Celebratio Mathematica

Fritz Hirzebruch played a ma­jor part in my life, par­tic­u­larly over the early form­at­ive peri­od. He be­came a close per­son­al friend, a long-term col­lab­or­at­or, and, through the Arbeit­sta­gung, my in­tro­duc­tion to the math­em­at­ic­al world. I learned a good deal from him on how to write pa­pers, how to present talks, and, most im­port­antly, how to handle people. In short, he was an ideal role mod­el.

I first met Fritz in 1954 when I was a young gradu­ate stu­dent and he vis­ited Cam­bridge at the in­vit­a­tion of Hodge, my su­per­visor. Hodge and Todd had been much im­pressed by what Fritz had been do­ing at Prin­ceton and were keen to be briefed on the Riemann–Roch The­or­em and Chern classes. What I re­mem­ber about the oc­ca­sion is how friendly and in­form­al Fritz was. Al­though he was already an as­sist­ant pro­fess­or at Prin­ceton and I was merely a gradu­ate stu­dent, there were no bar­ri­ers between us, and we quickly es­tab­lished a friend­ship which blos­somed over the sub­sequent years. We met again at the Am­s­ter­dam ICM of 1954 and then, for a longer peri­od, when I went on a postdoc­tor­al fel­low­ship to the In­sti­tute for Ad­vanced Study in Prin­ceton.

Those Prin­ceton years were, for me, for Fritz, and for many oth­ers the “golden years”. Al­geb­ra­ic geo­metry and to­po­logy were be­ing trans­formed by the new ideas of the French School. Sheaves and spec­tral se­quences from Leray com­bined with com­plex ana­lys­is by Henri Cartan pro­duced power­ful ma­chinery to tackle clas­sic­al prob­lems. This was taken up by Kodaira and Spen­cer, while Serre burst on the scene with spec­tac­u­lar ap­plic­a­tions to both al­geb­ra­ic to­po­logy and al­geb­ra­ic geo­metry. When I ar­rived in Septem­ber 1955, bril­liant young math­em­aticians were ab­sorb­ing the new ideas and carving out new routes for the fu­ture. I re­mem­ber in par­tic­u­lar the gang who reg­u­larly at­ten­ded Kodaira’s lec­tures: Fritz, Serre, Bott, Sing­er.

This had been for Fritz the ex­per­i­ence that trans­formed him from a prom­ising novice to a world fig­ure cap­able of com­pet­ing with the greatest tal­ents of the time. With­in a short peri­od of time he came up with two great tri­umphs. Both were based on the in­nov­at­ive way of as­so­ci­at­ing mul­ti­plic­at­ive classes to form­al power series in one vari­able. First there was his for­mula $$\mathrm{Sign}(M)={\int }_M\mathcal{L}(M)={\int }_M\prod _{i=1}^n\frac{x_i}{\tanh x_i},\text{where (formally) }p(M)=\prod _{i=1}^n(1+x_i^2)$$ for the sig­na­ture of a 4$n$-di­men­sion­al man­i­fold in terms of the Pon­trja­gin classes $p_j(M)$. This was a beau­ti­ful ap­plic­a­tion of Thom’s new cobor­d­ism the­ory. But Fritz’s second tri­umph, his gen­er­al­iz­a­tion (now known as “HRR” or Hirzebruch–Riemann–Roch) $$\chi (X,V)={\int }_X\mathrm{ch}(V)\mathrm{Td}(X)$$ of the Riemann–Roch The­or­em, was even more im­press­ive. Here $$\chi (X,V)=\sigma (\text{from }q=1\text{ to }q=m)\text{ of }(-1{)}^q\dim H^q(X,V)$$ is the holo­morph­ic Euler char­ac­ter­ist­ic $\sum _{q=0}^m\dim H^q(X,V)$ of the sheaf co­homo­logy groups of a holo­morph­ic vec­tor bundle $V$ (of di­men­sion $d$) over a com­plex pro­ject­ive al­geb­ra­ic man­i­fold $X$ of di­men­sion $m$. The Chern char­ac­ter is defined in terms of the total Chern class $c(V)$ by $$\mathrm{ch}(V)=\sum _{i=1}^d{e}^{x_i},\text{where (formally) }c(V)=\prod _{i=1}^d(1+{\chi }_i),$$ and the Todd class $Td(X)$ is defined sim­il­arly in terms of the total Chern class $c(X)$ of the tan­gent bundle of $X$ by $$\mathrm{Td}(X)=\prod _{j=1}^m\frac{y_j}{1-e^{-y_j}},\text{where (formally) }c(X)=\prod _{j=1}^m(1+y_j).$$ Of course HRR built on fun­da­ment­al work by Kodaira, Spen­cer, and Serre, but the proof was a tour de force that had the hall­mark of Fritz’s own math­em­at­ic­al style.

