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

1. 无限维流形与ANR理论。在这一部分中，我们在以下方面的研究取得了许多成果：(1)双拓扑无限维流形（Sakai-Banakh）的表征；(2)自由拓扑半格的研究（Sakai-Banakh）；(3) Banach-Mazur compacta （Sakai-Kawamura-Banakh）的直接极限；(4)关于同胚和嵌入空间的研究（Yagasaki）；(5) Peano和ANR连续空间（Yagasaki）；(6) ANR的特征（Sakai）。最近，我们在以下两方面的研究取得了一些进展，它们的发展是值得期待的：(7)从映射空间到超空间的映射（Yagasaki）；(8)非紧度量空间的闭集的超空间（Sakai-Kurihara-Yang）Menger流形与n形理论。在这一部分中，我们取得了以下研究成果：(1)Menger流形（kato - kawamura - tuncalii - tymchatyn）的动力学；(2) Menger compacta （Kawamura-Brechner）的同胚群的维数；(3)关于Menger流形的Lusternik-Schnirelmann型不变量（Kawamura）；(4)门格尔曲线上的群体作用（Kawamura）；(5)度量空间的一类闭象在泛空间中的应用（Kawamura-Tuda）；(6)适当n形理论研究（Sakai-Akaike）；(7)强n形的配方（Sakai-Iwamoto）。关于这个项目，我们邀请了Ageev教授（白俄罗斯）来学习他关于诺贝尔空间特征的研究。现在，我们已经做好了与他合作的准备，并期待与他进一步的共同研究。

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.