractable. The dual B of P {i|i 1}can be identified both with the subalgebra of A generated by the Steenrod operations P iand with the quotient of A by the two-sided ideal generated by the Bockstein fi. Note that, in quotie* *nt form, B also makes sense when p = 2. We shall come back to it later. Shortly after Adams' work, the techniques he developed were adapted to solve the analogue of the Hopf invariant one problem at odd primes p, showing that th* *ere can be a two-cell complex with P nconnecting the bottom cell to the top cell in mod p cohomology if and only if n = 1. This work was done independently by Liulevicius [Liu62a] and by Shimada and Yamanoshita [SY61 ]. Using the structure theory for mod p Hopf algebras of Milnor and Moore and Milnor's analysis of the Steenrod algebra, I later developed tools in homologic* *al algebra that allowed the use of the Adams spectral sequence for explicit comput* *a- tion of the stable homotopy groups of spheres in a range of dimensions consider* *ably greater than had been known previously [May65 , May65, May66]. Correspondence initiated in the course of this work led Adams and myself to a long friendship,* * and I have given a brief account of all of Adams' work in [May2 ] and a eulogy and personal reminiscences in [May1 ]. 10. S-duality and the introduction of spectra Setting up the Adams spectral sequence as Adams did it originally is a tedious business, the reason being that one is trying to do stable work with unstable o* *bjects: one should be using "spectra" rather than spaces. Similarly, the representabili* *ty of ordinary cohomology and the introduction of cobordism and K-theory must even- tually have forced the introduction of spectra, which appear naturally as seque* *nces of Eilenberg-Mac Lane spaces, as sequences of Thom spaces, and as sequences of spaces featuring in the Bott periodicity theorem. Nevertheless, the fact is that the introduction of spectra had nothing whatev* *er to do with these lines of work. Rather, it grew out of the work on S-duality of Spanier and Whitehead. I will be brief about this since it is also treated in [* *BG ] in this volume. In 1949, Spanier [Sp50] reconsidered Borsuk's cohomotopy groups [Bor36]. For a (nice) compact pair (X; A), where dim X < 2n - 1, Spanier defined ssn(X; A) to * *be the set of homotopy classes of maps (X; A) -! (Sn; *). As in Borsuk [Bor36], th* *ese are abelian groups, and Spanier showed that these cohomotopy groups satisfy all* * of the Eilenberg-Steenrod axioms for a cohomology theory, except that they are only defined in a range of non-negative degrees depending on the dimension of X. He * *also showed that the cohomotopy groups map naturally to the integral Cech cohomology groups and that, for a CW complex X with subcomplex A, ssn(Xm [ A; Xm-1 [ A) is isomorphic to the cellular cochain group Cm (X; A; ssm (Sn)). These were pu* *z- zling results. The real explanation, that these cohomotopy groups are the terms* * in a positive range of dimensions of a cohomology theory whose coefficients are no* *n- zero in negative dimensions, would come later. With hindsight, the cellular coc* *hain STABLE ALGEBRAIC TOPOLOGY, 1945-1966 19 isomorphism just mentioned is the first hint of the Atiyah-Hirzebruch spectral * *se- quence for stable cohomotopy theory. Spanier also observed that the Hurewicz isomorphism theorem for [Sn; X] and the Hopf classification theorem for [X; Sn] are dual to one another. To make a home for such duality phenomena in all dimensions, Spanier and Whitehead devised the S-category in [SW53 , SW57 ]. Its objects are based space* *s, and the set {X; Y }of S-maps X -! Y is {X; Y }= colimn0[nX; nY ]: That is, homotopy classes of based maps f : nX -! nY and g : qX -! qY define the same S-map if kf and n-q+kg are homotopic for some k 0. The S-category is additive, and : {X; Y }-! {X; Y } is a bijection. Although obscured by their language of "carriers", in retrospect a most unfor* *tu- nate choice of technical details, Spanier and Whitehead introduce graded morphi* *sms by setting {X; Y }q= {qX; Y } if q 0 and {X; -qY } if q < 0. They prove that, for CW complexes X and Y with X finite, the abelian groups {X; Y }qsatisfy all except the dimension axiom of the Eilenberg-Steenrod axioms for a homology theory in Y when X is fixed and for a cohomology theory in X when Y is fixed. They even set up the Atiyah-Hirzebruch spectral sequences for stable homotopy and stable cohomotopy. However, they do not take the step of describing their results in a language * *of homology and cohomology theories, and none of their later papers return to this point of view. With their definitions, the wedge axiom would not be satisfied * *in cohomology for infinite X, and only homology and cohomology theories represented by suspension spectra of spaces would be obtained. Thus this would not have been the right way to set up generalized homology and cohomology theories, and that was far from their intention. The useful version of the Spanier-Whitehead categ* *ory is its full subcategory of finite CW complexes. This category is far too small* * to form a satisfactory foundation for stable homotopy theory, but it is appropriat* *e for the study of duality between finite CW complexes, which is the main point of the papers [SW55 , SW58 ] and the expository notes [Whi56 , Sp56, Sp58]. The 1956 note [Sp56] of Spanier, reviewed by Hilton, gives a nice description of dual theorems in algebraic topology and seems to have been a forerunner of Eckmann-Hilton duality. The 1956 survey of Whitehead [Whi56 ] looks more to- wards the past, based as it was on Whitehead's presidential address to the Lond* *on Mathematical Society. Prior to this point, it had been common practice to discu* *ss duality in ordinary homology and cohomology in terms of Pontryagin duality of groups. Whitehead gives an interesting exposition of this point of view on dual* *ity, the role of colimits in understanding singular homology and Cech cohomology, and various other aspects of duality theory in algebraic topology. At that stage in* * our story, it is not very surprising that Whitehead understands the Eilenberg-Steen* *rod axioms solely in terms of ordinary homology and cohomology theories. In retrospect, it is more surprising that Spanier in his 1959 paper [Sp59b] s* *till understands the axioms this way. In a footnote, he refers to the Eilenberg-Stee* *nrod axioms to specify what he means by homology and cohomology, and of course he means all of the axioms. There is no hint of generalized homology and cohomol- ogy theories in the paper, although one of its main points is the convenience a* *nd importance of spectra in the study of duality theory. Nevertheless, the work of 20 J. P. MAY Spanier and Whitehead, especially the work in [Sp59b], was soon to lead to dual* *ity theorems in generalized homology and cohomology. Before saying more about [Sp59b], I should mention the interesting paper [Sp5* *9a] that Spanier wrote a year earlier. In it, he returns to the Dold-Thom descripti* *on [DT58 ] of integral homology as the homotopy groups of the infinite symmetric product, and he shows how this can be related to the S-category and Spanier- Whitehead duality. Function spaces are used heavily in the comparison, and it seems that their use may have led to the idea of spectra. In any case, Spanier's student Lima introduced spectra in his 1958 thesis, pu* *b- lished in [Lima59]. In Lima's work, a spectrum is a sequence of based finite CW complexes Li and S-maps i : Li -! Li+1. Lima also considers inverse spec- tra, with structure maps reversed. He uses spectra to give an extension of the S-category and an extension of Spanier-Whitehead duality from polyhedra embed- ded in spheres to general compact subspaces of spheres. In a sequel, Lima [Lima* *60] develops Postnikov systems in his category of spectra. He also gives a curious * *dual theory whose dual Postnikov invariants lie in homology groups with coefficients* * in cohomotopy groups. In Spanier's paper [Sp59a], he redefines spectra X to be sequences of based s* *paces Tiand based maps, not S-maps, oei: Ti- ! Ti+1that satisfy certain connectivity and convergence conditions. These conditions have the effect of giving