Studies on Infinite-Dimensional Manifolds and Menger Manifolds, and their Applications

无限维流形和Menger流形的研究及其应用

• 批准号: 10640060
• 负责人: SAKAI Katsuro
• 金额: $ 2.18万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640060/
• 关键词: infinite-dimenisonal manifold; absolute neighborhood retract (ANR); space of mappings; hyperspace; Menger manifold; universal space; proper n-shape theory; strong n-shape theory; Attouch-Wets位相; Fell位相; Hilbert空間; Hilbert cube; strong universa

1. Infinite-Dimensional Manifolds and ANR Theory.In this part, we have many results in the following researches :(1) Characterizations of bitopological infinite-dimensional manifolds (Sakai-Banakh) ;(2) Studies on free topological semilattices (Sakai-Banakh) ;(3) Direct limits of Banach-Mazur compacta (Sakai-Kawamura-Banakh) ;(4) Studies on spaces of homeomorphisms and embeddings (Yagasaki) ;(5) Spaces of Peano and ANR continua (Yagasaki) ;(6) Characterizations of ANR's (Sakai).Recently, we have made some progress in the following two studies, whose development are expected :(7) Maps from mapping spaces to a hyperspaces (Yagasaki) ;(8) Hyperspaces of closed sets of non-compact metric spaces (Sakai-Kurihara-Yang).2. Menger Manifolds and n-Shape Theory.In this part, we have many results in the following researches :(1) Dynamics on Menger manifolds (Kato-Kawamura-Tuncali-Tymchatyn) ;(2) Dimension of the homeomorphism group of Menger compacta (Kawamura-Brechner) ;(3) Lusternik-Schnirelmann type invariants concerning Menger manifolds (Kawamura) ;(4) Groupe actions on Menger curve (Kawamura) ;(5) An application to a universal space for a class of closed images of metric spaces (Kawamura-Tuda) ;(6) Studies on proper n-shape theory (Sakai-Akaike) ;(7) Formulation of strong n-shape (Sakai-Iwamoto).3. In relation to this project, we invited Prof. Ageev (Belorussia) to learn about his research on the characterization of Nobeling spaces. Now, we are ready to work together with him, and further joint studies with him are expected.

1.无限维流形与ANR理论，在这一部分中，我们主要研究了以下几个方面的问题：（1）双拓扑无限维流形的刻画（Sakai-Banakh）;（2）自由拓扑半格研究（Sakai-Banakh）;（3）Banach-Mazur的直接界限（Sakai-Kawamura-Banakh）;（4）同胚空间和嵌入空间的研究（Yagasaki）;（5）Peano和ANR连续空间（Yagasaki）;（6）ANR（Sakai）的刻画。最近，我们在以下两个方面的研究中取得了一些进展，并对其发展进行了展望：（7）从映射空间到超空间的映射（Yagasaki）;（8）非紧度量空间闭集的超空间（Sakai-Kurihara-Yang）。2. Menger流形与n形理论，在这一部分中，我们主要研究了以下几个方面：（1）Menger流形上的动力学（Kato-Kawamura-Tuncali-Tymchatyn）;（2）Menger-Kawamura同胚群的维数（3）关于Menger流形的Lusternik-Schnirelmann型不变量（4）Menger曲线上的群体作用（5）度量空间的一类闭象在泛空间中的应用（6）真n形理论的研究（Sakai-Akaike）;（7）强n形的公式化（Sakai-Iwamoto）。在这个项目中，我们邀请了白俄罗斯的Jakiev教授了解他对Nobeling空间特征的研究。现在，我们愿意与他合作，并期待与他进行进一步的联合研究。

项目成果

岩本豊,酒井克郎: "Strong n-shape theory"Topology and its Applications. (印刷中).

Yutaka Iwamoto、Katsuro Sakai：“强 n 形理论”拓扑及其应用（正在出版）。

T.Banakh and K.Sakai: "Free topological semilattices homeomorphic to P^∞ or Q^∞"Topology Appl.. 106. 135-147 (2000)

T.Banakh 和 K.Sakai：“同胚于 P^∞ 或 Q^∞ 的自由拓扑半格”拓扑应用.. 106. 135-147 (2000)

B.Brechner and K.Kawamura: "On the dimension of a homeomorphism group"Proc. Amer. Math. Soc.. 129. 617-620 (2000)

B.Brechner 和 K.Kawamura：“论同胚群的维数”Proc。

K.Sakai: "The completion of metric ANR's and homotopy dense subsets"J.Math. Soc. Japan. 52. 835-846 (2000)

K.Sakai：“度量 ANR 和同伦稠密子集的完成”J.Math。

T.Yagasaki: "The homeomorphism groups of noncompact 2-manifolds. Memoirs of the Faculty of Eng. and Design"Kyoto Institute of Technology. 47. 41-48 (1998)

T.Yagasaki：“非紧2-流形的同胚群。工程设计学院回忆录”京都工业大学。