The Cohomology Of S(3) And Nontrivial Products In The Stable Homotopy Ring Of Spheres

In[1],R.Kato and K.Shimomura use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres.In this paper,we follow their ideal to detect the new nonzero products in the homotopy groups:For p ≥ 7,0≠ξnγβ1∈π*S,if n≡2 mod 3,s(?)0,±1 mod p.And list all nonzero products,that can be detected by H*S(3),among α family,β family,γfamily,and R.Cohen’s elements ξn.In the homotopy ring of spheres,the products among elements as,αs,βs,γs,ξn can be detected by H*S(3)as nontrivial areⅰ)α1β1,ⅱ)α1β2,ⅲ)β1β2,ⅳ)α1γs,ifs(?)0,±1 mod p,ⅴ)β1γs,ifs≠0,±1 mod p,ⅵ)β2γs,ifs(?)0,±1 mod p,ⅶ)ξnγs,ifs(?)0,±1 mod p and n(?)1 mod 3,n ≥1,ⅷ)ξnβ1,if n(?)0 mod 3,andⅸ)ξnγsβ1,if n(?)2 mod 3,s(?)0,± 1 mod p.In fact most of above list were calculated.ⅰ)and ⅱ)can be found in[2].ⅳ),ⅷ)and part of ⅲ)——ⅸ)but ⅷ)were appeared in[3].It should be mention that Lee’s work has its own advantege in powers of β family.[1]displayed ⅳ),ⅴ),ⅵ)and part of ⅶ).[4]follows[1]and deals with ⅶ).