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Celebratio Mathematica

Michael F. Atiyah

Papers on K-theory (1959–1970)

Commentary by M. Atiyah

Early papers on K-theory (1959–1962)

These pa­pers, cov­er­ing the years 1959–62, con­sist mainly of my joint pa­pers with Hirzebruch on \( K \)-the­ory. However [3], my joint pa­per with Todd, is in a sense the start of this de­vel­op­ment. Hav­ing been ex­posed, at the Arbeit­sta­gung, to many hours of Grothen­dieck ex­pound­ing his gen­er­al­iz­a­tion of the Hirzebruch–Riemann–Roch the­or­em, I was play­ing about with his for­mu­lae for com­plex pro­ject­ive space. From Ioan James (at that time a col­league in Cam­bridge) I had heard about the Bott peri­od­icity the­or­ems and also about stun­ted pro­ject­ive spaces. I soon real­ized that Grothen­dieck’s for­mu­lae led to rather strong res­ults for the James prob­lems. Moreover the Bott peri­od­icity the­or­ems fit­ted in with the Grothen­dieck form­al­ism, so that one could draw genu­ine to­po­lo­gic­al con­clu­sions. It was this which con­vinced me that a to­po­lo­gic­al ver­sion of Grothen­dieck’s \( K \)-the­ory, based on the Bott peri­od­icity the­or­em, would be a power­ful tool in al­geb­ra­ic to­po­logy. In fact the purely al­geb­ra­ic prob­lem which emerged from ap­ply­ing these ideas to the James prob­lem on stun­ted pro­ject­ive spaces proved to be in­triguingly hard. It in­volved de­term­in­ing how large \( n \) has to be in or­der for the power series \( \log ((1+t)/t)^n \) to have its first \( r \) coef­fi­cients in­teg­ral. I showed this prob­lem to sev­er­al col­leagues and in due course Todd came up with the com­plete solu­tion, lead­ing to the joint pa­per [3]. Many years later Adams and Grant–Walk­er grat­i­fy­ingly proved our res­ults were best pos­sible.

Mo­tiv­ated by this prob­lem I was in the pro­cess of de­vel­op­ing my ideas on \( K \)-the­ory more sys­tem­at­ic­ally. Since most of the po­ten­tial ap­plic­a­tions in­volved co­homo­lo­gic­al cal­cu­la­tions with char­ac­ter­ist­ic classes and ho­mo­gen­ous spaces, in or­der to get in­teg­rabil­ity the­or­ems of the type pi­on­eered by Hirzebruch, it was nat­ur­al that I should get Hirzebruch’s as­sist­ance. In this way, our ex­tens­ive col­lab­or­a­tion began and we soon form­al­ized our ideas on \( K \)-the­ory as a gen­er­al­ized co­homo­logy the­ory. [1] gave the first gen­er­al ap­plic­a­tions of the the­ory.

I re­mem­ber that in­tro­du­cing the odd-di­men­sion­al \( K \)-groups seemed at the time a dar­ing gen­er­al­iz­a­tion, fol­low­ing the vogue set by Grothen­dieck. Nev­er­the­less all our early pa­pers on \( K \)-the­ory were aimed at con­crete ap­plic­a­tions. Ex­tract­ing the op­tim­al res­ults from these new meth­ods fre­quently in­volved some soph­ist­ic­ated al­gebra, as for ex­ample in [2]. Ori­gin­ally I had noted that the dif­fer­en­ti­able Riemann–Roch the­or­em of [1] had an “un­stable” ver­sion which could be used to prove strong non-em­bed­ding the­or­ems for man­i­folds. Hirzebruch, us­ing the el­eg­ant ap­par­at­us of “Hil­bert poly­no­mi­als” de­veloped in [2] vastly ex­ten­ded my ini­tial res­ults. This pa­per also in­tro­duced (through their char­ac­ters) the \( \Psi^k \)-op­er­a­tions which Adams was sub­sequently to ex­ploit in bril­liant fash­ion.

