The Lannes–Zarati homomorphism and decomposable elements

Abstract

Let $X$ be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism $H:{\pi }_{\ast }\left(Q_{0}X\right)\to H_{\ast }\left(Q_{0}X\right)$ vanishes on classes of ${\pi }_{\ast }\left(Q_{0}X\right)$ of Adams filtration greater than $2$. Let ${\phi }_s^M:{Ext}_{\mathcal{A}}^s\left(M,{\mathbb{F}}_2\right)\to {\left({\mathbb{F}}_2{\otimes }_{\mathcal{A}}{R}_{s}M\right)}^{\ast }$ denote the $s^{th}$ Lannes–Zarati homomorphism for the unstable $\mathcal{A}$–module $M$. When $M={\tilde{H}}^{\ast }\left(X\right)$, this homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the $s^{th}$ Lannes–Zarati homomorphism, ${\phi }_s^M$, vanishes in any positive stem for $s>2$ and for any unstable $\mathcal{A}$–module $M$.

We prove that, for $M$ an unstable $\mathcal{A}$–module of finite type, the $s^{th}$ Lannes–Zarati homomorphism, ${\phi }_s^M$, vanishes on decomposable elements of the form $\alpha \beta$ in positive stems, where $\alpha \in {Ext}_{\mathcal{A}}^p\left({\mathbb{F}}_2,{\mathbb{F}}_2\right)$ and $\beta \in {Ext}_{\mathcal{A}}^q\left(M,{\mathbb{F}}_2\right)$ with either $p\ge 2$, $q>0$ and $p+q=s$, or $p=s\ge 2$, $q=0$ and $stem\left(\beta \right)>s-2$. Consequently, we obtain a theorem proved by Hung and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for ${\tilde{H}}^{\ast }\left(ℝ{\mathbb{P}}^{\infty }\right)$ vanishes on decomposable elements in positive stems.