# Celebratio Mathematica

## Raoul H. Bott

### The life and works of Raoul Bott

#### by Loring W. Tu

Raoul Bott passed away on Decem­ber 20, 2005. In a ca­reer span­ning five dec­ades, Raoul Bott has wrought pro­found changes on the land­scape of geo­metry and to­po­logy. It is a daunt­ing task to im­prove upon his own re­min­is­cences [22], [24], [17] and com­ment­ar­ies on pa­pers [23], [25], [26], [27], punc­tu­ated as they are by in­sight, col­or­ful turns of phrases and amus­ing an­ec­dotes. This art­icle is an up­dated re­print of one that first ap­peared in the book The founders of in­dex the­ory: Re­min­is­cences of Atiyah, Bott, Hirzebruch, and Sing­er (ed­ited by S.-T. Yau, In­ter­na­tion­al Press, 2003). Tak­ing a per­son­al in­terest in the pro­ject, Raoul Bott in­tro­duced me to some of his friends and gave me ac­cess to his files. After the ori­gin­al art­icle was com­pleted in June 2000, he read it and veri­fied the es­sen­tial cor­rect­ness of the ac­counts of both his life and his works. In the in­terest of pre­serving its of­fi­cial im­prim­at­ur as an au­thor­ized bio­graphy, the only changes I made to the 2003 ver­sion con­sist of this in­tro­duct­ory para­graph, an up­date of the awards he re­ceived, and the sup­pres­sion of a bib­li­o­graphy of his works not ref­er­enced in the art­icle.

#### Early years

Raoul Bott was born in Bud­apest in 1923. His lin­eage fully re­flects the geo­pol­it­ic­al com­plex­ity of the re­gion at the time. His moth­er’s fam­ily was Hun­gari­an and Jew­ish, while his fath­er’s side was Aus­tri­an and Cath­ol­ic. His par­ents di­vorced soon after his birth, so he grew up with his moth­er and step­fath­er. Raised as a Cath­ol­ic, Raoul spent his child­hood and ad­oles­cence in Slov­akia, which sev­enty years later, after al­tern­at­ing between Hun­gary and Czechoslov­akia, is today an in­de­pend­ent coun­try.

In the first five years of school Raoul was not a good stu­dent. This should give com­fort to all par­ents of late bloom­ers. In fact, he did not earn a single A ex­cept in singing and in Ger­man. Non­ethe­less, he showed an early tal­ent for break­ing rules and for gen­er­at­ing sparks — elec­tric­al sparks, that is, rigged up with wires, fuse boxes, va­cu­um tubes, and trans­formers. The schools were form­al and strict, and one could get slapped or have one’s ears pulled for mis­be­hav­ing. For a bud­ding ori­gin­al thinker, Raoul sur­vived the schools re­l­at­ively un­scathed. He re­calls a fri­ar hit­ting him on the hand once and a teach­er cuff­ing his ear an­oth­er time, for hors­ing around too much.

It was by all ac­counts an idyll­ic ex­ist­ence, com­plete with a fam­ily villa, Eng­lish gov­ernesses, and mu­sic les­sons. This world came to an ab­rupt halt in 1935, when his moth­er died of can­cer. In time his step­fath­er re­mar­ried.

Raoul’s ex­per­i­ment­al tal­ent found its full flower­ing in ad­oles­cence. He and a kindred spir­it Tomy Hor­nak built a small box with a slit for coins. When someone dropped a coin through the slit, a dis­play lit up say­ing “Thank you.” In this way they fun­ded their early ex­per­i­ments.

Raoul struggled with some sub­jects in school, and a tu­tor was hired to help him a few hours a week in his house. At the time Raoul and Tomy had built a gad­get to com­mu­nic­ate by Morse code. As he was be­ing tutored, he would hold the gad­get un­der the table and Tomy would be sit­ting in the base­ment. Raoul re­ceived the code by get­ting short and long elec­tric shocks in his hand. He then re­spon­ded by press­ing a but­ton to light up a bulb in the base­ment. While the tu­tor be­lieved that his stu­dent was listen­ing in­tently to the les­son, Raoul was chat­ting away in Morse code un­der the table. In ret­ro­spect, Bott calls this his first at­tempt at e-mail.

In 1938, with Hitler’s as­cend­ancy and Ger­many’s march in­to Czechoslov­akia, Bott’s step­par­ents flew him to the safety of Eng­land and en­rolled him in an Eng­lish board­ing school. Since they had only trans­it visas for Eng­land, the fol­low­ing year they headed for Canada, a coun­try that to this day has been ex­traordin­ar­ily wel­com­ing to refugees and im­mig­rants from around the world.

In the fall of 1941, after a rig­or­ous year of pre­par­at­ory stud­ies in Ontario, Raoul Bott found him­self at Mc­Gill Uni­versity in Montreal. Giv­en his elec­tric­al know-how, he chose, not sur­pris­ingly, elec­tric­al en­gin­eer­ing as his ma­jor. His grades were re­spect­able, but as he re­calls in [24], he was more in­ter­ested in up­hold­ing the “en­gin­eer­ing tra­di­tion of hard drink­ing, loud, bois­ter­ous, mis­chiev­ous, and macho be­ha­vi­or.” Math­em­at­ics was his best sub­ject; still, it was math­em­at­ics in the en­gin­eer­ing sense, not the kind of pure reas­on­ing for which he be­came so well known years later.

With his European flair, his 6 ft. 2 in. frame, and the con­spicu­ous fur cap he of­ten wore, Bott stood out from the crowd at Mc­Gill. When friends asked him where he was from, he said from Di­oszeg, Czechoslov­akia, and he ad­ded fa­cetiously, where he “was a count.” After that, every­one called him the Count.

The Count some­times spoke a very for­eign tongue. In the street­cars of Montreal, Raoul and his room­mate Ro­dolfo Gur­d­i­an would oc­ca­sion­ally en­gage in a de­lib­er­ately loud and an­im­ated con­ver­sa­tion. Noth­ing they said made sense, for they were mak­ing up the lan­guage as they went along. From the corners of their eyes, they en­joyed watch­ing the quiz­zical ex­pres­sions on the faces of the sur­round­ing pas­sen­gers, who were try­ing hard to fig­ure out what lan­guage the two of them were speak­ing.

Bott loved the op­era, but as a pen­ni­less stu­dent how was he to af­ford it? One time the fam­ous ten­or Ezio Pinza came to sing in His Majesty’s Theat­er, the op­era house of Montreal in the Forties. For this oc­ca­sion, Bott dressed up in his Sunday best and went to the theat­er. When the man at the en­trance stopped him, Bott told him he couldn’t do this be­cause he was Ezio Pinza’s neph­ew. Bott said it with such as­sur­ance that the man let him in. After that, Bott could go to all the shows at this theat­er for free.

Bott’s room­mate Ro­dolfo, equally pen­ni­less, also loved the op­era. But Ro­dolfo did not have the nerve to sneak in­to the theat­er. When the op­era “Car­men” was play­ing, Ro­dolfo was very eager to at­tend. Bott mag­nan­im­ously in­vited him. By then, the tick­et-taker knew Bott very well, but he stopped Ro­dolfo at the en­trance. Bott turned around and in­toned in his au­thor­it­at­ive voice, “It’s all right. He can come in.” Without any hes­it­a­tion the tick­et taker obeyed the or­der of this “neph­ew” of Ezio Pinza.

One New Year’s Day, Raoul, Ro­dolfo, and some friends went to Mont Tremb­lant, a winter re­sort north of Montreal. In the most prom­in­ent and ex­pens­ive hotel, a big cel­eb­ra­tion was go­ing on. Some­how, to the envy of his friends, Raoul sneaked in. A little later, Raoul was stand­ing on the bal­cony, look­ing down con­temp­tu­ously at his friends and show­ing them a chick­en leg he was eat­ing. After he fin­ished it, he threw the bone, with dis­dain, to his hungry friends.

(Old habits die hard. In 1960, Bott, by then a full pro­fess­or at Har­vard, was in In­dia with Mi­chael Atiyah, both giv­ing lec­tures as guests of the Tata In­sti­tute of Math­em­at­ics. One day, as they walked in the streets of New Dehli, they passed by a big cel­eb­ra­tion. Bott de­cided to slip in un­in­vited, drag­ging Atiyah along with him. Atiyah, a pro­fess­or at Ox­ford who was later anoin­ted Sir Mi­chael by the Queen and elec­ted Pres­id­ent of the Roy­al So­ci­ety, was at first dis­com­fited, but soon joined whole-heartedly in the fest­iv­it­ies. They had a rous­ing time, shar­ing in the gen­er­al mer­ri­ment of com­plete strangers.)

Upon gradu­ation, Bott joined the army, but the atom­ic bomb at Hiroshi­ma put an end to his mil­it­ary ca­reer after only four months. He entered a one-year mas­ter’s pro­gram in the En­gin­eer­ing De­part­ment at Mc­Gill. Gradu­ally it dawned on him that his in­terest lay more in math­em­at­ics than in en­gin­eer­ing, and he pro­duced a very math­em­at­ic­al mas­ter’s thes­is on “im­ped­ance match­ing,” which he said, “the de­part­ment ac­cep­ted with some mis­giv­ings and about whose math­em­at­ic­al rig­or I have doubts to this very day.”

At Mc­Gill, Raoul met his fu­ture wife, Phyl­lis, an Eng­lish-lit­er­at­ure ma­jor from the West In­dies. Today, Phyl­lis re­mem­bers Raoul’s first mar­riage pro­pos­al. At the time he was do­ing his short stint in the army. In full uni­form, he said, “Would you marry me? Be­cause if you do, the army will pay me more money.” And then point­ing through the win­dow to his little room, he ad­ded, “And we could be liv­ing there.” The pro­pos­al was not ac­cep­ted. But two years later, they mar­ried. The Botts have been to­geth­er ever since, and now have four chil­dren and eight grand­chil­dren. They cel­eb­rated their golden an­niversary in 1997.

#### Sermon

While in the mas­ter’s pro­gram in en­gin­eer­ing at Mc­Gill, Bott floundered in try­ing to de­cide on the gen­er­al dir­ec­tion of his ca­reer. Thirty years later, Bott was asked to de­liv­er a ser­mon at Har­vard’s Me­mori­al Chapel. As he dis­cussed the bib­lic­al pas­sage of Eli, the wise man who counseled the young Samuel (1 Samuel 3:3–6, 8–10), he re­flec­ted on the pivotal mo­ment in his life that launched his math­em­at­ic­al ca­reer. His de­scrip­tion of his own Eli de­serves to be read in the ori­gin­al:

And so when I saw the two read­ings we just heard jux­ta­posed in a Scrip­ture Ser­vice, I could not res­ist them. For they are ap­pro­pri­ate to all of us, wheth­er called to high causes or to lowly ones. And they are maybe es­pe­cially ap­pro­pri­ate to the young people of today in their search of their des­tiny.

For surely there nev­er has been a time when our young people have been giv­en such free­dom and there­fore such re­spons­ib­il­ity to find this des­tiny.

But how are we to know where we are called? And how are we to know who is call­ing us? These are ques­tions bey­ond a math­em­atician’s ken. There are some who seem to have per­fect pitch in these mat­ters. There are many more who might think that they have. But with most of us, it is as it was with Samuel, and we are then truly blessed to have an ad­visor such as Eli. He stands for all of us Teach­ers as an ex­ample. For apart from com­mu­nic­at­ing our call to our stu­dents, we should try and help them above all to dis­cern theirs.

I well re­mem­ber my Eli. He was the Dean of the Med­ic­al School at Mc­Gill and I ap­proached him for help in en­ter­ing the med­ic­al school there, when in 1945 the atom­ic bomb un­ex­pec­tedly put an end to the war and to my four-month old ca­reer in the Ca­na­dian In­fantry.

The Army very wisely de­cided to get rid of such green re­cruits as soon as pos­sible, and so we all again found ourselves quite un­ex­pec­tedly in charge of our own lives. I had gradu­ated in en­gin­eer­ing earli­er that year but had already de­cided against that ca­reer.