Hirzebruch and I spent the Fall of 1959 at the In­sti­tute for Ad­vanced Study de­vel­op­ing \( K \)-the­ory and giv­ing vari­ous ap­plic­a­tions. The re­port of the Tuc­son con­fer­ence [5], based on our work at the time, was the first ma­jor ex­pos­i­tion of \( K \)-the­ory. It es­tab­lished the re­la­tion between the rep­res­ent­a­tion ring \( R(G) \) of a com­pact con­nec­ted Lie group and the \( K \)-the­ory of the clas­si­fy­ing space. The cru­cial fact that \( K(B_G)\to K(B_T) \) (where \( T\subset G \) is the max­im­al tor­us) is in­ject­ive (un­like co­homo­logy) de­pends on the ex­ist­ence of the dir­ect im­age map \( K(B_T)\to K(B_G) \). I clearly re­mem­ber how, in a brief walk round the In­sti­tute hous­ing pro­ject, this fact sud­denly dawned on me. I had been hav­ing lengthy dis­cus­sions with Borel and Hirzebruch on the top­ic and on re­turn­ing from the walk we were able to clinch the mat­ter.

Pa­per [6] ex­ten­ded the res­ults about \( R(G) \) to fi­nite groups \( G \), and it de­pended heav­ily on help from the strong team that year of Serre, Tate, and Borel who were then in­tro­du­cing us in­to the mys­tery of Grothen­dieck’s schemes.

In the early fifties Hirzebruch had dis­covered some in­triguing re­la­tions between Steen­rod op­er­a­tions and the Todd poly­no­mi­als. With the ad­vent of \( K \)-the­ory it was pos­sible to shed new light on these ques­tions and this led to [8]. The fact that it is in Ger­man clearly points to its au­thor­ship and re­flects the fact that the ideas ori­gin­ated from Hirzebruch’s early work.

Mil­nor’s work on the Steen­rod al­gebra was an im­port­ant in­gredi­ent in our pa­per. This in turn is re­lated to Quil­len’s work on form­al groups which led to an el­eg­ant new proof of Mil­nor’s the­or­em on the struc­ture of the unit­ary cobor­d­ism ring.

My work with Todd on com­plex Stiefel man­i­folds had of course a more fam­ous real coun­ter­part, es­sen­tially the vec­tor field prob­lem on spheres. In ret­ro­spect it is per­haps no sur­prise that, us­ing real \( K \)-the­ory, Adams was even­tu­ally able to solve the vec­tor-field prob­lem. Of course the real case is tech­nic­ally more subtle be­cause one can­not use real co­homo­logy and Adams had to use \( K \)-the­ory op­er­a­tions. Stim­u­lated by Adams’ work I showed in [15] how to ap­ply sim­il­ar ideas to the prob­lem of em­bed­ding real pro­ject­ive spaces \( P_n(R) \) in \( R^N \), i.e., get­ting lower bounds for \( N \) in terns of \( n \). The ana­log­ous com­plex prob­lem was one of the early ap­plic­a­tions Hirzebruch and I had made in [2].

Pa­pers [9] and [7] are not dir­ectly con­cerned with \( K \)-the­ory but in­volve re­lated ideas. I was struck by the res­ults in Wall’s thes­is on the ori­ented cobor­d­ism ring and I saw what they meant in terms of the ap­pro­pri­ate gen­er­al­ized co­homo­logy the­ory. I should say that gen­er­al ideas about co­homo­logy and ho­mo­topy at this time were very much in the air and I was cer­tainly in­flu­enced by the ex­pos­i­tions of Hilton and Eck­man, which were el­eg­ant and clear. I also learnt much from Mil­nor in Bonn about cobor­d­ism the­ory and this is re­flec­ted in [7], which ex­am­ines more sys­tem­at­ic­ally the James prob­lem of stun­ted pro­ject­ive spaces, the ori­gin­al stim­u­lus for \( K \)-the­ory.

Throughout this peri­od I also had ex­tens­ive cor­res­pond­ence with Bott. I needed his help, in the first in­stance, to cla­ri­fy cer­tain as­pects of his peri­od­icity the­or­ems. Once the form­al­ism of \( K \)-the­ory had been de­veloped it seemed clear that the Bott peri­od­icity maps should be in­duced by tensor products. For the com­plex case this could be de­duced by co­homo­lo­gic­al ar­gu­ments, but for the real case a more dir­ect veri­fic­a­tion was ne­ces­sary, and we ap­pealed to both to help out. Bott’s great­er ex­per­i­ence with Lie groups proved very help­ful in a num­ber of dir­ec­tions.

