On triviality of Dickson invariants in the homology of the Steenrod algebra

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 \[ \varphi_k:{\rm Ext}_{{\cal A}}^{k, k+i}({\bb F}_2,{\bb F}_2)\rightarrow ({\bb F}_2\otimes_{{\cal A}}D_k)^{*}_{i}, \] which correspond to an associated graded of the Hurewicz map $H:\pi^s_*(S^0)\cong\pi_*(Q_0S^0)\rightarrow H_*(Q_0S^0)$. 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 That the conjecture is no longer valid for $k = 1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes.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 Hu'ng ‘9’. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k > 2$ ‘14’ and on the image of Singer's algebraic transfer for $k > 2$ ‘9, 12’. 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 ${\rm Sq}^0$ on $({\bb F}_2\otimes_{{\cal A}}D_k)^*$, which commutes with the classical ${\rm Sq}^0$ on ${\rm Ext}_{{\cal A}}^k({\bb F}_2,{\bb F}_2)$ through the Lannes–Zarati homomorphism.