A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S³has the form F₂X_2n] E(Sq¹X_2n) where x_2n is in degree 2n and n = 2. If F denotes the homotopytheoretic fibre of the map S³ B²S¹ of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice(cf. Section 7 of the present paper) that the existence of spaceswhose cohomology has this form for high values of n would immediatelyprovide Arf invariant elements in the stable stem. Hence, itis worthwhile to determine for what values of n the above algebracan be realized as the mod2 cohomology of some space. The purposeof this paper is to construct a further example of a space withsuch a cohomology algebra for n = 4 and to show that no othervalues of n are admissible. More precisely, we prove the following.