his spec* *tra a stable range analogous to the one implied for the suspension spectrum iX of a based space X by the generalized Freudenthal suspension theorem, which was first proven in [SW57 ]. His intent is to recast Spanier-Whitehead duality in t* *erms of smash products X ^ Y and function spectra F(X; Y ), where X and Y are based spaces and F(X; Y ) has ith space the function space F (X; iY ). Curiously, he * *does not define general function spectra F(X; T ). He writes F(X) for F(X; S0) and c* *alls it the functional dual of X, and he observes that H-q(X) ~=Hq(F(X)). He defines stable maps {X; T }from a space to a spectrum and shows that there are canonical duality isomorphisms {X; F(Y; Sn)}~={X ^ Y; Sn}~={Y; F(X; Sn)}: (Actually, his statement of this has F(-; Sn) replaced with the n-fold suspensi* *on of the functional dual, but his definition of suspension disagrees with the mod* *ern one.) While the asymmetry between spaces and spectra is clearly unsatisfactory, this was a step from the S-category towards the true stable homotopy category. He then redefines what it means for spaces X and Y to be n-dual to one anothe* *r. Let in 2 H"n(Sn) be the fundamental class. A map " : Y ^ X -! Sn is said to be an n-duality map if the homomorphism f" : H"q(Y ) -! H"n-q(X) defined by f"(y) = "*(in)=y is an isomorphism, where = is the slant product. He proves that " determines and is determined by a weak equivalence from the suspension spectrum of Y to F(X; Sn) such that the following diagram of spaces commutes in the S-category: 0^id n Y ^ XG_____________//_F (X; S ) ^ X GGG qqqq GGG qqq"q " GG## xxqq Sn: This gives an intrinsic characterization of the n-dual of X that leads to all* * of the properties proven in the earlier work of Spanier and Whitehead [SW55 ]. The ear* *lier STABLE ALGEBRAIC TOPOLOGY, 1945-1966 21 work shows that if X is embedded in Sn+1 and Y is embedded in the complement of X in such a way that the inclusion Y -! Sn+1 - X induces an isomorphism of all homology groups, then there is a duality map " : Y ^X -! Sn. This unfortunately means that Spanier's new notion of an n-duality is what in the earlier work was called an (n+1)-duality. The new notion relegates the role of the embeddings to* * the verification of a more conceptual defining property and makes it much simpler to determine when spaces X and Y are n-dual to one another. It is equivalent to the modern homotopical definition of a duality map in the stable homotopy category. All of this work of Spanier and Whitehead was independent of the work on cobordism, integrality theorems, and K-theory that was going on at the same tim* *e. In [MS60 ], submitted a month after [Sp59a], Milnor and Spanier show that if a smooth compact n-manifold M is embedded in the pair Rn+k with normal bundle , then the Thom space T () is (n + k)-dual (new style) to M+ . Moreover, they show that if k is sufficiently large, then is fiber homotopy trivial if and on* *ly if there is an S-map Sn -! M of degree one. They also make the nice observation that Adams' solution to the Hopf invariant one problem implies that the tangent bundle of a homotopy n-sphere is fiber homotopy trivial if and only if n is 1, * *3, or 7. A year later, in [At61c], Atiyah made a systematic study of the relationship between Thom complexes and S-duality. In particular, he proved the Atiyah duali* *ty theorem, which identifies the (n+k)-dual of the cofibration sequence @M+ ! M+ ! M=@M of a smooth compact n-manifold M with boundary @M as the cofibration sequence T ((@M)) -! T ((M)) ! T ((M))=T ((@M)) associated to the normal bundles of a proper embedding of the pair (M; @M) in (Rn+k-1 x [0; 1); Rn+k-1 x {0}). He also proved that, for any bundle over a smooth compact manifold M without boundary, the Thom complex T () is S-dual to the Thom complex T (plus? ), where plus? is trivial. We will return to this paper when we discuss the J-homomorphism. 11.Oriented cobordism and complex cobordism With the aid of the Adams spectral sequence, the work of Thom on the oriented cobordism ring could be completed. Although slightly ahistorical, the language * *of spectra will clarify how this came about. Using the structural maps oe : Tn -! Tn+1, the homotopy, homology, and cohomology of a spectrum T = {Tn} can be defined as follows: (11.1) ssq(T ) = colimssn+q(Tn) (11.2) Hq(T ) = colim"Hn+q(Tn) and (11.3) Hq(T ) = lim"Hn+q(Tn); where the last definition is only correct when lim1"Hn+q-1(Tn) = 0. As Adams noted in 1959 [Ad59 ], the Adams spectral sequence generalizes readily to a spe* *ctral sequence for the computation of ss*(T ) in terms of the mod p cohomology H*(T ), regarded as a module over the Steenrod algebra A. The E2-term is given by Es;t2= Exts;tA(H*(T ); Zp); 22 J. P. MAY and everything said earlier applies, with simpler proofs, in this more general * *setting. For each of the familiar sequences of classical groups G(n), namely G = O, SO, U, SU, Sp, and Spin, the Thom spaces T G(n) of the universal bundles give a Thom spectrum MG. A uniform method of attack on the problem of computing ss*(MG) is to first compute the mod p cohomology of MG for each prime p and then compute the mod p Adams spectral sequence. A key reason that Thom was able to compute N* completely was that the mod 2 cohomology H*(MO) is a free module over the mod 2 Steenrod algebra A. A quick direct proof of this fact, using Hopf algebra techniques, was given by Liulevic* *ius [Liu62b] in 1962. For an abelian group ss, the sequence of spaces K(ss; n) gives a spectrum Hss such that ss0(Hss) = ss and the remaining homotopy groups of Hss are zero. The mod p cohomology of HZp is the mod p Steenrod algebra, as Cartan had implic- itly shown [Ca55 ]. The representation of cohomology (1.1) generalizes to spect* *ra. Representing generators of H*(MO) as maps from MO to suspensions of HZ2, one obtains a map from MO to a product of suspensions of HZ2 that induces an isomorphism on mod 2 cohomology. Since one knows that ss*(MO) is a Z2-vector space, one readily deduces that this map is an equivalence of spectra, allowing* * one to read off ss*(MO). However, a good homotopy category of spectra in which to make such a deduction only appeared later. Using spectra and the Adams spectral sequence, Milnor [Mil60] in 1959 proved that M* = ss*(MSO) has no odd torsion. This was proven independently by Averbuh [Av59 ] and, a little later, Novikov [Nov60 ]. These are announcements. Averbuh's proofs never appeared and Novikov's proofs [Nov62 ] seem never to have been translated from the Russian. Also in 1959 [Wall60], but without using spectra or the Adams spectral sequen* *ce, Wall determined the 2-torsion in M*. In particular, he proved that M* has no elements of order 4 and that two oriented manifolds are cobordant if and only if they have the same Stiefel-Whitney and Pontryagin numbers. These results were both conjectured by Thom [Thom54 ]. A nice deduction from the explicit form of the generators Wall found is that the square of any manifold is cobordant to an oriented manifold, and he remarked the desirability of a direct geometric proof* *; we shall return to this in Sections 16 and 17. After calculating the 2-torsion in M* by other means, Wall used this calculat* *ion to prove that the mod 2 cohomology H*(MSO) is the direct sum of suspensions of copies of A and of A=ASq1. He remarks "It seems that a direct proof ... would be extremely difficult", but he found such a direct proof not long afterwards [Wal* *l62]. That allows a more direct calculation of M*. In fact, the mod 2 cohomology of HZ is A=ASq1. As Browder, Liulevicius, and Peterson observed later [BLP66 ], it follows that there is a map f from the spectrum MSO to a product of suspensions of copies of HZ and HZ2 that induces an isomorphism on mod 2 cohomology. In a good homotopy category of spectra, one readily deduces that f is a 2-local equivalence. Of course, the foundations for such an argument only came later, b* *ut the calculation of homotopy groups is easily made by use of the Adams spectral sequence. Milnor [Mil60] and Novikov [Nov60 , Nov62] also introduced and calculated com- plex cobordism ss*(MU). Although the geometric interpretation was not included in Milnor [Mil60], this is the cobordism theory of weakly almost complex manifo* *lds, namely manifolds with a complex structure on their stable normal bundles. The STABLE ALGEBRAIC TOPOLOGY, 1945-1966 23 explicit calculation, carried out one prime at a time and then collated algebra* *ically, showed that ss*(MU) is a polynomial ring on one generator of degree 2i for each i 1. Interestingly, there is no known geometric reason why the complex cobor- dism ring should be concentrated in even degrees. The analogue for symplectic cobordism is false. The cited papers of Milnor and Novikov raise the question of determining ss*(MG) for other classical groups G and give some information. We will return to this in Sections 16 and 17. 