Freudenthal theorem and spherical classes in $H_*QS^0$

Abstract

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 Curtis conjecture. We apply Freudenthal theorem to prove a vanishing result for the Hurewicz image of elements in ${\pi_*^s}$ that factor through certain finite spectra. Either in $p$-local or $p$-complete settings, 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 integral unstable Hurewicz homomorphism $\pi_*^s\simeq\pi_*QS^0\to H_*(QS^0;\mathbb{Z})$ when restricted to the submodule of decomposable elements, is given by $\mathbb{Z}\{h(\eta^2),h(\nu^2),h(\sigma^2)\}$. We apply this 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 a Eccles' conjecture on spherical classes in $H_*QS^n$, $n>0$, on finite loop spaces associated to spheres.