Celebratio Mathematica

Paul Baum

Paul Baum: A personal and mathematical appreciation

by Joel Leonard Wolf


In the sum­mer of 2016, not long after I re­tired from IBM Re­search, I was in­vited to a con­fer­ence at the Fields In­sti­tute at the Uni­versity of Toronto. This con­fer­ence was to fo­cus on the Baum–Connes con­jec­ture, and would also serve as an 80th birth­day party for Paul Baum. Paul was my Ph.D. ad­visor at Brown Uni­versity back in the early 1970s. My ca­reer since my aca­dem­ic days had taken a very dif­fer­ent math­em­at­ic­al turn, but Paul and I had kept in touch from time to time via email. Still, I hadn’t ac­tu­ally seen Paul since the mid 70s, and it took me pre­cisely zero seconds to de­cide to at­tend. I have al­ways ad­ored Paul, and this was a chance to catch up, maybe even to learn a little about Baum–Connes.

It’s 494 miles from my home in Kato­nah (Westchester), New York to the Fields In­sti­tute. So I had plenty of drive time to com­pose my thoughts on the way there, in case I was asked to say a few words. And those thoughts were pretty straight­for­ward:

  • Paul has al­ways been a mag­ni­fi­cent and in­vent­ive math­em­atician, with a keen sense of what prob­lems are im­port­ant. His math­em­at­ics is just plain el­eg­ant. And some­how, at 80, he was still go­ing strong, with a world of ac­com­plish­ments.

  • Paul was a won­der­ful Ph.D. ad­visor, al­ways point­ing me in the right dir­ec­tion. He was a great ment­or, gen­er­ous with his thoughts and with his time.

  • Paul was a char­ac­ter in the best sense of the word. He was al­ways ex­tro­ver­ted, gregari­ous: a joy to be with.

  • Above all, he was kind to me, and to every­body with whom he came in con­tact. He was sup­port­ive and caring when I needed it most.

The con­fer­ence was lovely. Paul gave a talk, and it seemed like old times to me. There were many oth­er fine speak­ers. And there was a ban­quet in Paul’s hon­or on the last day of the con­fer­ence, amaz­ingly enough on his ac­tu­al 80th birth­day. At­tend­ance was huge — every­body wanted to be present. Paul was kind enough to seat me at his table, so we talked all night. After the din­ner many math­em­aticians got up to say a few words about him, and what did they say, pretty much in every case? They said the same things I was go­ing to say. Mag­ni­fi­cence and el­eg­ance? Check. Ment­or­ing and gen­er­os­ity? Check. A gregari­ous, fun char­ac­ter? Check. Kind­ness and sup­port­ive­ness, par­tic­u­larly to young­er math­em­aticians? Check. They stole my lines! It is un­usu­al for me to be tongue-tied, but I, as the last speak­er, could really only echo what had been said be­fore. There was uni­ver­sal ad­mir­a­tion and love for the man. What more could you ask for in a math­em­atician and a per­son?

Since I have the space here, I will now go in­to a few more de­tails. I’m go­ing to fo­cus on my time at Brown, em­phas­iz­ing of course the joys and in­ev­it­able ag­on­ies of work­ing on my thes­is. I’ll give cred­it to Paul for the joys. I’ll take the blame for the ag­on­ies. I look back at my gradu­ate ca­reer with genu­ine fond­ness. Then I’ll say just a few words about my ca­reer after Brown, which began as a Ben­jamin Peirce As­sist­ant Pro­fess­or at Har­vard Uni­versity, and then took a pretty sig­ni­fic­ant turn to­wards com­bin­at­or­i­al op­tim­iz­a­tion (par­tic­u­larly schedul­ing the­ory and re­source al­loc­a­tion prob­lems). I was at IBM’s T. J. Wat­son Re­search Cen­ter for 32 years, with a 7-year in­ter­mis­sion at Bell Labs. This “pa­per” is not about me, of course. I simply want to ex­plain why I left the gor­geous field of to­po­logy, and how Paul still had a ma­jor in­flu­ence on my math­em­at­ic­al think­ing and ap­proach. And fi­nally I’d like to give a brief over­view of the math­em­at­ics in my thes­is on the co­homo­logy of ho­mo­gen­eous spaces, again with a nod to Paul. (I wasn’t cer­tain, when I star­ted to write that sec­tion, that I could ac­tu­ally bring it off: I have been asked, on more than one oc­ca­sion, to an­swer ques­tions about my main res­ults. One Fri­day, a col­league at IBM Re­search asked me something very spe­cif­ic. I told him I could not re­mem­ber the de­tails, but I would try to read the two res­ult­ing pa­pers and/or the thes­is it­self over the week­end and get back to him on Monday. On Monday I had to say I had looked at them and was won­der­ing: Who wrote this stuff? That last word wasn’t the one I ac­tu­ally used. Forty years of work­ing on com­pletely dif­fer­ent math­em­at­ics can do that to you — or at least it can to me. So it has taken me a while, but I think I’ve now done the sec­tion justice. Lots of de­tails have come back to me.)