12. K-theory, cohomology, and characteristic classes In their 1960 paper [AH61a ], Atiyah and Hirzebruch explicitly introduce K- theory as a generalized cohomology theory. Whether or not the idea of taking a generalized cohomology theory seriously occurred to anyone before, this paper is the first published account. They restrict attention to finite CW complexes X f* *or convenience, but they are fully aware of both represented K-theory and inverse limit K-theory, namely the inverse limit of K*(Xn) as Xn runs over the skeleta of X. Using Bott periodicity, they prove that Z-graded K-theory satisfies all * *of the Eilenberg-Steenrod axioms except the dimension axiom and they introduce Z2- graded K-theory. Regarding ordinary rational cohomology as Z2-graded by sums of even and odd degree elements, they prove that the Chern character extends to a multiplicative map of cohomology theories ch : K*(X) -! H**(X; Q) which becomes an isomorphism when the domain is tensored with Q. They also introduce what is now called the Atiyah-Hirzebruch spectral sequenc* *e. It satisfies Ep;q2= Hp(X; Kq(pt)); and it converges to K*(X). Since it is compatible with Bott periodicity, it may be regraded so as to eliminate the grading q. It collapses, E2 = E1 , if H*(X; * *Z) is concentrated in even degrees or, using the Chern character, if H*(X; Z) has no torsion. They state without proof that d3 can be identified with the integr* *al operation Sq3, and they give partial information about the product structure. T* *hey also state without proof that the spectral sequence generalizes to a Serre type spectral sequence for the K-theory of fibre bundles. The Riemann-Roch theorem of their earlier paper [AH59 ] is generalized to the cohomology theory K*, but still with no hint of K-homology and a genuine push- forward map in K-theory. The theorem states that if f : M -! N is a continuous map betweeen compact oriented differentiable manifolds and if there is a given * *el- ement c1(f) 2 H2(M; Z) such that c1(f) w2(M) - f*w2(N) mod 2, then, for x 2 K*(M), (12.1) f!(ch(x)ec1(f)=2. ^A(M)) = ch(f!(x)) . ^A(N) in H*(N; Q). On the left f!is the pushforward in rational cohomology determined by Poincare duality and f*; a posteriori, f!is defined similarly in K-theory. Using both the Riemann-Roch theorem and the spectral sequence, they study the K-theory of certain differentiable fiber bundles and compute K*(G=H) explicitly when H is a closed connected subgroup of maximal rank in a compact connected Lie group G. Moreover, when H*(G; Z) has no torsion, they prove that the natural map R(H) -! K(G=H) is surjective. Calculations with the maximal rank condition dropped came much later. 24 J. P. MAY Taking K(BG) to be the inverse limit K-theory of BG, they define a homomor- phism ff : R(G)^I-! K(BG) and prove that it is an isomorphism when G is a compact connected Lie group. They also prove that K1(BG) = 0 for such G. The proof is by direct calculation when T is a torus and by comparison with the res* *ult for a maximal torus in general. They conjecture that this result remains true f* *or any compact Lie group G. In [At61b], which appeared in 1961, Atiyah proves the same result for finite groups G. The proof is by direct calculation when G is cyclic, by induction up a composition series when G is solvable, and by application of the Brauer induction theorem to pass from solvable groups to general finite groups. The second step depends on a Hochschild-Serre type spectral sequence that satisfies Ep;q2= Hp(G=N; Kq(BN)) and converges to K*(BG), where N is a normal sub- group of G. The last step depends on the transfer homomorphisms in K-theory associated to finite covers. Atiyah claims in a footnote that the result does r* *emain true for general compact Lie groups. However, a proof did not appear until the 1969 paper [AS69 ] of Atiyah and Segal, which is based on the use of equivariant K-theory. This was developed in lectures at Harvard and Oxford in 1965, but the first published accounts appeared later [At66a, Seg68]. In 1961 [AH61c ], Atiyah and Hirzebruch make use of real K-theory KO to obtain a number of interesting results on characteristic classes in ordinary mo* *d p cohomology. These are less well-known than they ought to be, perhaps because [AH61c ] is written in German; some of its results were later reworked by Dyer [Dyer69]. Atiyah and Hirzebruch greatly extend and clarify observations Hirzebr* *uch had already made in 1953 [Hirz53], and they improve results in the expository p* *aper [AH61c ], also in German, which was written a bit earlier and contains a nice g* *eneral overview of the authors' results on K-theory, including some that I will not di* *scuss here. In [AH61c ], using Milnor's analysis of the Steenrod algebra, Atiyah and Hirz* *e- bruch first determine the group of naturalPring isomorphisms : H**(X)P-! H**(X). The obvious examples are = Sq Sqr if p = 2 and = P P r if p > 2. For a Zp-oriented vector bundle with Thom isomorphism OE, they define __() = OE-1OE(1). Thus S_q is the total Stiefel-Whitney class and P_is the tot* *al Wu class. They observe that, for a finite CW complex X, __extends to a natural homomorphism from KO(X) to the group G**(X) of elements of H**(X) with zeroth component 1 and, if p > 2, odd components zero, where the multiplication in G**(X) is given by the cup product. Write W u(; ) = -1__(). Then, when p = 2, X W u(Sq; ) = 2iTi(w1(); . .;.wi()); i0 where the Ti are the Todd polynomials. Here the right side makes sense since 2i* *Ti is a rational polynomial with denominator prime to 2. When p > 2, let f = p1=p-1 and let Pi be the ith Pontryagin class. Then X X W u(P; ) = f2iLi(P1(); . .;.Pi()) = f2i^Ai(P1(); . .;.Pi()): i0 i0 In both cases, there is an implied analogue for complex bundles, with Chern cla* *sses appearing on the right-hand sides of the equations. These formulas suggest a relationship between the differential Riemann-Roch theorem and Wu's formulas for the characteristic classes of manifolds. Let f : STABLE ALGEBRAIC TOPOLOGY, 1945-1966 25 M -! N be a continuous map between differentiable manifolds M and N. Atiyah and Hirzebruch prove that, for any x 2 H*(M), (12.2) f!((x) . W u(-1; oM )) = (f!(x)) . W u(-1; oN ); where f!is the pushforward map determined by Poincare duality and f*. When N is a point, this reduces to <(y); [M]> = <(y . (W u(; oM ); [M]>: Taking = Sq if p = 2 or = P if p > 2, this is Wu's formula for the determinat* *ion of the Stiefel-Whitney or L-classes of M in terms of Steenrod operations and cup products in H*(M). It should be remarked at this point that Adams [Ad61b ] proved the Wu relatio* *ns for not necessarily differentiable manifolds in 1961. In 1960 [Ad61a ], he pro* *ved an integrality theorem for the Chern character. Atiyah and Hirzebruch [AH61c ] observe that (12.2) is an analogue of the differentiable Riemann-Roch theorem (12.1), and they show that this is more than just an analogy by using Adams' integrality theorem to derive important cases of (12.2) from (12.1). In a notew* *orthy remark, they point out that one can ask for such a Riemann-Roch type theorem whenever one has a natural transformation from one generalized cohomology theory to another, provided that both theories satisfy an analogue of Poincare duality* * that allows pushforwards to be defined. This still precedes Poincare duality in K-th* *eory. Even without K-homology, Atiyah in 1962 [At62] found an ingenious and influ- ential proof of a K"unneth theorem for K-theory, obtaining a short exact sequen* *ce of the expected form 0 -! K*(X) K*Y )-ff!K*(X x Y )-fi!Tor(K*(X); K*(Y )) -! 