On the Hurewicz homomorphism on the extensions of ideals in $\pi_*^s$ and spherical classes in $H_*Q_0S^0$

TL;DR

This paper investigates the Curtis conjecture related to the Hurewicz map on stable homotopy groups, providing conditions under which elements are non-decomposable and analyzing the behavior of spherical classes through algebraic and homotopical tools.

Contribution

It establishes new criteria for elements in stable homotopy groups to be non-decomposable and explores the impact of homotopy operations and the EHP sequence on spherical classes.

Findings

Elements of Adams filtration at least 3 with non-zero Hurewicz image are non-decomposable.

If the conjecture holds on a set S, it extends to the span of S.

The Hurewicz map acts trivially on extensions generated by certain homotopy operations.

Abstract

This is about Curtis conjecture on the image of the Hurewicz map $h:{_2\pi_*}Q_0S^0\to H_*(Q_0S^0;\Z/2)$. First, we show that if $f\in{_2\pi_*^s}$ is of Adams filtration at least $3$ with $h(f)\neq 0$ then $f$ is not a decomposable element in ${_2\pi_*^s}$. Moreover, it is shown if $k$ is the least positive integer that $f$ is represented by a cycle in $\mathrm{Ext}^{k,k+n}_A(\Z/2,\Z/2)$, then (i) if $e_*h(f)\neq 0$ then $n\geqslant 2^k-1$; (ii) if $e_*h(f)=0$ then $n\geqslant 2^k-2^t$ for some $t>1$. Second, for $S\subseteq{_2\pi_{*>0}^s}$ we show that: (i) if the conjecture holds on $S$, then it holds on $\la S\ra$; (ii) if $h(S)=0$ then $h$ acts trivially on any extension of $S$ obtained by applying homotopy operations arising from ${_2\pi_*}D_rS^n$ with $n>0$. We also provide partial results on the extensions of $\la S\ra$ by taking (possible) Toda brackets of its elements. We also discuss how the $EHP$-sequence information maybe applied to eliminate classes from being spherical.