On the Hurewicz homomorphism on the extensions of ideals in ${\pi }_{\ast }^s$ and spherical classes in $H_{\ast }{Q}_0{S}^0$

Abstract

This is about Curtis conjecture on the image of the Hurewicz map $h:{}_2{\pi }_{\ast }{Q}_0{S}^0\to H_{\ast }(Q_0{S}^0;\mathbb{Z}/2)$. First, we show that if $f\in {}_2{\pi }_{\ast }^s$ is of Adams filtration at least $3$ with $h(f)\ne 0$ then $f$ is not a decomposable element in ${}_2{\pi }_{\ast }^s$. Moreover, it is shown if $k$ is the least positive integer that $f$ is represented by a cycle in ${\mathrm{Ext}}_A^{k,k+n}(\mathbb{Z}/2,\mathbb{Z}/2)$, then (i) if $e_{\ast }h(f)\ne 0$ then $n⩾2^k-1$; (ii) if $e_{\ast }h(f)=0$ then $n⩾2^k-2^t$ for some $t>1$. Second, for $S\subseteq {}_2{\pi }_{\ast >0}^s$ 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 ${}_2{\pi }_{\ast }{D}_r{S}^n$ 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.