Application Of The Adams Spectral Sequence In The Stable Homotopy Groups Of Spheres

To determine the stable homotopy groups of spheres Ï€*(S) is one of the central problems in homotopy theory. The main tools to approach it is the classical Adams spectral sequence and Adams-Novikov spectral sequence. The classical Adams spectral sequence (ASS) whose i?2-term is given by E2s,t=ExtAs,t(Zp,Zp)â†'Ï€t-sS which is the cohomology of A. The Adams differential is given by dr:Ers,tâ†'Ers+r,t+r-1.In this paper, we will use the Adams spectral sequence and May spectral sequence get the following nontrivial product in the stable homotopy groups of spheres by an algebraic method. We give the proof of their in Chapter3to Chapter5.First, in the chapter3, we prove that when3≤s＜p, the homotopy element α1β1β2γs is nontrivial in Ï€2(p-a)(sp2+(s+2)p+s)-7S.Second, in the chapter4, we prove that the existence of nontrivial product of fil-tration p+2in the group Ï€(pn+p3-p-2)q-6S. So, this element can be represented, in the sense of inner product of vectors, by the following product in the Adams spectral sequence γp-1h0bn-1∈ExtAp+2,(pn+p3-p-2)q+p-4(Zp,Zp) where n>1and n≠4. And we prove that the product βsh0hn survives to E∞in the Adams spectral sequence and it converges to a nontrivial element in Ï€*S, where p≥5,n＞4, p+1＜s＜2p-l,t=q(pn+sp+w)+s-2.Finally,in the chapter5,we prove that α2bnp≠0∈ExtA2p+2,t(Zp,Zp),where p＞7,t=q(pn+2+2)+1,0≤n≤2.