I’ve writ­ten hun­dreds of pa­pers and pat­ents, but I’ve nev­er writ­ten any­thing quite like this. So buckle up and let’s see how it turns out. You’ll laugh; you’ll cry. I know I had fun writ­ing it. I hope you have fun read­ing it.

The Brown years — Paul and me

I did my un­der­gradu­ate work at MIT, gradu­at­ing in 1968. Cathy, my fiancée, was a year be­hind me. I came to Brown in part be­cause I wanted to stay in the Bo­ston area, where she was. We mar­ried that year, to­wards the end of my first semester, and we are still mar­ried 49 years later. Cathy fol­lowed me to Brown the next year, start­ing a Ph.D. pro­gram in ex­per­i­ment­al psy­cho­logy. We have al­ways been a team.

I had taken a won­der­ful sem­in­ar in ho­mo­topy the­ory with Frank Peterson dur­ing my ju­ni­or year at MIT, and I knew then and there that I wanted to be a to­po­lo­gist. I took the usu­al four courses my first year at Brown, in or­der to pre­pare for the qual­i­fy­ing ex­ams: to­po­logy, al­gebra, real ana­lys­is and com­plex ana­lys­is. Two of the courses were mem­or­able to me. The to­po­logy course was taught by Paul, and I liked him from the start. The real ana­lys­is course was taught by Yuji Ito, and the dif­fer­ence in teach­ing styles could not have been more stark. In those days, chil­dren, pro­fess­ors taught us­ing a prim­it­ive im­ple­ment called chalk. Yuji would write com­plete, per­fect sen­tences on the board, nev­er ever mak­ing even the slight­est mis­take. It was in­cred­ible. The res­ult­ing notes, taken ver­batim, could have formed a book, no edit­ing re­quired. I still have these notes, and also those from two courses on er­god­ic the­ory which I took in part to see if he could con­tin­ue this amaz­ing ma­gic trick. He could. Yuji was close to an auto­maton. Paul, on the oth­er hand, was big­ger on verbally ex­plain­ing what was im­port­ant. He would say something like, “Let’s con­sider a to­po­lo­gic­al space \( X \),” and write a huge \( X \) on the board. And we would, in­deed, con­sider it. With something that big, how could you not? Then he would ex­plain why we should con­sider it, per­haps de­scrib­ing the Lef­schetz fixed-point the­or­em, prov­ing it cleanly, mostly verbally, but primar­ily em­phas­iz­ing its beauty. Just like that \( X \), he gave us the big pic­ture. And it was won­der­ful, great fun.

By the end of 1970 I had com­pleted my qual­i­fy­ing ex­ams and was ready to start work on a thes­is. I knew it would be in to­po­logy, and I had sev­er­al pos­sible ad­visors from which to choose. Bob MacPh­er­son had ar­rived at Brown, and I knew he was out­stand­ing. But I really wanted to work with Paul, and thank­fully he agreed.

So in the be­gin­ning of 1971 we star­ted talk­ing about thes­is top­ics. The very first thing Paul did was in­sist that I not tackle the co­homo­logy of ho­mo­gen­eous spaces. This was a sub­ject of in­terest to math­em­at­ic­al lu­minar­ies such as Henri Cartan [e1], [e2] and Ar­mand Borel [e3]. It was also the sub­ject of Paul’s Prin­ceton thes­is, pub­lished as [1]. The nat­ur­al and de­sired “the­or­em” (which I will state pre­cisely later) seemed to have a curse on it, there be­ing sev­er­al false proofs in the lit­er­at­ure. Sadly, Paul had a gap in his own proof, though many ex­cel­lent res­ults and ideas from his thes­is have sur­vived in­tact.

