Title of Paper: On triviality of Dickson invariants in the
homology of the Steenrod algebra
Author: Nguy\^{e}n H. V. Hung
2000 Mathematics Subject Classification: Primary 55P47, 55Q45,
55S10, 55T15.
Address of Author:
Current Address: Department of Mathematics,
Johns Hopkins University, 3400 N. Charles Street, Baltimore MD
21218 - 2689 E-mail address: nhvhung@math.jhu.edu
Permanent Address: Department of Mathematics, Vietnam National
University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Abstract: Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$
the Dickson algebra of $k$ variables. We study the Lannes-Zarati
homomorphisms
$$
\varphi_k: Ext_{\cal A}^{k,k+i}(F_2,F_2)\to (F_2\otimes_{\cal A}
D_k)_i^*,
$$
which correspond to an associated graded of the Hurewicz map $
H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)$. An algebraic
version of the long-standing conjecture on spherical classes
predicts that $\varphi_k=0$ in positive stems, for $k>2$. That the
conjecture is no longer valid for $k=1$ and $2$ is respectively an
exposition of the existence of Hopf invariant one classes and
Kervaire invariant one classes.
This conjecture has been proved for $k=3$ by Hung [Trans AMS 349
(1997), 3893-3910]. It has been shown that $\varphi_k$ vanishes on
decomposable elements for $k>2$ [Hung and Peterson, Math. Proc.
Camb. Phil. Soc. 124 (1998), 253-264] and on the image of Singer's
algebraic transfer for $k>2$ [Hung, 1997; Hung and Nam, Trans AMS
353 (2001), 5029-5040]. In this paper, we establish the conjecture
for $k=4$. To this end, our main tools include (1) an explicit
chain-level representation of $\varphi_k$ and (2) a squaring
operation $Sq^0$ on $(F_2\otimes_{\cal A} D_k)^*$, which commutes
with the classical $Sq^0$ on $Ext_{\cal A}^k(F_2,F_2)$ through the
Lannes-Zarati homomorphism.
(To appear in Math. Proc. Camb. Phil. Soc. 134 (2003).)