The Dean greeted me very cor­di­ally and as­sured me that there was a great need for tech­nic­ally trained doc­tors. “But,” he said, seat­ing me next to him, “first tell me a little about your­self. Did you ever have any in­terest in bot­any, say, or bio­logy?” “Well, not really,” I had to ad­mit. “How about chem­istry” — “Oh, I hated that course.” And so it went. After a while he said, “Well, is it maybe that you want to do good for hu­man­ity?” And then, while I was cough­ing in em­bar­rass­ment, he went on, “Be­cause they make the worst doc­tors.”

I thanked him, and as I walked out of his door I knew that I would start afresh and with God’s grace try and be­come a math­em­atician.

#### Mathematical career

Ini­tially Bott wanted to stay at Mc­Gill to do a math­em­at­ics Ph.D. Be­cause of his sketchy back­ground, however, the Mc­Gill Math De­part­ment re­com­men­ded that he pur­sue a Bach­el­or’s de­gree in math­em­at­ics first. It would have taken an­oth­er three years. Sens­ing his dis­ap­point­ment, Pro­fess­or Wil­li­ams of Mc­Gill then sug­ges­ted Carne­gie Tech (now Carne­gie-Mel­lon Uni­versity) to Bott, where John Synge was just form­ing a new gradu­ate pro­gram and would need some stu­dents.

Synge re­ceived Bott warmly at Carne­gie Tech, but as they read the rules of the pro­gram to­geth­er, they found that Bott would have to spend three years tak­ing courses in the newly min­ted mas­ter’s pro­gram. In a flash of in­spir­a­tion, Synge said, “Let’s look at the Ph.D. pro­gram.” It turned out to have hardly any re­quire­ments at all! Nor­mally the mas­ter’s pro­gram is a pre­requis­ite to the Ph.D. pro­gram, but per­haps re­cog­niz­ing a spe­cial gift in Bott, Synge put him in the Ph.D. pro­gram. In just two years Bott would walk out with his de­gree.

Bott found the Carne­gie Tech at­mo­sphere ex­ceed­ingly sup­port­ive. The small co­ter­ie of math­em­at­ics stu­dents in­cluded Hans Wein­ber­ger, now at the Uni­versity of Min­nesota, and John Nash, an ad­vanced un­der­gradu­ate who after a thirty-year battle with schizo­phrenia re­ceived the No­bel prize in 1994. In later years Bott said of Carne­gie Tech, “Be­ing a brand new gradu­ate pro­gram, they hadn’t learned yet how to put hurdles in front of gradu­ate stu­dents.” Bott con­siders him­self very for­tu­nate to have an ad­visor in R. J. Duffin, for Duffin treated him as an equal from the very out­set and to­geth­er they pub­lished two pa­pers on the math­em­at­ics of elec­tric­al net­works.

The first of these two pa­pers, on im­ped­ance func­tions [1], so im­pressed Her­mann Weyl that he in­vited Bott to the In­sti­tute for Ad­vanced Study in 1949. Thus began Bott’s ini­ti­ation in­to the mys­ter­ies of al­geb­ra­ic to­po­logy. Apart from Weyl, among his main teach­ers were N. Steen­rod, E. Speck­er, K. Re­idemeister, and M. Morse. Of Ernst Speck­er, Bott said in [18], “I bom­barded Ernst with so many stu­pid ques­tions that in des­per­a­tion he fi­nally im­posed a fine of 25 cents on any con­jec­ture he could dis­prove in less than five minutes. This should give you some idea of the in­fla­tion of the past thirty years, and also help to ex­plain Ernst’s vast for­tune at this time.”

At the time Nor­man Steen­rod was writ­ing his clas­sic book on the to­po­logy of fiber bundles, and teach­ing a course based on it. This course greatly in­flu­enced Bott’s math­em­at­ic­al de­vel­op­ment.

Bott de­scribes Steen­rod with ad­mir­a­tion as someone who treated high and low alike, with equal re­spect. At Prin­ceton, the gradu­ate stu­dents could be in­tim­id­at­ing, be­cause they knew so much, and they let you know it. Steen­rod, on the oth­er hand, was dif­fer­ent. In spite of his stature in the math­em­at­ic­al com­munity, he put every­one at ease. In sem­inars Steen­rod did not hes­it­ate to ask the most ba­sic ques­tions. This was quite of­ten a boon to the oth­ers in the audi­ence, too in­tim­id­ated and too be­fuddled to ask the ques­tions them­selves.

After two years at the In­sti­tute, Bott went to the Uni­versity of Michigan. In 1959 he be­came a pro­fess­or at Har­vard, where he has re­mained since. In 1999 Bott form­ally re­tired from teach­ing. He is now Wil­li­am Casper Graustein Re­search Pro­fess­or at Har­vard.

#### Dunster House

An un­usu­al item in the cur­riculum vitae of Raoul Bott, for a math­em­atician at least, was his ten­ure as the Mas­ter of Dun­ster House in 1978–84. At Har­vard the un­der­gradu­ates live in so­cial units called “Houses,” mod­elled some­what after the Col­leges at Ox­ford and Cam­bridge. A House is more than a place to sleep; it is a way to cre­ate a sense of a small com­munity with­in a large uni­versity. Each House has its own din­ing hall, dorm­it­or­ies, so­cial activ­it­ies, and a staff headed by a Mas­ter. The aca­dem­ic staff con­sists of a bevy of res­id­ent and non-res­id­ent tu­tors.

Wheth­er out of a lack of in­terest or a per­ceived mis­match of tem­pera­ment, pure math­em­aticians are rarely called to be Mas­ters of the un­der­gradu­ate Houses. In 1978, in a break with tra­di­tion, the Pres­id­ent of Har­vard Uni­versity ap­poin­ted Bott the Mas­ter of Dun­ster House. This en­tailed liv­ing in the Mas­ter’s Res­id­ence in the midst of three hun­dred un­der­gradu­ates. Bott’s gregari­ous­ness was a good match for the post.

Every year the Houses com­pete in a wa­ter-raft race on the Charles River. This is no gen­tle­man’s ca­noe race as prac­ticed in Eng­land. In the Har­vard ver­sion, at­tacks on oth­er Houses’ rafts are con­doned, even en­cour­aged. One year the Low­ell House team had its Mas­ter at the helm, resplen­dent in an ad­mir­al’s hat. Bott, com­mand­ing the Dun­ster House ar­mada, saw the beau­ti­ful hat. He hollered, “Get me that hat!” Now, this is the sort of or­der un­der­gradu­ates love to obey. In no time the Dun­ster stu­dents had paddled to the Low­ell raft. A struggle en­sued, and like any good pir­ates, the Dun­ster con­tin­gent cap­tured the ad­mir­al’s hat. It was later hung, as a trophy, high in the ceil­ing of the Dun­ster House din­ing hall.

Show­ing true House spir­it, the Dun­ster House crew team had its of­fi­cial team T-shirt em­blazoned with “Dun­ster House,” a pair of oars, and the ex­horta­tion: “Raoul, Raoul, Raoul your Bott.”

The Har­vard Houses have coun­ter­parts at Yale, where they are called Col­leges. A friendly rivalry has al­ways ex­is­ted between these two au­gust in­sti­tu­tions, and it ex­tends to the Houses and Col­leges. Some of the Houses at Har­vard even have “sis­ter Col­leges” with which they are loosely af­fil­i­ated. They would, for ex­ample, vis­it each oth­er dur­ing the Har­vard–Yale foot­ball games.

In the af­ter­math of the Six­ties, many of the tra­di­tions at the Ivy League uni­versit­ies, such as the dress code and the pari­et­al rules, have gone by the way­side, and for a num­ber of years Dun­ster House had not had con­tact with Berke­ley Col­lege, its sis­ter Col­lege at Yale. One year the Berke­ley Col­lege Mas­ter, a dis­tin­guished his­tor­i­an, de­cided to re­vive the tra­di­tion. He wrote to Bott sug­gest­ing a vis­it to Dun­ster House dur­ing the week­end of the Har­vard–Yale foot­ball game. Bott read­ily agreed, but de­cided to make the oc­ca­sion a mem­or­able one. Why not fool the Yalies in­to think­ing that Har­vard has kept up, at least to a cer­tain point, the Oxbridge tra­di­tion of high table and aca­dem­ic gowns at din­ner? Why not show that, per­haps, Dun­ster House was more “civ­il­ized” than its Yale coun­ter­part? With en­thu­si­asm, the Dun­ster House un­der­gradu­ates all sup­por­ted the idea.

On the ap­poin­ted day, the Dun­ster House din­ing hall was trans­formed from a cafet­er­ia in­to a hal­lowed hall, com­plete with lin­en, waiters and wait­resses, and even a wine stew­ard wear­ing a large medal. Un­like on a nor­mal day, there were no T-shirts or cut-offs in sight. Every tu­tor was at­tired in a black aca­dem­ic gown. An or­ches­tra sat in wait­ing. When the Yale Mas­ter and his tu­tors ar­rived, Bott asked, with a straight face, “Where are your gowns?” Of course, they didn’t have any. “Well, no prob­lem, you could bor­row some of ours.” So the Dun­ster tu­tors led them to some gowns that had just been lent from Har­vard’s Chapel. As Bott entered the Din­ing Hall with his guests, trum­pets blared forth and the or­ches­tra star­ted play­ing. The un­der­gradu­ates were already seated, look­ing prim, prop­er, and ser­i­ous. Bott and his tu­tors dined with the Yale vis­it­ors at a high table, on a stage es­pe­cially set up for this oc­ca­sion. The or­ches­tra ser­en­aded the diners with mu­sic. Everything went ac­cord­ing to plan. But the Yale Mas­ter, ever sharp, had the last laugh. He opened his speech by say­ing, “I’m glad to see that cul­ture has fi­nally mi­grated from New Haven to Har­vard.”

#### Bott as a teacher

Bott’s lec­tures are le­gendary for their seem­ing ease of com­pre­hen­sion. His style is typ­ic­ally the an­ti­thes­is of the defin­i­tion-the­or­em-proof ap­proach so favored among math­em­at­ic­al speak­ers. Usu­ally he likes to dis­cuss a simple key ex­ample that en­cap­su­lates the es­sence of the prob­lem. Of­ten, as if by ma­gic, a con­crete for­mula with trans­par­ent sig­ni­fic­ance ap­pears.

At a re­cep­tion for new gradu­ate stu­dents at Har­vard, he once shared his view of the pro­cess of writ­ing a Ph.D. thes­is. He said it is like do­ing a home­work prob­lem; it’s just a harder prob­lem. You try to un­der­stand the prob­lem thor­oughly, from every con­ceiv­able angle. Much of the thes­is work is per­sever­ance, as op­posed to in­spir­a­tion. Above all, “make the prob­lem your own.”

Many of his stu­dents testi­fy to his warmth and hu­man­ity, but he also ex­pects the stu­dents to meet an ex­act­ing stand­ard. He once banned the word “ba­sic­ally” from an ad­visee’s vocab­u­lary, be­cause that word to Bott sig­ni­fies that some de­tails are about to be swept un­der a rug.

This in­sist­ence on thor­ough­ness and clar­ity ap­plies to his own work as well. I. M. Sing­er re­marked that in their young­er days, whenev­er they had a math­em­at­ic­al dis­cus­sion, the most com­mon phrase Bott uttered was “I don’t un­der­stand,” and that a few months later Bott would emerge with a beau­ti­ful pa­per on pre­cisely the sub­ject he had re­peatedly not un­der­stood.

Sem­in­ar speak­ers at Har­vard tend to ad­dress them­selves to the ex­perts in the audi­ence. But like Steen­rod, Bott of­ten in­ter­rupts the speak­ers with the most ba­sic ques­tions, with the salut­ary ef­fect of slow­ing down the speak­ers and mak­ing them more in­tel­li­gible to less­er mor­tals.

At Michigan and Har­vard, Bott dir­ec­ted over 36 Ph.D. theses. Some of his stu­dents have be­come lu­minar­ies in their own right: Steph­en Smale and Daniel Quil­len re­ceived the Fields Medal in 1966 and 1978 re­spect­ively, and Robert MacPh­er­son the Na­tion­al Academy of Sci­ence Award in Math­em­at­ics in 1992.

A com­plete list of his stu­dents can be found here.