Al­though \( K \)-the­ory was prov­ing to be a power­ful new tool in al­geb­ra­ic to­po­logy, Hirzebruch and I were still in­ter­ested in its al­gebro-geo­met­ric ori­gin in the Grothen­dieck–Riemann–Roch the­or­em. In [12] we ex­ten­ded Grothen­dieck’s the­or­em to the case of com­plex ana­lyt­ic em­bed­dings, show­ing that one could define com­pat­ible dir­ect im­age maps in \( K \)-the­ory both for ana­lyt­ic and to­po­lo­gic­al bundles. The same ideas were in­volved in [14], al­though the em­phas­is there was on sin­gu­lar vari­et­ies. We had no­ticed that the co­homo­logy class of an al­geb­ra­ic sub­vari­ety was an­ni­hil­ated by all dif­fer­en­tials in the spec­tral se­quence re­lat­ing co­homo­logy to \( K \)-the­ory. Us­ing a con­struc­tion of Serre, pro­du­cing al­geb­ra­ic vari­et­ies from fi­nite groups, it fol­lowed that not all the tor­sion in the (even-di­men­sion­al) co­homo­logy could be al­geb­ra­ic, dis­pos­ing of the in­teg­ral Hodge con­jec­tures.

My talk [16] at the Stock­holm Con­gress sum­mar­ized the ap­plic­a­tions of \( K \)-the­ory and con­cluded with the \( K \)-the­ory in­ter­pret­a­tion of the sym­bol of an el­lipt­ic op­er­at­or. From this pa­per on­wards the in­dex of el­lipt­ic op­er­at­ors be­comes a dom­in­at­ing theme and the in­ter­ac­tion with \( K \)-the­ory is a two-way pro­cess. The later pa­pers on \( K \)-the­ory there­fore ap­peared in­ter­spersed chro­no­lo­gic­ally with pa­pers on in­dex the­ory. In some cases a pa­per is sim­ul­tan­eously de­voted to both top­ics, and I have cat­egor­ized it ac­cord­ing to its main com­pon­ent, and al­though this is some­times rather ar­bit­rary.

Later papers on K-theory (1964–1970)

Pa­per [17] deal­ing with Clif­ford al­geb­ras and their re­la­tion to real \( K \)-the­ory ori­gin­ated with Bott and Sha­piro. After Sha­piro’s un­timely death I joined forces with Bott and we even­tu­ally pro­duced a rather care­ful treat­ment of the Thom iso­morph­ism in real \( K \)-the­ory, based on spinors and Clif­ford al­geb­ras. My col­lab­or­a­tion with Bott con­tin­ued with [18], our ele­ment­ary proof of the com­plex peri­od­icity the­or­em. This was strongly mo­tiv­ated by the ques­tion of well-posed bound­ary con­di­tions for the el­lipt­ic sys­tems, the top­ic of [e1] and [e2]. Ori­gin­ally the proof was modeled on some ideas of Grothen­dieck and for­mu­lated in terms of sheaf the­ory, but Bott per­suaded me to elim­in­ate all high-brow ma­chinery and so the “ele­ment­ary” proof emerged.

Just be­fore the ad­vent of \( K \)-the­ory Adams had ap­plied sec­ond­ary co­homo­logy op­er­a­tions to solve the long-stand­ing prob­lem of ele­ments of Hopf in­vari­ant one. His pa­per was long and dif­fi­cult and it was an in­dic­a­tion of the power of \( K \)-the­ory that it led to new and much short­er proofs of the Hopf in­vari­ant prob­lem. I was pleased to find a proof which could be writ­ten “on a post-card”. Ac­tu­ally this in­volved the \( \Psi^k \) op­er­a­tions which Adams had in­tro­duced in his sub­sequent work on the vec­tor-field prob­lem. I wrote to Adams ex­plain­ing the “post-card” proof and he im­me­di­ately saw how to gen­er­al­ize it to ob­tain new res­ults for the case of odd primes. This led to our joint pa­per [21].

