Spherical classes and the Lambda algebra

• Published in: Transactions of the American Mathematical Society 2001-11, Vol.353 (11), p.4447-4460
• Main Author: Nguyen H. V. Hu'ng
• Format: Article
• Language: English
• Subjects: Algebra; Functors; Homomorphisms; Mathematical sequences; Mathematical theorems
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-01-02766-0

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 \[ \varphi_k^*: \mathbb{F}_2\underset{\mathcal{A}}{\otimes} D_k \to Tor^{\mathcal{A}}_k(\mathbb{F}_2, \mathbb{F}_2) \cong H_k(\Gamma^{\wedge}). \] 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 \textit{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.