Im(J)-theory and the Kervaire invariant (Q1374874)

If \(p\) is a prime, let \(\ell\) denote the Adams summand of \((-1)\)-connective, \(p\)-local complex \(K\)-theory. Let \(\overline{\ell}\) denote the cofibre of the unit map, \(S^{\circ}\to \ell\). Since \(\pi_*(\ell)\) is torsion for every element of \(\pi_*(S^{\circ})\) lifts to \(\pi_{*+1} (\overline{\ell})\). If \(k\in (\mathbb{Z}/p^2)^*\) is a generator (\(p\) odd) or \(k=3\) if \(p=2\) then \(A\)-theory is the fibre of the lifting of \(\psi^k-1:\ell\to \ell\) to factorize through \(v_1: \Sigma^{2(p-1)} \ell\to\ell\). The author shows that there exists a stably spherical element in \(A_{2n-1} (\overline{\ell})\) if and only if there is an element of Kervaire invariant one in \(\pi^s_{2n-2} (S^{\circ})\) localized at \(p\). As a corollary it is shown that there exists a stably spherical element in \(\pi^s_{2n-2} (B\Sigma_p)\) if and only if there exists an element of Kervaire invariant one in \(\pi^s_{2n-2} (S^{\circ})_{(p)}\). The most important case is when \(p=2\). This result was conjectured in [\textit{M. G. Barratt}, \textit{J. D. S. Jones} and \textit{M. E. Mahowald}, Lect. Notes Math. 1286, 135-173 (1985; Zbl 0633.55011)] and ``proved'' in [\textit{J. Klippenstein} and \textit{V. Snaith}, Topology 27, No. 4, 387-392 (1988; Zbl 0676.55021)]. Unfortunately, Klippenstein's error (found by the author in 1990) was not corrected before he left the academic profession. In a footnote it is mentioned that H. Minami has a proof of the author's theorem, based on [\textit{K. Shimomura}, Hiroshima Math. J. 11, 499-513 (1981; Zbl 0485.55013)] and \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson}, Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022). The author's proof is also based on the latter paper. Incidentally, when \(p=2\), Huajian Yang and the reviewer have found an independent proof of the author's theorem, using BP-operations.