In at­tempt­ing to un­der­stand real­ity ques­tions of el­lipt­ic op­er­at­ors Sing­er and I were, for a long time, held up by the fact that real \( K \)-the­ory be­haves dif­fer­ently from com­plex \( K \)-the­ory. Even­tu­ally Sing­er poin­ted out that a dif­fer­ent no­tion of real­ity was needed. The es­sen­tial point is that the Four­i­er trans­form of a real-val­ued func­tion is not real but in­stead sat­is­fies the re­la­tion \( f(-x)=f(x) \). It was there­fore ne­ces­sary to de­vel­op a new ver­sion of real \( K \)-the­ory for spaces with in­vol­u­tion. This was car­ried out in [20]. In fact \( KR \)-the­ory, as I called it, turned out to provide a bet­ter ap­proach even for purely to­po­lo­gic­al pur­poses.

Adams had in 1961 proved a res­ult re­lat­ing Chern char­ac­ters and Steen­rod powers. Hirzebruch and I had used the res­ult in [8] in our work on the Todd genus, but I was still mys­ti­fied by the real sig­ni­fic­ance of Adams’ res­ult. Search­ing for the an­swer even­tu­ally led to [22] where I went back to first prin­ciples in study­ing power op­er­a­tions and the in­ter­ac­tion with the sym­met­ric group.

My col­lab­or­a­tion with Hirzebruch on \( K \)-the­ory was in­ten­ded to cul­min­ate in a book on the sub­ject. We had many meet­ings draw­ing up plans but alas we nev­er seemed to have the ne­ces­sary time. However, dur­ing my sab­bat­ic­al term at Har­vard in the Fall of 1964 I gave a course of lec­tures on \( K \)-the­ory. The notes of this course were even­tu­ally pub­lished [23], and this has had to act as a sub­sti­tute for the pro­jec­ted joint book. In fact this course was giv­en at a pro­pi­tious time since I was able to in­cor­por­ate the ele­ment­ary proof of the peri­od­icity the­or­em and to in­tro­duce the Adams op­er­a­tions. As a res­ult the book is pure \( K \)-the­ory without any use or men­tion of op­er­a­tions. As an ap­pendix I also ex­plained the role of the space of Fred­holm op­er­at­ors as a clas­si­fy­ing space for \( K \)-the­ory, a res­ult which had been in­de­pend­ently found by Jänich and made a nat­ur­al bridge to el­lipt­ic op­er­at­or the­ory.

The in­ter­play between in­dex the­ory and \( K \)-the­ory at this time was very ex­tens­ive and in [25] and [26] I gave what I con­sidered defin­it­ive treat­ments of this re­la­tion­ship. In the course of de­vel­op­ing this I was ex­tremely sur­prised to find the proof of the peri­od­icity the­or­em dis­in­teg­rat­ing in­to form­al­it­ies. All the work which Bott and I had put in­to [18] ap­peared to be un­ne­ces­sary! It was in this re­spect some­what re­min­is­cent of Quil­len’s mi­ra­cu­lous treat­ment of com­plex cobor­d­ism.

The in­tim­ate re­la­tion between group rep­res­ent­a­tions and \( K \)-the­ory, lead­ing even­tu­ally to the hy­brid of equivari­ant \( K \)-the­ory, was an­oth­er re­cur­rent theme. I was first in­tro­duced to such ideas by Serre who had come across them in the con­text of al­geb­ra­ic geo­metry or num­ber the­ory. In [6] I had es­tab­lished the the­or­em \( R(G)=K(B_G) \) identi­fy­ing the com­pleted rep­res­ent­a­tion ring of a fi­nite group with the \( K \)-group of its clas­si­fy­ing space. After Adams’ series of pa­pers on the \( J \)-ho­mo­morph­ism, de­term­in­ing the fibre ho­mo­topy clas­si­fic­a­tion of sphere bundles, it was nat­ur­al to look at the cor­res­pond­ing ques­tion for group rep­res­ent­a­tions. This was car­ried out in my joint pa­per [28] with my stu­dent Dav­id Tall. This in­volved a form­al treat­ment of Grothen­dieck’s \( \lambda \)-rings and was es­sen­tially an al­geb­ra­ic ver­sion of Adams’ work.

