Research for low-dimensional manifolds with various geometric structures

各种几何结构的低维流形研究

基本信息

批准号：
14540076
负责人：
UE Masaaki
金额：
$ 1.73万
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：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540076/
关键词：
3-manifolds 4-manifold spin structure Dirac operator Dehn surgery Seiberg-Witten theory V-manifold cone manifold 双曲多様体葉層構造

项目摘要

Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds by certain invariants originated from Seiberg-Witten theory. The contribution of the index of the Dirac operator to the isolated singularities of V 4-manifolds previously studied by him is an integer valued invariant for the pair of a spherical 3-manifold and its spin structure, which gives an integral lift of the Rochlin invariant (determined modulo 16), which coincides with the Neumann-Siebenmann invariant. He considered the case when a certain spherical 3-manifold is obtained by surgery on a knot and gave some constraints on its type in terms of the above invariant and also gave certain relations between the invariants of the spherical 3-manifolds in the case that they are obtained by simultanious surgery on a common knot. He also extended the results to the case of general Seifert 3-manifolds and gave some constraints of them to be obtained by surgery on a knot in terms of the Neumann-Siebenmann invariants. Recently some constraints for the Seifert 3-manifolds to be obtained by surgery on a knot are given by Ozsvath-Szabo's Floer homology. So our next task is to investigate the relations between the Floer homology and the above invariants. Fujii suceeded the study of the local transfromations of 3-dimension hyperbolic cone manifolds in terms of Gaussian hypergeometric functions. Imanishi suceeded the study of the cohomology of the the group of Lipschitz homeomorphisms preserving the differentiable foliations of codimension 1 by utilizing several results about the group of Lipschitz homeomorphisms of the interval.

Ue研究了源于Seiberg-Witten理论的某些不变量对3、4流形微分同胚类型的约束。狄拉克算子的指数对他之前研究的 V 4-流形的孤立奇点的贡献是球形 3-流形及其自旋结构对的整数值不变量，它给出了 Rochlin 不变量的积分升力（以模 16 确定），这与 Neumann-Siebenmann 不变量一致。他考虑了通过对结进行手术获得某个球形 3-流形的情况，并根据上述不变量对其类型给出了一些约束，并且还给出了在以下情况下球形 3-流形的不变量之间的某些关系：是通过在普通结上同时进行手术获得的。他还将结果扩展到一般 Seifert 3 流形的情况，并根据 Neumann-Siebenmann 不变量给出了通过对结进行手术获得的一些约束。最近，Ozsvath-Szabo 的 Floer 同调给出了通过结手术获得 Seifert 3 流形的一些约束。所以我们的下一个任务是研究Floer同源性与上述不变量之间的关系。 Fujii 继续用高斯超几何函数研究三维双曲锥流形的局部变换。今西利用区间 Lipschitz 同胚群的几个结果，继续研究了 Lipschitz 同胚群的上同调，保留了余维 1 的可微分叶。