The study on controlled surgery theory and its application

控制手术理论及其应用研究

• 批准号: 11640093
• 负责人: YAMASAKI Masayuki
• 金额: $ 2.24万
• 依托单位: Josai University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640093/
• 关键词: controlled surgery; stability; splitting; local simpleness; ホワイトヘッドのねじれ; マイヤー・ビートリス完全系列

We studied properties of surgery groups controlled over metric spaces. Such groups are supposed to appear in controlled surgery sequences. Controlled surgery sequences have been verified to be exact in the case of trivial local fundamental groups. A key ingredient of the proof was the stability of the controlled surgery groups. Our main objective was to prove the stability of the controlled surgery groups in a more general setting.To prove the stability, one needs a method to split controlled Poincare quadratic com-plexes. Splitting is always possible in the case of trivial local fundamental groups, because controlled Whitehead groups vanish. Unfortunately we cannot hope to acomplish splitting in general.During the period of this research project, we introduced the notion of local simpleness for controlled Poincare quadratic complexes and used this notion to give a certain sufficient condition to splitting. Although this condition is not satisfied in general, there is a hope that the complexes appearing in the proof of stability of controlled surgery groups. In fact we have succeeded to verify the stability using our splitting in the case when the control space is a subcomplex of the unit circle.We plan to continue this using some induction argument to establish the stability for control spaces embedded in higher dimensional spheres.

我们研究了对度量空间控制的手术组的特性。这些组应该以受控的手术序列出现。在琐碎的局部基本组的情况下，已证实受控手术序列是确切的。证明的关键要素是受控手术组的稳定性。我们的主要目的是在更一般的环境中证明受控手术组的稳定性。要证明稳定性需要一种分裂控制的繁殖性繁殖二次com-plexes的方法。在琐碎的本地基本群体的情况下，始终可能会分裂，因为受控的白头群体消失了。不幸的是，我们不能希望一般进行分裂。在该研究项目的时期内，我们介绍了当地简单性的概念，用于受控的Poincare二次复合物，并使用此概念给出了一定的分裂条件。尽管总体上不满足这种情况，但希望复合物在受控手术组的稳定性证明中出现。实际上，在控制空间是单位圆的子复合的情况下，我们已经成功地使用我们的分裂来验证稳定性。我们计划使用一些感应参数继续此操作，以确定嵌入更高维球体的控制空间的稳定性。

项目成果

Takahiro TSUCHIYA: "General saddlepoint approximations to distributions under an ellip-tical population"Communications in Statdstics. 28. 727-754 (1999)

Takahiro TSUCHIYA：“椭圆总体下分布的一般鞍点近似”统计通讯。

Qing-Ming Cheng: "Submanifolds with constant scalar curvature"Proceedings of the Royal Society of Edinburgh. (to appear).

Qing-Ming Cheng：“具有恒定标量曲率的子流形”爱丁堡皇家学会会议录。

Qing-Ming Cheng: "Hypersurfaces in a unit sphere S^<n+1>(1) with constant scalar curvature"Journal of the London Mathematical Society. 64. 755-768 (2001)

Qing-Ming Cheng：“具有恒定标量曲率的单位球体 S^<n 1>(1) 中的超曲面”伦敦数学会杂志。

西澤 清子: "Paramentrization by fixed points multipliers of the polynomials with degree n"数理解析研究所講究禄. 1199. 127-131 (2001)

Kiyoko Nishizawa：“n 次多项式的定点乘法参数化”数学科学研究所 Kokyuroku，1199. 127-131 (2001)。

Qing-Ming Cheng,et. al: "Conformally flat 3-manifolds with constant scalar curvature II"Japanese J. Math.. 26. (2000)

程庆明，等。