On the Hurewicz homomorphism on the extensions of ideals in π_*^s and spherical classes in H_*Q₀S⁰

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.