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The Adams spectral sequence for Minami’s theorem. (English) Zbl 0916.55011

Mahowald, Mark (ed.) et al., Homotopy theory via algebraic geometry and group representations. Proceedings of a conference on homotopy theory, Evanston, IL, USA, March 23–27, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 220, 143-177 (1998).
A fundamental problem in homotopy theory is to determine whether there are elements \(\theta_i\in\pi^S_{2^{i+1}-2}\) of Kervaire invariant one. \(\theta_1=\eta^2,\theta_2=\nu^2\) and \(\theta_3=\sigma^2\) are the squares of the Hopf maps. M. Mahowald and M. Tangora [Topology 6, 349-369 (1967; Zbl 0166.19004)] established the existence of \(\theta_4\) while M. G. Barratt, J. D. S. Jones, and M. E. Mahowald [J. Lond. Math. Soc., II. Ser. 30, 533-550 (1984; Zbl 0606.55010)] showed that \(\theta_5\) exists. It is not known whether \(\theta_i\) exists for \(i\geq 6\). Let \(t:P\to S^0\) denote the transfer map of the twofold covering \(P\to S^0\). The Kahn-Priddy Theorem says that if \(\theta_i\) exists, it must factor through \(t\). N. Minami [Topology 34, No. 2, 481-488 (1995; Zbl 0820.55008)] showed that if \(\theta_i\), for \(i\geq 5\), exists, then \(\theta_i\) can not factor through \(t\wedge t:P\wedge P\to S^0\). His proof uses a \(BP\)-Adams operation to show that the canonical element \(e_{2^i-1}\otimes e_{2^i-1}\in H_{2^{i+1}-2}(P\wedge P;\mathbb{Z}/2)\) is not in the image of the Hurewicz homomorphism for \(i\geq 5\). The paper under review studies the classical Adams spectral sequence of \(P\wedge P\). Let \(\overline e_i\in E_2^{0,2^{i+1}-2}\) denote the element determined by \(e_{2^i-1}\otimes e_{2^i-1}\in H_{2^{i+1}-2}(P\wedge P;\mathbb{Z}/2)\). If \(\overline e_i\) were an infinite cycle, it would have Hurewicz image \(e_{2^i-1}\otimes e_{2^i-1}\). The authors prove that \(d_3(\overline e_i)=h^2_0h_i\overline e_{i-1}\neq 0\) for \(i\geq 5\) which explains Minami’s result from this viewpoint. They also show that \(\overline e_4\) is an infinite cycle, and hence \(\theta_4\) factors through \(t\wedge t\).
For the entire collection see [Zbl 0901.00044].

MSC:

55Q45 Stable homotopy of spheres
55T15 Adams spectral sequences
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