1 results
Freudenthal's theorem and spherical classes in H*QS0
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 2 / May 2020
- Published online by Cambridge University Press:
- 20 December 2019, pp. 323-341
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- Article
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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.
