by Larry Smith
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 \( H \leq G \) 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 \( H^*(E) \) and whose \( E_2 \)-term is \( H^*(B; H^*(F)) \) (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 \( H^*(BG) \) and not the sought for \( H^*(G/H) \). 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 \( H^*(G/H) \) 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 \[ \mathrm{Tor}_{H^*(BG)}^* (\mathbb{F}, H^*(BH))_*, \] 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 \[ \mathrm{Tor}_{H^*(BG)}^* (\mathbb{F}, H^*(BH))_*, \] 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 \[ \mathrm{Tor}_{H^*(BG)}^* (\mathbb{F}, H^*(BH))_* \] 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.