#### Honors and awards

Throughout his ca­reer, Bott has been showered with hon­ors, awards, and prizes. The more note­worthy awards in­clude: Sloan Fel­low­ship (1956–60), Veblen Prize of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1964), Gug­gen­heim Fel­low­ship (1976), Na­tion­al Medal of Sci­ence (1987), Steele Prize for Life­time Achieve­ment of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1990), and the Wolf Prize in Math­em­at­ics (2000).

He was twice in­vited to ad­dress the In­ter­na­tion­al Con­gress of Math­em­aticians, in Ed­in­burgh in 1958 and in Nice in 1970.

He was elec­ted Vice Pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1974–75, Hon­or­ary Mem­ber of the Lon­don Math­em­at­ic­al So­ci­ety (1976), Hon­or­ary Fel­low of St. Cath­er­ine’s Col­lege, Ox­ford (1985), Hon­or­ary Mem­ber of the Mo­scow Math­em­at­ic­al So­ci­ety (1997), and For­eign Mem­ber of the Roy­al So­ci­ety (2005). He has been a mem­ber of the Na­tion­al Academy of Sci­ence since 1964, and the French Academy of Sci­ences since 1995.

In 1987 he gave the Con­voc­a­tion Ad­dress at Mc­Gill Uni­versity. He has also re­ceived Hon­or­ary De­grees of Doc­tor of Sci­ence from the Uni­versity of Notre Dame (1980), Mc­Gill Uni­versity (1987), Carne­gie Mel­lon Uni­versity (1989), and the Uni­versity of Leicester, Eng­land (1995).

#### Mathematical works

The bib­li­o­graphy in Raoul Bott’s “Col­lec­ted pa­pers” [23], [25], [26], [27] lists his pub­lic­a­tions, with some omis­sions, up to 1990.

When asked to single out the top three, in the man­ner of an Olympic con­test, he replied, “Can I squeeze in an­oth­er one?” But after list­ing four as the tops, he sighed and said, “This is like be­ing asked to single out the fa­vor­ites among one’s chil­dren.” In the end he came up with a top-five list, in chro­no­lo­gic­al or­der:

• [6] Ho­mo­gen­eous vec­tor bundles,

• [7] The peri­od­icity the­or­em,

• [15] To­po­lo­gic­al ob­struc­tion to in­teg­rabil­ity,

• [19] Yang–Mills equa­tions over Riemann sur­faces,

• [20] The loc­al­iz­a­tion the­or­em in equivari­ant co­homo­logy.

To dis­cuss only these five would not do justice to the range of his out­put. On the oth­er hand, it is evid­ently not pos­sible to dis­cuss every item in his ever-ex­pand­ing opus. As a com­prom­ise, I asked him to make a longer list of all his fa­vor­ite pa­pers, without try­ing to rank them. What fol­lows is a leis­urely romp through the nine­teen pa­pers he chose. My goal is to ex­plain, as simply as pos­sible, the main achieve­ment of his own fa­vor­ite pa­pers. For this reas­on, the the­or­ems, if stated at all, are of­ten not in their greatest gen­er­al­ity.

#### Impedance

The sub­ject of Raoul Bott’s first pa­per [1] dates back to his en­gin­eer­ing days. An elec­tric­al net­work de­term­ines an im­ped­ance func­tion $$Z(s)$$, which de­scribes the fre­quency re­sponse of the net­work. This im­ped­ance func­tion $$Z(s)$$ is a ra­tion­al func­tion of a com­plex vari­able $$s$$ and is pos­it­ive-real (p.r.) in the sense that it maps the right half-plane in­to it­self. An old ques­tion in elec­tric­al en­gin­eer­ing asks wheth­er con­versely, giv­en a pos­it­ive-real ra­tion­al func­tion $$Z(s)$$, it is pos­sible to build a net­work with $$Z(s)$$ as its im­ped­ance func­tion. In some sense O. Brune had solved this prob­lem in 1931, but Brune’s solu­tion as­sumes the ex­ist­ence of an “ideal trans­former,” which in prac­tice would have to be the size of, say, the Har­vard Sci­ence Cen­ter. The as­sump­tion of an ideal trans­former renders Brune’s al­gorithm not so prac­tic­al, and it was Raoul’s dream at Mc­Gill to re­move the ideal trans­former from the solu­tion.

At his first meet­ing with his ad­visor Richard Duffin at Carne­gie Tech, he blur­ted out the prob­lem right away. Many days later, after a par­tic­u­larly fruit­less and strenu­ous dis­cus­sion, Raoul went home and real­ized how to do it. He called Duffin. The phone was busy. As it turned out, Duffin was call­ing him with ex­actly the same idea! They wrote up the solu­tion to the long-stand­ing prob­lem in a joint pa­per, which amaz­ingly took up only two pages.

#### Morse theory

As men­tioned earli­er, the pa­per on im­ped­ance so im­pressed Her­mann Weyl that he in­vited Bott to the In­sti­tute for Ad­vanced Study at Prin­ceton in 1949. There Bott came in­to con­tact with Mar­ston Morse. Morse’s the­ory of crit­ic­al points would play a de­cis­ive role throughout Bott’s ca­reer, not­ably in his work on ho­mo­gen­eous spaces, the Lef­schetz hy­per­plane the­or­em, the peri­od­icity the­or­em, and the Yang–Mills func­tion­al on a mod­uli space.

In the Twen­ties, Morse had ini­ti­ated the study of the crit­ic­al points of a func­tion on a space and its re­la­tion to the to­po­logy of the space. A smooth func­tion $$f$$ on a smooth man­i­fold $$M$$ has a crit­ic­al point at $$p$$ in $$M$$ if there is a co­ordin­ate sys­tem $$(x_1, \ldots, x_n)$$ at $$p$$ such that all the par­tial de­riv­at­ives of $$f$$ van­ish at $$p$$: $\dfrac{\partial f}{\partial x_i} (p) =0 \qquad \text{for all } i= 1,\ldots, n.$ Such a crit­ic­al point is nonde­gen­er­ate if the mat­rix of second par­tials, called the Hes­si­an of $$f$$ at $$p$$, $H_p f= \left[ \dfrac{\partial^2 f}{\partial x_i \partial x_j}(p) \right],$ is nonsin­gu­lar. The in­dex $$\lambda (p)$$ of a nonde­gen­er­ate crit­ic­al point $$p$$ is the num­ber of neg­at­ive ei­gen­val­ues of the Hes­si­an $$H_p f$$; it is the num­ber of in­de­pend­ent dir­ec­tions along which $$f$$ will de­crease from $$p$$.

If a smooth func­tion has only nonde­gen­er­ate crit­ic­al points, we call it a Morse func­tion. The be­ha­vi­or of the crit­ic­al points of a Morse func­tion can be sum­mar­ized in its Morse poly­no­mi­al: $\mathcal{M}_t(f) := \sum t^{\lambda (p)},$ where the sum runs over all crit­ic­al points $$p$$.

A typ­ic­al ex­ample of a Morse func­tion is the height func­tion $$f$$ of a tor­us stand­ing ver­tic­ally on a table top (see the fig­ure). The height func­tion on this tor­us has four crit­ic­al points of in­dex 0, 1, 1, 2 re­spect­ively. Its Morse poly­no­mi­al is $\mathcal{M}_t (f) = 1 + 2t+t^2.$

For a Morse func­tion $$f$$ on a com­pact man­i­fold $$M$$, the fun­da­ment­al res­ults of Morse the­ory hinge on the fact that $$M$$ has the ho­mo­topy type of a CW com­plex with one cell of di­men­sion $$\lambda$$ for each crit­ic­al point of $$f$$ of in­dex $$\lambda$$. This real­iz­a­tion came about in the early Fifties, due to the work of Pitch­er, Thom, and Bott.

Two con­sequences fol­low im­me­di­ately:

1. The weak Morse in­equal­it­ies: $\#\{\text{critical points of index } i\} \ge i\text{-th Betti number}.$ If $P_t (M) = \sum \dim H_i(M)\, t^i$ is the Poin­caré poly­no­mi­al of $$M$$, the Morse in­equal­it­ies can be re­stated in the form $\mathcal{M}_t(f) \ge P_t(M),$ mean­ing that their dif­fer­ence $$\mathcal{M}_t(f) - P_t(M)$$ is a poly­no­mi­al with non­neg­at­ive coef­fi­cients. This in­equal­ity provides a to­po­lo­gic­al con­straint on ana­lys­is, for it says that the $$i$$-th Betti num­ber of the man­i­fold sets a lower bound on the num­ber of crit­ic­al points of in­dex $$i$$ that the func­tion $$f$$ must have.

2. The la­cun­ary prin­ciple: If no two crit­ic­al points of the Morse func­tion $$f$$ have con­sec­ut­ive in­dices, then $$\label{perfect} \mathcal{M}_t(f) = P_t(M).$$ The ex­plan­a­tion is simple: since in the CW com­plex of $$M$$ there are no two cells of con­sec­ut­ive di­men­sions, the bound­ary op­er­at­or is auto­mat­ic­ally zero. There­fore, the cel­lu­lar chain com­plex is its own ho­mo­logy.

A Morse func­tion $$f$$ on $$M$$ sat­is­fy­ing \eqref{perfect} is said to be per­fect. The height func­tion on the tor­us above is a per­fect Morse func­tion.

Clas­sic­al Morse the­ory deals only with func­tions all of whose crit­ic­al points are nonde­gen­er­ate; in par­tic­u­lar, the crit­ic­al points must all be isol­ated points. In many situ­ations, however, the crit­ic­al points form sub­man­i­folds of $$M$$. For ex­ample, if the tor­us now sits flat on the table, as a donut usu­ally would, then the height func­tion has the top and bot­tom circles as crit­ic­al man­i­folds (see the fig­ure).

One of Bott’s first in­sights was to see how to ex­tend Morse the­ory to this situ­ation. In [2] he in­tro­duced the no­tion of a nonde­gen­er­ate crit­ic­al man­i­fold: a crit­ic­al man­i­fold $$N$$ is nonde­gen­er­ate if at any point $$p$$ in $$N$$ the Hes­si­an of $$f$$ re­stric­ted to the nor­mal space to $$N$$ is nonsin­gu­lar. The in­dex $$\lambda (N)$$ of the nonde­gen­er­ate crit­ic­al man­i­fold $$N$$ is then defined to be the num­ber of neg­at­ive ei­gen­val­ues of this nor­mal Hes­si­an; it rep­res­ents the num­ber of in­de­pend­ent nor­mal dir­ec­tions along which $$f$$ is de­creas­ing. For sim­pli­city, as­sume that the nor­mal bundles of the nonde­gen­er­ate crit­ic­al man­i­folds are all ori­ent­able. To form the Morse poly­no­mi­al of $$f$$, each crit­ic­al man­i­fold $$N$$ is coun­ted with its Poin­caré poly­no­mi­al; thus, $\mathcal{M}_t(f) := \sum P_t(N)\, t^{\lambda (N)},$ summed over all crit­ic­al man­i­folds.

With this defin­i­tion of the Morse poly­no­mi­al, Bott proved in [2] that if a smooth func­tion $$f$$ on a smooth man­i­fold $$M$$ has only nonde­gen­er­ate crit­ic­al man­i­folds, then the Morse in­equal­ity again holds: $\mathcal{M}_t(f) \ge P_t(M).$

#### Lie groups and homogeneous spaces

In the Fifties, Bott ap­plied Morse the­ory with great suc­cess to the to­po­logy of Lie groups and ho­mo­gen­eous spaces. In [3] he showed how the dia­gram of a com­pact semisimple con­nec­ted and simply con­nec­ted group $$G$$ de­term­ines the in­teg­ral ho­mo­logy of both the loop space $$\Omega G$$ and the flag man­i­fold $$G/T$$, where $$T$$ is a max­im­al tor­us.

