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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利用Seiberg-Witten理论中的一些不变量研究了3，4-流形的非同态类型的约束。狄拉克算子的指数对V4-流形孤立奇点的贡献是球面3-流形及其自旋结构对的整数值不变量，它给出了Rochlin不变量的整数提升（确定模16），这与Neumann-Siebenmann不变量一致。他认为的情况下，当一定的球形3流形是通过外科手术的一个结，并给出了一些限制其类型方面的上述不变量，也给出了一定的关系之间的不变量的球形3流形的情况下，他们是通过外科手术的一个共同的结。他还扩大了结果的情况下，一般塞弗特3-流形，并给予了一些限制，他们得到的手术结方面的诺伊曼-西本曼不变量。最近，Ozsvath-Szabo的Floer同调给出了Seifert 3-流形在纽结上通过外科手术得到的一些约束条件。因此，我们的下一个任务是研究Floer同调与上述不变量之间的关系。Fujii成功地研究了三维双曲锥流形的局部变换。Imanishi利用区间上Lipschitz同胚群的几个结果，成功地研究了保余维为1的可微叶理的Lipschitz同胚群的上同调。