The main the­or­em of [6] for fi­nite groups \( G \) had also been es­tab­lished earli­er in [5] for com­pact con­nec­ted Lie groups, and it was nat­ur­al to con­jec­ture that there should be a uni­form treat­ment ap­ply­ing to all com­pact Lie groups. The proof in [6] I had long re­garded with mis­giv­ing: it was lengthy, highly tech­nic­al, and de­pended on many ap­par­ent pieces of good for­tune. I was un­happy with it and con­stantly searched for something more nat­ur­al. Even­tu­ally, work­ing sys­tem­at­ic­ally with equivari­ant \( K \)-the­ory gave the right ap­proach, and this was worked out in my joint pa­per [27] with Segal. In fact, Segal, who had been my stu­dent, wrote his thes­is on equivari­ant \( K \)-the­ory and then rap­idly be­came my col­lab­or­at­or in this field. [32] was an­oth­er joint pa­per es­tab­lish­ing some rather pe­cu­li­ar al­geb­ra­ic prop­er­ties of \( \lambda \)-rings.

The clas­sic­al the­or­em of Hopf identi­fy­ing the num­ber of zer­os of a vec­tor field on a man­i­fold with the Euler char­ac­ter­ist­ic has in­ter­est­ing gen­er­al­iz­a­tions to sev­er­al vec­tor fields. These are dis­cussed briefly in [29] and [31]. A much more ex­tens­ive treat­ment is giv­en in my joint pa­per [33] with Dupont, who was then my as­sist­ant at the In­sti­tute for Ad­vanced Study. These pa­pers straddle the \( K \)-the­ory/in­dex the­ory fron­ti­er in a fun­da­ment­al way. In par­tic­u­lar, they use rather re­fined in­dex the­or­ems for real op­er­at­ors where the in­dices are in­tegers mod­ulo high powers of 2. I found such links between ana­lys­is and ho­mo­topy the­ory par­tic­u­larly ap­peal­ing. Most of the time ana­lys­is is linked with real co­homo­logy, via dif­fer­en­tial forms, and it is a sur­pris­ing nov­elty to con­nect ana­lys­is with subtle tor­sion ef­fects.

My in­terest in ho­mo­topy the­ory was al­ways un­ortho­dox. I found the con­ven­tion­al ap­proach rather ugly and cum­ber­some and I was al­ways on the look-out for some in­dir­ect geo­met­ric or ana­lyt­ic­al ap­proach. This was the spir­it in which [34] was writ­ten. My hope was that the use of nat­ur­al framed man­i­folds, e.g., Lie groups, to rep­res­ent the stable ho­mo­topy of spheres might lead to a break-through. Al­though there has been some pro­gress on this point, my ex­pect­a­tions have not yet been real­ized.

The fi­nal \( K \)-the­ory pa­per [35] is a sur­vey of the told of \( K \)-the­ory in vari­ous branches of al­gebra, geo­metry, and ana­lys­is. It gives a brief but per­haps help­ful over­view.

Works

[1]M. F. Atiyah and F. Hirzebruch: “Riemann–Roch the­or­ems for dif­fer­en­ti­able man­i­folds,” Bull. Amer. Math. Soc. 65 : 4 (1959), pp. 276–​281. MR 0110106 Zbl 0142.​40901 article

[2]M. F. Atiyah and F. Hirzebruch: “Quelques théorèmes de non-plonge­ment pour les var­iétés différen­ti­ables” [Some the­or­ems of non-im­mer­sion for dif­fer­en­ti­able man­i­folds], Bull. Soc. Math. France 87 (1959), pp. 383–​396. See also Col­loques Int. Centre Nat. Rech. Sci. 89 (1960). MR 0114231 article

[3]M. F. Atiyah and J. A. Todd: “On com­plex Stiefel man­i­folds,” Proc. Cam­bridge Philos. Soc. 56 : 4 (1960), pp. 342–​353. MR 0132552 Zbl 0109.​16102 article

[4]M. F. Atiyah and F. Hirzebruch: “Quelques théoremes de non-plonge­ment pour les var­iétés différen­ti­ables” [Some non-em­bed­ding the­or­ems for dif­fer­en­ti­able man­i­folds], Col­loques Int. Centre Nat. Rech. Sci. 89 (1960), pp. 383–​396. See also Bull. Soc. Math. France 87 (1959). Zbl 0108.​18202 article

