Spectral Sequences And Related Results In Stable Homotopy Theory

An important topic in the homotopy theory is the stable homotopy groups π*S,and the most basic tool to study it is the classical Adams spectral sequence with E2-term:E2s,t≌ExtAs,t(Zp,Zp)(?)πt-sS,where S is the sphere spectrum,E2s,t is the cohomology of mod p Steenrod algebra A.Adams differential is dr:Ers,t→Ers+r,t+r-1(r≥2).This paper consists of three chapters.The first chapter is the introduction that briefly describes the development of stable homotopy theory and the overview of stable homotopy groups.The second chapter introduces the related knowledge of May spectral sequence,and use May spectral sequence as a tool to calculate the Ext group of Adams spectral sequence.The first section proves the product h0b0bn-1γp-2∈ExtAp+3,t(Zp,Zp)is nontrivial in the Adams spectral sequence,and converges to the element β1ζn-1γp-2 in the stable homotopy groups of spheres.The following section proves the nontriviality of the product elements h1b1δs+4 involving the fourth Greek letter family element.In the third chapter,using Adams spectral sequence and May spectral sequence,a new nonzero element is defected in the homotopy groups of Smith-Toda spectrum V(1),and the element is denoted by b13g1 in Adams spectral sequence.