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The study of low-dimensional manifolds with various geometric structures

各种几何结构的低维流形的研究

基本信息

批准号：
16540063
负责人：
UE Masaaki
金额：
$ 1.41万
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：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

The head investigator continued the research on the structures of 3 and 4-manifolds, in particular the diffeomorphism types of them. For the research of the structures of 4-manifolds with boundary, he generalized the Fukumoto-Furuta invariants to apply them to rational homology 3-spheres, which have been originally defined by using the index of the Dirac operator on V-manifolds based on the Seiberg-Witten theory. In particular he proved that the Fukumoto-Furuta invariant for Siefert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. He applied these results to the constraints for the intersection forms of 4-manifolds whose boundaries are Seifert manifolds, and to the conditions for the Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The constraints for the intersection forms of 4-manifolds with boundary or the conditions for the 3-manifolds to be obtained by Dehn surgery on knots also have been studied by 3-manifold invariants derived from the Heegaard Floer homology by Oszvath and Szabo, which are based on the different principle from ours. We investigated the relation between Oszvath-Szabo's invariant and the Fukumoto-Furuta invariant for the lens spaces via the eta invariant, but their relationship for more general cases is still open for the research in the future.The investigator Fujii found the relation between the deformation of the hyperbolic structure of the complement of a hyperbolic knot and the rational points of the elliptic curves. The investigator Kato, Ushiki, and Nishiwada studied the representation theory of p-adic symmetric spaces, two dimensional complex dynamical systems, Theorema elegantissimum by Gauss respectively, and Imanishi continued the study of foliations.

首席研究员继续研究3和4-流形的结构，特别是它们的同构类型。为了研究带边界的4-流形的结构，他将Fukumoto-Furuta不变量推广到有理同调3-球面，有理同调3-球面最初是基于Seiberg-Witten理论通过使用V-流形上的Dirac算子的指数定义的。特别是他证明了Fukumoto-Furuta不变量的Siefert 3-流形符合纽曼-Siebenmann不变量，也证明了其自旋有理同调配边不变性。他应用这些结果的限制交叉形式的4流形的边界是塞弗特流形，并为条件的塞弗特3流形获得Dehn手术结在3球。Oszvath和Szabo从Heegaard Floer同调导出的3-流形不变量，基于与我们不同的原理，研究了4-流形与边界的交形式的约束条件或3-流形通过Dehn对纽结的手术得到的条件.我们通过eta不变量研究了透镜空间的Oszvath-Szabo不变量和Fukumoto-Furuta不变量之间的关系，但它们在更一般情况下的关系仍有待于今后的研究。研究者Fujii发现了双曲纽结的补的双曲结构的变形与椭圆曲线的有理点之间的关系。调查员加藤，牛木，和西和田研究了代表性理论的p-adic对称空间，二维复杂的动力系统，定理elegantissimum高斯分别，和今继续研究叶。

项目成果

期刊论文数量（26）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Theorem a elegantissimum by Gauss

高斯的优雅定理

DOI：
发表时间：
2004
期刊：
Tsudajukudaigaku Sugaku keisankikagaku kenkyushohou 25
影响因子：
0
作者：
A.Kono;H.Hamanaka;Kimimasa Nishiwada
通讯作者：
Kimimasa Nishiwada

The Neumann-Siebenmann invariant and Seifert surgery

Neumann-Siebenmann 不变量和 Seifert 手术

DOI：
发表时间：
2005
期刊：
Math.Z. 250
影响因子：
0
作者：
Michihiko Fujii;Masaaki Ue
通讯作者：
Masaaki Ue

Degeneration of hyperbolic structures on the figure-eight knot complement and oints of finite order on the ellintic curve

八字结补上的双曲结构的退化和椭圆曲线上的有限阶点

DOI：
发表时间：
2005
期刊：
J.Math.Kyoto Univ. 45-2
影响因子：
0
作者：
J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;阿賀岡芳夫;Y.Agaoka;Y.Agaoka;阿賀岡芳夫;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Masaaki Ue;Akira Kono;Michihiko Fujii
通讯作者：
Michihiko Fujii

Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve

椭圆曲线上八字结补和有限阶点上双曲结构的退化