A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabil...A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabilization problem,reducibilities,minimal Heegaard splitting and Heegaard distance.展开更多
Let Mi be a compact orientable 3-manifold, and Fi be an incompressible surface on δMi, i -= 1,2. Let f ： F1 →F2 be a homeomorphism, and M = M1 UI M2. In this paper, under certain assumptions for the attaching surfa...Let Mi be a compact orientable 3-manifold, and Fi be an incompressible surface on δMi, i -= 1,2. Let f ： F1 →F2 be a homeomorphism, and M = M1 UI M2. In this paper, under certain assumptions for the attaching surface Fi, we show that if both M1 and M2 have Heegaavd splittings with distance at least 2（g（M1）＋ g（M2））＋ 1, then g（M） = g（M1）＋g（M2）.展开更多
We prove that for any integer n≥2 and g ≥ 2, there are bounded 3-manifolds admitting distance n, genus g Heegaard splittings with any given bound-aries.
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
For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut ope...For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut open along these tori,one resulting piece is hyperbolic while the others are small Seifert fibered spaces;(3)it provides a substantial result to the rank versus genus problem.These generalize a result in Qiu and Zou(2019).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
It is Thurston's result that for a hyperbolic knot K in S^3, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the s...It is Thurston's result that for a hyperbolic knot K in S^3, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in S^3, denoted by E(K), admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings. In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of E(K) is at least 3, then for any two degenerating slopes on ?E(K), there is a universal bound of their distance in the curve complex of ?E(K).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
基金This work was partially supported by Science and Technology Commission of Shanghai Municipality(STCSM),(18dz2271000)and NSFC(12131009)。
文摘A Heegaard splitting is a type of combinatorial structure on an orientable compact 3-manifold.We will give a survey on Heegaard spliting and its applications,including those pertaining to the classification and stabilization problem,reducibilities,minimal Heegaard splitting and Heegaard distance.
基金Supported by the National Natural Science Foundation of China(Grant No.11226084)
文摘Let Mi be a compact orientable 3-manifold, and Fi be an incompressible surface on δMi, i -= 1,2. Let f ： F1 →F2 be a homeomorphism, and M = M1 UI M2. In this paper, under certain assumptions for the attaching surface Fi, we show that if both M1 and M2 have Heegaavd splittings with distance at least 2（g（M1）＋ g（M2））＋ 1, then g（M） = g（M1）＋g（M2）.
文摘We prove that for any integer n≥2 and g ≥ 2, there are bounded 3-manifolds admitting distance n, genus g Heegaard splittings with any given bound-aries.
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 National Natural Science Foundation of China(Grant Nos.12131009 and 12326601)Science and Technology Commission of Shanghai Municipality(Grant No.22DZ2229014)。
文摘For any pair of integers g and n with g≥3 and 1≤n≤g,we build a 3-manifold with a distance-2,genus-g Heegaard splitting so that(1)it contains n pairwise disjoint and nonisotopic essential tori;(2)after it is cut open along these tori,one resulting piece is hyperbolic while the others are small Seifert fibered spaces;(3)it provides a substantial result to the rank versus genus problem.These generalize a result in Qiu and Zou(2019).
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.11371094,11571110 and 11601065)
文摘It is Thurston's result that for a hyperbolic knot K in S^3, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in S^3, denoted by E(K), admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings. In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of E(K) is at least 3, then for any two degenerating slopes on ?E(K), there is a universal bound of their distance in the curve complex of ?E(K).
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）.
