Two New Families Of Elements In The Stable Homotopy Groups Of Spheres

The stable homotopy group ?*S is the central problem in homotopy theory.A main tool for this problem is the Adams spectral sequence with E2-term:E2s,t?ExtAs,t(Zp,Zp)(?)?t-sS,where E2s,t is the cohomology of mod p Steenrod algebra A.dr:Ers,t?Ers+r,t+r-1 is its Adams differential.In this paper,we use Adams and May spectral sequence to discuss the nontrivial products in the stable homotopy groups of spheres.Firstly,in chapter 1,we introduce the research survey about ?*S and give the main results of this paper.Secondly,in chapter 2,we give some relevant notions and results for the convenience of the reader.In chapter 3,we prove that there is a new family of nontrivial elements in ?*S which is represented by 0 ??s+3lng0?ExtAs+8,qn+1q+2pn+q+()s+3)pq+(s+3)q+s(Zp,Zp)in the Adams spectral sequence,where p?7,n>3,0?s<p-3.In chapter 4,when n?3,p?5,p+1<s<2p-1,we prove that the product element 0 ??sh0bn?ExtAs+3,q(pn+1+sp+s)+s-2,*(Zp,Zp)survives to E? in Adams spectral sequence and it converges to a nontrivial element in ?*S.