Spherical classes and the Lambda algebra

Let $\Gamma^{\wedge}= \bigoplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\mathcal{A}}(\mathbb{F} _2, \mathbb{F} _2)$, where $\mathcal{A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, $D_k$, into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes-Zarati dual homomorphism

The Lannes-Zarati homomorphisms themselves, $\varphi_k$, correspond to an associated graded of the Hurewicz map

Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in $D_k$, of positive degree represents the homology class $0$ in $Tor^{\mathcal{A}}_k(\mathbb{F} _2,\mathbb{F} _2)$ for $k>2$.

We also show that $\varphi_k^*$ factors through $\Fd\underset{\mathcal{A}}{\otimes} Ker\partial_k$, where $\partial_k : \Gamma^{\wedge}_k \to \Gamma^{\wedge}_{k-1}$ denotes the differential of $\Gamma^{\wedge}$. Therefore, the problem of determining $\mathbb{F} _2 \underset{\mathcal{A}}{\otimes} Ker\partial_k$ should be of interest.