A conjecture of Barratt-Jones-Mahowald concerning framed manifolds having Kervaire invariant one (Q1122827)

This paper is concerned with the following reformulation of the Kervaire invariant one problem (see also [\textit{M. G. Barratt}, \textit{J. D. S. Jones} and \textit{M. E. Mahowald}, Lect. Notes Math. 1286, 135-173 (1987; Zbl 0633.55011)]). The \({\mathbb{Z}}/2\)-transfer map t: \(\pi^ s_ n({\mathbb{P}}_{\infty}{\mathbb{R}})\to \pi^ s_ n(S^ 0)_{(2)}\) from stable homotopy of the real projective space to the stable homotopy of spheres is onto by the Kahn-Priddy theorem \((n>0)\). So if there exist elements \(\Theta_ N\) \((N=2^{k+1}-2)\) of Kervaire invariant 1 they have preimages \({\hat \Theta}{}_ N\) on \(\pi^ 2_*({\mathbb{P}}_{\infty}{\mathbb{R}})\). It is well known that any \({\hat \Theta}{}_ N\) is detected by \(Sq^{2^ k}\). Let \(jo_*\) be the generalized homology Theory defined by the cofibration \(jo\to bo_{(2)}^{\psi^ 3-1}\to bspin_{(2)}\) and h: \(\pi^ s_*(x)\to jo_*(x)\) the associated Hurewicz map (bo is the spectrum of connective KO-theory and bspin its 3-connected cover). The main result of the paper is that there exists \(y\in \pi^ s_ N({\mathbb{P}}_{\infty}{\mathbb{R}})\) detected by \(Sq^{2^ k}\) if and only if h(y)\(\neq 0\). The condition h(y)\(\neq 0\) is also reformulated in terms of the classical complex and real e-invariant. So the problem of detecting \(\Theta_ N\) is translated from using secondary cohomology operations to the computation of a degree. The proof given by the authors is short, direct and quite nice. The main ingredient is a factorization of the stable bo-operation \((\psi^ 3-1)(\psi^ 3-9)...(\psi^ 3-9^ m)\) through an operation appearing in the splitting of bo\(\wedge bo\) (e.g. \textit{R. J. Milgram} [Conf. homotopy theory, Evanston 1974, 127-158 (1975; Zbl 0333.55009)]). That any \({\hat \Theta}{}_ N\) is detectable by jo-theory is to a large extent a folklore type result without a published proof. As far as I know, M. Mahowald first found it, but did not publish a proof. Remarks on the problem and a proof of one implication (see also the first citation) may be found in \textit{M. Mahowald}'s paper in Ann. Math., II. Ser. 116, 65-112 (1982; Zbl 0504.55010).