 
Summary: L'Enseignement Math´ematique (2) 54 (2008), 34
1
ON COHOMOLOGY ALGEBRAS
by Jaume AGUAD´E
When I met Guido Mislin at the ETH in Zurich for the first time  in the
late seventies of the past century  I was fascinated by the many subtle and
mysterious conditions that a graded commutative Fp algebra has to satisfy in
order to be the mod p cohomology algebra of some space. Back in those days,
the AdamsWilkerson embedding ([1]) was hot news, the Sullivan conjecture
was not yet a theorem and new ideas were appearing quickly. Since then, we
have learned a lot about the big questions of yore, like polynomial algebras
which are cohomology algebras, the injective objects in the category of unstable
modules over the Steenrod algebra, the type of finite loop spaces, the mod p
homotopy uniqueness of classifying spaces of finite loop spaces and so on.
It is not unusual that concentration on the major problems of a certain
subject leaves unsolved many questions which just fail to be in the mainstream
of mathematical progress. I wish to mention here two problems in realizability
of algebras as cohomology rings that were considered around 1980 and, to the
best of my knowledge, were forgotten and have remained unanswered. I do
not think they have ever appeared in print but I clearly remember how Alex