In­deed, Morse the­ory gives a beau­ti­ful CW cell struc­ture on $$G/T$$, up to ho­mo­topy equi­val­ence. To ex­plain this, re­call that the ad­joint ac­tion of the group $$G$$ on its Lie al­gebra $$\mathfrak{g}$$ re­stricts to an ac­tion of the max­im­al tor­us $$T$$ on $$\mathfrak{g}$$. As a rep­res­ent­a­tion of the tor­us $$T$$, the Lie al­gebra $$\mathfrak{g}$$ de­com­poses in­to a dir­ect sum of ir­re­du­cible rep­res­ent­a­tions $\mathfrak{g} = \mathfrak{t} \oplus \sum E_{\alpha} ,$ where $$\mathfrak{t}$$ is the Lie al­gebra of $$T$$ and each $$E_{\alpha}$$ is a 2-di­men­sion­al space on which $$T$$ acts as a ro­ta­tion $$e^{2\pi i \alpha (x)}$$, cor­res­pond­ing to the root $$\alpha (x)$$ on $$\mathfrak{t}$$. The dia­gram of $$G$$ is the fam­ily of par­al­lel hy­per­planes in $$\mathfrak{t}$$ where some root is in­teg­ral. A hy­per­plane that is the zero-set of a root is called a root plane.

For ex­ample, for the group $$G = \mathrm{SU} (3)$$ and max­im­al tor­us $T=\left\{ \left. \begin{bmatrix} e^{2\pi i x_1} & & \\ & e^{2\pi i x_2} & \\ & & e^{2\pi i x_3} \end{bmatrix} \ \right| \ x_1+x_2+x_3 = 0, x_i \in \mathbb{R}\right\} .$ the roots are $$\pm(x_1 - x_2), \pm(x_1 - x_3) , \pm(x_2 - x_3)$$, and the dia­gram is the col­lec­tion of lines in the plane $$x_1+x_2+x_3=0$$ in $$\mathbb{R}^3$$ as in the fig­ure. In this fig­ure, the root planes are the thickened lines.

For $$G=\mathrm{SU} (2)$$ and $T= \left\{ \left. \begin{bmatrix} e^{2\pi i x} & 0 \cr 0 & e^{-2\pi i x} \end{bmatrix} \ \right| \ x \in\mathbb{R}\right\},$ the Lie al­gebra $$\mathfrak{t}$$ is $$\mathbb{R}$$, the roots are $$\pm 2x$$, and the ad­joint rep­res­ent­a­tion of of $$G$$ on $$\mathfrak{g} = \mathbb{R}^3$$ cor­res­ponds to ro­ta­tions. The root plane is the ori­gin.

A point $$B$$ in $$\mathfrak{t}$$ is reg­u­lar if its nor­mal­izer has min­im­al pos­sible di­men­sion, or equi­val­ently if its nor­mal­izer is $$T$$. It is well known that a point $$B$$ in $$\mathfrak{t}$$ is reg­u­lar if and only if it does not lie on any of the hy­per­planes of the dia­gram. If $$B$$ is reg­u­lar, then the sta­bil­izer of $$B$$ un­der the ad­joint ac­tion of $$G$$ is $$T$$ and so the or­bit through $$B$$ is $$G/T$$.

Choose an­oth­er reg­u­lar point $$A$$ in $$\mathfrak{t}$$ and define the func­tion $$f$$ on $\operatorname{Orbit} (B) = G/T$ to be the dis­tance from $$A$$; here, the dis­tance is meas­ured with re­spect to the Killing form on $$\mathfrak{g}$$. Let $$\{ B_i\}$$ be all the points in $$\mathfrak{t}$$ ob­tained from $$B$$ by re­flect­ing about the root planes. Then Bott’s the­or­em as­serts that $$f$$ is a Morse func­tion on $$G/T$$ whose crit­ic­al points are pre­cisely all the $$B_i$$’s. Moreover, the in­dex of a crit­ic­al point $$B_i$$ is twice the num­ber of times that the line seg­ment from $$A$$ to $$B_i$$ in­ter­sects the root planes. This cell de­com­pos­i­tion of Morse the­ory fits in with the more group-the­or­et­ic Bruhat de­com­pos­i­tion.

For $$G=\mathrm{SU} (3)$$ and $$T$$ the set of di­ag­on­al matrices in $$\mathrm{SU}(3)$$, the or­bit $$G/T$$ is the com­plex flag man­i­fold $$\operatorname{Fl} (1,2,3)$$, con­sist­ing of all flags $V_1 \subset V_2 \subset \mathbb{C}^3, \qquad \dim_{\mathbb{C}} V_i = i.$ Bott’s re­cipe gives 6 crit­ic­al points of in­dex $$0,2,2,4,4,6$$ re­spect­ively on $$G/T$$ (See the fig­ure). By the la­cun­ary prin­ciple, the Morse func­tion $$f$$ is per­fect. Hence, the flag man­i­fold $$\operatorname{Fl} (1,2,3)$$ has the ho­mo­topy type of a CW com­plex with one 0-cell, two 2-cells, two 4-cells, and one 6-cell. Its Poin­caré poly­no­mi­al is there­fore $P_t(\operatorname{Fl} (1,2,3)) = 1+2t^2 +2t^4 +t^6.$

#### Index of a closed geodesic

For two points $$p$$ and $$q$$ on a Rieman­ni­an man­i­fold $$M$$, the space $$\Omega_{p,q}(M)$$ of all paths from $$p$$ to $$q$$ on $$M$$ is not a fi­nite-di­men­sion­al man­i­fold. Non­ethe­less, Morse the­ory ap­plies to this situ­ation also, with a Morse func­tion on the path space $$\Omega_{p,q}$$ giv­en by the en­ergy of a path: $E(\mu )=\int_a^b \Bigl\langle \frac{d\mu}{dt}, \frac{d\mu}{dt} \Bigr\rangle\, dt.$ The first res­ult of this in­fin­ite-di­men­sion­al Morse the­ory as­serts that the crit­ic­al points of the en­ergy func­tion are pre­cisely the geodesics from $$p$$ to $$q$$.

Two points $$p$$ and $$q$$ on a geodes­ic are con­jug­ate if keep­ing $$p$$ and $$q$$ fixed, one can vary the geodes­ic from $$p$$ to $$q$$ through a fam­ily of geodesics. For ex­ample, two an­ti­pod­al points on an $$n$$-sphere are con­jug­ate points. The mul­ti­pli­city of $$q$$ as a con­jug­ate point of $$p$$ is the di­men­sion of the fam­ily of geodesics from $$p$$ to $$q$$. On the $$n$$-sphere $$\mathbb{S}^n$$, the mul­ti­pli­city of the south pole as a con­jug­ate point of the north pole is there­fore $$n-1$$.

If $$p$$ and $$q$$ are not con­jug­ate along the geodes­ic, then the geodes­ic is nonde­gen­er­ate as a crit­ic­al point of the en­ergy func­tion on $$\Omega_{p,q}$$. Its in­dex, ac­cord­ing to the Morse in­dex the­or­em, is the num­ber of con­jug­ate points from $$p$$ to $$q$$ coun­ted with mul­ti­pli­cit­ies.

On the $$n$$-sphere let $$p$$ and $$p^{\prime}$$ be an­ti­pod­al points and $$q \ne p^{\prime}$$. The geodesics from $$p$$ to $$q$$ are $$pq$$, $$pp^{\prime}q$$, $$pqp^{\prime}pq$$, $$pp^{\prime}qpp^{\prime}q$$, …, of in­dex $$0, n-1$$, $$2(n-1)$$, $$3(n-1), \dots$$, re­spect­ively. By the Morse in­dex the­or­em, the en­ergy func­tion on the path space $$\Omega_{p,q}(\mathbb{S}^n)$$ has one crit­ic­al point each of in­dex $$0, n-1$$, $$2(n-1)$$, $$3(n-1), \dots$$. It then fol­lows from Morse the­ory that $$\Omega_{p,q}(\mathbb{S}^n)$$ has the ho­mo­topy type of a CW com­plex with one cell in each of the di­men­sions $$0, n-1$$, $$2(n-1)$$, $$3(n-1), \dots$$.

Now con­sider the space $$\Omega M$$ of all smooth loops in $$M$$, that is, smooth func­tions $$\mu: \mathbb{S}^1 \to M$$. The crit­ic­al points of the en­ergy func­tion on $$\Omega M$$ are again the geodesics, but these are now closed geodesics. A closed geodes­ic is nev­er isol­ated as a crit­ic­al point, since for any ro­ta­tion $$r: \mathbb{S}^1 \to \mathbb{S}^1$$ of the circle, $$\mu \circ r: \mathbb{S}^1 \to M$$ is still a geodes­ic. In this way, any closed geodes­ic gives rise to a circle of closed geodesics. When the Rieman­ni­an met­ric on $$M$$ is gen­er­ic, the crit­ic­al man­i­folds of the en­ergy func­tion on the loop space $$\Omega$$ will all be circles.

Morse had shown that the in­dex of a geodes­ic is the num­ber of neg­at­ive ei­gen­val­ues of a Sturm dif­fer­en­tial equa­tion, a bound­ary-value prob­lem of the form $$Ly=\lambda y$$, where $$L$$ is a self-ad­joint second-or­der dif­fer­en­tial op­er­at­or. For cer­tain bound­ary con­di­tions, Morse had ex­pressed the in­dex in terms of con­jug­ate points, but this pro­ced­ure does not ap­ply to closed geodesics, which cor­res­pond to a Sturm prob­lem with peri­od­ic bound­ary con­di­tions.

In [5] Bott found an al­gorithm to com­pute the in­dex of a closed geodes­ic. He was then able to de­term­ine the be­ha­vi­or of the in­dex when the closed geodes­ic is it­er­ated. Bott’s meth­od is in fact ap­plic­able to all Sturm dif­fer­en­tial equa­tions. And so in his pa­per he also gave a geo­met­ric for­mu­la­tion and new proofs of the Sturm–Morse sep­ar­a­tion, com­par­is­on, and os­cil­la­tion the­or­ems, all based on the prin­ciple that the in­ter­sec­tion num­ber of two cycles of com­ple­ment­ary di­men­sions is zero if one of the cycles is ho­mo­log­ous to zero.

#### Homogeneous vector bundles

Let $$G$$ be a con­nec­ted com­plex semisimple Lie group, and $$P$$ a para­bol­ic sub­group. Then $$G$$ is a prin­cip­al $$P$$-bundle over the ho­mo­gen­eous man­i­fold $$X= G/P$$. Any holo­morph­ic rep­res­ent­a­tion $$\varphi : P \to \operatorname{Aut} (E)$$ on a com­plex vec­tor space $$E$$ in­duces a holo­morph­ic vec­tor bundle $$\mathbb{E}$$ over $$X$$: $\mathbb{E}:= G \times_{\varphi} E := (G \times E)/ \sim,$ where $$(gp,e)\sim (g, \varphi(p) e)$$. Then $$\mathbb{E}$$ is a holo­morph­ic vec­tor bundle over $$X= G/P$$. A vec­tor bundle over $$X$$ arising in this way is called a ho­mo­gen­eous vec­tor bundle. Let $$\mathcal{O} (\mathbb{E})$$ be the cor­res­pond­ing sheaf of holo­morph­ic sec­tions. The ho­mo­gen­eous vec­tor bundle $$\mathbb{E}$$ in­her­its a left $$G$$-ac­tion from the left mul­ti­plic­a­tion in $$G$$: $h.(g,e)=(hg, e) \quad\quad \text{for } h,g \in G, e \in E.$ Thus, all the co­homo­logy groups $$H^q(X, \mathcal{O}(\mathbb{E}))$$ be­come $$G$$-mod­ules.

In [6] Bott proved that if the rep­res­ent­a­tion $$\varphi$$ is ir­re­du­cible, the co­homo­logy groups $$H^q(X, \mathcal{O}(\mathbb{E}))$$ all van­ish ex­cept pos­sibly in one single di­men­sion. Moreover, in the non­van­ish­ing di­men­sion $$q$$, $$H^q(X, \mathcal{O}(\mathbb{E}))$$ is an ir­re­du­cible rep­res­ent­a­tion of $$G$$ whose highest weight is re­lated to $$\varphi$$.

This the­or­em gen­er­al­izes an earli­er the­or­em of Borel and Weil, who proved it for a pos­it­ive line bundle.

In Bott’s pa­per one finds a pre­cise way of de­term­in­ing the non­van­ish­ing di­men­sion in terms of the roots and weights of $$G$$ and $$P$$. Thus, on the one hand, Bott’s the­or­em gives a geo­met­ric real­iz­a­tion of in­duced rep­res­ent­a­tions, and on the oth­er hand, it provides an ex­tremely use­ful van­ish­ing cri­terion for the co­homo­logy of ho­mo­gen­eous vec­tor bundles.

