Some New Elements In The Homotopy Groups Of Spectrum V(n) And A Note On The Cohomology Of Mod P Steenrod Algebra

Let A be the mod p Steenrod algebra,S the sphere spectrum localized at an odd prime p,and P be the subalgebra of A generated by reduced powers pⁱ(i?0).To determine the homotopy groups of spectrum V(n)and sphere spectrum is one of the important problem in homotopy theory in algebraic topology,the main tool is the Adams spectral sequence.For connected finite type spectral X,Y,there is an Adams spectral sequence{E_r^s,t,d_r}satisfied:where E₂^s,t is the cohomology of A.When X is sphere spectrum S,Moore spectrum M,Toda-Smith spectrum V(1),V(2)respectively,?_t-s(X)_p is the stable homotopy group of them.In this paper,the Adams spectral sequence is used as a tool to detect the existence of some nontrivial elements in the homotopy group of Toda-Smith spectrum,and a nontrivial product element will be given in the E₂^-term of the Adams spectral sequence.This paper consists of three chapters.In the first chapter,we will give introduction including an overview of the background,the progress and the conclusions about homotopy group,and then,some preliminary and main conclusions will be given.In chapter two,by using the May spectral sequence and combined with the algebraic and number-theoretical methods,we proved that b₁h₁,b₁g₀,h₂ and h₂ are permanent cycles in the Adams spectral sequence and not d_r boundaries,thus they converge to the corresponding nonzero elements of the homotopy group ?_*V(1).The following section of this chapter,we detected two nontrivial elements in the homotopy groups of spectrum V(n)(n = 1,2),which are represented by (i₁i₀)_*(b₀g₀) and (i₂i₁i₀)*(b₀g₀) in the Adams spectral sequence.In chapter three,the May spectral sequence is used to prove the nontriviality of the fourth Greek letter family product elementein the E₂^- term of the Adams spectral sequence,where t(s)=q[(s+4)p³+(s+3)p²+(s+4)p+(s+3)]+s,p?11.0?s<p-4.