Detection of some elements in the stable homotopy groups of spheres

Abstract

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π _* S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres $\pi _{p^n q + 2pq + q - 3} S$ which is of order p and is represented by k ₀ h _n ∈ $Ext_A^{3,p^n q + 2pq + q} $(ℤ _{p, ℤp}) in the Adams spectral sequence, where p ≥ 5 is an odd prime, n ≥ 3 and q = 2(p − 1). In the course of the proof, a new family of homotopy elements in $\pi _{p^n q + (p + 1)q - 1} V(1)$ which is represented by β _* i′_* i _*(h _n) ∈ $Ext_A^{2,p^n q + (p + 1)q + 1} $(H ^* V(1), ℤ _p) in the Adams sequence is detected.

Keywords

Stable homotopy groups of spheres / Adams spectral sequence / May spectral sequence / Steenrod algebra

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