#### The periodicity theorem

Ho­mo­topy groups are no­tori­ously dif­fi­cult to com­pute. For a simple space like the $$n$$-sphere already, the high­er ho­mo­topy groups ex­hib­it no dis­cern­ible pat­terns. It was there­fore a com­plete sur­prise in 1957 when Raoul Bott com­puted the stable ho­mo­topy groups of the clas­sic­al groups and found a simple peri­od­ic pat­tern for each of the clas­sic­al groups [7].

We first ex­plain what is meant by the stable ho­mo­topy group. Con­sider the unit­ary group $$U(n+1)$$. It acts trans­it­ively on the unit sphere $$\mathbb{S}^{2n+1}$$ in $$\mathbb{C}^{n+1}$$, with sta­bil­izer $$U(n)$$ at the point $$(1,0,\dots,0)$$. In this way, the sphere $$\mathbb{S}^{2n+1}$$ can be iden­ti­fied with the ho­mo­gen­eous space $$U(n+1)/U(n)$$, and there is a fiber­ing $$U(n+1) \to \mathbb{S}^{2n+1}$$ with fiber $$U(n)$$. By the ho­mo­topy ex­act se­quence of a fiber­ing, the fol­low­ing se­quence is ex­act: $\cdots \to \pi_{k+1}(\mathbb{S}^{2n+1}) \to \pi_k (U(n)) \to \pi_k (U(n+1)) \to \pi_k (\mathbb{S}^{2n+1}) \to \cdots.$ Since $$\pi_k (\mathbb{S}^m) = 0$$ for $$m > k$$, it fol­lows im­me­di­ately that as $$n$$ goes to in­fin­ity (in fact for all $$n > k/2$$), the $$k$$-th ho­mo­topy group of the unit­ary group sta­bil­izes: $\pi_k (U(n)) = \pi_k (U(n+1)) = \pi_k(U(n+2)) = \cdots.$ This com­mon value is called the \emph{$$k$$-th stable ho­mo­topy group} of the unit­ary group, de­noted $$\pi_k (U)$$.

In the ori­gin­al proof of the peri­od­icity the­or­em [7], Bott showed that in the loop space of the spe­cial unit­ary group $$\mathrm{SU}(2n)$$, the man­i­fold of min­im­al geodesics is the com­plex Grass­man­ni­an $G(n,2n) =\frac{U(2n)}{U(n)\times U(n)}.$ By Morse the­ory, the loop space $$\Omega \mathrm{SU} (2n)$$ has the ho­mo­topy type of a CW com­plex ob­tained from the Grass­man­ni­an $$G(n,2n)$$ by at­tach­ing cells of di­men­sion $$\ge 2n+2$$: $\Omega \mathrm{SU} (2n) \sim G(n,2n) \cup e_{\lambda} \cup \cdots \qquad\text{with }\dim e_{\lambda} \ge 2n+2.$ It fol­lows that $\pi_k (\Omega \mathrm{SU} (2n)) = \pi_k (G(n,2n)) \qquad\text{for } n\gg k.$

It is eas­ily shown that $\pi_k (\Omega \mathrm{SU} (2n)) = \pi_{k+1}(\mathrm{SU}(2n)) = \pi_{k+1}(U(2n)).$ Us­ing the ho­mo­topy ex­act se­quence of the fiber­ing $U(n) \to U(2n)/U(n) \to G(n,2n),$ one gets $\pi_k (G(n,2n)) = \pi_{k-1} (U(n)).$ Put­ting all this to­geth­er, for $$n$$ large re­l­at­ive to $$k$$, we get $\pi_{k-1} (U(n)) = \pi_k (G(n,2n)) = \pi_k(\Omega\mathrm{SU}(2n)) = \pi_{k+1}(U(2n)).$ Thus, the stable ho­mo­topy group of the unit­ary group is peri­od­ic of peri­od 2: $\pi_{k-1} (U) = \pi_{k+1} (U).$

Ap­ply­ing the same meth­od to the or­tho­gon­al group and the sym­plect­ic group, Bott showed that their stable ho­mo­topy groups are peri­od­ic of peri­od 8.

#### Clifford algebras

The Clif­ford al­gebra $$C_k$$ is the al­gebra over $$\mathbb{R}$$ with $$k$$ gen­er­at­ors $$e_1, \dots, e_k$$ and re­la­tions \begin{align*} e_i^2 &= -1 \quad \quad \text{for } i=1,\dots, k, \cr e_ie_j &= - e_je_i \quad \quad \text{for all } i \ne j. \end{align*}

The first few Clif­ford al­geb­ras are easy to de­scribe $C_0 = \mathbb{R},\qquad C_1 = \mathbb{C},\qquad C_2 = \mathbb{H} =\{ \text{quaternions} \}.$ If $$\mathbb{F}$$ is a field, de­note by $$\mathbb{F} (n)$$ the al­gebra of all $$n{\times}n$$ matrices with entries in $$\mathbb{F}$$. We call $$\mathbb{F} (n)$$ a full mat­rix al­gebra. It turns out that the Clif­ford al­geb­ras are all full mat­rix al­geb­ras or the dir­ect sums of two full mat­rix al­geb­ras:

 $$k$$ $$C_k$$ $$k$$ $$C_k$$ $$k$$ $$C_k$$ 0 $$\mathbb{R}$$ 8 $$\mathbb{R}(16)$$ 16 $$\mathbb{R}(2^8)$$ 1 $$\mathbb{C}$$ 9 $$\mathbb{C}(16)$$ 17 $$\mathbb{C}(2^8)$$ 2 $$\mathbb{H}$$ 10 $$\mathbb{H}(16)$$ 18 $$\vdots$$ 3 $$\mathbb{H}\oplus\mathbb{H}$$ 11 $$\mathbb{H}(16)\oplus\mathbb{H}(16)$$ 4 $$\mathbb{H}(2)$$ 12 $$\mathbb{H}(32)$$ 5 $$\mathbb{C}(4)$$ 13 $$\mathbb{C}(64)$$ 6 $$\mathbb{R}(8)$$ 14 $$\mathbb{R}(128)$$ 7 $$\mathbb{R}(8)\oplus\mathbb{R}(8)$$ 15 $$\mathbb{R}(128)\oplus\mathbb{R}(128)$$

This table ex­hib­its clearly a peri­od­ic pat­tern of peri­od 8, ex­cept for the di­men­sion in­crease after each peri­od. The 8-fold peri­od­icity of the Clif­ford al­geb­ras, long known to al­geb­ra­ists, is re­min­is­cent of the 8-fold peri­od­icity of the stable ho­mo­topy groups of the or­tho­gon­al group.

In the early Six­ties, Mi­chael Atiyah, Raoul Bott, and Arnold Sha­piro found an ex­plan­a­tion for this tan­tal­iz­ing con­nec­tion. The link is provided by a class of lin­ear dif­fer­en­tial op­er­at­ors called the Dir­ac op­er­at­ors. The link between dif­fer­en­tial equa­tions and ho­mo­topy groups first came about as a res­ult of the real­iz­a­tion that el­lipt­i­city of a dif­fer­en­tial op­er­at­or can be defined in terms of the sym­bol of the dif­fer­en­tial op­er­at­or.

Sup­pose we can find $$k$$ real matrices $$e_1, \dots, e_k$$ of size $$n{\times}n$$ sat­is­fy­ing $e_i^2 = -1, \qquad e_ie_j=-e_je_i \quad \text{for } i \ne j.$ This cor­res­ponds to a real rep­res­ent­a­tion of the Clif­ford al­gebra $$C_k$$. The as­so­ci­ated Dir­ac op­er­at­or $$D=D_{k,n}$$ is the lin­ear first-or­der dif­fer­en­tial op­er­at­or $D= I \frac{\partial}{\partial x_0} + e_1 \frac{\partial}{\partial x_1} + \dots + e_k \frac{\partial}{\partial x_k},$ where $$I$$ is the $$n{\times}n$$ iden­tity mat­rix. Such a dif­fer­en­tial op­er­at­or on $$\mathbb{R}^{k+1}$$ has a sym­bol $$\sigma^{}_D(\xi)$$ ob­tained by re­pla­cing $$\partial/\partial{x_i}$$ by a vari­able $$\xi_i$$: $\sigma^{}_D(\xi) = I \xi_0 + e_1 \xi_1 + \dots + e_k \xi_k.$ The Dir­ac op­er­at­or $$D$$ is read­ily shown to be el­lipt­ic; this means its sym­bol $$\sigma^{}_D(\xi)$$ is nonsin­gu­lar for all $$\xi \ne 0$$ in $$\mathbb{R}^{k+1}$$. There­fore, when re­stric­ted to the unit sphere in $$\mathbb{R}^{k+1}$$, the sym­bol of the Dir­ac op­er­at­or gives a map $\sigma^{}_D(\xi): \mathbb{S}^k \to \mathrm{GL} (n, \mathbb{R}).$ Since $$\mathrm{GL} (n,\mathbb{R})$$ has the ho­mo­topy type of $$O(n)$$, this map giv­en by the sym­bol of the Dir­ac op­er­at­or defines an ele­ment of the ho­mo­topy group $\pi_k(\mathrm{GL} (n,\mathbb{R})) = \pi_k (O(n)) .$

The pa­per [8] shows that the min­im­al-di­men­sion­al rep­res­ent­a­tions of the Clif­ford al­geb­ras give rise to Dir­ac op­er­at­ors whose sym­bols gen­er­ate the stable ho­mo­topy groups of the or­tho­gon­al group. In this way, the 8-fold peri­od­icity of the Clif­ford al­geb­ras re­appears as the 8-fold peri­od­icity of the stable ho­mo­topy groups of the or­tho­gon­al group.

#### The index theorem for homogeneous differential operators

The Six­ties was a time of great fer­ment in to­po­logy, and one of its crown­ing glor­ies was the Atiyah–Sing­er in­dex the­or­em. In­de­pend­ently of Atiyah and Sing­er’s work, Bott’s pa­per [10] on ho­mo­gen­eous dif­fer­en­tial op­er­at­ors ana­lyzes an in­ter­est­ing ex­ample where the ana­lyt­ic­al dif­fi­culties can be avoided by rep­res­ent­a­tion the­ory.

Sup­pose $$G$$ is a com­pact con­nec­ted Lie group and $$H$$ a closed con­nec­ted sub­group. As in our earli­er dis­cus­sion of ho­mo­gen­eous vec­tor bundles, a rep­res­ent­a­tion $$\rho$$ of $$H$$ gives rise to a vec­tor bundle $$G \times_{\rho} H$$ over the ho­mo­gen­eous space $$X=G/H$$. Now sup­pose $$E$$ and $$F$$ are two vec­tor bundles over $$G/H$$ arising from rep­res­ent­a­tions of $$H$$. Since $$G$$ acts on the left on both $$E$$ and $$F$$, it also acts on their spaces of sec­tions, $$\Gamma (E)$$ and $$\Gamma (F)$$. We say that a dif­fer­en­tial op­er­at­or $$D: \Gamma (E) \to \Gamma (F)$$ is ho­mo­gen­eous if it com­mutes with the ac­tions of $$G$$ on $$\Gamma (E)$$ and $$\Gamma (F)$$. If $$D$$ is el­lipt­ic, then its in­dex $\operatorname{index} (D) = \dim \ker D - \dim \operatorname{coker} D$ is defined.

Atiyah and Sing­er had giv­en a for­mula for the in­dex of an el­lipt­ic op­er­at­or on a man­i­fold in terms of the to­po­lo­gic­al data of the situ­ation: the char­ac­ter­ist­ic classes of $$E$$, $$F$$, the tan­gent bundle of the base man­i­fold, and the sym­bol of the op­er­at­or. In [10] Raoul Bott veri­fied the Atiyah–Sing­er in­dex the­or­em for a ho­mo­gen­eous op­er­at­or by in­tro­du­cing a re­fined in­dex, which is not a num­ber, but a char­ac­ter of the group $$G$$. The usu­al in­dex may be ob­tained from the re­fined in­dex by eval­u­at­ing at the iden­tity. A sim­il­ar the­or­em in the in­fin­ite-di­men­sion­al case has re­cently been proven in the con­text of phys­ics-in­spired math­em­at­ics.

