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 k0hn ∈ \( Ext_A^{3,p^n q + 2pq + q} \)(ℤp, 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*(hn) ∈ \( Ext_A^{2,p^n q + (p + 1)q + 1} \)(H*V(1), ℤp) in the Adams sequence is detected.