This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z}, \mathbb{Z}/p$, with a special focus on the case of p=2 related to the Curtis conjecture. We apply Freudenthal's theorem to prove a vanishing result for the unstable Hurewicz image of elements in ${\pi _*^s}$ that factor through certain finite spectra. After either p-localization or p-completion, this immediately implies that elements of well-known infinite families in ${_p\pi _*^s}$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${_p\pi _*^s}\simeq {_p\pi _*}QS^0\to H_*(QS^0;\mathbb{Z} /p)$. We also observe that the image of the submodule of decomposable elements under the integral unstable Hurewicz homomorphism $\pi _*^s\simeq \pi _*QS^0\to H_*(QS^0;\mathbb{Z} )$ is given by $\mathbb{Z} \{h(\eta ^2),h(\nu ^2),h(\sigma ^2)\}$. We apply the latter to completely determine spherical classes in $H_*(\Omega ^dS^{n+d};\mathbb{Z} /2)$ for certain values of n>0 and d>0; this verifies Eccles' conjecture on spherical classes in $H_*QS^n$, n>0, on finite loop spaces associated with spheres.

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H*QS0 中的弗洛伊登塔尔定理和球类

这篇笔记是关于球类的$H_*(QS^0;k)$什么时候$k=\mathbb{Z}, \mathbb{Z}/p$, 特别关注以下情况p=2 与柯蒂斯猜想有关。我们应用弗洛伊登塔尔定理证明了元素的不稳定 Hurewicz 图像的消失结果${\pi _*^s}$该因素通过某些有限光谱。之后p-本地化或p-完成，这立即暗示着名的无限家庭的元素${_p\pi _*^s}$，例如 Mahowaldean 族，在不稳定的 Hurewicz 同态下平凡映射${_p\pi _*^s}\simeq {_p\pi _*}QS^0\to H_*(QS^0;\mathbb{Z} /p)$. 我们还观察到积分不稳定 Hurewicz 同态下可分解元素子模的图像$\pi _*^s\simeq \pi _*QS^0\to H_*(QS^0;\mathbb{Z} )$是（谁）给的$\mathbb{Z} \{h(\eta ^2),h(\nu ^2),h(\sigma ^2)\}$. 我们应用后者来完全确定球类$H_*(\Omega ^dS^{n+d};\mathbb{Z} /2)$对于某些值n>0 和d>0; 这验证了 Eccles 关于球类的猜想$H_*QS^n$,n>0，在与球体相关的有限循环空间上。