0: 13. Generalized homology and cohomology theories The work of G.W. Whitehead [Wh60 , Wh62a ] and Brown [Br63, Br65] defined and characterized represented generalized homology and cohomology theories in close to their modern forms. We have seen that K-homology is nowhere men- tioned in the work of Atiyah and Hirzebruch. However, Whitehead's announce- ment [Wh60 ] of his definition of represented homology was already submitted in February, 1960, and appeared that year, although the full paper [Wh62a ] was not submitted until May, 1961, and appeared in 1962. More surprisingly, [Wh62a ] makes no mention of either K-theory or bordism and contains no references to Atiyah and Hirzebruch, although the Bott spectrum is mentioned briefly. There seems to have been little mutual influence. It seems that the main influence on Whitehead was his own earlier work on the homotopy groups of smash products of spaces [Wh56 ] and the work on duality of Spanier and J.H.C. Whitehead [SW55 ] and its further development by Spanier [Sp59b]. Whitehead defines a spectrum E to be a sequence of spaces Ei and maps oei: Ei- ! Ei+1, dropping the convergence conditions that Spanier imposed. He says that E is an M-spectrum if the adjoint maps "oe: Ei- ! MEi+1are homotopy equivalences. Actually, he insists on spaces Ei for all integers i, rather tha* *n for i 0 as is now more usual. He defines a map f : E -! E0 to be a sequence of 26 J. P. MAY maps fi: Ei- ! E0isuch that the diagrams Ei __oe//i_Ei+1 (13.1) fi || |fi+1| |fflffloe0fflffl|i E0i ____//_E0i+1 commute up to homotopy, and he says that two maps f and g are homotopic if fi' gi for all i. Taking the obvious steps beyond Spanier [Sp59b], Whitehead defines the functi* *on spectrum F(X; E) and the smash products E ^X ~=X ^E between a based space X and a spectrum E. As an unfortunate choice, he restricts X to be compact in the* *se definitions, and his homology and cohomology theories are only defined on finite CW complexes. Remember that the additivity axiom came a bit later. In particula* *r, these definitions give ME = F(S1; E) and E = E ^ S1 (except that he writes the suspension coordinate on the left). Defining the homotopy groups of spectra as * *in (11.1), he proves that suspension gives an isomorphism * : ssq(E) -! ssq+1(E). For finite based CW complexes X and a spectrum E, Whitehead defines (13.2) "Hq(X; E) = ssq(E ^ X): This is suggested by the more obvious cohomological analogue (13.3) H"q(X; E) = ss-q(F (X; E)): In retrospect, this definition of homology is correct for general CW complexes * *X, but this definition of cohomology is only correct for general CW complexes X wh* *en E is an M-spectrum. Much of [Wh62a ] is concerned with products in generalized homology and coho- mology theories. These are induced by pairings (D; E) -! F of spectra, which are specified by maps Dm ^ En -! Fm+n that are suitably compatible up to homotopy with the structure maps oe of D, E, and F . Starting from such pairings of spectra, Whitehead defines and studies t* *he properties of external products "Hm(X; D) "Hn(Y ; E) -! "Hm+n(X ^ Y ; F ) H"m (X; D) "Hn(Y ; E) -! "Hm+n(X ^ Y ; F ) and slant products \ : "Hn(X ^ Y ; D) "Hm(X; E) -! "Hn-m(Y ; F ) = : "Hn(X ^ Y ; D) "Hm(Y ; E) -! "Hn-m(X; F ): He obtains cup and cap products by pulling back along diagonal maps. By now, all of this is familiar standard practice. Similarly, the familiar duality theorems are proven. Whitehead defines a ring spectrum E in terms of a product (E; E) -! E and unit S -! E, where S is the sphere spectrum, namely the suspension spectrum of S0. He defines an E- orientation of a compact connected n-manifold M in terms of a fundamental class in "Hn(M; E), and he proves a version of Alexander duality for dual pairs embed* *ded in M. This specializes to give Poincare duality for M. Taking M = Sn+1, which is E-oriented for any E, it specializes to give Spanier-Whitehead duality in any theory. STABLE ALGEBRAIC TOPOLOGY, 1945-1966 27 When [Wh62a ] was written, Brown [Br63] had already proven his celebrated representation theorem. That paper also gave an incorrect first approximation to Milnor's additivity axiom [Mil62]. In fact, James and Whitehead [JW58 ] had exhibited homology theories that fail to satisfy the additivity axiom and whose existence contradicted one of Brown's results. The correction of [Br63] noted t* *his and pointed out simpler axioms for the representability theorem. Brown later published the improved version in a general categorical setting [Br65]. That ve* *rsion is one of the foundation stones of modern abstract homotopy theory. Let k be a contravariant set-valued homotopy functor defined on based CW complexes. The functor k is said to satisfy the Mayer-Vietoris axiom if, for a * *pair of subcomplexes A and B of a CW complex X with union X and intersection C, the natural map from k(X) to the pullback of the pair of maps k(A) -! k(C) and K(B) -! k(C) is surjective; k is said to satisfy the wedge axiom if it converts wedges to products. Brown in [Br65] proves that k(X) is then naturally isomorph* *ic to [X; Y ] for some CW complex Y . If k is only defined on finite CW complexes, Brown reaches the same conclusion but with a countability assumption on the k(S* *q). Adams [Ad71a ] later showed that the countability assumption can be removed when the functor k is group-valued. Applied to the term "kn(-) of a (reduced) generalized cohomology theory "k*, Brown's theorem gives a CW complex En such that "kn(X) ~=[X; En] for all CW complexes X. The suspension axiom on the theory leads to homotopy equivalences En -! MEn+1. Thus a cohomology theory "k*gives rise to an M-spectrum E. Whitehead [Wh62a ] followed up by using Spanier's version [Sp59b] of duality th* *eory to show that a homology theory gives rise to a cohomology theory on finite CW complexes. Applying Brown's theorem for finite CW complexes (and using Adams' variant to avoid countability hypotheses), it follows that a homology theory on finite CW complexes is also representable by a spectrum. Since the Brown representation is natural, a map of cohomology theories gives rise to a map of M-spectra. Defining the category of cohomology theories on spa* *ces in the evident way, we see that it is equivalent to the homotopy category of M- spectra E whose spaces En are homotopy equivalent to CW complexes. We call this the Whitehead category of M-spectra. Milnor's basic result [Mil59] that the loop space of a space of the homotopy type of a CW complex has the homotopy type of a CW complex is relevant here. Via the suspension spectrum functor and a functor that converts spectra to M-spectra, one can check that the S-category of finite CW complexes embeds as a full subcategory of the Whitehead category. Thus the Whitehead category is an approximation to stable homotopy theory that substantially improves on the S- category by providing the proper home for cohomology theories on spaces. Howeve* *r, as we shall see in Section 21, this is not yet the genuine stable homotopy cate* *gory. In the summer of 1962, there was an International Congress in Stockholm, pre- ceded by a colloquium on algebraic topology at Aarhus. The proceedings of the latter contain brief expositions of generalized cohomology by Dold [Dold62], Dy* *er [Dyer62], and Whitehead [Wh62b ]. Dold was the first to make the important ob- servation that rational cohomology theories are products of ordinary cohomology theories, and he gave the first general exposition of the Atiyah-Hirzebruch spe* *ctral sequence. Making systematic use of Brown's representability theorem, his later book [Dold66], in German, gave a complete treatment of these matters and much 28 J. P. MAY else. Dyer was the first to write down a general treatment of the Riemann-Roch theorem, although already in 1962 he described the result as a folk theorem kno* *wn to Adams, Atiyah, Hirzebruch and others. His later book [Dyer69] gave a complete treatment, along with an exposition of much of the work of Atiyah and Hirzebruch described in the previous section. He still avoids use of K*, but this appears * *im- plicitly in the form of Atiyah duality, which allows an appropriate definition * *of pushforward maps. Not everything in cohomology theory was to be done using its represented form. For example, working directly from the axioms, Araki and Toda [AT65 ] made a sy* *s- tematic study of products in mod q cohomology theories and of Bockstein spectral sequences in generalized cohomology. Nevertheless, most work was to be simplifi* *ed and clarified by working with represented theories. 