Research on 3-manifolds by combinatorial and constructive methods

在这个研究项目中，我们为三维流形引入了一个新的拓扑不变量，称为“块数”，它估计了三维流形的某种复杂性，就像Heegaard亏格一样。块数是用流脊定义的，它使我们能够对3-流形进行分类，并在每类中用100个整数来参数化3-流形。利用这一不变量，我们得到了以下结果：1.块数为0的唯一的梳状3-流形是2-球面与圆的乘积上的正则流形，块数为1的梳状3-流形是透镜空间（包括3-球面）上的正则流形. 2.给出了块数为2的梳状3-流形的参数化.利用Reidemeister挠率，我们得到了一些结果，这些结果表明我们的参数化一致化了一个梳状3-流形的表示。003.我们给出了一个计算所有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：4.如果闭辛流形的泛覆盖空间是可收缩的，则该流形不允许任何正曲率的黎曼度量。

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.