The weak conjecture on spherical classes
This paper is a contribution to the study of the stable Hurewicz map, \(H\colon \pi^S_\ast(S^0) \cong \pi_\ast(Q_0S^0) \to H_\ast(Q_0S^0)\), here understood with mod 2 coefficients. It was conjectured that only the Hopf invariant one and Kervaire invariant one elements are detected by this mapping. The target \(H_\ast(Q_0S^0)\) is well understood in terms of the Araki-Kudo-Dyer-Lashof operations, and in work of I.~Madsen, in terms of invariant theory and the Dickson invariants. \textit{W. M. Singer} has taken this theme very far in his analysis of the \(\Lambda\)-algebra in terms of invariants [Trans. Am. Math. Soc. 280, 673-693 (1983; Zbl 0533.55013)] and an algebraic transfer homomorphism [Math. Z. 202, No.~4, 493-523 (1989; Zbl 0687.55014)]. In this note the author formulates a weak conjecture about spherical classes in terms of Singer's transfer, that there are no spherical classes for \(Q_0S^0\) detected by the algebraic transfer except Hopf and Kervaire invariant one elements. The main theorem is an equivalence of this conjecture with the vanishing of the homomorphism induced by inclusion in positive dimensions: \[ j_k\colon \mathbf{F}_2 \otimes_{{\mathcal A}} (\text\textbf{F}_2[x_1, \ldots, x_k]^{Gl_k}) \to (\text\textbf{F}_2 \otimes_{{\mathcal A}} \text\textbf{F}_2[x_1, \ldots, x_k])^{Gl_k}. \] Here \({\mathcal A}\) denotes the mod~2 Steenrod algebra and \(\mathbf{F}_2[x_1, \ldots, x_k]\) is isomorphic to \(H^\ast(B(\mathbf{F}_2)^k; \text\textbf{F}_2)\) on which \(Gl_k\) acts. The analysis is based on the author's work on the action of the Steenrod algebra on modular invariants [Proc. Am. Math. Soc. 113, No.~4, 1097-1104 (1991; Zbl 0736.55014)].
scientific article; zbMATH DE number 1336872
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