Freudenthal theorem and spherical classes in H_*QS⁰

2018/01/18 by Zare, Hadi
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.06427

Abstract

This note is on spherical classes in H_*(QS⁰;k) when k=ℤ,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 π_*^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 _pπ_*^s, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism _pπ_*^s≃_pπ_*QS⁰→ H_*(QS⁰;ℤ/p). We also observe that the image of the integral unstable Hurewicz homomorphism π_*^s≃π_*QS⁰→ H_*(QS⁰;ℤ) when restricted to the submodule of decomposable elements, is given by ℤ\h(η²),h(ν²),h(σ²)\. We apply this latter to completely determine spherical classes in H_*(Ω^dS^n+d;ℤ/2) for certain values of n>0 and d>0; this verifies a Eccles' conjecture on spherical classes in H_*QSⁿ, n>0, on finite loop spaces associated to spheres.

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