Char­ac­ter­ist­ic classes had been de­vel­op­ing for many years, from the al­geb­ra­ic geo­metry of the Itali­an School through sig­ni­fic­ant ad­vances by Todd and the later to­po­lo­gic­al ap­proach of Steen­rod and Chern. But all this em­phas­ized their geo­met­ric­al ori­gin and sig­ni­fic­ance. It was Fritz, in col­lab­or­a­tion with Borel, who took the dual route of co­homo­logy and, con­nect­ing it to the the­ory of Lie groups, gave Chern classes their form­al al­geb­ra­ic set­ting, which has now be­come stand­ard. With his com­mand of this al­gebra and with his in­sight in­to the right al­geb­ra­ic frame­work, Fritz had de­veloped his the­ory of mul­ti­plic­at­ive se­quences, which provided the right tools to tame the hor­rendous-look­ing for­mu­lae.

When Todd, no slouch at al­geb­ra­ic com­pu­ta­tions, had com­puted the first half dozen “Todd poly­no­mi­als”, it had been a mat­ter of brute force. In the hands of Fritz as “ma­gi­cian” the cal­cu­la­tions be­came el­eg­ant and trans­par­ent. After see­ing this, Todd re­marked that he now had to re­verse the earli­er view he had held of the “Prin­ceton School” that, while they might be good at gen­er­al the­ory, they were not ad­ept at cal­cu­la­tions. The old maes­tro con­ceded de­feat to the young con­tender.

It was for­tu­nate for the new gen­er­a­tion like me, eager to learn about the great ad­vances in al­geb­ra­ic geo­metry, that Fritz was also a bril­liant ex­pos­it­or. His book Neue To­po­lo­gis­che Meth­oden in der Al­geb­rais­chen Geo­met­rie (Spring­er, 1956) gave an im­pec­cably clear ac­count of sheaf the­ory, Chern classes, and all the new ma­chinery that cul­min­ated in the Hirzebruch–Riemann–Roch The­or­em. The book and its sub­sequent Eng­lish edi­tion To­po­lo­gic­al Meth­ods in Al­geb­ra­ic Geo­metry (with ap­pen­dices by R. L. E. Schwar­zen­ber­ger, one of my early stu­dents) has re­mained the stand­ard work for over fifty years.

When Fritz re­turned to Ger­many as a full pro­fess­or at the Uni­versity of Bonn, a new day dawned for Ger­man math­em­at­ics. With his en­thu­si­asm, abil­ity, ef­fi­ciency and drive, Fritz soon trans­formed Bonn in­to a ma­jor cen­ter of the math­em­at­ic­al world. Mod­elled on Prin­ceton, it aimed to in­tro­duce in­to Europe the fea­tures that had so at­trac­ted Fritz and oth­ers across the At­lantic.

