The mod two cohomology of the three connective covering of
S3 has the form
formula here
where x2n is in degree 2n and
n = 2. If F denotes the homotopy theoretic 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 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.