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
In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A^(2)_(3)-complexes.The main technique used is analysing the homotopy property of J(A,X),defined by Gray(1973) for a CW-pair(X,A),which...In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A^(2)_(3)-complexes.The main technique used is analysing the homotopy property of J(A,X),defined by Gray(1973) for a CW-pair(X,A),which is homotopy equivalent to the homotopy fibre of the pinch map X∪CA→∑A.展开更多
A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator...A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.展开更多
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 hom...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.展开更多
This note shows the nullity of homotopy groups for complete three dimensional manifolds with Ricci0 under some growth condition of the geodesic ball. The author also gives some examples which show the growth condition...This note shows the nullity of homotopy groups for complete three dimensional manifolds with Ricci0 under some growth condition of the geodesic ball. The author also gives some examples which show the growth condition here is optimal in some sense.展开更多
Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply conne...Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.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.
The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group ...The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group transformation is used to get the invariants. Using the invariants, a third-order ordinary differential equation corresponding to the momentum is obtained. An analytical solution is obtained in a series form using a homotopy analysis method. Reliability and efficiency of series solutions are shown by the good agreement with numerical results presented in the literature. The effects of the slip parameter, the magnetic field parameter, the velocity ratio parameter, the suction velocity parameter, and the power law exponent on the flow are investigated. The results show that the velocity and shear stress profiles are greatly influenced by these parameters.展开更多
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
In this paper,we study the homotopy classification of continuous maps between two r-1 connected 2r dimensional topological manifolds M,N.If we assume some knowledge on the homotopy groups of spheres,then the complete ...In this paper,we study the homotopy classification of continuous maps between two r-1 connected 2r dimensional topological manifolds M,N.If we assume some knowledge on the homotopy groups of spheres,then the complete classification can be obtained from the homotopy invariants of M,N.We design an algorithm and compose a program to give explicit computations.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
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
Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bo...Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n,and upper bounds on the absolute value of sectional curvature and diameter of M.In this paper,we will generalize thisπ2-diffeomorphism finiteness by removing the condition thatπ1(M)-0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M.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
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...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.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,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
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supported by National Natural Science Foundation of China (Grant No. 11701430), supported by National Natural Science Foundation of China (Grant No. 11971461)。
文摘In this paper,we calculate the 2-local unstable homotopy groups of indecomposable A^(2)_(3)-complexes.The main technique used is analysing the homotopy property of J(A,X),defined by Gray(1973) for a CW-pair(X,A),which is homotopy equivalent to the homotopy fibre of the pinch map X∪CA→∑A.
基金The authors would like to thank Joan Birman and Haynes Miller for their encouragements and helpful suggestions on this project.Fuquan Fang and Fengchun Lei supported in part by a Key Grant(No.11431009)an Overseas-Collaboration Grant(No.11329101)of NSFC of ChinaResearch is supported by the Singapore Ministry of Education research Grant(AcRF Tier 1 WBS No.R-146-000-190-112)and a Grant(No.11329101)of NSFC of China.
文摘A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.
基金supported by the National Natural Science Foundation of China(Nos.12001474,12171165)Guangdong Natural Science Foundation(Nos.2020A1515011008,2021A1515010374)the“13th Five-Year”Science and Technology Project of Jilin Department of Education(No.JJKH20200508KJ)
文摘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.
文摘This note shows the nullity of homotopy groups for complete three dimensional manifolds with Ricci0 under some growth condition of the geodesic ball. The author also gives some examples which show the growth condition here is optimal in some sense.
基金supported by Postdoctoral International Exchange Program for Incoming Postdoctoral Students under Chinese Postdoctoral Council and Chinese Postdoctoral Science Foundation, Chinese Postdoctoral Science Foundation (Grant No.2018M631605)National Natural Science Foundation of China (Grant No.11801544)。
文摘Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.
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Project supported by the National Natural Science Foundation of China (No. 50936003)the Open Project of State Key Laboratory for Advanced Metals and Materials and the Research Foundation of Engineering Research Institute of University of Science and Technology Beijing (No. 2009Z-02)
文摘The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group transformation is used to get the invariants. Using the invariants, a third-order ordinary differential equation corresponding to the momentum is obtained. An analytical solution is obtained in a series form using a homotopy analysis method. Reliability and efficiency of series solutions are shown by the good agreement with numerical results presented in the literature. The effects of the slip parameter, the magnetic field parameter, the velocity ratio parameter, the suction velocity parameter, and the power law exponent on the flow are investigated. The results show that the velocity and shear stress profiles are greatly influenced by these parameters.
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the National Natural Science Foundation of China(Grant No.10671018)
文摘In this paper,we study the homotopy classification of continuous maps between two r-1 connected 2r dimensional topological manifolds M,N.If we assume some knowledge on the homotopy groups of spheres,then the complete classification can be obtained from the homotopy invariants of M,N.We design an algorithm and compose a program to give explicit computations.
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.
文摘Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n,and upper bounds on the absolute value of sectional curvature and diameter of M.In this paper,we will generalize thisπ2-diffeomorphism finiteness by removing the condition thatπ1(M)-0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M.
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.
基金Supported by NSFC(Grant Nos.11671154,11761072,12001474 and 11871284)Guangdong Natural Science Foundation(Grant No.2020A1515011008)“13th Five-Year”Science and Technology Project of Jilin Department of Education(Grant No.JJKH20200508KJ)。
文摘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.
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.