Topology of Infinite-Dimensional Manifolds and Universal Spaces

无限维流形和宇宙空间的拓扑

基本信息

批准号：
14540059
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

Throughout three years, making our efforts to achieve the following three intensions in this subject, we have many results relating to each item.1.Characterizing universal spaces for various classes of non-separable spaces which had not been investigated ;2.In order to enrich the theory of infinite-dimensional manifolds, finding natural examples of non-separable infinite-dimensional manifolds and investigating their topological and geometrical structures ;3.Studying other subjects related to this subject and applying each others.For the first item, by the joint work of Sakai and his student Yaguchi and the work of the other student Mine, Bestvina-Mogilski's theory of absorbing sets have been extended to non-separable absolute Borel classes. By Iwamoto's studies, it have been clear that Ageev's proof of the characterization of Nobeling spaces contains gaps. The complete proof is excepted.For the second item, we have many results concering hyperspaces and mapping spaces. The hyperspace of non-empty closed sets in a non-compact metric space is studied by introducing various topologies instead of the classical Vietoris topology, that is, the Hausdorff metric topology, the Fell topology, the Attouch-Wets topology, the Wijsman topology, etc. Sakai has studied with Banakh, Yang, Kubis and his students, Kurihara, Yaguchi and Mine, and obtained results that such hyperspaces are homeomorphic to infinite-dimensional universal spaces for some classes and that they are AR's even if they are not known about the above. On the other hand, Yagasaki have been working on the spaces of embeddings and homeomorphisms and he have many results. Moreover, Uehara had a result concerning the spaces of upper semi-ccontinuous set-valued functions.Concering the third item, there are results by Sakai and Sakai-Iwamoto and Kawamura and Yamazaki have many interesting results.

在三年中，我们努力实现以下三个问题，我们有许多与每个项目有关的结果。1。尚未研究的各种非分离空间的通用空间来示意； 2.丰富了无限二维流形的理论，找到了自然的无分离无限二维流形的例子，并研究了它们的拓扑结构和几何结构； 3.研究与该主题相关的其他主题，并应用彼此相关的其他主题。 Sakai和他的学生Yaguchi的作品以及另一个学生矿山的作品，Bestvina-Mogilski的吸收套装理论已扩展到不可分割的绝对Borel类。根据iWamoto的研究，很明显，Ageev证明了诺贝林空间的表征的证据含有差距。完整的证明是不例外的。对于第二项，我们有许多有关超空间和映射空间的结果。通过引入各种拓扑而不是经典的越野拓扑，即Hausdorff度量拓扑，秋季拓扑，Attouch-Wets-Wets Topology，Wijsman拓扑，Wijsman拓扑，，Wijsman拓扑， Sakai曾与Banakh，Yang，Kubis和他的学生Kurihara，Yaguchi和Mine一起学习，并获得了结果，这些超空间对某些班级而言是同构的无限二维通用空间，即使他们不知道AR以上。另一方面，Yagasaki一直在嵌入嵌入和同构的空间，他有很多结果。此外，UEHARA的结果是关于上半连续的设置值函数的空间。第三个项目，Sakai和Sakai-Iwamoto和Kawamura和Kawamura和Yamazaki的结果有许多有趣的结果。