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Spherical classes and the Lambda algebra


Author: Nguyen H. V. Hu'ng
Journal: Trans. Amer. Math. Soc. 353 (2001), 4447-4460
MSC (2000): Primary 55P47, 55Q45, 55S10, 55T15.
DOI: https://doi.org/10.1090/S0002-9947-01-02766-0
Published electronically: May 22, 2001
MathSciNet review: 1851178
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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, $D_k$, 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 $D_k$, 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.


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Additional Information

Nguyen H. V. Hu'ng
Affiliation: Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
Email: nhvhung@hotmail.com

DOI: https://doi.org/10.1090/S0002-9947-01-02766-0
Keywords: Spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra.
Received by editor(s): February 4, 1999
Received by editor(s) in revised form: November 4, 1999
Published electronically: May 22, 2001
Additional Notes: The research was supported in part by the National Research Project, No. 1.4.2.
Article copyright: © Copyright 2001 American Mathematical Society

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