#### Nevanlinna theory and the Bott–Chern classes

Nevan­linna the­ory deals with the fol­low­ing type of ques­tions: Let $$f:\mathbb{C} \to \mathbb{C}P^1$$ be a holo­morph­ic map. Giv­en $$a$$ in $$\mathbb{C}P^1$$, what is the in­verse im­age $$f^{-1}(a)$$? Since $$\mathbb{C}$$ is non­com­pact, there may be in­fin­itely many points in the pre-im­age $$f^{-1}(a)$$. Some­times $$f^{-1}(a)$$ will be empty, mean­ing that $$f$$ misses the point $$a$$ in $$\mathbb{C}P^1$$.

The ex­po­nen­tial map $$\exp : \mathbb{C}\to \mathbb{C}P^1$$ misses ex­actly two points, 0 and $$\infty$$, in $$\mathbb{C}P^1$$. Ac­cord­ing to a clas­sic­al the­or­em of Pi­card, a non­con­stant holo­morph­ic map $$f: \mathbb{C}\to \mathbb{C}P^1$$ can­not miss more than two points.

Nevan­linna the­ory re­fines Pi­card’s the­or­em in a beau­ti­ful way. To each $$a \in \mathbb{C}P^1$$, it at­taches a real num­ber $$\delta (a)$$ between 0 and 1 in­clus­ive, the de­fi­ciency in­dex of $$a$$. The de­fi­ciency in­dex is a nor­mal­ized way of count­ing the num­ber of points in the in­verse im­age. If $$f^{-1}(a)$$ is empty, then the de­fi­ciency in­dex is 1.

In this con­text the first main the­or­em of Nevan­linna the­ory says that a non­con­stant holo­morph­ic map $$f: \mathbb{C}\to \mathbb{C}P^1$$ has de­fi­ciency in­dex 0 al­most every­where. The second main the­or­em yields the stronger in­equal­ity: $\sum_{a\in \mathbb{C}P^1} \delta (a) \le 2.$

Ahlfors gen­er­al­ized these two the­or­ems to holo­morph­ic maps with val­ues in a com­plex pro­ject­ive space $$\mathbb{C}P^n$$.

In [9] Bott and Chern souped up Nevan­linna’s hard ana­lys­is to give a more con­cep­tu­al proof of the first main the­or­em.

A by-product of Bott and Chern’s ex­cur­sion in Nevan­linna the­ory is the no­tion of a re­fined Chern class, now called the Bott–Chern class, that has since been trans­formed in­to a power­ful tool in Arakelov geo­metry and oth­er as­pects of mod­ern num­ber the­ory.

Briefly, the Bott–Chern classes arise as fol­lows. On a com­plex man­i­fold $$M$$ the ex­ter­i­or de­riv­at­ive $$d$$ de­com­poses in­to a sum $$d= \partial + \bar{\partial}$$, and the smooth $$k$$-forms de­com­pose in­to a dir­ect sum of $$(p,q)$$-forms. Let $$A^{p,p}$$ be the space of smooth $$(p,p)$$-forms on $$M$$. Then the op­er­at­or $$\partial \bar{\partial}$$ makes $$\bigoplus A^{p,p}$$ in­to a dif­fer­en­tial com­plex. Thus, the co­homo­logy $$H^*\{ A^{p,p}, \partial \bar{\partial} \}$$ is defined.

A Her­mitian struc­ture on a holo­morph­ic rank-$$n$$ vec­tor bundle $$E$$ on $$M$$ de­term­ines a unique con­nec­tion and hence a unique curvature tensor. If $$K$$ and $$K^{\prime}$$ are the curvature forms de­term­ined by two Her­mitian struc­tures on $$E$$, and $$\varphi$$ is a $$\mathrm{GL} (n,\mathbb{C})$$-in­vari­ant poly­no­mi­al on $$\mathfrak{gl} (n,\mathbb{C})$$, then it is well known that $$\varphi (K)$$ and $$\varphi(K^{\prime})$$ are glob­al closed forms on $$M$$ whose dif­fer­ence is ex­act: $\varphi (K) - \varphi(K^{\prime}) = d \alpha$ for a dif­fer­en­tial form $$\alpha$$ on $$M$$. This al­lows one to define the char­ac­ter­ist­ic classes of $$E$$ as co­homo­logy classes in $$H^*(M)$$.

In the holo­morph­ic case, $$\varphi (K)$$ and $$\varphi (K^{\prime})$$ are $$(p,p)$$-forms closed un­der $$\partial \bar{\partial}$$. Bott and Chern found that in fact, $\varphi (K) - \varphi(K^{\prime}) = \partial \bar{\partial} \beta$ for some $$(p-1,p-1)$$-form $$\beta$$. For a holo­morph­ic vec­tor bundle $$E$$, the Bott–Chern class of $$E$$ as­so­ci­ated to an in­vari­ant poly­no­mi­al $$\varphi$$ is the co­homo­logy class of $$\varphi (E)$$, not in the usu­al co­homo­logy, but in the co­homo­logy of the com­plex $$\{ A^{p,p}, \partial \bar{\partial} \}$$.

#### Characteristic numbers and the Bott residue

Ac­cord­ing to the cel­eb­rated Hopf in­dex the­or­em, the Euler char­ac­ter­ist­ic of a smooth man­i­fold is equal to the num­ber of zer­os of a vec­tor field on the man­i­fold, each coun­ted with its in­dex. In [13] and [12], Bott gen­er­al­ized the Hopf in­dex the­or­em to oth­er char­ac­ter­ist­ic num­bers such as the Pontry­agin num­bers of a real man­i­fold and the Chern num­bers of a com­plex man­i­fold.

We will de­scribe Bott’s for­mula only for Chern num­bers. Let $$M$$ be a com­pact com­plex man­i­fold of di­men­sion $$n$$, and $$c_1(M), \dots, c_n(M)$$ the Chern classes of the tan­gent bundle of $$M$$. The Chern num­bers of $$M$$ are the in­teg­rals $\int_M \varphi (c_1(M), \dots , c_n(M)) ,$ as $$\varphi$$ ranges over all weighted ho­mo­gen­eous poly­no­mi­als of de­gree $$n$$. Like the Hopf in­dex the­or­em, Bott’s for­mula com­putes a Chern num­ber in terms of the zer­os of a vec­tor field $$X$$ on $$M$$, but the vec­tor field must be holo­morph­ic and the count­ing of the zer­os is a little more subtle.

For any vec­tor field $$Y$$ and any $$C^{\infty}$$ func­tion $$f$$ on $$M$$, the Lie de­riv­at­ive $$\mathcal{L}_{X}$$ sat­is­fies: $\mathcal{L}_{X} (fY)= (Xf)Y+f\mathcal{L}_{X} Y.$ It fol­lows that at a zero $$p$$ of $$X$$, $(\mathcal{L}_{X} fY)_p = f(p) (\mathcal{L}_{X} Y)_p.$ Thus, at $$p$$, the Lie de­riv­at­ive $$\mathcal{L}_{X}$$ in­duces an en­do­morph­ism $L_p : T_pM \to T_p M$ of the tan­gent space of $$M$$ at $$p$$. The zero $$p$$ is said to be nonde­gen­er­ate if $$L_p$$ is nonsin­gu­lar.

For any en­do­morph­ism $$A$$ of a vec­tor space $$V$$, we define the num­bers $$c_i (A)$$ to be the coef­fi­cients of its char­ac­ter­ist­ic poly­no­mi­al: $\det (I+tA) = \sum c_i (A)\, t^i.$

Bott’s Chern num­ber for­mula is as fol­lows. Let $$M$$ be a com­pact com­plex man­i­fold of com­plex di­men­sion $$n$$ and $$X$$ a holo­morph­ic vec­tor field hav­ing only isol­ated nonde­gen­er­ate zer­os on $$M$$. For any weighted ho­mo­gen­eous poly­no­mi­al $$\varphi (x_1, \dots, x_n)$$ with $$\deg x_i = 2i$$, $$\label{characteristic} \int_M \varphi (c_1(M), \dots, c_n(M)) = \sum_p \dfrac{\varphi (c_1 (L_p), \dots, c_n(L_p))}{c_n(L_p)},$$ summed over all the zer­os of the vec­tor field. Note that by the defin­i­tion of a nonde­gen­er­ate zero, $$c_n(L_p)$$, which is $$\det L_p$$, is nonzero.

In Bott’s for­mula, if the poly­no­mi­al $$\varphi$$ does not have de­gree $$2n$$, then the left-hand side of \eqref{characteristic} is zero, and the for­mula gives an iden­tity among the num­bers $$c_i(L_p)$$. For the poly­no­mi­al $$\varphi (x_1, \dots, x_n) = x_n$$, Bott’s for­mula re­cov­ers the Hopf in­dex the­or­em: $\int_M c_n (M) = \sum_p \dfrac{c_n(L_p)}{c_n(L_p)} = \#\{\text{zeros of } X\}.$

Bott’s for­mula \eqref{characteristic} is re­min­is­cent of Cauchy’s residue for­mula, and so the right-hand side of \eqref{characteristic} may be viewed as a residue of $$\varphi$$ at $$p$$.

In [12] Bott gen­er­al­ized his Chern num­ber for­mula \eqref{characteristic}, which as­sumes isol­ated zer­os, to holo­morph­ic vec­tor fields with high­er-di­men­sion­al zero-sets and to bundles oth­er than the tan­gent bundle (a vec­tor field is a sec­tion of the tan­gent bundle).

#### The Atiyah–Bott fixed point theorem

A con­tinu­ous map of a fi­nite poly­hed­ron, $$f:P \to P$$, has a Lef­schetz num­ber: $L(f) = \sum (-1)^i \operatorname{tr} f^* |_{H^i (P)},$ where $$f^*$$ is the in­duced ho­mo­morph­ism in co­homo­logy and $$\operatorname{tr}$$ de­notes the trace. Ac­cord­ing to the Lef­schetz fixed point the­or­em, if the Lef­schetz num­ber of $$f$$ is not zero, then $$f$$ has a fixed point.

In the smooth cat­egory, the Lef­schetz fixed point the­or­em has a quant­it­at­ive re­fine­ment. A smooth map $$f: M\to M$$ from a com­pact man­i­fold to it­self is trans­vers­al if its graph is trans­vers­al to the di­ag­on­al $$\Delta$$ in $$M \times M$$. Ana­lyt­ic­ally, $$f$$ is trans­vers­al if and only if at each fixed point $$p$$, $\det (1- f_{*,p}) \ne 0,$ where $$f_{*,p}: T_pM \to T_pM$$ is the dif­fer­en­tial of $$f$$ at $$p$$.

The $$C^{\infty}$$ Lef­schetz fixed point the­or­em states that the Lef­schetz num­ber of a trans­vers­al map $$f$$ is the num­ber of fixed points of $$f$$, coun­ted with mul­ti­pli­city $$\pm 1$$ de­pend­ing on the sign of the de­term­in­ant $$\det (1- f_{*,p})$$: $L(f) = \sum_{f(p)=p} \pm 1.$

In the Six­ties, Atiyah and Bott proved a far-reach­ing gen­er­al­iz­a­tion of the Lef­schetz fixed point the­or­em [11], [14]. This type of res­ult, re­lat­ing a glob­al in­vari­ant to a sum of loc­al con­tri­bu­tions, is a re­cur­ring theme in some of Bott’s best work.

To ex­plain it, re­call that the real sin­gu­lar co­homo­logy of $$M$$ is com­put­able from the de Rham com­plex $\textstyle \Gamma \bigl(\bigwedge^0\bigr) \stackrel{d}{\longrightarrow} \Gamma \bigl(\bigwedge^1\bigr) \stackrel{d}{\longrightarrow} \Gamma \bigl(\bigwedge^2\bigr) \stackrel{d}{\longrightarrow} \cdots,$ where $$\bigwedge^q = \bigwedge^q T^*M$$ is the $$q$$-th ex­ter­i­or power of the co­tan­gent bundle. The de Rham com­plex is an ex­ample of an el­lipt­ic com­plex on a man­i­fold.

