1998 Fiscal Year Final Research Report Summary
The Classification of Manifolds based on K-Theory
| Project/Area Number |
09640103
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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 |
YOSHIMURA Zen-ichi Nagoya Institute of Technology, Technology, Professor, 工学部, 教授 (70047330)
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| Co-Investigator(Kenkyū-buntansha) |
SAEKI Akihiro Nagoya Institute of Technology, Technology, Lecturer, 工学部, 講師 (50270997)
OHYAMA Yoshiyuki Nagoya Institute of Technology, Technology, Assist.Prof., 工学部, 助教授 (80223981)
ADACHI Tosiaki Nagoya Institute of Technology, Technology, Assist.Prof., 工学部, 助教授 (60191855)
MINAMI Norihiko Nagoya Institute of Technology, Technology, Assist.Prof., 工学部, 助教授 (80166090)
NATSUME Toshikazu Nagoya Institute of Technology, Technology, Professor, 工学部, 教授 (00125890)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Real K-Theory / Complex K-Theory / K-Localization Theory / KO-Homology Equivalence / Mod 8 Lens Space / CW-Spetrum / Weighted Projective Space / Thom Complex |
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| Research Abstract |
The head investigator introduced the concept of "quasi K0*-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. He has already obtained some satisfactory results, and moreover obtained two new results mentioned below during the period of Scientific Research Project in 1997- 1998.
The first subject is to classify CW-complexes (or CW-spectra) X by the quasi KO_*-equivalence when KU_<xxxxx>*X has a simple form. In 1997 he established the classification when KU_*X is isomorphic to the direct sum of a free abelian group and one cyclic 2-torsion group 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 considering a pair of X and its dual DX simultaneously.
The second subject is to determine the quasi K0_*-equivalent types of some known manifolds (or CW-complexes). In 1998 he determined the quasi KO_*-equivalent types of the stunted mod 8 lens spaces L_n/L_m as a joint work with Y.Nishimura(Osaka City Univ.). It is very bothersome to determine their quasi K0_*-equivalent types in the usual way because the behavior of the conjugations on KU_*L_n/L_mis very complicated. To avoid a wasted effort he investigated several basic properties about the quasi K0_*-equivalent types of stunted mod q lens spaces by using Thorn complexes and weighted projective spaces. Consequently he suceeded to limit the investigation of the conjugations to a minimum.
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