The Nontriviality And Convergence Of Some Products In E₂ Term Of The Adams Spectral Sequence

The stable homotopy group of sphere occupies an important position in algebraic topology.The main tools to approach it are the Adams spectral sequence.For connected finite type spectral M,N,there is an Adams spectral sequence {Ers,t,dr}satisfied:(1)dr:Ers,t?Ers+r,t+r-1(r?2)is Adams differential,(2)E2s,t(?)ExtAs,t(H*(M),H*(N))(?)[?t-sN,M]p,where ExtAs,t(Zp,Zp)is the cohomology of A.If a family of homotopy generators xi in E2s,*converges nontrivially in the Adams spectral sequence,then we get a family of homotopy elements fi in ?*S and we say that fi is represented by xi?E2s,*and has filtration s in the Adams spectral sequence.This paper consists of three chapters.In the first chapter,we will give introduction,which is an overview of the background to this issues,the progress and the conclusions involved in this paper.Then,for the convenience of reader,some knowledge and the computing method of E1 term about the May spectral sequence are given.In the second chapter,we will give the nontriviality of the product elements b1?s+4 related to the fourth Greek letter family element.Let p?11,0?s<p-4,q=2(p-1),we prove that the products 0?b1?s+4 ? ExtAs+6,q[(s+4)p3+(s+4)p2+(s+2)p+(s+1)]+s(Zp,Zp)in the Adams spectral sequence,where ?s+4 is the fourth Greek letter family element,and A is the mod p Steenrod algebra.In the third chapter,we will prove that there is a new family of nontrivial elements in ?*S which is represented by 0??s+3hng0?ExtAs+6,q[pn+(s+3)p2+(s+3)p+s+3]+s(Zp,Zp)in the Adams spectral sequence,where p>7,n>5,p<s<2p-4,g=2(p-1).