Celebratio Mathematica

The Ei­len­berg–Moore spec­tral se­quence (EMSS for short) was dis­covered by J. C. Moore and S. Ei­len­berg some­time be­fore 1960 and would seem to have first been made widely avail­able in the art­icle by J. C. Moore in the 1959/1960 Cartan Sem­in­ar [e4]. At this time Paul was a gradu­ate stu­dent at Prin­ceton Uni­versity with N. E. Steen­rod as ad­visor and had the in­geni­ous idea to ex­ploit the EMSS to ex­tend the work of A. Borel on the co­homo­logy of ho­mo­gen­eous spaces (see, e.g., [e2] and [e3]). The start­ing point here is the fibra­tion $$G/H\hookrightarrow BH\downarrow BG,$$ where $G$ is a com­pact (con­nec­ted) Lie group and $H\le G$ a closed (con­nec­ted) sub­group. At the time A. Borel did his work, the stand­ard tool for deal­ing with the co­homo­logy of fibra­tions was the Serre spec­tral se­quence, which for a fibra­tion $$F\hookrightarrow E\downarrow B$$ is a spec­tral se­quence1 con­ver­ging to $H^{\ast }(E)$ and whose $E_2$-term is $H^{\ast }(B;H^{\ast }(F))$ (see e.g., [e1]). As one sees, the tar­get of this spec­tral se­quence ap­plied to $$G/H\hookrightarrow BH\downarrow BG,$$ the case stud­ied by A. Borel, is the known2 $H^{\ast }(BG)$ and not the sought for $H^{\ast }(G/H)$. This forced A. Borel to in­vent a num­ber of cun­ning and tech­nic­al tools that al­low ex­trac­tion of in­form­a­tion con­cern­ing the co­homo­logy of the fiber term from the Serre spec­tral se­quence. The EMSS however ap­plied in the same con­text provides a spec­tral se­quence con­ver­ging dir­ectly to $H^{\ast }(G/H)$ just as one wants. But there is a prob­lem.

The catch here is that the $E_2$-term of the EMSS is not so simple as that of the Serre spec­tral se­quence; for the EMSS in the con­text of the fibra­tion $$G/H\hookrightarrow BH\downarrow BG,$$ it is the bi­graded al­gebra $${\mathrm{Tor}}_{H^{\ast }(BG)}^{\ast }(\mathbb{F},H^{\ast }(BH){)}_{\ast },$$ where $\mathbb{F}$ is a field and co­homo­logy is taken with coef­fi­cients in $\mathbb{F}$. At the time Paul was work­ing on his thes­is, the lit­er­at­ure on the com­pu­ta­tion of tor­sion products was just be­gin­ning to take form, and the first two chapters of Paul’s thes­is are de­voted to de­vel­op­ing tools from com­mut­at­ive and ho­mo­lo­gic­al al­gebra for do­ing so. the The main res­ult of his thes­is then be­ing that un­der ap­pro­pri­ate re­stric­tions on $G$, $H$, and $\mathbb{F}$, the EMSS col­lapses at the $E_2$-term.3 The struc­ture of this $E_2$-term, namely the tor­sion product $${\mathrm{Tor}}_{H^{\ast }(BG)}^{\ast }(\mathbb{F},H^{\ast }(BH){)}_{\ast },$$ was also in­vest­ig­ated and to this day some points con­cern­ing it re­main ob­scure.

After fin­ish­ing his Ph.D. at Prin­ceton Uni­versity, Paul went to Eng­land on a postdoc­tor­al fel­low­ship and it was there that my ad­visor W. S. Mas­sey met him and learned of his work. Upon re­turn­ing to Yale Uni­versity (where I be­came W. S. Mas­sey’s stu­dent in the winter of 1963/1964) he presen­ted me with notes on Paul’s work, en­cour­aged me to buy a copy of Paul’s thes­is from Uni­versity Mi­cro­films, and con­fron­ted me with [e4] and sev­er­al thes­is prob­lems where the Ei­len­berg–Moore spec­tral se­quence was either the sub­ject per se or seemed ex­actly the right tool to deal with the prob­lem. The res­ult was [e6].

Dur­ing the years 1964–1966, while I was still a stu­dent, I con­tac­ted Paul first by mail, and, after he had re­turned to the U.S., in per­son at the In­sti­tute for Ad­vanced Stud­ies. The over­lap between [1] and [e6] be­ing ob­vi­ous to both of us, par­tic­u­larly con­cern­ing the case of real coef­fi­cients, then res­ul­ted in our col­lab­or­a­tion and the pub­lic­a­tion of [2]. Dur­ing this time Paul was most gen­er­ous in shar­ing ideas with me. As a con­sequence I was able to bounce many of my half-baked ideas off him and one res­ult for me was [e5].

After re­ceiv­ing my Ph.D. I had the enorm­ous good luck that J. C. Moore was taken by my work and with his re­com­mend­a­tion I ob­tained an in­struct­or­ship at Prin­ceton Uni­versity where Paul had be­come an as­sist­ant pro­fess­or. We con­tin­ued our dis­cus­sions and had plans to work on sev­er­al dif­fer­ent pro­jects to­geth­er, but time and our chan­ging in­terests (Paul was mov­ing un­der the in­flu­ences of M. F. Atiyah and R. Bott and I un­der that of R. E. Stong and P. E. Con­ner) led to our re­search agen­das branch­ing away from each oth­er.

In 1968 I was awar­ded a postdoc­tor­al fel­low­ship which al­lowed me to spend a year at the IHES in Bures-sur-Yvette and con­tin­ue my joint work with J. C. Moore who was on sab­bat­ic­al at Par­is V. The spa­tial and chro­no­lo­gic­al dis­tances4 to­geth­er with the changed in­terests between Paul and me con­spired to di­min­ish our com­mu­nic­a­tions and we lost con­tact for many dec­ades. Thanks to Paul’s col­lab­or­a­tion with T. Schick in re­cent times we have after all those in­ter­ven­ing years be­come reac­quain­ted as Paul has spent some time in Göttin­gen. He has been very help­ful to me in listen­ing to my ideas on how to prove a con­jec­ture of B. Kostant con­cern­ing the struc­ture of the tor­sion product $${\mathrm{Tor}}_{H^{\ast }(BG)}^{\ast }(\mathbb{F},H^{\ast }(BH){)}_{\ast }$$ oc­cur­ring in the EMSS; as far as I know this con­jec­ture only ap­pears in the un­pub­lished ver­sion of Paul’s thes­is \cite[page 3.27 et seq.] {mr:2613915} With his help and cri­ti­cism, as well as from oth­er friends, many cases of this con­jec­ture are proven and it would be nice if a new col­lab­or­a­tion would res­ult out of this.