Non-triviality of Two Elements β1λs and α1λs in the Stable Homotopy of Spheres

In this paper, it is proved that for p≥7 an arbitrary odd prime and 3≤s 〈 p, the homotopy elements β1λs and α1λs are nontrivial in the stable homotopy groups of spheres π＊S.

SOME NEW FAMILIES OF FILTRATION FIVE IN THE STABLE HOMOTOPY OF SPHERES

This study proves a general result on convergence of α2x ∈ ExtA^s＋2.tq＋2q＋1 （Zp, Zp） in the Adams spectral sequence and as a consequence, the study detects some new families in the stable homotopy groups of spheres πtq＋2q-4S which is represented in the Adams spectral sequence by α2fn,α2fn,α2huhmhn ∈ ExtA^5,tq＋2q＋1（Zp,Zp） with tq=p^n＋1q＋2p^nq,2p^n＋1q_P^nq,p^uq＋p^mq＋p^nq,respectively, where α2∈Extα^2,2q＋1（Zp,Zp）,fn∈ExtA^3,p^n＋1q＋2p^nq（Zp,Zp）,fn∈ExtA^3,2p^n＋2q＋p^nq（Zp,Zp）,hn∈ExtA^1,p^nq（Zp,Zp）and p≥5 is a prime,q=2（p=-1）,n≥2.

New Families in the Stable Homotopy of Spheres Revisited

This paper constructs a new family in the stable homotopy of spheres π_(t-)S represented by h_ngoY_3 ∈E_2~6(,t) in the Adams spctral sequence which revisits the b_(n-1goY3)-elements ∈π_(-7)S con- structced in[3]. where t=2p^(11)(p-1)+6(p^2+p+1)(p-1)and p≥7 is a prime. 11≥4

A Nontrivial Product of Filtration s+5 in the Stable Homotopy of Spheres

In this paper, some groups Ext A^s.t （Zp, Zp） with specialized s and t are first computed by the May spectrM sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s＋5 in the stable homotopy groups of spheres πpnq＋（s＋3）pq＋（s＋1）q-5S which is represented （up to a nonzero scalar） by β＋2bohh∈ExtA^s＋5,P^nq＋（n＋3）pq＋（n＋1）q＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 5 is a prime number, n ≥3, 0≤ s 〈 p - 3, q = 2（p - 1）.

A NEW FAMILY OF FILTRATION S+5 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

In this article, by the algebraic method, the author proves the existence of a new nontrivial family of filtration s ＋ 5 in the stable homotopy groups of spheres πrS,which is represented by 0 ≠γ^-s＋3hnhm∈Ext^s＋5,A ^t（Zp,Zp）in the Adams spectral sequence,where r=q（p^m＋p^n＋（s＋3）p^2＋（s＋2）p＋（s＋1））-5,t=p^mq＋p^nq＋（s＋3）p^2q＋（s＋2）pq＋（s＋1）q＋s,p≥7,m≥n＋2〉5,0≤s〈p-3,q=2（p-1）.

SOME NEW FAMILIES OF FILTRATION FOUR IN THE STABLE HOMOTOPY OF SPHERES

This paper proves the existence of 4 families of nontrivial homotopy elements in the stablehomotopy of spheres which are represented by in the ternis of the Adams spectral sequence respectively, where and are theknown generators in the terms whose internal degree are 4(p - 1) + 1, 2pn+1(p-1),

A NON-TRIVIAL PRODUCT OF FILTRATION s+6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

By a method improving that of [1], the authors prove the existence of a non-trivial product of filtration, s ＋ 6, in the stable homotopy groups of sphere, πt-6S, which is represented up to non-zero scalar by β^-s＋2ho（hmbn-1 -hnbm-1） ∈ ExtA^s＋6,t＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 7, n ≥ m ＋ 2 ≥ 5, q = 2（p- 1）, 0 ≤ s 〈 p - 2, t= （s ＋ 2 ＋ （s ＋ 2）p ＋ p^m ＋ p^n）q. The advantage of this method is to extend the range of s without much complicated argument as in [1].

A NEW FAMILY OF FILTRATION s+6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

In the year 2002, Lin detected a nontrivial family in the stable homotopy groups of spheres ;π-6S which is represented by hngoγ^-3 ∈ Ext^6tA（Zp, Zp） in the Adams spectral sequence, where t = 2p^n（p- 1）＋ 6（p^2 ＋p ＋ 1）（p- 1） and p ≥ 7 is a prime number. This article generalizes the result and proves the existence of a new nontrivial family of filtration s ＋ 6 in the stable homotopy groups of spheres πt1-8-6S which is represented by bygoγ^s＋3 ∈ Ext^s＋6＋t1Atl （Zp, Zp） in the Adams spectral sequence, where n≥ 4, 0 ≤ s 〈 p - 4, t1 = 2p^n（p- 1） ＋ 2（p- 1）（（s ＋ 3）p^2 ＋ （s ＋ 3）p ＋ （s ＋ 3）） ＋ s.

