1998 Fiscal Year Final Research Report Summary
Transformation Group Theory and Critical Point Theory
| Project/Area Number |
09640113
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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 Univ., Faculty of Sci., Professor, 理学部, 教授 (00034744)
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| Co-Investigator(Kenkyū-buntansha) |
HATAYA Yasushi Yamaguchi Univ., Faculty of sci., Assistant, 理学部, 助手 (20294621)
SATO Yoshihisa Yamaguchi Univ., Faculty of Edu., Lecturer, 教育学部, 講師 (90231349)
NAKAUCHI Nobumitsu Yamaguchi Univ., Faculty of Sci., Associate Professor, 理学部, 助教授 (50180237)
WATANABE Tadashi Yamaguchi Univ., Faculty of Edu., Professor, 教育学部, 教授 (10107724)
KATO Takao Yamaguchi Univ., Faculty of Sci., Professor, 理学部, 教授 (10016157)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Borsuk-Ulam theorem / equivariant K-theory / representation ring / Euler class / equivariant maps / mapping degree |
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| Research Abstract |
The Borsuk-Ulam theorem is useful and attractive in the study of Topology, Global Analysis and other areas in Mathematics. The theorem has a long history since it was published in 1933. For more than 60 years many researchers has been contributing to various kind of applications and generalizations of the theorem.
The classical Borsuk-Ulam theorem is 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 SU and SW of unitary representations U and W of a more general compact Lie group G.
In 1997 we mainly concerned with the case in which G is a torus. Observing the algebraic structure of the equivariant K-ring of a representation sphere, we obtained a natural generalization of the classical Borsuk-Ulam theorem. Moreover, under some conditions on representations we showed U must be a subrepresentation of W if there exists an equivariant maps ftom SU to SW.
In 1998 we obtained results on the degrees of equivariant maps between representation spheres of a compact Lie group C.The method heavily depends on algebraic observation of the Euler class of representation which is defined in the representation ring R(G) of G.
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