关于$π_*^s$中理想扩张与$H_*Q_0S^0$中球面类扩张的Hurewicz同态
On the Hurewicz homomorphism on the extensions of ideals in $π_*^s$ and spherical classes in $H_*Q_0S^0$
摘要：
这是关于Hurewicz映射$h:{_2π_*}Q_0S^0\to H_*(Q_0S^0;\Z/2)$的图像上的Curtis猜想。首先，我们证明了，如果$f\in{_2π_*^s}$是亚当斯过滤至少$3$与$h(f)\neq 0$，那么$f$是不是${_2π_*^s}$中的可分解元素。此外，它表明，如果$k$是最小的正整数，$f$表示为一个循环在$\mathrm{Ext}^{k,k+n}_A(\Z/2,\Z/2)$，那么（i）如果$e_*h(f)\neq 0$，然后$n\geqslant 2^k-1$ ;（ii）如果$e_*h(f)=0$，然后$n\geqslant 2^k-2^t$的一些$t>1$。其次，对于$S\subseteq{_2π_{*>0}^s}$，我们证明了：（i）如果猜想在$S$上成立，那么它在$\la S\ra$上成立;（ii）如果$h(S)=0$，那么$h$平凡地作用于$S$的任何扩展，该扩展是通过应用从${_2π_*}D_rS^n$与$n>0$产生的同伦运算而获得的。我们还提供了部分结果$\la S\ra$的扩展采取（可能的）户田括号的元素。我们还讨论了如何$EHP$序列的信息可能会被应用到消除类是球形的。
英文摘要：
This is about Curtis conjecture on the image of the Hurewicz map $h:{_2π_*}Q_0S^0\to H_*(Q_0S^0;\Z/2)$. First, we show that if $f\in{_2π_*^s}$ is of Adams filtration at least $3$ with $h(f)\neq 0$ then $f$ is not a decomposable element in ${_2π_*^s}$. 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 $S\subseteq{_2π_{*>0}^s}$ 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 ${_2π_*}D_rS^n$ 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.
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