Suppose V∪S W is a genus-g weakly reducible Heegaard splitting of a closed 3-manifold with finitely many pairs of disjoint compression disks on distinct sides up to isotopy and g>2.We show V∪_(S) W admits an unte...Suppose V∪S W is a genus-g weakly reducible Heegaard splitting of a closed 3-manifold with finitely many pairs of disjoint compression disks on distinct sides up to isotopy and g>2.We show V∪_(S) W admits an untelescoping:(V_(1)∪_(S1) W_(1))∪F(W_(2)∪_(S2) V_(2))such that Wi has a unique separating compressing disk and d(S_(i))≥2,for i=1,2.If there exist more than one but finitely many pairs of disjoint compression disks,at least one of d(S_(i))is 2 and S is a critical Heegaard surface.展开更多
The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus...The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.展开更多
Let M be a compact orientable 3-manifold with 0M connected. If V Us W is a Heegaard splitting of M with distance at least 6, then the 0-stabilization of V Us W along OM is unstabilized. Hence M has at least two unstab...Let M be a compact orientable 3-manifold with 0M connected. If V Us W is a Heegaard splitting of M with distance at least 6, then the 0-stabilization of V Us W along OM is unstabilized. Hence M has at least two unstabilized Heegaard splittings with different genera. The basic tool is a result on disk complex given by Masur and Schleimer.展开更多
Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the dec...Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim（Cy ∩ Cw）.展开更多
Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittin...Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.展开更多
In the paper,we give two conditions that the Heegaard splitting admits the disjoint curve property.The main result is that for a genus g(g■2)strongly irreducible Heegaard splitting(C_1,C_2;F),let D_i be an essential ...In the paper,we give two conditions that the Heegaard splitting admits the disjoint curve property.The main result is that for a genus g(g■2)strongly irreducible Heegaard splitting(C_1,C_2;F),let D_i be an essential disk in C_i,i=1,2,satisfying(1)at least one of ■D_1 and ■D_2 is separating in F and|■D_1∩■D_2|■2g-1;or(2)both ■D_1 and ■D_2 are non-separating in F and|■D_1∩■D_2|■2g-2,then(C_1,C_2;F)has the disjoint curve property.展开更多
In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says ...In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M, then there exists a complete surface system S for M, which is equivalent to S′, such that D is disjoint from S. We also obtain some properties of such 3-manifolds which can be embedded in S;.展开更多
An essential annulus A properly imbedded in a 3-manifold M is said to be spanning with respect to a Hereaard splitting VUFW of M if the two components of A are lying in A and separately and A intersects the Heegaard s...An essential annulus A properly imbedded in a 3-manifold M is said to be spanning with respect to a Hereaard splitting VUFW of M if the two components of A are lying in A and separately and A intersects the Heegaard surface F in one circle. In this paper,a sufficient and necessary condition that there is a spanning annulus A in a 3-manifold M with respect to the Heegaard splitting VUFW is given.展开更多
Let F=F(X) be a free group of rand n, be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of if x is in the normal closure of in F(X). Final...Let F=F(X) be a free group of rand n, be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of if x is in the normal closure of in F(X). Finally, an application of t he theorem in Heegaard splitting of 3-manifolds is given.展开更多
Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result...Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result has applications in Floer theories of 3-manifolds.The special case where F=D^(2) or F=S^(2) has been a folklore theorem,which has already been used by experts before.展开更多
Let M be a connected orientable compact irreducible 3-manifold. Suppose that αM consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M, F1) = g(M) + g(...Let M be a connected orientable compact irreducible 3-manifold. Suppose that αM consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M, F1) = g(M) + g(F1), where g(M, F1)is the Heegaard genus of M relative to F1. Let Mfbe the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(Mf) = g(M, αM) + 1.展开更多
In this paper, we show the following result： Let Ki be a knot in a closed orientable 3- manifold Mi such that （Mi,Ki） is not homeomorphic to （S^2 × S^1,x0 × S^1）, i = 1,2. Suppose that the Euler Charact...In this paper, we show the following result： Let Ki be a knot in a closed orientable 3- manifold Mi such that （Mi,Ki） is not homeomorphic to （S^2 × S^1,x0 × S^1）, i = 1,2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E（Ki） is less than the difference of one and twice of the tunnel number of Ki. Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. We further show that if the distance of a Heegaard splitting of each knot complement is strictly bigger than twice the tunnel number of the knot （twice the sum of the tunnel number of the knot and one, respectively）, then the tunnel number of connected sum of two such knots is additive （super additive, respectively）.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11671064)。
文摘Suppose V∪S W is a genus-g weakly reducible Heegaard splitting of a closed 3-manifold with finitely many pairs of disjoint compression disks on distinct sides up to isotopy and g>2.We show V∪_(S) W admits an untelescoping:(V_(1)∪_(S1) W_(1))∪F(W_(2)∪_(S2) V_(2))such that Wi has a unique separating compressing disk and d(S_(i))≥2,for i=1,2.If there exist more than one but finitely many pairs of disjoint compression disks,at least one of d(S_(i))is 2 and S is a critical Heegaard surface.
