 
Summary: A MOD TWO ANALOGUE OF A CONJECTURE OF COOKE
J. Aguad´e, C. Broto and D. Notbohm
§0. Introduction.
The mod two cohomology of the three connective covering of S3
has the form
F2[x2n] E(Sq1
x2n)
where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the
map S3
B2
S1
of degree 2, then the mod 2 cohomology of F is also of the same form
for n = 1. Notice (cf. section 7 of the present paper) that the existence of spaces whose
cohomology has this form for high values of n would immediately provide Arf invariant
elements in the stable stem. Hence, it is worthwhile to determine for what values of n the
above algebra can be realized as the mod 2 cohomology of some space. The purpose of
this paper is to construct a further example of a space with such a cohomology algebra for
n = 4 and to show that no other values of n are admissible. More precisely, we prove:
Theorem 1. There is a space X such that H
(X; F2) = F2[x2n] E(Sq1