14.Vector fields on spheres and J(X) In the proceedings of the 1962 Aarhus and Stockholm conferences, Adams [Ad62d* * ] described his solution of the vector fields on spheres problem [Ad62b , Ad62c] * *and outlined his work on the groups J(X), which appeared gradually in [Ad63 , Ad65a, Ad65b, Ad66a]. I summarized these papers in [May2 ], emphasizing their impact on later work and the reformulations that became possible with later technology. These applications of K-theory have been of central importance to the developme* *nt of stable algebraic topology. The key new idea was the introduction of the Adams operations k in real and complex K-theory. These play a role in K-theory that is of comparable importance to the role played by Steenrod operations in ordinary mod p cohomology. It was clear from Grothendieck's work [Gro57] how to extend the exterior power operati* *ons k from vector bundles to K-theory. The "Newton polynomials" Qk that express the power operations xk1+. .+.xknin a polynomial ring Z[x1; : :;:xn] as polynom* *ials in the elementary symmetric polynomials oek were familiar to topologists from t* *heir role in the study of characteristic classes. Adams' ingenious idea was to define k(x) = Qk(1(x); : :;:n(x)): Here X is a finite CW complex, x 2 K(X), and n is large. Either by a representation theoretical argument, as in [Ad62c ], or by use of* * the splitting principle and reduction to the case of line bundles, one finds that t* *he k are natural ring homomorphisms that commute with each other. They are easily evaluated on line bundles and on the K-theory of spheres, and their relationshi* *p to the Chern character and the Bott isomorphism are easily determined. They greatly enhance the calculational power of K-theory. Adams discovered these operations after first trying to solve the vector fiel* *ds on spheres problem by use of secondary and higher operations in ordinary cohomology in [Ad62a ], a paper that was obsolete by the time it appeared. The idea that a problem that required higher order operations in ordinary cohomology could be solved using primary operations in K-theory had a strong impact on the directio* *ns taken by stable algebraic topology. The vector fields on spheres problem asks how many linearly independent vector fields there are on Sn-1. The answer is ae(n) - 1. Here ae(n) = 2c + 8d, where n = (2a + 1)2b and b = c + 4d, 0 c 3. It had long been known [Eck42] that there exist ae(n) - 1 such fields. Adams proved that there are no more. Work of James [Ja58a, Ja58b, Ja59] had reduced the problem to a question about the STABLE ALGEBRAIC TOPOLOGY, 1945-1966 29 reducibility of a certain complex. Up to suspension, Atiyah [At61c] identified * *the S-dual of that complex with a stunted projective space. This reduced the problem to the question of the coreducibility of X = RPm+ae(n)=RPm-1 for a suitable m. Here coreducibility means that there is a map f : X -! Sm that has degree 1 when restricted to the bottom cell Sm of X. Adams proves that X is not coreducible, thus solving the problem. For the proof, Adams starts with the calculation of K(CPn) and K(CPn=CPm ), which was first carried out by Atiyah and Todd [AT60 ]. He next calculates K(RP* *n) and K(RPn=RPm ). Finally he calculates KO(RPn) and KO(RPn=RPm ). In each case, he obtains complete information on the ring structure and the Adams op- erations. The main tools are just the Atiyah-Hirzebruch spectral sequence and the Chern character. For X as above, the existence of a coreduction f and the naturality relation f* k = kf* lead to a contradiction. For a connected finite CW complex X, define J(X) to be ZplusJ"(X), where "J(X) is the quotient of "K(X) obtained by identifying two stable equivalence c* *lasses of vector bundles if they are stably fiber homotopy equivalent. Let J : K(X) -! J(X) be the evident quotient map. Atiyah in [At61b] (where J(X) means what we and Adams call "J(X)) proved that the bundle O(n)=O(n - k) -! Sn-1, n 2k, admits a section if and only if n is a multiple of the order of J(1-), where i* *s the canonical line bundle over RPk-1. Thus the vector fields problem can be viewed * *as a special case of the problem of determining J(X). In fact, as Bott first obser* *ved [Bott62, Bott63], Adams' calculations in [Ad62c ] imply that KO(RPn) ~=J(RPn). While Adams was aware of the relationship between the vector fields problem and the study of J, he chose not to discuss this in [Ad62c ]; he published a proof * *of the cited isomorphism in [Ad65a ]. The results just discussed have complex analogues, using U(n)=U(n - k) and CPk-1. The bundle ssn;k: U(n)=U(n - k) -! S2n-1 admits a section if and only if n is divisible by a certain number Mk. The necessity was proven first, by Atiya* *h and Todd [AT60 ], and the sufficiency was then proven by Adams and Walker [AW64 ]. For the proof, they compute KO(CPn) and KO(CPn=CPm ), use the methods and results of [Ad63 , Ad65a] to study J : KO(CPn) -! J(CPn), and deduce that the order of J(1 - ) is Mk, where is the canonical line bundle over CPk-1. Many of the results of Atiyah [At61b] and Adams [Ad62c ] on stunted projective spaces have analogues for stunted lens spaces, and these were worked out by Kam* *be, Matsunaga, and Toda [Ka66 , KMT66 ]. The papers [Ad63 , Ad65a, Ad65b, Ad66a] carry out the general study of J(X) for a connected finite CW complex X. The overall plan is to define two further, more computable, quotients J0(X) and J00(X) of K(X) such that the quotient homomorphisms from K(X) factor to give epimorphisms J00(X) -! J(X) -! J0(X) and then to prove that J0(X) = J00(X). Thus J0(X) is a lower bound and J00(X) an upper bound for J(X), and these two bounds coincide. That J00(X) really is an upper bound depends on the Adams conjecture: "If k is an integer, X is a finite CW complex and y 2 KO(X), then there exists a non- negative integer e = e(k; y) such that ke( k - 1)y maps to zero in J(X)." Adams [Ad63 ] proved this when y is a linear combination of O(1) or O(2) bundles and when X = S2n and y is a complex bundle. His proof is based on the "Dold theorem mod k", which asserts that if f : j -! is a fiberwise map of sphere bundles of 30 J. P. MAY degree k on each fiber, then kej and ke are fiber homotopy equivalent for some e > 0. For k = 1, this is a result of Dold [Dold63]. The groups J0(X) and J00(X) are defined and calculated in favorable cases in [Ad65a ]. In particular, the image of J in sss4k-1is shown to be either the den* *om- inator of Bk=4k, as expected, or twice it; the expected answer would follow from the Adams conjecture. The group J00(X) is KO(X)=W (X), where W (X) is the subgroup generated by all elements ke(k)( k - 1)y for a suitable function e. The content of the Adams conjecture is that J00(X) is indeed an upper bound for J(X* *). To define J0(X), Adams needs certain operations aek which he calls "cannibali* *stic classes". They are related to the k as the Stiefel-Whitney classes are related* * to the Steenrod operations. That is, aek = OE-1 kOE(1) where OE is the KO-theory T* *hom isomorphism. This definition and calculations based on it require good control * *on KO-orientations of vector bundles. While Adams developed some of this himself, the published version of [Ad65a ] relies on the paper [ABS64 ] of Atiyah, Bott,* * and Shapiro, and I shall say more about that in the next section. This definition o* *nly works for Spin(8n)-bundles, in which case the operations aek were introduced by Atiyah (unpublished) and Bott [Bott62, Bott63], who denoted them k. Adams shows that the operations can be extended to all of KO(X) if one localizes the target groups away from k. If sphere bundles j and are fiber homotopy equivale* *nt, then aek() = aek(j)[ k(1 + y)=(1 + y)] for some y 2 "KO(X), independent of k. T* *he group J0(X) is KO(X)=V (X), where V (X) is the subgroup