文摘The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.
基金supported by the National Natural Science Foundation of China(Nos.11271058,11171108)
文摘Let M be a compact orientable 3-manifold with 0M connected. If V Us W is a Heegaard splitting of M with distance at least 6, then the 0-stabilization of V Us W along OM is unstabilized. Hence M has at least two unstabilized Heegaard splittings with different genera. The basic tool is a result on disk complex given by Masur and Schleimer.
基金supported by National Natural Science Foundation of China(Grant Nos.10931005 and 11101058)the National Science Foundation for Post-doctoral Scientists of China(Grant No.2011M500049)
文摘Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim（Cy ∩ Cw）.
基金the Specialized Research Fund for the Doctoral Program of Higher Education(No.200801411069)
文摘Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.
基金Foundation item： the National Natural Science Foundation of China （No. 10571034）
文摘In the paper,we give two conditions that the Heegaard splitting admits the disjoint curve property.The main result is that for a genus g(g■2)strongly irreducible Heegaard splitting(C_1,C_2;F),let D_i be an essential disk in C_i,i=1,2,satisfying(1)at least one of ■D_1 and ■D_2 is separating in F and|■D_1∩■D_2|■2g-1;or(2)both ■D_1 and ■D_2 are non-separating in F and|■D_1∩■D_2|■2g-2,then(C_1,C_2;F)has the disjoint curve property.
基金Supported in part by NSFC(12071051)the National Science Foundation of Liaoning Province of China(2020-MS-244)the Fundamental Research Funds for the Central Universities(DUT21LAB302)。
文摘In this paper,we will give a sufficient condition for the self-amalgamation of a handlebody to be strongly irreducible.
基金The NSF(11329101,11431009,11329101,11471151 and 11401069)of Chinathe Fundamental Research Funds(DUT16LK40)for the Central Universities
文摘In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M, then there exists a complete surface system S for M, which is equivalent to S′, such that D is disjoint from S. We also obtain some properties of such 3-manifolds which can be embedded in S;.
文摘An essential annulus A properly imbedded in a 3-manifold M is said to be spanning with respect to a Hereaard splitting VUFW of M if the two components of A are lying in A and separately and A intersects the Heegaard surface F in one circle. In this paper,a sufficient and necessary condition that there is a spanning annulus A in a 3-manifold M with respect to the Heegaard splitting VUFW is given.
文摘Let F=F(X) be a free group of rand n, be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of if x is in the normal closure of in F(X). Finally, an application of t he theorem in Heegaard splitting of 3-manifolds is given.
基金supported by U.S.National Science Foundation(Grant No.DMS-1811900).
文摘Given a compact,oriented,connected surface F,we show that the set of connected sutured manifolds(M,γ)with R±(γ)■F is generated by the product sutured manifold(F,∂F)×[0,1]through surgery triads.This result has applications in Floer theories of 3-manifolds.The special case where F=D^(2) or F=S^(2) has been a folklore theorem,which has already been used by experts before.
基金supported by the National Natural Science Foundation of China(No.11271058)The second author is supported by the National Natural Science Foundation of China(No.11171108)
文摘Let M be a connected orientable compact irreducible 3-manifold. Suppose that αM consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M, F1) = g(M) + g(F1), where g(M, F1)is the Heegaard genus of M relative to F1. Let Mfbe the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(Mf) = g(M, αM) + 1.
基金The first author is supported by Development Program for Outstanding Young Teachers in Harbin Institute of Technology （HITQNJS.2009.029） the second author is supported by National Natural Science Foundation of China （Grant No. 15071034）
文摘In this paper, we show the following result： Let Ki be a knot in a closed orientable 3- manifold Mi such that （Mi,Ki） is not homeomorphic to （S^2 × S^1,x0 × S^1）, i = 1,2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E（Ki） is less than the difference of one and twice of the tunnel number of Ki. Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. We further show that if the distance of a Heegaard splitting of each knot complement is strictly bigger than twice the tunnel number of the knot （twice the sum of the tunnel number of the knot and one, respectively）, then the tunnel number of connected sum of two such knots is additive （super additive, respectively）.