[5]M. F. Atiyah and F. Hirzebruch: “Vec­tor bundles and ho­mo­gen­eous spaces,” pp. 7–​38 in Dif­fer­en­tial geo­metry. Edi­ted by C. B. Al­lendo­er­fer. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 3. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1961. MR 0139181 Zbl 0108.​17705 incollection

[6]M. F. Atiyah: “Char­ac­ters and co­homo­logy of fi­nite groups,” Inst. Hautes Études Sci. Publ. Math. 9 : 1 (1961), pp. 23–​64. MR 0148722 Zbl 0107.​02303 article

[7]M. F. Atiyah: “Thom com­plexes,” Proc. Lon­don Math. Soc. (3) 11 : 1 (1961), pp. 291–​310. MR 0131880 Zbl 0124.​16301 article

[8]M. F. Atiyah and F. Hirzebruch: “Co­homo­lo­gie-Op­er­a­tion­en und charak­ter­istische Klassen,” Math. Z. 77 : 1 (1961), pp. 149–​187. MR 0156361 Zbl 0109.​16002 article

[9]M. F. Atiyah: “Bor­d­ism and cobor­d­ism,” Proc. Cam­bridge Philos. Soc. 57 (1961), pp. 200–​208. MR 0126856 Zbl 0104.​17405 article

[10]M. F. Atiyah and F. Hirzebruch: “Bott peri­od­icity and the par­al­lel­iz­ab­il­ity of the spheres,” Proc. Cam­bridge Philos. Soc. 57 (1961), pp. 223–​226. MR 0126282 Zbl 0108.​35902 article

[11]M. F. Atiyah and F. Hirzebruch: “Charak­ter­istische Klassen und An­wendun­gen,” En­sei­gne­ment Math. (2) 7 (1961), pp. 188–​213. MR 0154294 Zbl 0104.​39801 article

[12]M. F. Atiyah and F. Hirzebruch: “The Riemann–Roch the­or­em for ana­lyt­ic em­bed­dings,” To­po­logy 1 : 2 (1962), pp. 151–​166. MR 0148084 Zbl 0108.​36402 article

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

[14]M. F. Atiyah and F. Hirzebruch: “Ana­lyt­ic cycles on com­plex man­i­folds,” To­po­logy 1 (1962), pp. 25–​45. MR 0145560 Zbl 0108.​36401 article

[15]M. F. Atiyah: “Im­mer­sions and em­bed­dings of man­i­folds,” To­po­logy 1 : 2 (April–June 1962), pp. 125–​132. MR 0145549 Zbl 0109.​41101 article

[16]M. F. Atiyah: “The Grothen­dieck ring in geo­metry and to­po­logy,” pp. 442–​446 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians 1962 (Stock­holm, 15–22 Au­gust 1962), vol. 1. Inst. Mit­tag-Leffler (Djur­sholm), 1963. MR 0180975 Zbl 0121.​39702 incollection

[17]M. F. Atiyah, R. Bott, and A. Sha­piro: “Clif­ford mod­ules,” To­po­logy 3 : Supplement 1 (July 1964), pp. 3–​38. MR 0167985 Zbl 0146.​19001 article

[18]M. Atiyah and R. Bott: “On the peri­od­icity the­or­em for com­plex vec­tor bundles,” Acta Math. 112 : 1 (1964), pp. 229–​247. MR 0178470 Zbl 0131.​38201 article

[19]M. F. Atiyah: “On the \( K \)-the­ory of com­pact Lie groups,” To­po­logy 4 : 1 (1965), pp. 95–​99. MR 0178092 Zbl 0136.​21001 article

[20]M. F. Atiyah: “\( K \)-the­ory and real­ity,” Quart. J. Math. Ox­ford Ser. (2) 17 : 1 (1966), pp. 367–​386. MR 0206940 Zbl 0146.​19101 article

[21]J. F. Adams and M. F. Atiyah: “\( K \)-the­ory and the Hopf in­vari­ant,” Quart. J. Math. Ox­ford Ser. (2) 17 (1966), pp. 31–​38. MR 0198460 Zbl 0136.​43903 article

