The study of Low-dimensicnal manifolds with various geometric structures

各种几何结构低维流形的研究

基本信息

批准号：
18540081
负责人：
UE Masaaki
金额：
$ 1.54万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

项目摘要

The head investigator Ue continued the research of 3 and 4-manifolds. In particular he considered Fukumoto-Furuta invariant for rational homology 3-spheres coming from Seiberg-Witten theory (which coincides with the Neumann-Siebenmann invariant in case of plumbed 3-manifolds) and Ozsvath-Szabo's d-invariant defined by Heegaard Floer homology. He showed that these two invariants coincide for spherical 3-manifolds, and also for certain plumbed 3-manifolds. This implies that under such conditions both invariants are the integral 〓 of the classicalRochlin invariant and also homology cobordism invariants. But they are not the sane in general, and it is sill open whether there exists an invariant satisfying the above two conditions. He also gave certain constraits for the signature of 4-manifolds bounding Seifert rational homology 3-spheres in terms of the above invariants.The investigator Tsuyoshi Kato estimated the growth of the Casson handles embedded in the K3 surface by Yang-Milis Gauge theory, and also analized the entropy of iterations by families of maps. Fujii found the confluence phenomena of singular points of ordinary differential equations induced by deformations of hyperbolic 2-cone-manifolds, and descried harmonic vector fields on hyperbola 3-cone-manifolds in terms of hypergeometric functions. Shin'ichi Kato established the relative = symmetric space version of Jaquet's theorem, which claims that every irreducible admissible representation of p-adic reductive groups is embedded to an induced representation for irreducible cusp representation of a parabolic subgroup. Ushiki considered transgression operators over the space of distributions induced by complex dynamical systems over the Riemann sphere and showed that the Fredholm determinant is represented by Artin-Mazurzeta function.

首席研究员继续研究 3 流形和 4 流形，特别是他认为来自 Seiberg-Witten 理论的有理同源 3 流形的 Fukumoto-Furuta 不变量（与管道 3 流形的情况下的 Neumann-Siebenmann 不变量一致）。和 Heegaard Floer 定义的 Ozsvath-Szabo d 不变量他证明了这两个同源性。对于球形 3 流形，以及某些管道 3 流形，不变量是一致的，这意味着在这种情况下，两个不变量都是经典 Rochlin 不变量的积分〓，也是同调协边不变量，但它们通常不是理智的。是否存在满足上述两个条件的不变量仍是未知的。他还对4-流形的签名给出了一定的约束。根据上述不变量限制 Seifert 有理同调 3-球体。研究者 Tsuyoshi Kato 通过 Yang-Milis 规范理论估计了嵌入 K3 表面的 Casson 柄的增长，并通过 Fujii 族分析了迭代的熵。发现了双曲二锥流形变形引起的常微分方程奇异点汇合现象，并描述了双曲线上的调和向量场根据超几何函数的 3-锥流形，Shin'ichi Kato 建立了 Jaquet 定理的相对 = 对称空间版本，该定理声称 p-adic 还原群的每个不可约的可接受表示都嵌入到不可约尖点表示的诱导表示中。 Ushiki 考虑了黎曼球面上复杂动力系统引起的分布空间上的海侵算子，并表明 Fredholm 行列式由下式表示： Artin-Mazurzeta 函数。