2006 Fiscal Year Final Research Report Summary
The topology of the space of submanifolds and its geometric applications
| Project/Area Number |
17540080
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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 | Okayama University |
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Principal Investigator |
SHIMAKAWA Kazuhisa Okayama University, Graduate School of Natural Science and Technology, Professor, 大学院自然科学研究科, 教授 (70109081)
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| Co-Investigator(Kenkyū-buntansha) |
MIMURA Mamoru Okayama University, Professor Emeritus, 名誉教授 (70026772)
YAMAGUCHI Kohhei The University of Electro-Communications Department of Electro-Communications, Professor, 電気通信学部, 教授 (00175655)
MURAYAMA Mitutaka Tokyo Institute of Technology, Graduate School of Science and Technology, Associate Professor, 大学院理工学研究科, 助教授 (40157805)
OKUYAMA Shingo Takuma National College of Technology, Department of Control Engineering, Assistant Professor, 電子制御工学科, 講師 (50290812)
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| Project Period (FY) |
2005 – 2006
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| Keywords | configuration space / generalized homology theory / infinite loop space / equivariant homotopy |
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| Research Abstract |
We studied the homotopy type of the labeled configuration space of the countable dimensional Euclidean space and showed that its homology-theoretic group-completion can be obtained by algebraically completing the label space. Among applications, this implies the result that the homology theory coming from arbitrary finite subset of the additive group of integers coincides with the stable homotopy theory. This is the vast improvements of the results Caruso that the space of "positive and negative particles in the countable dimensional Euclidean space" is homotopic the infinite loop space QS^0.
We also established the method for constructing generalized homology and cohomology theories by using the notion of continuous functors, instead of spectra. This enables us to uniformize the definitions of various (co)homology theories in a simple and efficient manner. Furthermore, this methods is suitable for equivariantly generalize the construction of theories.
In addition to the fundamental studies stated above, we applied our theory of configuration spaces to geometric problems and obtained several interesting results concerning the topology of the space of holomorphic maps.
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