The Method Of Infinite Descent In The Stable Homotopy Groups Of Spheres

To calculate the stable homotopy groups of spheres has always been the impor-tant open problem in homotopy theory.In this paper, we make use of Adams spectral sequence and the Adams-Novikov spectral se-quence to determine the p-component of the stable homotopy groups of the spheres.In this paper,we introduce a spectral sequence to compute E2-term of the Adams-Novikov spectral sequence-the small descent spectral sequence By this spectral sequence,we calculate ExtBP*BPS,T(BP*,BP*)at the some certain dimen-sion t-s.Making use of thi s calculation,we prove that the secondary periodic element βp2/p2-1survives to π*(S0),so is h0h3in the Adams spectral sequence. For n≥4, convergence of h0hn is still unresolved.M.Hovey listed it as one of the most important problems in algebraic topology on his home page.We also find an interesting element βp/p-1γ3in π*(S0).This element is trivial in E2-term of the Adams-Novikov spectral sequence,but it is not t rivial in π*(S0) and is represented by the element α1β1p-1h2.0γ3with higher filtration.E2S,*-terms of Adams spectral sequence are important data to detect the homotopy elements,and are the fist step to compute the stable homotopy groups of spheres.It is hard to calculate completely E2S,*.When s>3,we have no complete results for E2S,*. therefore,so we focus our attention on the product of the known elements and consider their convergence.In the chapter4,we use the May spectral sequence to determine that the non-triviality of the composite map ζn-1β2γs+3in π*(S0) and the non-triviality of the two product elements g0δs and b0h0hnδs in E2-term of Adams spectral sequence.In the chapter5,we use the May spectral sequence to determine some nontrivial elements in ExtA4;(z/P,z/P))and compute these elements’secondary differentials by the matric Massey product(see Theorem5.6and Theorem5.9).