Ac­cord­ingly, Paul sug­ges­ted oth­er top­ics which would be sim­pler and more suit­able for me. And, of course, the more he pushed me away from the co­homo­logy of ho­mo­gen­eous spaces, the more I wanted to tackle it. I’m stu­pid that way. Clearly, Paul thought it was im­port­ant, and I wanted to do something im­port­ant. By the end of the spring semester Paul had re­luct­antly agreed, and I began.

The first thing Paul sug­ges­ted was that I read his full Prin­ceton thes­is, in those days avail­able in book form from the Uni­versity of Michigan mi­cro­films. Cathy and I had planned a one-month va­ca­tion dur­ing the sum­mer of 1971. It was to be our first trip to Europe, and the only home­work Paul gave me for that month was to read the thes­is. Cathy had read the book Europe on \$5 a Day but ap­par­ently thought it was Europe on \$5 a Month. So in Au­gust 1971 we flew (the cheap) Iceland­ic Air­lines to Lux­em­bourg. On the next day we took the train to Cal­ais, then the boat and the train to Lon­don. We re­gistered at our (very cheap) bed and break­fast hotel, and I put the thes­is on the bed so I could start to read it that even­ing. And off we went out to see the sights of Lon­don. You get one guess as to what I dis­covered when we re­turned: the maid had de­cided to clean up from the pre­vi­ous oc­cu­pants, and mis­takenly thrown out the thes­is. I spent sev­er­al hours look­ing for it in the trash, to no avail.

So here I was on day two of a one-month va­ca­tion, and the dog had already eaten my home­work. In those days there was no fix for this: I was toast. I ex­pec­ted Paul to be furi­ous, but when I re­turned I quickly learned one of his best char­ac­ter traits — kind­ness and un­der­stand­ing. No prob­lem at all, he said. Just or­der an­oth­er copy and start again. He said I was now res­ted and ready. And in­deed I was. Res­ted, ready and ex­tremely re­lieved.

And so I star­ted to make some pro­gress. Paul was great at mak­ing sug­ges­tions, and I fol­lowed them up. I re­mem­ber with great fond­ness the oc­ca­sion­al trips to his home in Provid­ence. He did his work in an at­tic, and I was honored to be al­lowed in. (Paul said that his wife and kids were not al­lowed in, mak­ing it that much more of an hon­or.) In those days Paul and Raoul Bott of Har­vard were just fin­ish­ing up their great and fam­ous work on the sin­gu­lar­it­ies of holo­morph­ic fo­li­ations [2]. Ac­cord­ingly, Paul was ex­traordin­ar­ily busy. But I al­ways felt I had full ac­cess to him, and he al­ways seemed to find the time to dis­cuss whatever idea I was think­ing about. And, of course, to point me in the right dir­ec­tion.

When we wer­en’t talk­ing math­em­at­ics Paul and I talked about math­em­at­ics. One thing he said was that I shouldn’t be afraid to make mis­takes. He claimed that all great math­em­aticians ex­cept Gauss made mis­takes — if you wer­en’t mak­ing mis­takes from time to time you wer­en’t try­ing hard enough. This turned out to be rel­ev­ant to me, be­cause I was quick to have false Eureka mo­ments. I would think of an idea that al­most worked and tri­umphantly an­nounce my pro­gress — be­fore I had writ­ten it down and checked it care­fully. Truth­fully, I’ve done this throughout my ca­reer: It’s solved! Oops, it’s not. I have nev­er in­ferred from this trait that I was a great math­em­atician. Un­like Gauss, I just peri­od­ic­ally made mis­teaks. More on this shortly.

An­oth­er metamathem­at­ic­al dis­cus­sion we used to have was the re­l­at­ive im­port­ance of the vari­ous branches of our field. Paul al­ways claimed that the geo­met­ric areas (al­geb­ra­ic and dif­fer­en­tial to­po­logy, al­geb­ra­ic and dif­fer­en­tial geo­metry) were the top­most level, and I, of course, agreed. At the oth­er end of the spec­trum were the more some­how dis­crete areas, such as com­bin­at­or­ics. I agreed with this too. More on this shortly as well.