A ■_n-Related Family of Homotopy Elements in the Stable Homotopy of Spheres

To determine the stable homotopy groups of spheres π＊（S） is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a Bn-related family of homotopy elements, β1ωnγs, in the stable homotopy groups of spheres, where Bn〉 3, 3≤s〈 p-2 and the Bn-element was detected by X. Liu.

A nontrivial product in the stable homotopy groups of spheres

Let A be the mod p Steenrod algebra for p an arbitrary odd prime. In 1962, Liulevicius described h i and b k in Ext* A ’*(Zp,Zp) having bigrading (1, sui— 1) and (2, 2p k+1 x(p— 1)), respectively. In this paper we prove that for p ≥ 7, n ≥ 4 and $3 \leqslant s < p - 1, h_0 b_{n - 1} \tilde \gamma _s \in Ext_A^{s + 3,p^n q + sp^2 q + (s - 1)pq + (s - 1)q + s - 3} (Z_p ,Z_p )$ survives to E∞ in the Adams spectral sequence, where q = 2(p — 1).

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.

A Relation in the Stable Homotopy Groups of Spheres

Let p ≥ 7 be an odd prime. Based on the Toda bracket 〈α1β1^p-1, α1β1,p, γs〉, the authors show that the relation α1β1^P-1h2,0γs= βp/p-1γ/s holds. As a resulL they can obtain α1β1^ph2,0γs= 0 ∈ π＊（S^0） for 2 ≤ s ≤ p - 2, even though α1h2,0γs and β1α1h2,0γs are not trivial. They also prove that β1^p-1 α1h2,0γ3 is nontrivial in π＊（S^0） and conjecture β1^p-1 α1h2,0γs is nontrivial in π＊（S^0） for 3 ≤s ≤ p - 2. Moreover, it is known that βp/p-1γ3 = 0 ∈ EXtBP＊Bp^5,＊（BP＊, BP＊）, but βp/p-1γ3 is nontrivial in π＊（S^0） and represents the element β1^p-1α1h2,0γ3.

Four Families of Nontrivial Product Elements in the Stable Homotopy Groups of Spheres

In this paper,the authors introduce a new effective method to compute the generators of the E-term of the May spectral sequence.This helps them to obtain four families of non-trivial product elements in the stable homotopy groups of spheres.

Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree p^nq ＋p^mq ＋ q- 4 and a nontrivial element in the stable homotopy group of Moore spectum of degree p^nq ＋ p^mq ＋ q - 3 which are represented by h0（hmbn-1 - hnbm-1） and ie（hohnhm） in the E2-terms of the Adams spectral sequence respectively, where p ≥ 7 is a prime, n ≥ m ＋ 2 ≥ 4, q = 2（p - 1）.

ON AN INFINITE FAMILY IN π_S*

In this paper, a new family of homotopy elements in the stable homotopy groups of spheres represented by h1hnhm γ？s in the Adams spectral sequence is detected, where n-2≥m≥5 and 3≤s 〈p.

On the Convergence of Products _sh_1h_n in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.

Some Secondary Differentials on the Fourth Line of the Adams Spectral Sequence

In this paper,we determine some nontrivial secondary Adams differentials on the fourth line Ext^(4,*)_A(Z/p,Z/p)of the classical Adams spectral sequence.Specially,among these differentials,two of them are obtained via the matrix Massey products.

A Four-filtrated May Spectral Sequence and Its Applications

In this paper, we introduce a four-filtrated version of the May spectral sequence (MSS), from which we study the general properties of the spectral sequence and give a collapse theorem. We also give an efficient method to detect generators of May E 1-term E 1 s,t,b,* for a given (s, t, b, *). As an application, we give a method to prove the non-triviality of some compositions of the known homotopy elements in the classical Adams spectral sequence (ASS).

On a family involving R.L. Cohen's ζ-element(II)

In 1981, Cohen constructed an infinite family of homotopy elements ζk∈ π＊（S） represented by h0bk ∈ ExtA3,2（p-1）（pk＋1＋1）（z/p,Z/p） in the Adams spectral sequence, where p 〉 2 and k ≥ 1. In this paper, we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs＋3 is nontrivial in the stable homotopy groups of spheres πt（s,n）-s-8（S）, where p ≥7, n 〉 3, 0≤s 〈p-5 andt（s,n） =2（p-1）[pn＋（s＋3）p2＋（s＋4）p＋（s＋3）]＋s.

A Pull Back Theorem in the Adams Spectral Sequence

This paper proves that, for any generator x∈ExtA^s,tq（Zp,Zp）, if （1L ∧i）＊Ф＊（x）∈ExtA^s＋1,tq＋2q（H＊L∧M, Zp） is a permanent cycle in the Adams spectral sequence （ASS）, then b0x ∈ExtA^s＋1,tq＋q（Zp, Zp） also is a permenent cycle in the ASS. As an application, the paper obtains that h0hnhm∈ExtA^3,pnq＋p^mq＋q（Zp, Zp） is a permanent cycle in the ASS and it converges to elements of order p in the stable homotopy groups of spheres πp^nq＋p^mq＋q-3S, where p ≥5 is a prime, s ≤ 4, n ≥m＋2≥4 and M is the Moore spectrum.