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Spherical classes and the Lambda algebra

Authors:	Nguyen H V Hu'ng

Institution:	Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam

Abstract:	Let $\Gamma^{\wedge}= \bigoplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\mathcal{A}}(\mathbb{F} _2, \mathbb{F} _2)$ , where $\mathcal{A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, , into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes-Zarati dual homomorphism $\begin{displaymath}\varphi_k^: \mathbb{F} _2\underset{\mathcal{A}}{\otimes} D_k... ..._k(\mathbb{F} _2, \mathbb{F} _2) \cong H_k(\Gamma^{\wedge})\,. \end{displaymath}$ The Lannes-Zarati homomorphisms themselves, $\varphi_k$ , correspond to an associated graded of the Hurewicz map $\begin{displaymath}H:\pi_^s(S^0)\cong \pi_(Q_0S^0)\to H_(Q_0S^0)\,. \end{displaymath}$ 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 , of positive degree represents the homology class in $Tor^{\mathcal{A}}_k(\mathbb{F} _2,\mathbb{F} _2)$ for . 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.

Keywords:	Spherical classes loop spaces Adams spectral sequences Steenrod algebra lambda algebra invariant theory Dickson algebra

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