2007 Fiscal Year Final Research Report Summary
Topology of hyperspaces, mapping spaces and universal spaces
| Project/Area Number |
17540061
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | University of Tsukuba |
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Principal Investigator |
SAKAI Katsuro University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor (50036084)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAMURA Kazuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor (40204771)
KATO Hisao University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor (70152733)
YAGASAKI Tatsuhiko Kyoto Institute of Technology, Graduate School of Science and Technology, Associate Professor (40191077)
YAMAZAKI Kaori Takasaki City University of Economics, Faculty of Economics, Associate Professor (80301076)
AKAIKE Yuji Kure National College of Technology, Department of General Education, Associate Professor (70311074)
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| Project Period (FY) |
2005 – 2007
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| Keywords | hyperspace / mapping space / universal space / ANR / Hilbert space / Hilbert cube / the direct limit of Euclidean spaces / LF space |
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| Research Abstract |
To achieve three objects mentioned at the beginning of this project, the head investigator have been making researches in cooperation with investigators and invited W. Kubis from Poland and T. Banakh from Ukraine to do joint works and to exchange information. Finally, we obtained many good results. Concerning the first object to specify under what conditions and to what spaces each of various hyperspaces is homeomorphic (even in non-separable case), by many joint works conducted by the head, we had such results as: specifying hyperspaces on Banach spaces with the Wijsman topology; finding conditions of metric spaces whose hyperspaces of closed sets are ANR's; proving that the hyperspace of closed convex sets in a normed space is an ANR with respect to Hausdorff uniformity and Attouch-Wets topology and for a finite-dimensional normed space it is homeomorphic to the product of the base space and the Hilbert cube; specifying hyperspaces consisting of compacta of various types; proving tha
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t the hyperspaces of bounded closed sets in the space of irrationals and the Noebeling spaces. Concerning the second object to find mapping spaces being infinite-dimensional manifold and to clarify their topological structure, Yagasaki classified the connected component of the inclusion in the space of embeddings of a subpolyhedron into a surface and generalized Berlange's result on the group of measure-preserving homeo-morphisms to non-compact case. The head collaborated Uehara on clarifying topological structure of the space of lower semi-continuous functions. Moreover, in the joint work with Banakh, Mine and Yagasaki, he proved that the homeomorphism group of non-compact surfaces with the Whitney topology can be embedded in the product of the Hilbert space and the direct limit of Euclidean spaces as open sets. Concerning the object to enrich studies on non-separable infinite-dimensional universal spaces and to complete the proof of characterization of Noebeling spaces, the latter was done by Nagorko and we could not give any contribution but the head and Mine could obtain the classification theorem on open sets in LF-spaces, which can be a foothold on studying LF-manifolds. Less
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