Dur­ing my en­tire time at Brown I can re­mem­ber only one col­loqui­um giv­en by a non­aca­dem­ic. Roy Adler from the IBM T. J. Wat­son Re­search Cen­ter was an er­godi­cian. Both he and Yuji were stu­dents of Shizuo Kak­utani at Yale, and Roy’s talk de­scribed his ex­cel­lent work on tor­al auto­morph­isms. There was a party that even­ing at Yuji’s house, and I re­mem­ber ask­ing Roy how in the world he wound up at IBM Re­search. He claimed not to really know, but he said that Wat­son was ef­fect­ively a world class uni­versity in which you didn’t have to teach. I’ll come back to this also.

I have one oth­er aside from my gradu­ate days, one which in­volved Paul and Wal­ter Strauss, not me. But it’s too funny not to men­tion. Wal­ter, who worked on dif­fer­en­tial equa­tions, was also at Brown. And both he and Paul had old cars of the same mod­el and year. Paul re­minds me that they were Ply­mouth Vali­ants. (They prob­ably wer­en’t all that vali­ant.) Nat­ur­ally, Paul and Wal­ter would com­pare notes on their cars from time to time, and one day Wal­ter offered Paul four tires. So Paul asked the ob­vi­ous ques­tion: why didn’t Wal­ter want them? Well, it seems that Wal­ter had gone to New York for a week-long meet­ing at NYU. And he had found, to his (brief) de­light, a great park­ing spot on a street right near the uni­versity. Now New York was not in great shape in the early 70s. Even briefly aban­doned cars were of­ten stripped by thieves for parts. The city, in an ef­fort to lim­it this prob­lem, would quickly tow the cars to the pound. Then, if the car was not worth much, they would as­sume it had been left there to rot, and would com­pact it. Thus when Wal­ter went back to where he had left his car, it wasn’t there. After fi­nally find­ing the right agency, he was told the bad news: his car had been com­pacted two hours earli­er. I guess he could have got­ten his car back if he had wanted it — ba­sic­ally in a dog­gie bag. I have nev­er be able to un­see this im­age. I don’t re­call how Wal­ter got back to Provid­ence, but it’s a good bet that a Grey­hound bus was in­volved. Of course, he had no fur­ther use for those tires.

And now back to my thes­is. By the middle of 1972, fol­low­ing Paul’s sug­ges­tions, I had a proof of the main the­or­em on the co­homo­logy of ho­mo­gen­eous spaces — but for the easy vari­ants of the prob­lem — real and ra­tion­al coef­fi­cients. And by the end of 1972 I had found what seemed like a beau­ti­ful, short proof of the more im­port­ant vari­ants — fields of char­ac­ter­ist­ic 0 or (un­der a mild re­stric­tion) of char­ac­ter­ist­ic \( p \). I was very ex­cited and Paul said he was as well. All that was left was to write it up and cel­eb­rate. I had made Paul proud.

I star­ted ap­ply­ing for aca­dem­ic po­s­i­tions on the United States Elec­tion Day, in Novem­ber of 1972 — Nix­on won 49 out of 50 states, which should have been an omen. A num­ber of of­fers came in quite quickly. I can re­call Vander­bilt, Penn and Brit­ish Columbia by early 1973. There may have been oth­ers. But the of­fer that in­ter­ested me the most was from Har­vard, which had a four-year non­ten­ur­able po­s­i­tion for young math­em­aticians. I was offered a Ben­jamin Peirce As­sist­ant Pro­fess­or­ship. There is no doubt that Paul was in­stru­ment­al in get­ting me this and the oth­er po­s­i­tions. And, of course, he was work­ing closely with Raoul, the Har­vard chair­man at that time. That surely did not hurt. Now hav­ing no chance of ten­ure wasn’t good — but stay­ing in the Bo­ston area was ex­actly what I needed: my wife was again one year be­hind me, and wouldn’t get her Ph.D. at Brown un­til 1974. Har­vard wasn’t too bad a uni­versity, of course. That didn’t hurt either.

