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研究了某些源自塞伯格（Seiberg-Witten）理论的某些不变式的3，4个manifolds的差异类型的约束。 DIRAC操作员对他先前研究的V 4-Manifolds孤立的奇异性的索引的贡献是一对球形3型Manifold及其旋转结构的整数不变性，它与Rochlin Invariant（确定的Modulo 16）的整体升降（与Neumann-Siebenman-Siebenmann Mannmannewiantiant vishiant visheriant seventiant and旋转结构相吻合。他认为，当一个结上的手术获得一定的球形3个序列时，就上述不变性而对其类型产生了一些限制，并且在球形3个manifolds的不变剂之间也给出了某些关系，因为它们是在普通结上同时通过同时的手术获得的。他还将结果扩展到了塞弗特（Seifert）一般的3个manifolds的情况下，并给出了一些限制因素，可以通过诺伊曼·塞本曼（Neumann-Siebenmann）不变式来通过手术来获得。最近，Ozsvath-Szabo的Floer同源性给出了通过手术获得的Seifert 3 manifolds的一些限制。因此，我们的下一个任务是研究浮子同源性与上述不变性之间的关系。 Fujii在高斯高几幅功能方面进行了研究3维双曲线歧管的局部转移的研究。 Imanishi对Lipschitz同构同构的共同体学进行了研究，该研究通过利用有关该间隔的Lipschitz同构同构的几个结果，从而保留了Codimension 1的可区分叶子。