of these elements x such that aek(x) = k(1 + y)=(1 + y) in KO(X) Z[1=k] for all k 6= 0 and some y 2 "KO(X). Adams gives the proof that J0(X) = J00(X) in [Ad65b ]. This entails a good deal of representation theory, some of it involving the extension to the real c* *ase of arguments used by Atiyah and Hirzebruch [AH61a ] in their comparison between R(G)^Iand K(BG) for a compact connected Lie group G. This is used to construct a certain diagram between K-groups, the motivation for which is the heuristic i* *dea that 1 + y = aekx is a solution of the equation ae`( k - 1)x = `(1 + y)=(1 + y* *). This diagram is then proven to be a weak pullback by calculational analysis. To get a more precise hold on J0(X), Adams proves that the k are periodic in the sense that, for any positive integer m, there is an exponent e, depending only * *on X, such that, for any x 2 KO(X), k(x) `(x) modQm if k ` mod me. He uses this to characterize which elements (k) 2 k6=0(1 + "KO(X)[1=k]) are of t* *he form k = aek(x) k[(1 + y)=(1 + y)] for some x 2 "KSpin(X) and y 2 "KO(X). Modulo the Adams conjecture, Adams proves in [Ad66a ] that J(Sn) is a direct summand of sssn. He does this by studying invariants d and e that are associate* *d to maps f : Sq+r -! Sq; there are two variants, real and complex. The real invaria* *nt dR(f) is just the induced homomorphism f* on K"O, and it is zero unless r 1 or 2 mod 8, when it detects certain well-known direct summands Z2 of ssS*. When dR(f) = 0 and dR(f) = 0, the cofiber sequence Sq -! Cf -! Sq+r+1 gives a short exact sequence on application of "KO, and eR(f) is the resulting element * *of the appropriate Ext1 group of extensions. Here Ext1 is taken with respect to an abelian category of abelian groups with Adams operations that commute with each other and satisfy the periodicity relations. Building in that much structure al* *lows direct computation of the relevant Ext1 group, which in the cases of interest i* *s an explicitly determined subgroup of Q=Z. Adams' algebraic formalism leads to an STABLE ALGEBRAIC TOPOLOGY, 1945-1966 31 analysis of how eR relates Toda brackets in homotopy theory to Massey products * *in Extgroups, and these relations are the key to many of Adams' detailed calculati* *ons. The real e-invariant is essential to the proof of the splitting of sss*. The * *complex e-invariant eC admits a more elementary description in terms of the Chern chara* *cter and was introduced and studied independently by Dyer [Dyer63] and Toda [To63]. Adams, Dyer, and Toda all show that eC can be used to reprove the Hopf invariant one theorem, at all primes p. Adams [Ad66a ] also uses eC to prove that if Y is* * the mod pf Moore space, p odd, with bottom cell in a suitable odd dimension, and if r = 2(p - 1)pf-1, then there is a map A : rY -! Y that induces an isomorphism on "K. Iterating A s times, by use of suspensions, and first including the bott* *om cell and then projecting on the top cell, there result elements ffs 2 sssrs-1, and A* *dams uses eC to prove that these maps are all essential. This generalized and clarif* *ied a construction of Toda [To58a] and was a forerunner of a great deal of recent work on periodicity phenomena in stable homotopy theory. When f = 1, Toda himself [To63] showed how to use eC to detect these elements as Toda brackets. Once the Adams conjecture was proven, various classifying spaces not available to Adams were constructed, and the theories of localization and completion were developed, the proof that J0(X) = J00(X) could later be carried out in a more conceptual homotopy theoretic way. The speculative last section of [Ad65b ] an- ticipated much of this. Adams showed that, once appropriate foundations were in place, one would be able to deduce that, for any KO-oriented spherical fibration of dimension 8n, the sequence aek() = OE-1 kOE(1) would be of the form cited ab* *ove. This would imply that, for any x in the group "K(F ; KO)(X) of KO-oriented stab* *le spherical fibrations, there is an element x02 "KSpin(X) such that aek(x) = aek(* *x0) for all k. In retrospect, this was headed towards localized splittings of the c* *lassi- fying space for KO-oriented spherical fibrations, with one factor being BSpin a* *nd the other a space BCokerJ whose homotopy groups are essentially the cokernel of J : ss*(BSpin) -! sss*. Adams asked, among other things, whether or not the J(X) specify a natural direct summand of some other functor of X, and he observed that, since the J(X) do not give a term in a cohomology theory on X, they cannot be direct summands of a term of a cohomology theory. We now fully understand the answers to his questions. The process of reaching that understanding was to have major impact on geometric topology and algebraic K-theory, as well as on many topics within algebraic topology. 15. Further applications and refinements of K-theory The need for K and KO orientations of suitable vector bundles was apparent from the moment K-theory was introduced. Such orientations were essential to the work of Adams just discussed and were first studied in detail by Bott [Bott* *62, Bott63]. However, the definitive treatment was given in the beautiful paper [AB* *S64 ] of Atiyah, Bott, and Shapiro, which was written by the first two authors after Shapiro's untimely death. The authors first give a comprehensive algebraic treatment of Clifford algebr* *as and their relationship to spinor groups. LetPCk be the Clifford algebra of the standard negative definite quadratic form - x2ion Rk and let M(Ck) be the free abelian group generated by the irreducible Z2-graded Ck-modules. The inclusion * *of Ck in Ck+1 induces a homomorphism M(Ck+1) -! M(Ck). Let Ak be its cokernel. 32 J. P. MAY Then the groups Ak are periodic of period 8 and are isomorphic to the homotopy groups ssk(BO). Their complex analogues Ackare isomorphic to the homotopy groups of BU. Under tensor product, the Ak and Ackform graded rings isomorphic to the positive dimensional homotopy groups of KO and KU. These facts are far too striking to be mere coincidences. They next give an account of relative K-theory in bundle theoretic terms, pro* *ving that, for any n, a suitably defined set Ln(X; Y ) of equivalence classes of seq* *uences of vector bundles over X, exact over Y and of length any fixed n 1, maps isomorphically to K(X; Y ) under an Euler characteristic they construct. The pr* *oof depends on a difference bundle construction that is important in many applicati* *ons. Combining ideas, they view the algebraic theory as a theory of bundles over a point and generalize it to a theory of bundles over X. Starting from a fixed Euclidean vector bundle V over X, they construct an associated Clifford bundle C(V ) over X whose fiber over x is the Clifford algebra C(Vx). They define M(V ) to be the Grothendieck group of Z2-graded C(V )-modules over X and define A(V ) to be the cokernel of the homomorphism M(V plus1) -! M(V ). Using their explicit description of relative K-theory, an elementary construction gives a n* *atural homomorphism OV : A(V ) -! "KO(B(V ); S(V )) ~="KO(T V ): It is multiplicative on external sums of bundles in the sense that OV (E) . OW (F ) = OV plusW(E F ): If V is the associated bundle V = P xSpin(k)Rk of a principal Spin(k)-bundle P and M is a Ck-module, then E = P xSpin(k)M is a C(V )-module. This gives a homomorphism fiP : Ak -! A(V ) and thus a composite homomorphism ffP = V fiP : Ak -! "KO(T V ). Taking X to be a point and P to be trivial, there resu* *lts a homomorphism of rings X ff : A* -! KO-k(pt): k0 The beautiful theorem now is that ff and its complex analogue are isomorphisms of rings. This suggests that a proof of Bott periodicity based on the use of Cl* *ifford algebras should be possible. Using Banach algebras, Wood [Wood65 ] and Karoubi [Kar66, Kar68] later found such proofs.. Now consider a Spin-bundle V ~= P xSpin(n)Rn, where n = 8k. Define V = ffP (k) 2 K"O(T V ). Then V restricts on fibers to the canonical generator of the free KO*(pt)-module KO*(Sn). That is, it is an orientation of V , and so it induces a Thom isomorphism OE : KO*(X) -! K"O*(T V ). It follows that a Spin(8k)-bundle V is KO-orientable if and only w1(V ) = 0 and w2(V ) = 0. The orientation is multiplicative