And so I began the thes­is write-up in earn­est. I knew it wouldn’t take long: the proof was so beau­ti­fully short. And then, poof! As I typed, the proof of the key part of the the­or­em evap­or­ated be­fore my hor­ri­fied eyes. I had found one very subtle, but ter­rible mis­take. The curse had struck again! To quote Verdi’s Rigo­letto, la mal­ed­iz­ione! I was mor­ti­fied. Would I still have a job? Would I even gradu­ate? I wondered briefly wheth­er I should even point out the mis­take. Nobody else, not even Paul had no­ticed it. But I only wondered very briefly. I knew I had to fess up, whatever the cost. And so I did.

And that brought out Paul’s ab­so­lute best. Well, maybe not his first com­ment, which was ex­ist­en­tial in nature. He poin­ted out that if he hadn’t had a gap in his thes­is, I wouldn’t have a thes­is at all. (I ad­mit this was amus­ing, though of scant com­fort.) But then he re­minded me of what he had told me be­fore: that all great math­em­aticians ex­cept Gauss made mis­takes, and now I had qual­i­fied. I had to re­mind him that I wasn’t ex­actly feel­ing like a great math­em­atician at that mo­ment, let alone a Gauss. He said that I would fix the prob­lem and all would be well again. That was more com­fort­ing. He said that I would laugh about this little crisis one day. (Note to Paul: not yet.) He told me that I had enough to gradu­ate any­way, and after call­ing Raoul he told me that my po­s­i­tion at Har­vard was safe. He nev­er re­minded me that he had warned me of the curse. In­stead, his con­fid­ence gave me the cour­age to con­tin­ue. And once I had calmed down enough to work, he gave me an ex­cel­lent math­em­at­ic­al idea. Spe­cific­ally, he sug­ges­ted a slightly dif­fer­ent tack at the proof, one which made use of so-called strongly ho­mo­topy mul­ti­plic­at­ive (SHM) maps. And he poin­ted me to Jim Stasheff, who was at Temple dur­ing those days. Jim was an SHM ex­pert, and I made a pil­grim­age to Phil­adelphia. One in­cred­ibly bru­tal month later I had salvaged my the­or­em us­ing the SHM concept, and this time the write-up pro­cess went well. The proof was longer than it had been, but I did have a proof, a thes­is, a job and a Ph.D. Also, a fu­ture. Thank you, Paul.

The years after Brown

As I have noted already, I’m go­ing to keep this sec­tion brief. I mainly want to point out how Paul in­flu­enced my ca­reer heav­ily in terms of philo­sophy, if not in math­em­at­ic­al sub­stance. Said dif­fer­ently, I hope Paul did not waste his time by be­ing my ad­visor, at least not com­pletely.

At Har­vard I fi­nally got to teach ad­vanced courses, not just the cal­cu­lus sec­tions I had been giv­en at Brown. I taught to­po­logy at both the un­der­gradu­ate and gradu­ate levels, and I had great fun do­ing it. My fa­vor­ite course was modeled after Mil­nor’s great book about Smale’s work on the Poin­caré con­jec­ture [e6]. These courses were quite well re­ceived, and that brought me joy. On the re­search side I was less pro­duct­ive. I know the hor­ror of the in­cred­ible van­ish­ing thes­is had taken its toll on me. But there was also a really long re­cov­ery from a bout of pneu­mo­nia, the very sud­den death of my fath­er, my moth­er’s hor­rible ill­ness, and (at last something good!) the birth of our first daugh­ter. The Har­vard years were cer­tainly my least math­em­at­ic­ally cre­at­ive.

