Title of Paper: Spherical classes and the Lambda algebra
Author: Nguy\^{e}n H. V. Hung
2000 Mathematics Subject Classification: Primary 55P47, 55Q45,
55S10, 55T15.
Address: Department of Mathematics, Vietnam National University,
Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Abstract: Let $\Gamma^{\wedge}= \oplus_k \Gamma_k^{\wedge}$ be
Singer's invariant-theoretic model of the dual of the Lambda
algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\cal A}(F_2, F_2)$,
where ${\cal A}$ 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
$$
\varphi_k^*: F_2\otimes_{\cal A} D_k \to Tor^{\cal A}_k(F_2,F_2)
\cong H_k(\Gamma^{\wedge}).
$$
The Lannes--Zarati homomorphisms themself, $\varphi_k$, 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 {\it
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
$Tor^{\cal A}_k(F_2, F_2)$ for $k>2$.