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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.

三年来，我们努力实现本课题的以下三个内涵，每一项都取得了许多成果：1.对尚未研究的各类不可分空间刻画普适空间；2.为了丰富无限维流形理论，寻找不可分无限维流形的自然例子并研究其拓扑和几何结构；3.其他学科的研究对于第一项，通过 Sakai 和他的学生 Yaguchi 以及另一位学生 Mine 的共同工作，Bestvina-Mogilski 的吸收集理论已扩展到不可分绝对 Borel 类。通过岩本的研究，很明显阿吉耶夫对诺贝尔空间特征的证明存在缺陷。完整的证明除外。对于第二项，我们有很多关于超空间和映射空间的结果。研究非紧度量空间中非空闭集的超空间，引入各种拓扑代替经典的Vietoris拓扑，即Hausdorff度量拓扑、Fell拓扑、Attouch-Wets拓扑、Wijsman拓扑等。Sakai与Banakh、Yang、Kubis及其学生Kurihara、Yaguchi和Mine一起研究，并得到了结果对于某些类来说，这样的超空间与无限维通用空间是同构的，并且即使它们不知道上述情况，它们也是 AR 的。另一方面，Yagasaki一直致力于嵌入和同胚空间的研究，并取得了很多成果。此外，上原对于上半连续集值函数的空间也有一个结果。关于第三项，Sakai和Sakai-Iwamoto有结果，Kawamura和Yamazaki有许多有趣的结果。