My wife gradu­ated in 1974, tak­ing a one-year job at Welles­ley and then be­com­ing a postdoc at MIT. We had an early case of the two-body prob­lem, and in those days uni­versit­ies had an­ti­nepot­ism rules — quite the re­verse of the situ­ation today. We were de­term­ined to find po­s­i­tions to­geth­er when my Peirce ended in 1977. One of the gradu­ate stu­dents at Har­vard was the bril­liant Don Cop­per­smith, who had won the Put­nam ex­am all four years as an un­der­gradu­ate at MIT. And we be­came good friends. He had been go­ing to IBM Re­search every sum­mer, and he told us that we would love it. I re­membered Roy Adler. Don said the same thing that Roy had said: it was a uni­versity without teach­ing re­spons­ib­il­it­ies, filled with won­der­ful math­em­aticians and oth­er sci­ent­ists. (At that time the re­search staff could even take sab­bat­ic­als at uni­versit­ies.) Cathy and I de­cided to try it. So in the sum­mer of 1976 we went to Wat­son as vis­it­ing fac­ulty. And we did in­deed en­joy it. In 1977 we each had a few aca­dem­ic op­por­tun­it­ies, but nev­er in the same place. We had to ul­ti­mately de­cide between (a) tak­ing po­s­i­tions with half the se­cur­ity, half the salar­ies and half of us, or (b) leav­ing aca­dem­ics. We went with heresy, plan (b). And more or less, we nev­er looked back.

While nobody at IBM Re­search forced me, it was clear my ca­reer would do bet­ter with a change in spe­cial­ties. And so I drif­ted quite quickly in­to com­bin­at­or­i­al op­tim­iz­a­tion, ac­tu­ally still quite geo­met­ric­al in nature — though at the lower end of the math­em­at­ic­al to­tem pole in the dis­cus­sions Paul and I used to have. It amuses me these days to see the close re­la­tion­ship between to­po­logy and quantum com­put­ing — but this is now, and that was then. My wife moved from ex­per­i­ment­al psy­cho­logy (lan­guage ac­quis­i­tion) to speech re­cog­ni­tion and then to hu­man com­puter in­ter­ac­tion. We prospered, both at Wat­son and dur­ing our short in­ter­mis­sion at Bell Labs. We al­ways were a team — no more two-body prob­lem wor­ries.

I tried hard in my ca­reer to fo­cus on the­or­et­ic­al prob­lems I thought were im­port­ant, tak­ing my cue from Paul. I can claim a few suc­cesses. In [e13] we pro­duced the first poly­no­mi­al al­gorithm for a nat­ur­al prob­lem in what is now known as mold­able par­al­lel makespan schedul­ing. In [e15] we cre­ated the first poly­no­mi­al-time al­gorithm for mold­able par­al­lel re­sponse time schedul­ing. In [e12] we de­signed the second poly­no­mi­al-time al­gorithm for what are known as class-con­strained re­source al­loc­a­tion prob­lems. (I say second be­cause we were just beaten to the punch by Fed­er­gru­en and Groenevelt, a fact we dis­covered while fin­ish­ing the proofs of our own pa­per. Sigh.) I am proud of sev­er­al pa­pers which found el­eg­ant op­tim­iz­a­tion tech­niques for com­puter sci­ence prob­lems which were im­port­ant to IBM. Two of these are [e14] and [e16]. A fi­nal pa­per of mine has both prop­er­ties: math­em­at­ic­ally a first and a schedul­ing res­ult which is of value to IBM. This pa­per [e17] ef­fect­ively sub­sumes and gen­er­al­izes (to mal­le­able par­al­lel schedul­ing and a wide range of per­form­ance met­rics) a great deal of my schedul­ing the­ory work.

I was a Dis­tin­guished Mem­ber of Tech­nic­al Staff at Bell Labs and a Prin­cip­al Re­search Staff Mem­ber at Wat­son. I am an IEEE Fel­low. I have over 12,000 cita­tions. So I’ve done ok. Is it to­po­logy, Paul? No, I went to the dark side. I went rogue. Mea culpa. But all math­em­at­ics can be beau­ti­ful if you look for it. And I’ve had my fun. No com­plaints.

I’ve tried in my way to emu­late Paul in oth­er ways as well. I have dis­covered and ment­ored many fine young math­em­aticians. I’ve tried hard to nur­ture them and get them through their oc­ca­sion­al but in­ev­it­able rough patches. I’ve spent a ca­reer with many friends and no real en­emies. And I’m pretty sure I was gregari­ous and fun enough.

Thanks, Paul.

The cohomology of homogeneous spaces

The main the­or­em of [e10] has a con­cep­tu­ally easy proof, though the de­tails are com­plic­ated and reas­on­ably lengthy. We will re­strict ourselves to an over­view here. For a com­plete proof see [e10]. The the­or­em is as fol­lows.