Let $$E$$ and $$F$$ be vec­tor bundles of ranks $$r^{}_E$$ and $$r^{}_F$$ re­spect­ively over $$M$$. An $$\mathbb{R}$$-lin­ear map $D: \Gamma (E) \to \Gamma (F)$ is a dif­fer­en­tial op­er­at­or if about every point in $$M$$ there is a co­ordin­ate chart $$(U, x_1, \dots, x_n)$$ and trivi­al­iz­a­tions for $$E$$ and $$F$$ re­l­at­ive to which $$D$$ can be writ­ten in the form $D=\sum_{|\alpha| \le m} A^{\alpha}(x) \frac{\partial^\alpha}{\partial x^{\alpha}}$ for $$x \in U$$, $$\alpha=(\alpha_1,\dots,\alpha_n)$$, and $$\partial^\alpha/\partial x^{\alpha} = \partial^{\alpha_1}/\partial x_1^{\alpha_1} \cdots \partial^{\alpha_n}/\partial x_n^{\alpha_n}$$, where $$|\alpha|=\sum \alpha_i$$ and $$A^{\alpha}(x)$$ is an $$r^{}_F{\times}r^{}_E$$ mat­rix that de­pends on $$x$$. The or­der of $$D$$ is the highest $$| \alpha |$$ that oc­curs.

Giv­en a co­tan­gent vec­tor $$\xi = \sum \xi_i dx_i \in T_x^* M$$, we write $\xi^{\alpha} = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}$ and define the sym­bol of a dif­fer­en­tial op­er­at­or $$D$$ of or­der $$m$$ to be $\sigma (D, \xi)_x = \sum_{|\alpha| = m} A^{\alpha} (x)\, \xi^{\alpha} \,\in \operatorname{Hom} (E_x, F_x).$ In oth­er words, the sym­bol of $$D$$ is ob­tained by first dis­card­ing all but the highest-or­der terms of $$D$$ and then re­pla­cing $$\partial^\alpha / \partial x^{\alpha}$$ by $$\xi^{\alpha}$$. Be­cause $$\xi^\alpha$$ trans­forms like $$\partial^\alpha / \partial x^\alpha$$ un­der a change of co­ordin­ates, it is not dif­fi­cult to show that the sym­bol is well defined, in­de­pend­ent of the co­ordin­ate sys­tem.

Let $$E_i$$ be vec­tor bundles over a man­i­fold $$M$$. A dif­fer­en­tial com­plex $$\label{complex} \mathcal{E}: 0 \to \Gamma(E_0) \stackrel{D}{\longrightarrow} \Gamma(E_1) \stackrel{D}{\longrightarrow} \Gamma(E_0) \stackrel{D}{\longrightarrow} \cdots, \qquad D^2=0,$$ is el­lipt­ic if for each nonzero co­tan­gent vec­tor $$\xi \in T_x^*M$$, the as­so­ci­ated sym­bol se­quence $0 \to E_{0,x} \xrightarrow{\sigma(D,\xi)} E_{1,x} \xrightarrow{\sigma(D,\xi)} E_{2,x} \xrightarrow{\sigma(D,\xi)} \cdots$ is an ex­act se­quence of vec­tor spaces. A fun­da­ment­al con­sequence of el­lipt­i­city is that all the co­homo­logy spaces $$H^i = H^i(\Gamma (E_{*}))$$ are fi­nite-di­men­sion­al.

An en­do­morph­ism of the com­plex \eqref{complex} is a col­lec­tion of lin­ear maps $$T_i: \Gamma (E_i) \to \Gamma (E_i)$$ such that $T_{i+1} \circ D = D \circ T_i$ for all $$i$$. Such a col­lec­tion $$T=\{T_i\}$$ in­duces maps in co­homo­logy $$T_i^*:H^i \to H^i$$. The Lef­schetz num­ber of $$T$$ is then defined to be $L(T) = \sum (-1)^i \operatorname{tr} T_i^*.$

A map $$f: M \to M$$ in­duces a nat­ur­al map $\Gamma_f : \Gamma (E) \to \Gamma(f^{-1} E)$ by com­pos­i­tion: $$\Gamma_f (s) = s \circ f$$. There is no nat­ur­al way to in­duce a map of sec­tions: $$\Gamma (E) \to \Gamma (E)$$. However, if there is a bundle map $$\varphi: f^{-1} E \to E$$, then the com­pos­ite $\Gamma (E) \stackrel{\Gamma_f}{\longrightarrow} \Gamma(f^{-1} E) \stackrel{\tilde{\varphi}}{\longrightarrow} \Gamma (E)$ is an en­do­morph­ism of $$\Gamma (E)$$. Any bundle map $$\varphi : f^{-1} E \to E$$ is called a lift­ing of $$f$$ to $$E$$. At each point $$x\in M$$, a lift­ing $$\varphi$$ is noth­ing oth­er than a lin­ear map $$\varphi_x: E_{f(x)} \to E_x$$.

In the case of the de Rham com­plex, a map $$f:M\to M$$ in­duces a lin­ear map $f_x^*: T_{f(x)}^*M \to T_x^*M$ and hence a lin­ear map $\textstyle\bigwedge^q f_x^* : \bigwedge^q T_{f(x)}^*M \to \bigwedge^q T_x^* M,$ which is the lift­ing that fi­nally defines the pull­back of dif­fer­en­tial forms $\textstyle f^* : \Gamma\bigl(\bigwedge^q T^* M\bigr) \to \Gamma \bigl(\bigwedge^q T^* M\bigr) .$

Atiyah–Bott fixed point the­or­em:  Giv­en an el­lipt­ic com­plex \eqref{complex} on a com­pact man­i­fold $$M$$, sup­pose $$f: M \to M$$ has a lift­ing $$\varphi_i : f^{-1} E_i \to E_i$$ for each $$i$$ such that the in­duced maps $$T_i: \Gamma(E_i) \to \Gamma (E_i)$$ give an en­do­morph­ism of the el­lipt­ic com­plex. Then the Lef­schetz num­ber of $$T$$ is giv­en by $L(T)= \sum_{f(x)=x} \dfrac{\sum (-1)^i \operatorname{tr} \varphi_{i,x}} {|\det(1- f_{*,x})|}.$

As evid­ence of its cent­ral­ity, the Atiyah–Bott fixed point the­or­em has an as­ton­ish­ing range of ap­plic­ab­il­ity.

Here is an eas­ily stated co­rol­lary in al­geb­ra­ic geo­metry: Any holo­morph­ic map of a ra­tion­al al­geb­ra­ic man­i­fold to it­self has a fixed point.

Spe­cial­iz­ing the Atiyah–Bott fixed point the­or­em to the de Rham com­plex, one re­cov­ers the clas­sic­al Lef­schetz fixed point the­or­em. When ap­plied to oth­er geo­met­ric­ally in­ter­est­ing el­lipt­ic com­plexes, Atiyah and Bott ob­tained new fixed point the­or­ems, such as a holo­morph­ic Lef­schetz fixed point the­or­em in the com­plex ana­lyt­ic case and a sig­na­ture for­mula in the Rieman­ni­an case. In the ho­mo­gen­eous case, the fixed point the­or­em im­plies the Weyl char­ac­ter for­mula.

#### Obstruction to integrability

A sub­bundle $$E$$ of the tan­gent bundle $$TM$$ of a man­i­fold $$M$$ as­signs to each point $$x$$ of the man­i­fold a sub­space $$E_x$$ of the tan­gent space $$T_xM$$. An in­teg­rable man­i­fold of the sub­bundle $$E$$ is a sub­man­i­fold $$N$$ of $$M$$ whose tan­gent space $$T_xN$$ at each point $$x$$ in $$N$$ is $$E_x$$. The sub­bundle $$E$$ is said to be in­teg­rable if for each point $$x$$ in $$M$$, there is an in­teg­rable man­i­fold of $$E$$ passing through $$x$$.

By the Frobeni­us the­or­em, of­ten proven in a first-year gradu­ate course, a sub­bundle $$E$$ of the tan­gent bundle $$TM$$ is in­teg­rable if and only if its space of sec­tions $$\Gamma (E)$$ is closed un­der the Lie brack­et.

The Pontry­agin ring $$\operatorname{Pont}(V)$$ of a vec­tor bundle $$V$$ over $$M$$ is defined to be the sub­ring of the co­homo­logy ring $$H^*(M)$$ gen­er­ated by the Pontry­agin classes of the bundle $$V$$. In [15] Bott found an ob­struc­tion to the in­teg­rabil­ity of $$E$$ in terms of the Pontry­agin ring of the quo­tient bundle $$Q:=TM/E$$. More pre­cisely, if a sub­bundle $$E$$ of the tan­gent bundle $$TM$$ is in­teg­rable, then the Pontry­agin ring $$\operatorname{Pont} (Q)$$ van­ishes in di­men­sions great­er than twice the rank of $$Q$$.

What is so strik­ing about this the­or­em is not only the sim­pli­city of the state­ment, but also the sim­pli­city of its proof. It spawned tre­mend­ous de­vel­op­ments in fo­li­ation the­ory in the Sev­en­ties, as re­coun­ted in [e5] and [e4].

#### The cohomology of the vector fields on a manifold

For a fi­nite-di­men­sion­al Lie al­gebra $$L$$, let $$A^q(L)$$ be the space of al­tern­at­ing $$q$$-forms on $$L$$. Tak­ing cues from the Lie al­gebra of left-in­vari­ant vec­tor fields on a Lie group, one defines the dif­fer­en­tial $d: A^q(L) \to A^{q+1}(L)$ by $$\label{differential} (d\omega)(X_0, \dots, X_q) = \sum_{i < j} (-1)^{i+j} \omega\bigl([X_i, X_j], X_0, \dots, \hat{X}_i, \dots, \hat{X}_j, \dots, X_q\bigr).$$ As usu­al, the hat over $$X_i$$ means that $$X_i$$ is to be omit­ted. This makes $$A^*(L)$$ in­to a dif­fer­en­tial com­plex, whose co­homo­logy is by defin­i­tion the co­homo­logy of the Lie al­gebra $$L$$.

When $$L$$ is the in­fin­ite-di­men­sion­al Lie al­gebra $$L(M)$$ of vec­tor fields on a man­i­fold $$M$$, the for­mula \eqref{differential} still makes sense, but the space of all al­tern­at­ing forms $$A^*(L(M))$$ is too large for its co­homo­logy to be com­put­able. Gel­fand and Fuks pro­posed put­ting a to­po­logy, the $$C^{\infty}$$ to­po­logy, on $$L(M)$$, and com­put­ing in­stead the co­homo­logy of the con­tinu­ous al­tern­at­ing forms on $$L(M)$$. The Gel­fand–Fuks co­homo­logy of $$M$$ is the co­homo­logy of the com­plex $$\{ A_c^*(L(M)), d\}$$ of con­tinu­ous forms. They hoped to find in this way new in­vari­ants of a man­i­fold. As an ex­ample, they com­puted the Gel­fand–Fuks co­homo­logy of a circle.

It is not clear from the defin­i­tion that the Gel­fand–Fuks co­homo­logy is a ho­mo­topy in­vari­ant. In [16] Bott and Segal proved that the Gel­fand–Fuks co­homo­logy of a man­i­fold $$M$$ is the sin­gu­lar co­homo­logy of a space func­tori­ally con­struc­ted from $$M$$. Hae­fli­ger [e2] and Trauber gave a very dif­fer­ent proof of this same res­ult. The ho­mo­topy in­vari­ance of the Gel­fand–Fuks co­homo­logy fol­lows. At the same time it also showed that the Gel­fand–Fuks co­homo­logy pro­duces no new in­vari­ants.

#### Localization in equivariant cohomology

Just as sin­gu­lar co­homo­logy is a func­tor from the cat­egory of to­po­lo­gic­al spaces to the cat­egory of rings, when a group $$G$$ acts on a space $$M$$ one seeks a func­tor that would in­cor­por­ate both the to­po­logy of the space and the ac­tion of the group.