Be­cause of the friend­ship I had forged with Fritz in Prin­ceton and be­cause of the prox­im­ity of Cam­bridge to Bonn, I was for­tu­nate to have been in­vited to the very first of the an­nu­al meet­ings that be­came the fam­ous Arbeit­sta­gung. I went on at­tend­ing these meet­ings for al­most thirty years. It be­came an ob­lig­at­ory part of the aca­dem­ic cal­en­dar where new res­ults were an­nounced, many fam­ous math­em­aticians reg­u­larly at­ten­ded, and the whole event was un­der the care­ful but lov­ing care of the “maes­tro”. Fritz’s tal­ents were fully ex­ploited, but not ex­posed, in these an­nu­al gath­er­ings. With their re­laxed at­mo­sphere, the Rhine cruises and the skill­ful se­lec­tion of speak­ers by what has been de­scribed as “guided demo­cracy”, the Arbeit­sta­gungs were unique. Happy fam­ily gath­er­ings they may have been, but much ser­i­ous math­em­at­ics was al­ways be­ing presen­ted and fostered. Ideas flowed, col­lab­or­a­tions emerged, and suc­cess­ive years re­flec­ted the latest move­ments.

Moreover, as the years passed, Fritz was al­ways keen to at­tract new tal­ent, and he en­cour­aged me to send prom­ising gradu­ate stu­dents to at­tend. I was happy to re­spond, and, over the years, my stu­dents were in­tro­duced to the in­ter­na­tion­al scene through the Arbeit­sta­gung. Graeme Segal, Nigel Hitchin, Si­mon Don­ald­son, Frances Kir­wan, and many oth­ers came and be­came, in their turn, reg­u­lar par­ti­cipants.

But if the en­tire series of Arbeit­sta­gungs be­came high­points of the aca­dem­ic cal­en­dar, the ini­tial one (in 1957) on a very mod­est scale was par­tic­u­larly note­worthy for launch­ing Grothen­dieck. He had just de­veloped his bril­liant new ap­proach to the Hirzebruch–Riemann–Roch The­or­em, based on $K$-the­ory. I re­mem­ber him lec­tur­ing for many hours on his ideas. In fact he seemed al­most to mono­pol­ize the timetable, but the nov­elty and im­port­ance of his work fully jus­ti­fied the time de­voted to it. The fact that the pro­gram was suf­fi­ciently gen­er­ous and flex­ible to al­low this to hap­pen was an early in­dic­a­tion of the way Fritz wanted the Arbeit­sta­gungs to work. No set plans, and full steam ahead for nov­el and ex­cit­ing math­em­at­ics.

Grothen­dieck’s ex­plos­ive entry on the scene was a hard act to fol­low, but the Arbeit­sta­gungs in those early years saw a suc­ces­sion of new and ex­cit­ing res­ults, in­clud­ing Mil­nor’s dis­cov­ery of exot­ic spheres and their sub­sequent real­iz­a­tion by Brieskorn (a stu­dent of Fritz) via isol­ated sin­gu­lar­it­ies of al­geb­ra­ic vari­et­ies (a study ini­ti­ated by Fritz). In fact, so many new ideas filled the Arbeit­sta­gung air that most of my own work (and prob­ably that of many oth­ers) emerged from this back­ground. We learned many new things from dis­par­ate fields, and cross-fer­til­iz­a­tion be­came the norm. I will elab­or­ate on this in the next sec­tion.

In the three years 1959 to 1962 Fritz and I wrote eight joint pa­pers, all con­cerned with to­po­lo­gic­al $K$-the­ory and its ap­plic­a­tions. This had emerged nat­ur­ally from the early Arbeit­sta­gungs and in par­tic­u­lar from Grothen­dieck’s $K$-the­ory in al­geb­ra­ic geo­metry, as ex­pounded in the very first Arbeit­sta­gung. But there were many oth­er in­gredi­ents in the back­ground, not­ably the Bott peri­od­icity the­or­em.

To­po­lo­gic­al $K$-the­ory was mainly de­veloped by Fritz and me in 1959 when we both had a sab­bat­ic­al term at the IAS in Prin­ceton. A pre­lim­in­ary ac­count ap­pears in [1], and we planned to write an ex­pan­ded ver­sion in book form. In fact, we nev­er had time for this pro­ject, but a book [e1] did even­tu­ally ap­pear un­der my name based on a Har­vard course of lec­tures.

