Freudenthal theorem and spherical classes in $H_{\ast }Q{S}^0$

Abstract

This note is on spherical classes in $H_{\ast }(Q{S}^0;k)$ 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 ${\pi }_{\ast }^s$ 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 ${}_p{\pi }_{\ast }^s$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${}_p{\pi }_{\ast }^s\simeq {}_p{\pi }_{\ast }Q{S}^0\to H_{\ast }(Q{S}^0;\mathbb{Z}/p)$. We also observe that the image of the integral unstable Hurewicz homomorphism ${\pi }_{\ast }^s\simeq {\pi }_{\ast }Q{S}^0\to H_{\ast }(Q{S}^0;\mathbb{Z})$ 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 $H_{\ast }({\Omega }^d{S}^{n+d};\mathbb{Z}/2)$ for certain values of $n>0$ and $d>0$; this verifies a Eccles' conjecture on spherical classes in $H_{\ast }Q{S}^n$, $n>0$, on finite loop spaces associated to spheres.

This is an "under review" version of the paper, omments are welcome. arXiv admin note: substantial text overlap with arXiv:1609.03143