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 πpnq + 2pq + q - 3 S which is of order p and is represented by k0hn ε Ext A3,pn 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 πpnq + (p + 1)q-1 V(1) which is represented by β*i1*i*(hn) ε Ext 2,pn q+(p+ 1)q + 1 (H*V(1), ℤp ) in the Adams sequence is detected. © 2008 Editorial Office of CAM (Fudan University) and Springer-Verlag Berlin Heidelberg.