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 spheresπ*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 spheresπpnq+2pq+q-3S which is of order p and is represented by kohn∈Ext3,Pnq+2pq+qA（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 inπpnq+（p+1）q-1V（1）which is represented byβ*i’*i*（hn）∈EXT2,pnq+（p+1）q+1A（H*V（1）,Zp）in the Adams sequence is detected.