Topology of Infinite-Dimensional Manifolds and Universal Spaces

三年来，我们努力实现这一主题的以下三个意图，在每个项目上都取得了许多成果。对尚未研究过的各种不可分空间的全称空间进行刻画；2 .为了丰富无穷维流形的理论，寻找不可分无穷维流形的自然例子，研究其拓扑结构和几何结构；学习与本学科相关的其他学科，并相互应用。对于第一个项目，通过Sakai和他的学生Yaguchi以及另一个学生Mine的共同工作，Bestvina-Mogilski的吸收集理论已经推广到不可分的绝对Borel类。通过Iwamoto的研究，我们可以清楚地看到Ageev对诺贝尔空间特征的证明中包含了缺口。完整的证据除外。对于第二项，我们有许多关于超空间和映射空间的结果。通过引入各种拓扑来代替经典的Vietoris拓扑，即Hausdorff度量拓扑、Fell拓扑、Attouch-Wets拓扑、Wijsman拓扑等，研究了非紧度量空间中非空闭集的超空间。Sakai与Banakh， Yang， Kubis以及他的学生Kurihara， Yaguchi和Mine进行了研究，并得到了一些类的超空间与无限维泛空间同纯的结果，并且即使他们不知道上述情况，它们也是AR的。另一方面，Yagasaki一直在研究嵌入空间和同胚，并取得了许多成果。此外，Uehara还得到了关于上半连续集值函数空间的一个结果。关于第三项，有酒井和酒井-岩本的研究结果川村和山崎有很多有趣的研究结果。

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.