Spherical classes and the lambda algebra (Q2731951)

Let \(A\) be the mod \(2\) Steenrod algebra and \(D_k\) denote the Dickson algebra of \(k\) variables defined by the algebra of invariants, \(D_k=\mathbb F_2 [x_1,\cdots ,x_k]^{\text{GL}_k(\mathbb F_2)}\) with \(\deg x_i=1\). In this paper the author studies the Lannes-Zarati homomorphism [\textit{J. Lannes} and \textit{S. Zarati}, Math. Z. 194, 25-59 (1987; Zbl 0627.55014)] \(\phi_k:\text{Ext}_{A}^{k,k+i}(\mathbb F_2,\mathbb F_2)\to (\mathbb F_2\otimes_AD_k)^*_i\) and he shows that the inclusion \(D_k\to\Gamma^{\wedge}_k\) is a chain-label representation of the dual \(\phi_k^*\), where \(\Gamma^{\wedge}=\bigoplus_k\Gamma^{\wedge}_k\) denotes the Singer's invariant-theoretic model of the dual of the lambda algebra with \(H_*(\Gamma^{\wedge})\cong \text{Tor}_*^A(\mathbb F_2,\mathbb F_2)\). Because \(\phi_k\)'s correspond to an associated graded of the Hurewicz homomorphism \(H:\pi_*(Q_0(S^0))=\pi_*^S(S^0)\to H_*(Q_0(S^0))\), as an application, he also investigates the algebraic version of the classical conjecture on spherical classes, which states that only the Hopf invariant one class and the Kervaire invariant classes are detected by \(H\). Furthermore, he also proves that \(\phi_k^*\) factors through \(\mathbb F_2\otimes_A \text{Ker }\partial_k\), where \(\partial_k:\Gamma^{\wedge}_k\to \Gamma^{\wedge}_{k-1}\) denotes the differential of \(\Gamma^{\wedge}\).