Let \( G \) be a com­pact, con­nec­ted Lie group and \( H \) be a com­pact, con­nec­ted sub­group of \( G \). Form the ho­mo­gen­eous space \( G/H \). Sup­pose that either
  1. \( K \) is a field of char­ac­ter­ist­ic 0, or
  2. \( K \) is a field of char­ac­ter­ist­ic \( p \) and both \( H_{*}(G;Z) \) and \( H_{*}(H;Z) \) have no \( p \)-tor­sion.

Then, re­gard­ing \( H^{*}(BH;K) \) as a left \( H^{*}(BG;K) \)-mod­ule via the map \( f^{*} \) and \( K \) as a right \( H^{*}(BG;K) \)-mod­ule via aug­ment­a­tion, there ex­ists a mod­ule iso­morph­ism \[ H^{*}(G/H;K) \approx \mathrm{tor}_{H^{*}(BG;K)}(K,H^{*}(BH;K)). \]

Let \( T \) be the max­im­al tor­us of \( H \). By a res­ult of Paul’s [1] it is suf­fi­cient to prove the the­or­em in the spe­cial case \( G/T \). So, re­strict­ing ourselves to \( T \), the proof is based on the fol­low­ing dia­gram: \[ \begin{CD} C^{*}(BT;K) @<{{{f^{\#}} }}<{{}}< C^{*}(BG;K) \\ @V{\alpha}VV @AA{\theta_1}A\\ H^{*}(BT;K) @<{{}}<{{{f^{*}} }}< H^{*}(BG;K) \end{CD} \]

The left-hand side of the dia­gram is easy. By May [e8] there is a mul­ti­plic­at­ive ho­mo­logy iso­morph­ism \[ \alpha: C^{*}(BT;K) \rightarrow H^{*}(BT;K) \] which an­ni­hil­ates \( \cup_1 \) products. The right-hand side is more prob­lem­at­ic. Now \( H^{*}(BG;K) \) is a poly­no­mi­al al­gebra, but \( C^{*}(BG;K) \) is not ne­ces­sar­ily graded com­mut­at­ive. For­tu­nately, \( C^{*}(BG;K) \) is ho­mo­topy com­mut­at­ive (via \( \cup_1 \) products) in a very strong way. We take ad­vant­age of this to con­struct an ad­dit­ive ho­mo­logy iso­morph­ism \[ \theta_1: H^{*}(BG;K) \rightarrow C^{*}(BG;K) \] which is the first term in a strongly ho­mo­topy mul­ti­plic­at­ive (SHM) se­quence \( (\theta_1, \theta_2, \ldots) \) whose high­er terms are writ­ten in terms of \( \cup \) and \( \cup_1 \) products.

(Briefly, sup­pose \( A \) and \( B \) are dif­fer­en­tial graded al­geb­ras over \( K \), and \( (f_1, f_2,\ldots ) \) is a se­quence of \( K \)-mod­ule ho­mo­morph­isms such that \[ f_n : A \otimes \cdots (n) \cdots \otimes A \rightarrow B \] for each pos­it­ive in­teger \( n \). Sup­pose \( f_n \) has de­gree \( 1-n \) for each \( n \), and \[ \displaylines{ df_n(a_1 \otimes \cdots \otimes a_n ) - \sum_{i=1}^{n} \sigma(i-1)f_n(a_1 \otimes \cdots \otimes da_i \otimes \cdots \otimes a_n ) \hfill\cr\hfill = \sum_{i=1}^{n-1} \sigma(i) \bigl[ f_{n-1} (a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n ) \hfill\cr\hfill {}- f_i (a_1 \otimes \cdots \otimes a_i) f_{n-i} (a_{i+1} \otimes \cdots \otimes a_n)\bigr]. } \] Such a se­quence is said to be SHM. See [e5] and [e9] for more de­tails.)

The dia­gram above is, in fact, com­mut­at­ive. Fur­ther­more, the ef­fect of the high­er-or­der SHM maps is nul­li­fied by \( \alpha \).

