A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S 3 has the form F 2 [X 2n ] ⊗ E(Sq 1 X 2n ) where x 2 n is in degree 2 n and n = 2. If F denotes the homotopy theoretic fibre of the map S 3 → B 2 S 1 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 mod2 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 the following.