 
Summary: HOMOTOPY CLASSIFICATION OF SPACES
WITH INTERESTING COHOMOLOGY
AND A CONJECTURE OF COOKE,
PART I
J. Aguad´e, C. Broto and D. Notbohm
1. Introduction. The title of this paper is reminiscent of the title of one of the last papers
by George Cooke ([7]). In that paper, Cooke observes that if X is a pcomplete loop space
then there is an action of [S1
^p, S1
^p] = ^Zp on X. In particular, there is an action of the p1
roots of unity on X and by taking the quotients of appropriate loop spaces by this action
he obtains spaces with "interesting" cohomology, i.e. spaces whose cohomology algebras
have quite few generators and relations and whose attaching maps represent interesting
elements in the stable homotopy of spheres. By applying this technique to S3
3 , the
3connective covering of S3
, and to the fibre of the map S3
K(Z, 3) of degree p, Cooke
constructs spaces realizing the cohomology algebras (subscripts denote degrees)
(1) Fp[x2n] E(x2n),