By the res­ults of Ei­len­berg and Moore [e4], [e7] there is an al­gebra iso­morph­ism \[ H^{*}(G/H;K) \approx \mathrm{Tor}_{C^{*} (BG;K} (K, C^{*} (BH;K)), \] which we shall ap­ply to \( H=T \). Note that this \( \mathrm{Tor} \) is more gen­er­al (and vastly less com­put­able) than \( \mathrm{tor} \), our de­sired tor­sion product. To prove The­or­em 1 we will ac­tu­ally need to cre­ate a yet more gen­er­al and less com­put­able tor­sion product, namely \( \mathrm{TOR} \). It is de­scribed in terms of SHM se­quences, and we will not go in­to those de­tails here. Suf­fice it to say that it provides an iso­morph­ism \[ \mathrm{Tor}_{C^{*} (BG;K)} (K,H^{*}(BH;K)) \approx \mathrm{TOR}_{H^{*}(BG;K)} (K,H^{*}(BH;K)), \] which we will again ap­ply to \( H=T \).

Wav­ing our hands a bit more (to avoid many pages of ex­pos­i­tion), the key cal­cu­la­tion shows that the com­pos­ite SHM se­quence \[ (\alpha f^{\#}, 0, \ldots) \circ (\theta_1, \theta_2, \ldots) \] is ac­tu­ally and ma­gic­ally equal to the strictly mul­ti­plic­at­ive se­quence \[ (f^{*}, 0, \ldots), \] mean­ing that the most com­plex \( \mathrm{TOR} \) col­lapses in one fell swoop to \( \mathrm{tor} \). The res­ult­ing se­quence of mod­ule iso­morph­isms \[ \eqalign{ H^{*}(G/T) &\approx \mathrm{Tor}_{C^{*} (BG;K} (K, C^{*} (BT;K) \cr &\approx \mathrm{Tor}_{C^{*} (BG;K} (K,H{*}(BT;K)) \cr &\approx \mathrm{TOR}_{H^{*}(BG;K)} (K,H^{*}(BT;K)) \cr &= \mathrm{tor}_{H^{*}(BG;K)} (K,H^{*}(BT;K)) } \] proves the the­or­em. For the full de­tails, see [e10].

The main the­or­em in [e11] is a sim­pler and less im­port­ant res­ult, namely:

Giv­en a dif­fer­en­tial fiber bundle \[ \sigma = (E, \pi, X, G/H, G), \] where \( K \) is either the reals or ra­tion­als, and \( X=G^{\prime}/H^{\prime} \) is a ho­mo­gen­eous space formed as the quo­tient of a com­pact, con­nec­ted Lie group \( G^{\prime} \) by a com­pact, con­nec­ted sub­group \( H^{\prime} \) of de­fi­ciency 0 in \( G^{\prime} \), then there is an al­gebra iso­morph­ism \[ H^{*}(E;K) = \mathrm{tor}_{H^{*}(BG;K)} (H^{*}(X;K), H^{*}(BH;K). \]

We omit the de­tails, but they are avail­able in [e11].

Last words

So here’s to Paul Baum, the new­est mem­ber of Cel­eb­ra­tio Math­em­at­ica. What a great hu­man be­ing. And what a mag­ni­fi­cent ca­reer he has had — so far, at age 82. I am 12 years young­er than Paul, and yet I have re­tired and Paul is still at the top of his game. In 2017 he was the Gauss Pro­fess­or at the Uni­versity of Göttin­gen. How lovely and ap­pro­pri­ate is that?

At the end of the 2016 ban­quet at the Fields In­sti­tute, Paul para­phrased Churchill. He said he didn’t think it was the be­gin­ning of the end. He said it might be the end of the be­gin­ning. I don’t think it’s even that. I’ll para­phrase Jerome Kern in Show­boat: like Old Man River, Paul just keeps rolling along. Be afraid, un­solved to­po­lo­gic­al prob­lems. Be very afraid.


[1] P. F. Baum: “On the co­homo­logy of ho­mo­gen­eous spaces,” To­po­logy 7 : 1 (January 1968), pp. 15–​38. MR 219085 Zbl 0158.​42002 article

[2] P. Baum and R. Bott: “Sin­gu­lar­it­ies of holo­morph­ic fo­li­ations,” J. Diff. Geom. 7 : 3–​4 (1972), pp. 279–​342. To S. S. Chern and D. C. Spen­cer on their 60th birth­days. MR 377923 Zbl 0268.​57011 article