#### Volume 19, issue 3 (2019)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
The Lannes–Zarati homomorphism and decomposable elements

### Ngô A Tuấn

Algebraic & Geometric Topology 19 (2019) 1525–1539
##### 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 ${\phi }_{s}^{M}:{Ext}_{\mathsc{A}}^{s}\left(M,{\mathbb{F}}_{2}\right)\to {\left({\mathbb{F}}_{2}{\otimes }_{\mathsc{A}}{R}_{s}M\right)}^{\ast }$ denote the ${s}^{th}$ Lannes–Zarati homomorphism for the unstable $\mathsc{A}$–module $M\phantom{\rule{0.3em}{0ex}}$. When $M={\stackrel{̃}{H}}^{\ast }\left(X\right)$, 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, ${\phi }_{s}^{M}\phantom{\rule{0.3em}{0ex}}$, vanishes in any positive stem for $s>2$ and for any unstable $\mathsc{A}$–module $M\phantom{\rule{0.3em}{0ex}}$.

We prove that, for $M$ an unstable $\mathsc{A}$–module of finite type, the ${s}^{th}$ Lannes–Zarati homomorphism, ${\phi }_{s}^{M}\phantom{\rule{0.3em}{0ex}}$, vanishes on decomposable elements of the form $\alpha \beta$ in positive stems, where $\alpha \in {Ext}_{\mathsc{A}}^{p}\left({\mathbb{F}}_{2},{\mathbb{F}}_{2}\right)$ and $\beta \in {Ext}_{\mathsc{A}}^{q}\left(M,{\mathbb{F}}_{2}\right)$ 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 Hưng and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for ${\stackrel{̃}{H}}^{\ast }\left(ℝ{ℙ}^{\infty }\right)$ vanishes on decomposable elements in positive stems.

##### Keywords
spherical classes, loop spaces, Adams spectral sequences, Steenrod algebra, lambda algebra, invariant theory, Dickson algebra
##### Mathematical Subject Classification 2010
Primary: 55P47, 55Q45, 55S10, 55T15
##### Publication
Received: 2 May 2018
Revised: 19 September 2018
Accepted: 15 October 2018
Published: 21 May 2019
##### Authors
 Ngô A Tuấn Department of Mathematics Vietnam National University, Hanoi Hanoi Vietnam