Detection of Some Elements in the Stable Homotopy Groups of Spheres

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 which is of order p and is represented by k ₀ h _n ∈ (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 which is represented by β _* i′_* i _*(h _n ) ∈ (H ^* V(1), ℤ _p ) in the Adams sequence is detected. Project supported by the National Natural Science Foundation of China (Nos. 10501045, 10771105) and the Fund of the Personnel Division of Nankai University (No. J02017).