The Nontriviality Of Some Elements In The Homotopy Groups Of Sphere Spectrum And Related Spectrum

Abstract:To determine the stable homotopy groups of sphere spectrum and related spectrum is one of the important problems in homotopy theory. The main tools for this problem are the Adams spectral sequence and Adams-Novikov spectral sequence. For the Adams spectrum E2-term:E2s,t=ExtAs,t(Zp,Zp)=> πt-sS, where, A is mod p steenrod algebra, E2s,t is the cohomology of A. The Adams differential is given by dr:Ers,t→Ers+r,t+r-1. In this paper, we will use the Adams spectral sequence and May spectral sequence to detect the nontriviality of some elements in the homotopy groups of sphere spectrum and related spectrum by the algebraic method.First, in the chapter one, we will give introduction including an overview of the background, the progress and the conclusions involved in this paper, and then, some prior knowledge to make it easier to read will be given at last.Second, In chapter two, we will prove that, in π*S, there is a new family of elements in the Adams spectral sequence which is represented by γsh0gn∈ExtAs+3,pn+1q+2pnq+sp2q+(s-1)pq+(s-1)q+s-3(Zp,Zp), where3≤s<p-1, p≥7, n>3, q=2(p-1).Finally, in the chapter three, let p≥11, q=2(p-1), we will prove that, in Adams spectral sequence, l1g0∈ExtA5,p2q+3pq+2q(H*(V(1)),Zp) is a permanent cycle, not the dr-boundary, and then converges to an element in π*(V(1)).