Spherical classes and the Lambda algebra

Abstract:

Let $\Gamma ^{\wedge }= \bigoplus _k \Gamma _k^{\wedge }$ be Singer’s invariant-theoretic model of the dual of the lambda algebra with $H_k(\Gamma ^{\wedge })\cong \operatorname {Tor}_k^{\mathcal {A}}(\mathbb {F}_2, \mathbb {F}_2)$, 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 \[ \varphi _k^*: \mathbb {F}_2 \underset {\mathcal {A}}{\otimes } D_k \to \operatorname {Tor}^{\mathcal {A}}_k(\mathbb {F}_2, \mathbb {F}_2) \cong H_k(\Gamma ^{\wedge }) . \] The Lannes–Zarati homomorphisms themselves, $\varphi _k$, correspond to an associated graded of the Hurewicz map \[ H:\pi _*^s(S^0)\cong \pi _*(Q_0S^0)\to H_*(Q_0S^0) . \] 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 $\operatorname {Tor}^{\mathcal {A}}_k(\mathbb {F}_2,\mathbb {F}_2)$ for $k>2$. We also show that $\varphi _k^*$ factors through $\mathbb {F}_2\underset {\mathcal {A}}{\otimes } Ker\partial _k$, 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.