Research for low-climensional manifolds with various geometric structures

各种几何结构低维流形的研究

• 批准号: 12640068
• 负责人: UE Masaaki
• 金额: $ 2.3万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640068/
• 关键词: 3-manifold; 4-manifold; spin structure; Dirac operator; intersection form; Seiberg-Witten theory; V-manifold; cone manifold; 交叉形式; ホモロジー3球面; ザイフェルト多様体; ホモロジーコボルディズム

Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds coming from the various geometric structures by certain invariants, in particular the one originated from Seiberg-Witten theory. First in the joint work with Mikio Furuta and Yoshihiro Fukumoto, he studied the W invariants of homology 3-spheres, and showed that in case of Seifert homology 3-spheres this invariant coincides with the Neumann-Siebenmann invariant, and that it is homology cobordism invariant under certain extra conditions. Saveliev used our key result about the estimates on the index of the Dirac operator over V 4-manifolds, and extended our results. Also Ue determined all the contributions from the isolated singularities to the index of the Dirac operator over V 4-manifolds. This contribution itself is an invariant for a pair of spherical 3-manifold and its spin structure. As its application, he gave certain constraints on the intersection forms of definite spin 4-manifolds bounded by spherical 3-manifolds, and also new estimates on the normal Euler number of the real projective plane embedded in 4-manifolds. These results are applicable to a wider class of 3-manifolds. Fujii studied 3-dimension cone manifolds with only simple loops as their singlar sets, and succeeded to give explicit solutions in terms of Gaussian hypergeometric functions to the system of ordinary differential equations for the harmonic vector field, which is a key to describe the harmonic 1-form on the tubular neighborhood of the singular set. Also in the joint work with Kazuhiko Fukui, Imanishi determined the first 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.

我们研究了基于不同几何结构的3,4流形的微分同胚型的约束条件，特别是源自Seiberg-Witten理论的约束条件。首先在与Mikio Furuta和Yoshihiro Fukumoto的联合工作中，他研究了同调3球的W不变量，并证明了在Seifert同调3球的情况下，这个不变量与Neumann-Siebenmann不变量是一致的，并且在一定的附加条件下它是同调协不变量。Saveliev使用了我们关于v4流形上Dirac算子索引估计的关键结果，并扩展了我们的结果。我们还确定了孤立奇点对v4流形上狄拉克算子指标的所有贡献。这一贡献本身是一对球面3流形及其自旋结构的不变量。作为应用，他给出了以球面3-流形为界的确定自旋4-流形的交点形式的若干约束条件，并给出了嵌入在4-流形中的实射影平面的法向欧拉数的新估计。这些结果适用于更广泛的3流形。Fujii研究了仅以简单环为其奇异集的三维锥流形，成功地给出了调和向量场的常微分方程组的高斯超几何函数的显式解，这是描述奇异集管状邻域上调和1型的关键。在与Kazuhiko Fukui的合作中，Imanishi利用区间的Lipschitz同纯群的几个结果，确定了保持余维数为1的可微叶的Lipschitz同纯群的第一上同调。

项目成果

藤井道彦: "On strong convergence of hyperbolic 3-cone-manitolds whose singular sets have uniformly thick tubular neighborhoods"J. Math. Kyoto Univ.. 41巻2号. 421-428 (2001)

藤井道彦：“关于具有均匀厚管状邻域的双曲 3 锥体的强收敛”，京都大学数学杂志，第 41 卷，第 2 期，421-428（2001 年）

Shinichi Kato: "Whittaker-Shintani functions for orthogonal groups"(to appear).

Shinichi Kato：“正交群的 Whittaker-Shintani 函数”（即将出现）。

Michihiko, Fujii: "On strong convergence of hyperbolic 3-cone-manifolds whose singular sets have uniformly thick tubular-neighborhoods"J. Math. Kyoto Univ.. Vol. 41, No. 2. 424-428 (2001)

Michihiko, Fujii：“关于双曲 3-锥流形的强收敛性，其奇异集具有均匀厚的管状邻域”J.

加藤信一: "Whittaker-Shintani fuctions for orthogonal groups"(未定).

Shinichi Kato：“正交群的 Whittaker-Shintani 函数”（待定）。

Masaaki Ue: "On the intersection forms of spin 4-manifolds bounded by spherical 3-manifolds"Algebraic and Geometric Topology. Vol. 1. 549-578 (2001)

Masaaki Ue：“论以球形 3 流形为界的自旋 4 流形的交集形式”代数和几何拓扑。