Research on 3-manifolds by combinatorial and constructive methods

3-流形的组合和构造方法研究

• 批准号: 15540091
• 负责人: ISHLI Ippei
• 金额: $ 1.98万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540091/
• 关键词: 3-manifold; spine; Heegaard splitting; Seifert fibered space; hyperbolic manifold; topological invariant; symplectic manifold; エゴード分解; グラフ

In this research project, we have introduced a new topological invariant, called the "block number", for 3-manifolds, which estimates some kind of complexity of a 3-manifold just like as the Heegaard genus. The block number is defined by means of a flow-spine, and is enable us to classify 3-manifolds, and to parameterize 3-manifolds in each class by finitely many integers. Moreover the block number can be defined not only for a 3-manifold but also for a combed 3-manifold, a pair of a 3-manifold and a non-singular vector field on it. Using this invariant, we have gotten the following results :1.The only combed 3-manifold having 0 as its block number is the canonical one on the product of the 2-sphere and the circle, and combed 3-manifolds with the block number 1 are canonical ones on lens spaces (including the 3-sphere).2.The parameterization for 3-manifolds with the block number 2 was given. And, using the Reidemeister torsion, we have shown some results which imply that our parameterization uniformize the presentation of a combed 3-manifold.003.We have given a formula for calculating the value of the Thraev-Viro invariant for all Seifert fibered 3-manifolds.On symplectic manifolds, we have gotten the following result :4.If the universal covering space of a clsed symplectic manifold is contractible, the manifold does not admit any Riemannian metric with positive curvature.

在本研究项目中，我们为3流形引入了一个新的拓扑不变量，称为“块数”，它估计了3流形的某种复杂性，就像Heegaard属一样。利用流脊来定义块数，使我们能够对3流形进行分类，并用有限多个整数来参数化每一类中的3流形。此外，块数不仅可以定义3-流形，还可以定义梳状3-流形、3-流形对及其上的非奇异向量场。使用这个不变量，我们得到了以下结果：唯一块数为0的精梳3流形是2球与圆积上的正则流形，块数为1的精梳3流形是透镜空间（包括3球）上的正则流形。给出了块数为2的3流形的参数化。并且，利用Reidemeister扭转，我们给出了一些结果，这些结果表明我们的参数化统一了梳状3流形的表示。我们给出了计算所有Seifert纤维3-流形的Thraev-Viro不变量值的公式。在辛流形上，我们得到了以下结果：如果闭辛流形的普适覆盖空间是可缩并的，则该流形不存在任何正曲率的黎曼度规。

项目成果

Vertex‐disjoint cycles containing specified vertices in a bipartite graph

• DOI: 10.1002/jgt.10159
• 发表时间: 2004-07
• 期刊: Journal of Graph Theory
• 影响因子: 0.9
• 作者: Guantao Chen;H. Enomoto;K. Kawarabayashi;K. Ota;Dingjun Lou;Akira Saito

量子的な微分幾何

量子微分几何

• 发表时间: 2004
• 作者: 大森英樹;前田吉昭
• 通讯作者: 前田吉昭

R.Mori, A.Nakamoto, K.Ohta: "Diagonal flips in Hamiltonian triangulations on the sphere"Graphs and Combinatorics. 19. 413-418 (2003)

R.Mori、A.Nakamoto、K.Ohta：“球体上哈密顿三角剖分中的对角线翻转”图和组合学。

Meromorphic solutions of Riccati differential equations with doubly periodic coefficients

具有双周期系数的Riccati微分方程的亚纯解

• 期刊: J.Math.Anal.Appl. (to appear)
• 作者: G.Chen;H.Enomoto 他;K.Ota;S.Shimomura;S.Shimomura
• 通讯作者: S.Shimomura

K.Kawarabayashi, A.Nakamoto, K.Ohta: "2-connected 7-covering of 3-connected graphs on surfaces"J.Graph Theory. 43. 26-36 (2003)

K.Kawarabayashi、A.Nakamoto、K.Ohta：“曲面上 3 连通图的 2 连通 7 覆盖”J. 图论。