[22]M. F. Atiyah: “Power op­er­a­tions in \( K \)-the­ory,” Quart. J. Math. Ox­ford Ser. (2) 17 : 1 (1966), pp. 165–​193. Rus­si­an trans­la­tion pub­lished in Math­em­atika 14:2 (1970). MR 0202130 Zbl 0144.​44901 article

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

[24]M. F. Atiyah and D. W. An­der­son: Lekcii po \( K \)-teorii [Lec­tures on \( K \)-the­ory]. Mir (Mo­scow), 1967. Rus­si­an trans­la­tion of \( K \)-the­ory (1967). Zbl 0159.​53401 book

[25]M. F. Atiyah: “Bott peri­od­icity and the in­dex of el­lipt­ic op­er­at­ors,” Quart. J. Math. Ox­ford Ser. (2) 19 (1968), pp. 113–​140. MR 0228000 Zbl 0159.​53501 article

[26]M. F. Atiyah: “Al­geb­ra­ic to­po­logy and op­er­at­ors in Hil­bert space,” pp. 101–​121 in Lec­tures in mod­ern ana­lys­is and ap­plic­a­tions, vol. I. Edi­ted by C. T. Taam. Lec­ture Notes in Math­em­at­ics 103. Spring­er (Ber­lin), 1969. MR 0248803 Zbl 0177.​51701 incollection

[27]M. F. Atiyah and G. B. Segal: “Equivari­ant \( K \)-the­ory and com­ple­tion,” J. Dif­fer­en­tial Geo­metry 3 (1969), pp. 1–​18. MR 0259946 Zbl 0215.​24403 article

[28]M. F. Atiyah and D. O. Tall: “Group rep­res­ent­a­tions, \( \lambda \)-rings and the \( J \)-ho­mo­morph­ism,” To­po­logy 8 : 3 (July 1969), pp. 253–​297. MR 0244387 Zbl 0159.​53301 article

[29]M. F. Atiyah: Vec­tor fields on man­i­folds. Arbeits­ge­meinsch­aft für Forschung des Landes Nordrhein-West­falen 200. West­deutscher Ver­lag (Co­logne), 1970. MR 0263102 Zbl 0193.​52303 book

[30]M. F. Atiyah: “Power op­er­a­tions in \( K \)-the­ory,” Matem­atika 14 : 2 (1970), pp. 35–​65. Rus­si­an trans­la­tion of art­icle in Q. J. Math., Oxf. 17:1 (1966). Zbl 0208.​51503 article

[31]M. F. Atiyah: “El­lipt­ic op­er­at­ors and sin­gu­lar­it­ies of vec­tor fields,” pp. 207–​209 in Act­es du Con­grès In­ter­na­tion­al des Math­ématiciens (Nice, 1–10 Septem­ber 1970), vol. 2. Gau­th­i­er-Vil­lars (Par­is), 1971. MR 0415688 Zbl 0222.​58004 incollection

[32]M. F. Atiyah and G. B. Segal: “Ex­po­nen­tial iso­morph­isms for \( \lambda \)-rings,” Quart. J. Math. Ox­ford Ser. (2) 22 : 3 (1971), pp. 371–​378. MR 0291250 Zbl 0226.​13008 article

[33]M. F. Atiyah and J. L. Dupont: “Vec­tor fields with fi­nite sin­gu­lar­it­ies,” Acta Math. 128 : 1 (1972), pp. 1–​40. MR 0451256 Zbl 0233.​57010 article

[34]M. F. Atiyah and L. Smith: “Com­pact Lie groups and the stable ho­mo­topy of spheres,” To­po­logy 13 : 2 (1974), pp. 135–​142. MR 0343269 Zbl 0282.​55008 article

[35]M. F. Atiyah: “A sur­vey of \( K \)-the­ory,” pp. 1–​9 in \( K \)-the­ory and op­er­at­or al­geb­ras (Uni­versity of Geor­gia, Athens, GA, 21–25 April 1975). Edi­ted by B. B. Mor­rel and I. M. Sing­er. Spring­er (Ber­lin), 1977. MR 0474299 Zbl 0345.​55005 incollection

[36]M. Atiyah: Col­lec­ted works, vol. 2: \( K \)-the­ory. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951893 Zbl 0724.​55001 book

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