These joint pa­pers are a mix­ture of gen­er­al the­ory and con­crete prob­lems. For ex­ample, [3] showed that the fam­ous Hodge con­jec­tures were false for in­teger co­homo­logy (still leav­ing the case of ra­tion­al co­homo­logy as one of the Clay In­sti­tute Mil­len­ni­um Prize prob­lems). Oth­er pa­pers were re­lated to some of Fritz’s earli­er Prin­ceton peri­od, such as his dis­cov­ery of a re­la­tion between Steen­rod squares and the Todd poly­no­mi­als [2]. Some of our joint pa­pers ap­peared in Ger­man (writ­ten by Fritz), while oth­ers ap­peared in Eng­lish (writ­ten by either of us), but one ap­peared in French (writ­ten by neither of us!). That one gave bounds on the smal­lest di­men­sion in which vari­ous man­i­folds could be em­bed­ded. While a prim­it­ive ver­sion was an idea of mine, the fi­nal very pol­ished ver­sion was an ex­quis­ite il­lus­tra­tion of Fritz’s el­eg­ance with al­geb­ra­ic for­mu­lae. But my math­em­at­ic­al in­ter­ac­tion with Fritz ex­ten­ded far bey­ond these joint pub­lic­a­tions and the three years they cov­er. Much of my work was in­flu­enced in one way or an­oth­er by Fritz, and a later pub­lic­a­tion [4] is one of my fa­vor­ites. Here we proved that a spin man­i­fold that ad­mits a non­trivi­al circle ac­tion has van­ish­ing $\hat{A}$-genus. This emerged as a new ap­plic­a­tion of in­dex the­ory, which first ap­peared in the Arbeit­sta­gung pro­gram of 1962. Fritz took great in­terest in the de­vel­op­ment of in­dex the­ory, which owes so much to his pi­on­eer­ing work.

While our later math­em­at­ic­al paths may ap­pear to have di­verged, this is only su­per­fi­cially true. We met fre­quently in Bonn and else­where, and we fol­lowed each oth­er’s work with great in­terest. One not­able ex­ample is Fritz’s beau­ti­ful res­ults on the res­ol­u­tion of the cusp sin­gu­lar­it­ies of Hil­bert mod­u­lar sur­faces (as ex­plained by Don Za­gi­er). His key res­ult gave the sig­na­ture de­fect of such a cusp sin­gu­lar­ity as the value of a suit­able $L$-func­tion of the num­ber field. He then con­jec­tured that this res­ult would con­tin­ue to hold in high­er di­men­sions for ar­bit­rary real num­ber fields. This was one of the main sources of in­spir­a­tion that even­tu­ally led to the in­dex the­or­em for man­i­folds with bound­ary [e2] and its ap­plic­a­tion [e3] to prove Fritz’s con­jec­ture.

Fritz also fol­lowed with great in­terest the ex­cit­ing in­ter­ac­tion between geo­metry and phys­ics of re­cent dec­ades. He or­gan­ized sev­er­al meet­ings of math­em­aticians and phys­i­cists (in Bad Hon­nef in 1980 and in Schloss Ring­berg in 1988, 1989, and 1993). He also ex­ten­ded [5] the work of Wit­ten and oth­ers on the el­lipt­ic genus, a sub­ject close to his heart.

A close math­em­at­ic­al part­ner­ship leads to a close per­son­al friend­ship and also evolves from it. This ex­tends to fam­il­ies on both sides. Lily and I got to know Fritz and Inge in Prin­ceton when we both had small chil­dren, and we have re­mained close friends ever since, meet­ing oc­ca­sion­ally in Bonn, Ox­ford, Ed­in­burgh, Bar­celona, and else­where. In Bonn, at all the Arbeit­sta­gungs, Inge was al­ways a wel­com­ing host­ess, and the friendly at­mo­sphere of the Hirzebruch fam­ily was an im­port­ant in­gredi­ent in the suc­cess of both the Arbeit­sta­gungs and the MPI.