#### 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.