in the sense that V plusW= V . W . The authors prove that the orientation they construct coincides with that constructed earli* *er by Bott [Bott62, Bott63]. Similarly, they obtain an orientation cV2 K"U(T V ) for a Spinc-bundle of dimension n = 2k. They state that the agreement of their orientations with Bott's gives additional good properties, but they do not say * *what these properties are. In [Ad65a ], Adams explained some of these properties, since he needed them f* *or computation. Note first that, since U(k) -! SO(2k) lifts canonically to Spinc(2* *k), STABLE ALGEBRAIC TOPOLOGY, 1945-1966 33 the orientations of Spinc-bundles give orientations of complex bundles. The com- plexification of the orientation of a Spin-bundle V is the orientation of V C. According to Adams, the Todd and ^Aclasses are given in terms of the K-theory and rational cohomology Thom isomorphisms by the formulas ec1(VT)-1() = OE-1chcV for a complex bundle V and A^-1(V ) = OE-1chcV C for a Spin-bundle V . According to Adams "It is well known that this is the way* * ^A enters the theory of characteristic classes". That is, ^A(M) ^A(o) = OE-1chcC , where o is the tangent bundle of a manifold M with normal bundle . We have noted the analogy between Adams operations and Steenrod operations. In the 1966 paper [At66a], Atiyah went further and showed that this analogy cou* *ld be made into a precise mathematical relationship, at least for complex K-theory. He redefined the Adams operations by constructing a homomorphism of rings X j : R* = Hom Z(R(k); Z) -! Op(K): k Here k is the kth symmetric group, R(k) its character ring, and Op(K) is the ri* *ng of natural transformations from the functor K to itself. This makes essential u* *se of equivariant K-theory and the isomorphism KG (X) ~=K(X) R(G) for a finite group G and a space X regarded as a G-space with trivial action. The kth tensor power of a vector bundle over X is a k-bundle over X, and this gives a kth power map K(X) -! K(X) R(k); composing with homomorphisms R(k) -! Z, we obtain the kth component of j. As a matter of algebra, there is a copy of the polynomial algebra generated by certain elements that deserve to be denoted k sitting inside R*, and the images of the k under j are the Adams operations. Making essential use of the construction of relative K-theory in [ABS64 ], th* *is allows Atiyah to relate the Adams operations to Steenrod operations by a di- rect comparison of definitions. The K-theory of a CW complex X is filtered by Kq(X) = Ker(K(X) -! K(Xq)) with associated graded group E*0K(X). Sup- pose that H*(X) has no torsion and let p be a prime. The Atiyah-Hirzebruch spectral sequence implies an isomorphism H2q(X; Zp) ~=E2q0K(X) Zp. Atiyah proves that,Pfor x 2 K2q(X), there are elements xi 2 K2q+2i(p-1)(X) such that p(x) = qi=0pq-ixi. Writing xfor the mod p reduction of x and letting P i= Sq* *2i when p = 2, he then proves the remarkable formula P i(x) = xi. The idea of intr* *o- ducing Steenrod operations into generalized homology theories along the lines t* *hat Atiyah worked out in the case of K-theory has had many subsequent applications. In another influential 1966 paper, Atiyah [At66b] introduced Real K-theory KR, which must not be confused with real K-theory KO. In the paper, real vector bundles mean one thing over "real spaces" and another thing over "spaces", which has bedeviled readers ever since: we distinguish Real from real, never starting* * a sentence with either. A Real space is just a space with a Z2-action, or involut* *ion, denoted x ! x. A Real vector bundle p : E -! X is a complex vector bundle E with involution such that verlinecy = verlinec verliney and verlinep(y) = p(verliney) for c 2 C and y 2 E. There is a Grothendieck ring KR(X) of Real vector bundles over a compact Real space X. 34 J. P. MAY Atiyah shows that the elementary proof of the periodicity theorem in complex K-theory that he and Bott gave in [AB64 ] transcribes directly to give a period* *icity theorem in KR-theory. The wonderful thing is that this general theorem speciali* *zes and combines with information on coefficient groups deduced from Clifford algeb* *ras to give a new proof of the periodicity theorem for real K-theory. An essential * *point is to introduce a bigraded version of KR-theory, as was first done by Karoubi [Kar66] in a more general context. In more modern terms, KR is a theory graded on the real representation ring RO(Z2), and it is the first example of an RO(G)- graded cohomology theory. Such theories now play a central role in equivariant algebraic topology. In Atiyah's notation, define groups KRp;q(X; A) = KR(X x Bp;q; X x Sp;q[ A x Bp;q); where Bp;qand Sp;qare the unit disk and sphere in RqplusiRp. In the abso- lute case, these are the components of a bigraded ring. There is a Bott ele- ment fi 2 KR1;1(B1;1; S1;1), and multiplication by fi is an isomorphism. Setting KRp(X; A) = KRp;0(X; A), it follows that KRp;q(X; A) ~=KRp-q(X; A), and it turns out that this is periodic of period 8. When the involution on X is trivi* *al, KR(X) ~=KO(X), and this gives real Bott periodicity. Complex K-theory K and self-conjugate K-theory KSC, which is defined in terms of complex bundles E with an isomorphism from E to its conjugate, are also obtained from KR-theory by sui* *t- able specialization. This leads to long exact sequences relating real, complex,* * and self-conjugate K-theory that have been of considerable use ever since. The sel* *f- conjugate theory had been introduced by Green [Gr64] and Anderson [An64 ], who first discovered these exact sequences. The ideas in [At66b] have found a varie* *ty of recent applications. This is the paper of which Adams wrote in his review: "This is a paper of 19 pages that cannot adequately be summarized in less than 20". In contrast, we come now to the definitive proof by K-theory of the Hopf inva* *ri- ant one theorem, for all primes p, that was given in the paper [AA66 ] of Adams* * and Atiyah. They give a complete proof of the Hopf invariant one theorem for p = 2 in just over a page (see also [May1 ]). The essential idea is to apply the rel* *ation 2 3 = 3 2 in the K-theory of a two-cell complex Sn [f e2n, n even. If the Hopf invariant of f is one, then a simple calculation shows that this relation leads* * to a contradiction unless n is 2, 4, or 8. The proof at odd primes takes only a lit* *tle longer. 16.Bordism and cobordism theories We now back up and return to the story of cobordism. Immediately after the introduction of K-theory, in 1960, Atiyah [At61a] introduced the oriented bordi* *sm and cobordism theories, denoted MSO*(X) and MSO*(X), for finite CW com- plexes X. Just as K* was the first explicitly specified generalized cohomology theory, MSO* was the first explicitly specified generalized homology theory. For a finite CW pair (X; A) and any integer q, Atiyah defines (16.1) MSOq(X; A) = colim[n-qX=A; T SO(n)] and verifies that these groups satisfy all of the Eilenberg-Steenrod axioms exc* *ept the dimension axiom. This is the theory represented by the spectrum MSO, but Atiyah's work precedes Whitehead's paper [Wh62a ], and that language was not yet available. STABLE ALGEBRAIC TOPOLOGY, 1945-1966 35 He defines oriented bordism groups geometrically. He proceeds a little more generally than is currently fashionable, but with good motivation. He considers the category B of pairs (X; ff), where X is a finite CW complex (say) and ff is a principal Z2-bundle over X, that is, a not necessarily connected double cover. Maps and homotopies of maps in B are bundle maps and bundle homotopies. For a smooth manifold M (with boundary), let fl denote the orientation bundle of M. Then MSOq(X; ff) is defined to be the set of "bordism classes" of maps f : (M; fl) -! (X; ff), where M is a q-dimensional closed manifold. Here f is borda* *nt to f0 : (M0; fl) -! (X; ff) if there is a manifold W such that @W = M qM0togeth* *er with a map g : (W; fl) -! (X; ff) that restricts to f on M and to f0 on M0. When ff is trivial, f is just a map M -! X, where M is an oriented q-manifold, and Atiyah writes MSOq(X) for the resulting oriented bordism group. He observes that MGq(X) can be defined similarly for the other classical groups G. One virtue of the