The na­ive con­struc­tion of tak­ing the co­homo­logy of the quo­tient space $$M/G$$ is un­sat­is­fact­ory, be­cause for a non­free ac­tion the to­po­logy of the quo­tient can be quite bad. A solu­tion is to find a con­tract­ible space $$EG$$ on which $$G$$ acts freely, for then $$EG \times M$$ will have the same ho­mo­topy type as $$M$$ and the group $$G$$ will act freely on $$EG \times M$$ via the di­ag­on­al ac­tion. It is well known that such a space is the total space of the uni­ver­sal $$G$$-bundle $$EG\to BG$$, whose base space is the clas­si­fy­ing space of $$G$$. The ho­mo­topy the­or­ists have defined the ho­mo­topy quo­tient $$M_G$$ of $$M$$ by $$G$$ to be the quo­tient space $$(EG \times M)/G$$, and the equivari­ant co­homo­logy $$H_G^*(M)$$ to be the or­din­ary co­homo­logy of its ho­mo­topy quo­tient $$M_G$$.

The equivari­ant co­homo­logy of the simplest $$G$$-space, a point, is already quite in­ter­est­ing, for it is the or­din­ary co­homo­logy of the clas­si­fy­ing space of $$G$$: $H_G^*(\mathrm{pt})= H^*((EG\times \mathrm{pt})/G )= H^*(EG/G)= H^*(BG).$

Since equivari­ant co­homo­logy is a func­tor of $$G$$-spaces, the con­stant map $$M \to \mathrm{pt}$$ in­duces a ho­mo­morph­ism $$H_G^*(\mathrm{pt}) \to H_G^*(M)$$. Thus, the equivari­ant co­homo­logy $$H_G^*(M)$$ has the struc­ture of a mod­ule over $$H^*(BG)$$.

Char­ac­ter­ist­ic classes of vec­tor bundles over $$M$$ ex­tend to equivari­ant char­ac­ter­ist­ic classes of equivari­ant vec­tor bundles.

When $$M$$ is a man­i­fold, there is a push-for­ward map $\pi_*^M: H_G^*(M) \to H_G^*(\mathrm{pt}) ,$ akin to in­teg­ra­tion along the fiber.

Sup­pose a tor­us $$T$$ acts on a com­pact man­i­fold $$M$$ with fixed point set $$F$$, and $$\varphi \in H_T^*(M)$$ is an equivari­antly closed class. Let $$P$$ be the con­nec­ted com­pon­ents of $$F$$ and let $$\iota_P:P\to M$$ be the in­clu­sion map, $$\nu_P$$ the nor­mal bundle of $$P$$ in $$M$$, and $$e(\nu_P)$$ the equivari­ant Euler class of $$\nu_P$$. In [20] Atiyah and Bott proved a loc­al­iz­a­tion the­or­em for the equivari­ant co­homo­logy $$H_T^*(M)$$ with real coef­fi­cients: $\pi_*^M \varphi = \sum_P \pi_*^P \left( \dfrac{\iota_P^* \varphi}{e(\nu_P)} \right).$ It should be noted that Ber­line and Vergne [e3] in­de­pend­ently proved the same the­or­em at about the same time.

This loc­al­iz­a­tion the­or­em has as con­sequences the fol­low­ing res­ults of Duister­maat and Heck­man on a sym­plect­ic man­i­fold $$(M,\omega)$$ of di­men­sion of $$2n$$:

1. If a tor­us ac­tion on $$M$$ pre­serves the sym­plect­ic form and has a mo­ment map $$f$$, then the push-for­ward $$f_{*} (\omega^n)$$ of the sym­plect­ic volume un­der the mo­ment map is piece­wise poly­no­mi­al.

2. Un­der the same hy­po­theses, the sta­tion­ary-phase ap­prox­im­a­tion for the in­teg­ral $\int_M e^{-itf} \dfrac{\omega^n}{n!}$ is ex­act.

In case the vec­tor field on the man­i­fold is gen­er­ated by a circle ac­tion, the loc­al­iz­a­tion the­or­em spe­cial­izes to Bott’s Chern num­ber for­mu­las [13] of the Six­ties, thus provid­ing an al­tern­at­ive ex­plan­a­tion for the Chern num­ber for­mu­las.

#### Yang–Mills equations over Riemann surfaces

In al­geb­ra­ic geo­metry it is well known that, for any de­gree $$d$$, the set of iso­morph­ism classes of holo­morph­ic line bundles of de­gree $$d$$ over a Riemann sur­face $$M$$ of genus $$g$$ forms a smooth pro­ject­ive vari­ety which is to­po­lo­gic­ally a tor­us of di­men­sion $$g$$. This space is called the mod­uli space of holo­morph­ic line bundles of de­gree $$d$$ over $$M$$.

For holo­morph­ic vec­tor bundles of rank $$k \ge 2$$, the situ­ation is far more com­plic­ated. First, in or­der to have an al­geb­ra­ic struc­ture on the mod­uli space, it is ne­ces­sary to dis­card the so-called “un­stable” bundles in the sense of Mum­ford. It is then known that for $$k$$ and $$d$$ re­l­at­ively prime, the iso­morph­ism classes of the re­main­ing bundles, called “semistable bundles,” form a smooth pro­ject­ive vari­ety $$N(k,d)$$. In [e1] News­tead com­puted the Poin­caré poly­no­mi­al of $$N(2,1)$$. Apart from this, the to­po­logy of $$N(k,d)$$ re­mained mys­ter­i­ous.

In [19] Atiyah and Bott in­tro­duced the new and power­ful meth­od of equivari­ant Morse the­ory to study the to­po­logy of these mod­uli spaces.

Let $$P= M \times U(n)$$ be the trivi­al prin­cip­al $$U(n)$$-bundle over the Riemann sur­face $$M$$, $$\mathcal{A}=\mathcal{A}(P)$$ the af­fine space of con­nec­tions on $$P$$, and $$\mathcal{G}=\mathcal{G}(P)$$ the gauge group, i.e., the group of auto­morph­isms of $$P$$ that cov­er the iden­tity. Then the gauge group $$\mathcal{G} (P)$$ acts on the space $$\mathcal{A} (P)$$ of con­nec­tions and there is a Yang–Mills func­tion­al $$L$$ on $$\mathcal{A}(P)$$ in­vari­ant un­der the ac­tion of the gauge group.

Equivari­ant Morse the­ory harks back to Bott’s ex­ten­sion of clas­sic­al Morse the­ory to nonde­gen­er­ate crit­ic­al man­i­folds three dec­ades earli­er. The key res­ult of Atiyah and Bott is that the Yang–Mills func­tion­al $$L$$ is a per­fect equivari­ant Morse func­tion on $$\mathcal{A} (P)$$. This means the equivari­ant Poin­caré series of $$\mathcal{A} (P)$$ is equal to the equivari­ant Morse series of $$L$$: $$\label{eqmorse} P_t^{\mathcal{G}} (\mathcal{A}(P)) = \mathcal{M}_t^{\mathcal{G}}(L).$$

Once one un­ravels the defin­i­tion, the left-hand side of \eqref{eqmorse} is simply the Poin­caré series of the clas­si­fy­ing space of $$\mathcal{G} (P)$$, which is com­put­able from ho­mo­topy con­sid­er­a­tions. The right-hand side of \eqref{eqmorse} is the sum of con­tri­bu­tions from all the crit­ic­al sets of $$L$$. By the work of Narasim­han and Se­shadri, the min­im­um of $$L$$ is pre­cisely the mod­uli space $$N(k,d)$$. It con­trib­utes its Poin­caré poly­no­mi­al to the equivari­ant Morse series of $$L$$. By an in­duct­ive pro­ced­ure, Atiyah and Bott were able to com­pute the con­tri­bu­tions of all the high­er crit­ic­al sets. They then solved \eqref{eqmorse} for the Poin­caré poly­no­mi­al of $$N(k,d)$$.

#### Witten’s rigidity theorem

Let $$E$$ and $$F$$ be vec­tor bundles over a com­pact man­i­fold $$M$$. If a dif­fer­en­tial op­er­at­or $$D: \Gamma (E) \to \Gamma (F)$$ is el­lipt­ic, then $$\ker D$$ and $$\operatorname{coker} D$$ are fi­nite-di­men­sion­al vec­tor spaces and we can define the in­dex of $$D$$ to be the vir­tu­al vec­tor space $\operatorname{index} D = \ker D - \operatorname{coker} D.$

Now sup­pose a Lie group $$G$$ acts on $$M$$, and $$E$$ and $$F$$ are $$G$$-equivari­ant vec­tor bundles over $$M$$. Then $$G$$ acts on $$\Gamma (E)$$ by $(g\mathbin{.}s)(x)= g\mathbin{.}(s(g^{-1}\! \mathbin{.}x)),$ for $$g \in G$$, $$s\in \Gamma(E), x\in M$$. The $$G$$-ac­tion is said to pre­serve the dif­fer­en­tial op­er­at­or $$D$$ if the ac­tions of $$G$$ on $$\Gamma (E)$$ and $$\Gamma (F)$$ com­mute with $$D$$. In this case, $$\ker D$$ and $$\operatorname{coker} D$$ are rep­res­ent­a­tions of $$G$$, and so $$\operatorname{index} D$$ is a vir­tu­al rep­res­ent­a­tion of $$G$$. We say that the op­er­at­or $$D$$ is ri­gid if its in­dex is a mul­tiple of the trivi­al rep­res­ent­a­tion of di­men­sion 1. The ri­gid­ity of $$D$$ means that any non­trivi­al ir­re­du­cible rep­res­ent­a­tion of $$G$$ in $$\ker D$$ oc­curs in $$\operatorname{coker} D$$ with the same mul­ti­pli­city and vice versa.

If the mul­tiple $$m$$ is pos­it­ive, then $$m\mathbin{.}1= 1\oplus \dots \oplus 1$$ is the trivi­al rep­res­ent­a­tion of di­men­sion of $$m$$. If $$m$$ is neg­at­ive, the $$m\mathbin{.}1$$ is a vir­tu­al rep­res­ent­a­tion, and the ri­gid­ity of $$D$$ im­plies that the trivi­al rep­res­ent­a­tion 1 oc­curs more of­ten in $$\operatorname{coker} D$$ than in $$\ker D$$.

For a circle ac­tion on a com­pact ori­ented Rieman­ni­an man­i­fold, it is well known that the Hodge op­er­at­or $$d+d*: \Omega^{\text{even}} \to \Omega^{\text{odd}}$$ and the sig­na­ture op­er­at­or $$d_s=d+d^*:\Omega^+ \to \Omega^-$$ are both ri­gid.

An ori­ented Rieman­ni­an man­i­fold of di­men­sion $$n$$ has an at­las whose trans­ition func­tions take val­ues in $$\mathrm{SO} (n)$$. The man­i­fold is called a spin man­i­fold if it is pos­sible to lift the trans­ition func­tions to the double cov­er $$\mathrm{Spin} (n)$$ of $$\mathrm{SO} (n)$$.

In­spired by phys­ics, Wit­ten dis­covered in­fin­itely many ri­gid el­lipt­ic op­er­at­ors on a com­pact spin man­i­fold with a circle ac­tion. They are typ­ic­ally of the form $$d_s \otimes R$$, where $$d_s$$ is the sig­na­ture op­er­at­or and $$R$$ is some com­bin­a­tion of the ex­ter­i­or and the sym­met­ric powers of the tan­gent bundle. In [21] Bott and Taubes found a proof, more ac­cess­ible to math­em­aticians, of Wit­ten’s res­ults, by re­cast­ing the ri­gid­ity the­or­em as a con­sequence of the Atiyah–Bott fixed point the­or­em.

The idea of [21] is as fol­lows. To de­com­pose a rep­res­ent­a­tion, one needs to know only its trace, since the trace de­term­ines the rep­res­ent­a­tion. By as­sump­tion, the ac­tion of $$G$$ on the el­lipt­ic com­plex $$D: \Gamma (E) \to \Gamma (F)$$ com­mutes with $$D$$. This means each ele­ment $$g$$ in $$G$$ is an en­do­morph­ism of the el­lipt­ic com­plex. It there­fore in­duces an en­do­morph­ism $$g^*$$ in the co­homo­logy of the com­plex. But $$H^0 = \ker D$$ and $$H^1=\operatorname{coker} D$$. The al­tern­at­ing sum of the trace of $$g^*$$ in co­homo­logy is pre­cisely the left-hand side of the Atiyah–Bott fixed point the­or­em. It then stands to reas­on that the fixed point the­or­em could be used to de­com­pose the in­dex of $$D$$ in­to ir­re­du­cible rep­res­ent­a­tions.

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