2002 Fiscal Year Final Research Report Summary
Research of the Stable Homoyopy Theory from a view point of K-Theory
| Project/Area Number |
12640067
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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 | Nagoya Institute of Technology |
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Principal Investigator |
YOSIMURA Zen-ichi Nagoya Institute of Technology, Technology, Professor, 工学部, 教授 (70047330)
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| Co-Investigator(Kenkyū-buntansha) |
SHIMOMURA Katsumi Kochi University, Science, Professor, 理学部, 教授 (30206247)
MINAMI Norihiko Nagoya Institute of Technology, Technology, Professor, 工学部, 教授 (80166090)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Real K-Theory / Complex K-Theory / K-Localizatlon Theory / KO-Homology Equivalence / Adams Operation / Cw-Complex / Cw-Spectrum / Bousfield's Category |
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| Research Abstract |
The head investigator introduced the concept of "quasi KO_*-equivalence" in 1990, in order to give a certain classification of CW-complexes or manifolds, which is weaker than the classification based on the K_*-localization. Since then, he has continued to research mainly the following two subjects concerned with the quasi KO_*-equivalence and the K_*-localization. He has already obtained some satisfactory results, and moreover obtained three new results mentioned below during the period of Scientific Research Project in 2000-2002.
The first subject is to classify CW-complexes (or CW-spectra) X by the quasi KO_*-equivalence when KU_*X has a simple form such as a free abelian group, a direct sum of a free abelian group and a 2-cyclic group and so on. In 2000 he established the classification when KU_*X is isomorphic to the direct sum of two cyclic 2-torsion groups without the assumption that KU_1X = 0. Although there is an obstruction to establish our classification unless KU_1X = 0, he succeeded to overcome its obstruction by an algebraic method enploying the Bousfield's category different from the previous geometrical one constructing small cells complexes.
The second subject is to classify Cw-complexes (or CW-spectra) X by the K_*-localization when once X has been classified by the quasi KO_*-equivalence. For our purpose it is necessary to investigate the behaviour of the Adams operation on KO_*X. However it is never easy to determine their K_*-local types because the behaviour of the Adams operation on KO_*X is very complicated. In 2001-2002 he determined the K_*-local type of X when KU_*X is isomorphic to the free abelian group of rank 2 or the direct sum of the free abelian group of rank 1 and a 2-cyclic group.
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