more general definition is the observation that, for large * *n, (16.2) MSOq(RPn; ) ~=Nq; where : Sn -! RPn is the canonical double cover. More deeply, Atiyah proves that, for an n-manifold M without boundary MSOq(M; fl) is isomorphic in the stable range 2q < n to a certain group Lq(M) introduced by Thom[Thom54 ] and used in the proof of his "theoreme fondamental". This allows Atiyah to show that Thom's theorem directly implies Poincare duality: for a finite CW pair (X; A) s* *uch that X - A is a closed oriented n-manifold (16.3) MSOq(X; A) ~=MSOn-q(X - Y; fl): Taking Y to be empty and X to be oriented, this specializes to MSOq(X) ~=MSOn-q(X): Although he doesn't go into detail, Atiyah was aware of the expected interpreta* *tion in terms of cup and cap products induced from the maps T SO(m) ^ T SO(n) -! T SO(m + n): For n large and even, so that fl = , (16.2) and (16.3) imply that (16.4) Nq ~=MSO2n-q(RP2n): One of Atiyah's main motivations was to understand certain exact sequences relating oriented and unoriented cobordism groups, in particular the exact sequ* *ence (16.5) Mn-2!Mn -! Nn; due originally to Rohlin [Ro53 , Ro58] and also proven by Dold [Dold60]. These exact sequences play a central role in Wall's computation of M*. Using (16.4), Atiyah shows that they are just long exact sequences obtained by applying the theory MSO* to pairs of projective spaces. Conner and Floyd [CF64a ] followed up Atiyah's work with a thorough exposition and many interesting applications of the theories MO* and MSO*. Atiyah did not give a geometric definition of the relative groups MSO*(X; A). Conner and Floyd do so carefully, and they prove that MSO*(X; A) so defined satisfies MSOq(X; A) ~=ssn+q(X=A ^ T SO(n)) ifn q + 2: 36 J. P. MAY This shows that the geometrically defined theory agrees with the theory given by Whitehead's prescription. They construct the bordism Atiyah-Hirzebruch spec- tral sequence converging from H*(X; A; M*) to MSO*(X; A). For the unoriented theory, they show that (16.6) MO*(X; A) ~=H*(X; A; Z2) N*; as we see from the splitting of MO as a product of Eilenberg-Mac Lane spectra. Similarly, they show that, modulo the Serre class of odd order abelian groups, MSO*(X; A) ~=H*(X; A; M*): Using this, they reinterpret and generalize Thom's work on the Steenrod represe* *n- tation problem. For example, they show that the natural map MSO*(X; A) -! H*(X; A; Z) is an epimorphism if and only if the oriented bordism spectral sequ* *ence for (X; A) collapses and that this holds if H*(X; A; Z) has no odd torsion. They also generalize (16.5) to an exact sequence MSOn(X; A)-2!MSOn(X; A) -! MOn(X; A): However, the main point of Conner and Floyd's monograph [CF64a ] was the use of cobordism for the study of transformation groups of manifolds. The cohomolog* *i- cal study of group actions was initiated in the remarkable early work of P.A. S* *mith [Sm38 ]. The use of cohomological methods in the study of transformation groups was systematized in the seminar [Bo60 ] of Borel and others, including Floyd. I* *n its introduction, Borel had pointed out the desirability of making more effective u* *se of differentiability assumptions than had been possible previously. Conner and Flo* *yd introduced equivariant cobordism as a follow up, and they found many very inter- esting applications of it to the study of fixed point spaces of differentiable * *group actions. I shall only indicate a little of what they do. They define oriented and unoriented geometric equivariant cobordism groups for any finite group G with respect to group actions on manifolds with isotropy groups constrained to lie in any set of subgroups of G closed under conjugacy. Write N*G and MG*for these groups when all subgroups are allowed as isotropy groups. Conner and Floyd focus on the case of free actions (trivial isotropy gr* *oup). Here the geometric description of bordism theory directly implies that the cobo* *r- dism groups of smooth compact manifolds with free G-actions are isomorphic to the bordism groups MO*(BG). Restricting to oriented manifolds and orientation preserving actions, the resulting cobordism groups are isomorphic to the bordism groups MSO*(BG). This opens the way to calculations. As in Atiyah's work on K*(BG), transfer homomorphisms play a significant role. In the unoriented case, MO*(BG) is calculated in terms of H*(G; Z2) by (16.6). As an elementary application, Conner and Floyd give a geometric proof of Wall's observation that the square of a manifold is cobordant to an oriented manifold. However, the main applications concern the fixed point space F of a non-trivial smooth involution on a closed n-manifold M, which for clarity we assume to be connected. Let F m be the union of the components of F of dimension m. If the Stiefel-Whitney classes of the normal bundle of F m in M are trivial for 0 m <* * n, then F m is a boundary for 0 m < n. This is a substantial generalization of the fact that F cannot have exactly one fixed point, a fact that, with its odd prim* *ary analogue, motivated their entire study. STABLE ALGEBRAIC TOPOLOGY, 1945-1966 37 Remarkably, although they did not have a description of N*Z2as the homotopy groups of a space, Conner and Floyd were able to compute these cobordism groups in terms of bordism groups; precisely, they obtained a split short exact sequen* *ce Xn 0 -! NnZ2-! MOm (BO(n - m)) -! MOn-1(BZ2) -! 0: m=0 For an odd prime p, Conner and Floyd calculate the bordism groups MSO*(BZp) completely and give partial information on MSO*(BZpk) for k > 1. They also study MSO*((B(Zp)k)), ending with a conjecture on annihilator ideals that was only proven much later. In this connection, they obtained partial information on a K"unneth theorem for the computation of MSO*(X x Y ). Landweber [Lan66] later gave the complete result, along with the easier analogue for MU*. Conner and Floyd went on to study the equivariant complex bordism groups MU*(BG) for free G-actions in [CF64b ]. This work has been very influential in the developm* *ent of both equivariant geometric topology and equivariant stable algebraic topolog* *y, which recently has become a major subject in its own right. 17.Further work on cobordism and its relation to K-theory We have seen that Milnor [Mil60, Mil62] and Novikov [Nov60 , Nov65] raised the problem of determining the cobordism groups MG*~=ss*(MG) of G-manifolds for G = SU, Sp and Spin. They were aware that only the question of 2-torsion was at issue. Liulevicius [Liu64] described H*(MG; Z2) as a coalgebra over the Steenrod algebra for various G and began the study of the relevant mod 2 Adams spectral sequences. In particular, he calculated E2 and showed that E2 6= E1 for MSU and MSp. He also computed ss*(MSp) in low dimensions. The calculation of the 2-torsion in ss*(MSp) has been studied extensively over the last 30 years, and a complete answer is still out of sight. I shall say no more about that here. How* *ever, the remaining cases were all completely understood by the end of 1966. The lite* *ra- ture in this area burgeoned in the mid 1960's, and I will mention only some of * *the main developments. Stong [Sto68], unfortunately out of print, gives an excelle* *nt and thorough survey of results through 1967, with a complete bibliography. Foun- dationally, he starts from the systematic treatment of the geometric interpreta* *tion of ss*(MG) that was given by Lashof in 1963 [Las63]. As a preamble to explicit calculations, Milnor [Mil65] and others gave some a* *t- tractive conceptual results concerning the squares of manifolds. As a consequen* *ce of their work on fixed points of involutions in [CF64b ], Conner and Floyd had * *ob- served that if VR is the real form of a complex algebraic variety VC and both a* *re non-singular, then VC is unoriented cobordant to VR x VR. Milnor [Mil65] showed that this implies that an unoriented cobordism class contains a complex manifol* *d if and only if it contains a square. He also explained in terms of Stiefel-Whitney* * num- bers when a manifold is unoriented cobordant to a complex manifold. Further, he conjectured and proved in low dimensions that the square of an orientable manif* *old is unoriented cobordant to a Spin-manifold. P.G. Anderson [And66 ] proved that