The Lannes–Zarati homomorphism and decomposable elements

Let $X$ be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism $H : π_{*} (Q_{0} X) \to H_{*} (Q_{0} X)$ vanishes on classes of $π_{*} (Q_{0} X)$ of Adams filtration greater than $2$ . Let $φ_{s}^{M} : {Ext}_{A}^{s} (M, F_{2}) \to {(F_{2} \otimes_{A} R_{s} M)}^{*}$ denote the $s^{th}$ Lannes–Zarati homomorphism for the unstable $A$ –module $M$ . When $M = {\tilde{H}}^{*} (X)$ , 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, $φ_{s}^{M}$ , vanishes in any positive stem for $s > 2$ and for any unstable $A$ –module $M$ .

We prove that, for $M$ an unstable $A$ –module of finite type, the $s^{th}$ Lannes–Zarati homomorphism, $φ_{s}^{M}$ , vanishes on decomposable elements of the form $α β$ in positive stems, where $α \in {Ext}_{A}^{p} (F_{2}, F_{2})$ and $β \in {Ext}_{A}^{q} (M, F_{2})$ with either $p \geq 2$ , $q > 0$ and $p + q = s$ , or $p = s \geq 2$ , $q = 0$ and $stem (β) > s - 2$ . Consequently, we obtain a theorem proved by Hưng and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for ${\tilde{H}}^{*} (ℝ ℙ^{\infty})$ 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

References

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

Volume 19, issue 3 (2019)

Ngô A Tuấn

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors