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 That the conjecture is no longer valid for $k=1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes.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).)