2001 Fiscal Year Final Research Report Summary
Study on Transformation Group Theory and Equivariant K-theory
| Project/Area Number |
12640075
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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 | Yamaguchi University |
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Principal Investigator |
KOMIYA Katsuhiro Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (00034744)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAZAWA Yasuyuki Yamaguchi University, Faculty of Science, Assistant, 理学部, 助手 (60263761)
NAITOH Hiroo Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10127772)
ANDO Yoshifumi Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (80001840)
SATO Yoshihisa Yamaguchi University, Faculty of Education, Lecturer, 教育学部, 講師 (90231349)
WATANABE Tadashi Yamaguchi University, Faculty of Education, Professor, 教育学部, 教授 (10107724)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Borsuk-Ulam theorem / linear representation / equivariant map / equivariant K-theory / cutting and pasting / SK-equivalence / SK-group / Euler characteristic |
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| Research Abstract |
[1] The classical Borsuk-Ulam theorem is only concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres of unitary representations of a compact abelian group. We obtain a necessary and sufficient condition for the existence of such equivariant maps in terms of the Euler classes of unit spheres, and also obtain the result concerning with the degrees of equivariant maps. The method depends heavily on the equivariant K-theory.
[II] Cuttings and pastings of manifolds lead to the so-called SK-groups of manifolds. Various kinds of such groups are found in the literature. In our study we investigate the relation among them, and show that the following three SK-groups are essentially isomorphic to each other : (i) the SK-group of pairs of manifolds, (ii) the SK-group of manifolds with boundary, and (iii) the SK-group of manifolds with involution.
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