The purpose of this paper is to give a combinatorial characterization and also construct representations of the fundamental group of the submanifolds of elastic Klein Bottle by using some geometrical transformations. ...The purpose of this paper is to give a combinatorial characterization and also construct representations of the fundamental group of the submanifolds of elastic Klein Bottle by using some geometrical transformations. The homotopy groups of the limit elastic Klein Bottle are presented. The fundamental groups of some types of geodesics in elastic Klein Bottle are discussed. New types of homotopy maps are deduced. Theorems governing this connection are achieved.展开更多
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 metho...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).展开更多
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＋...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].展开更多
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...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.展开更多
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-...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.展开更多
In this paper,we prove the non-triviality of the product h 0 k o δ s＋4 ∈ Ext s＋6,t（s） A （Z p ,Z p ） in the classical Adams spectral sequence,where p ≥ 11,0 ≤ s p-4,t（s） = （s ＋ 4）p 3 q ＋ （s ＋ 3）p 2 q...In this paper,we prove the non-triviality of the product h 0 k o δ s＋4 ∈ Ext s＋6,t（s） A （Z p ,Z p ） in the classical Adams spectral sequence,where p ≥ 11,0 ≤ s p-4,t（s） = （s ＋ 4）p 3 q ＋ （s ＋ 3）p 2 q ＋ （s ＋ 4）pq ＋ （s ＋ 3）q ＋ s with q = 2（p-1）.The elementary method of proof is by explicit combinatorial analysis of the （modified） May spectral sequence.展开更多
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, Z...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.展开更多
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.
Let p be an odd prime.The authors detect a nontrivial element p of order p^2 in the stable homotopy groups of spheres by the classical Adams spectral sequence.It is represented by a_0^(p-2)h_1 ∈ Ext_A^(p-1,pq+p-2)(...Let p be an odd prime.The authors detect a nontrivial element p of order p^2 in the stable homotopy groups of spheres by the classical Adams spectral sequence.It is represented by a_0^(p-2)h_1 ∈ Ext_A^(p-1,pq+p-2)(Z/p,Z/p) in the E_2-term of the ASS and meanwhile p · p is the first periodic element αp.展开更多
Let p be an odd prime and q = 2（p- 1）. Up to total degree t - s 〈 max（（5p^3 ＋ 6p^2 ＋6p ＋4）q- 10,p^4q}, the generators of H^s,t（U（L））, the cohomology of the universal enveloping algebra of a bigraded Lie a...Let p be an odd prime and q = 2（p- 1）. Up to total degree t - s 〈 max（（5p^3 ＋ 6p^2 ＋6p ＋4）q- 10,p^4q}, the generators of H^s,t（U（L））, the cohomology of the universal enveloping algebra of a bigraded Lie algebra L, are determined and their convergence is also verified. Furthermore our results reveal that this cohomology satisfies an analogous Poindare duality property. This largely generalizes an earlier classical results due to J. P. May.展开更多
文摘The purpose of this paper is to give a combinatorial characterization and also construct representations of the fundamental group of the submanifolds of elastic Klein Bottle by using some geometrical transformations. The homotopy groups of the limit elastic Klein Bottle are presented. The fundamental groups of some types of geodesics in elastic Klein Bottle are discussed. New types of homotopy maps are deduced. Theorems governing this connection are achieved.
基金the National Natural Science Foundation of China (Nos.10501045,10771105)the Fund of the Personnel Division of Nankai University
文摘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).
基金supported by the National Natural Science Foundation of China (10501045, 10771105)the NCET and the Fund of the Personnel Division of Nankai University.
文摘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].
基金Supported by the National Natural Science Foundation of China (1051045)the Youth Project of Tianyuan Foundation of China (10426028)the China Postdoctoral,Science Foundation and Fund of the Personnel Division of Nankai University
文摘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.
基金the National Natural Science Foundation of China(No.10501045,10426028)the China Postdoc toral Science Foundationthe Fund of the Personnel Division of Nankai University
文摘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.
基金Supported by the National Natural Science Foundation of China (Grant No. 10361005)
文摘In this paper,we prove the non-triviality of the product h 0 k o δ s＋4 ∈ Ext s＋6,t（s） A （Z p ,Z p ） in the classical Adams spectral sequence,where p ≥ 11,0 ≤ s p-4,t（s） = （s ＋ 4）p 3 q ＋ （s ＋ 3）p 2 q ＋ （s ＋ 4）pq ＋ （s ＋ 3）q ＋ s with q = 2（p-1）.The elementary method of proof is by explicit combinatorial analysis of the （modified） May spectral sequence.
文摘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.
基金partially supported by NSFC(11171161)NCET+1 种基金SRF for ROCSSEM
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11761072,11671154)the China Postdoctoral Science Foundation Special Funded Project(No.2015T80909)
文摘Let p be an odd prime.The authors detect a nontrivial element p of order p^2 in the stable homotopy groups of spheres by the classical Adams spectral sequence.It is represented by a_0^(p-2)h_1 ∈ Ext_A^(p-1,pq+p-2)(Z/p,Z/p) in the E_2-term of the ASS and meanwhile p · p is the first periodic element αp.
基金supported by NSFC(Grant Nos.11671154 and 11761072)General Financial Grant from the China Postdoctoral Science Foundation(Grant No.2017M622721)
文摘Let p be an odd prime and q = 2（p- 1）. Up to total degree t - s 〈 max（（5p^3 ＋ 6p^2 ＋6p ＋4）q- 10,p^4q}, the generators of H^s,t（U（L））, the cohomology of the universal enveloping algebra of a bigraded Lie algebra L, are determined and their convergence is also verified. Furthermore our results reveal that this cohomology satisfies an analogous Poindare duality property. This largely generalizes an earlier classical results due to J. P. May.
