2001 Fiscal Year Final Research Report Summary
Research on homotopy groups of localized finite complexes
| Project/Area Number |
12640077
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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 | KOCHI UNIVERSITY |
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Principal Investigator |
SHIMOMURA Katsumi Faculty of Science, Kochi University, Professor, 理学部, 教授 (30206247)
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| Co-Investigator(Kenkyū-buntansha) |
YOSIMURA Zen-ichi Nagoya Institute of Technology, Professor, 工学部, 教授 (70047330)
KOMATSU Kazushi Faculty of Science, Kochi University, Associate Professor, 理学部, 助教授 (00253336)
HEMMI Yutaka Faculty of Science, Kochi University, Professor, 理学部, 教授 (70181477)
OHKAWA Tetsusuke Hiroshima Institute of Technology, Associate Professor, 工学部, 助教授 (60116548)
YAGITA Nobuaki Ibaraki University Faculty of Education, Professor, 教育学部, 教授 (20130768)
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| Project Period (FY) |
2000 – 2001
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| Keywords | homotopy groups / spheres / finite complexes / spectra / Morava K-theories / Bousfield localization / Johnson-Wilson spectrum / Adams-Novikov spectral sequence |
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| Research Abstract |
In this research, we aimed two subjects. One is to make a more deep understanding of finite complexes itself and the Bousfield localization of finite complexes with respect to the Morava K-theories, and the other is to determie the homotopy groups π_*(L_<K(2)>S^O) of the Bousfield localized sphere spectrum L_2S^O with respect to K(2).
For the first one, Hemmi showed that even dimensional generator of the cohomology ring of a finite H-space appears only at dimension 8 and 20. This reflects an important feature of finite complexes. Komatsu studied finite real projective spaces through the bundle structure. Some information on the localization with respect to K(1) was obtained by Yosimura from the view point of KO_*.-quasi equivalence. Yagita obtained a result on the non-commutativity of the homotopy groups, and Ohkawa studied the Bousfield classes in stable homotopy categories.
For the second, we determined the homotopy groups π_*(L_K (2)S^O) at the prime 3 in the first year. The groups for the prime p > 3 was determined before. We also determine the E_2-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_*(L_2 S^O) at the prime 2. Since the computation of the differentials of the spectral sequence is too difficult to make, we studied, in the second year, the homotopy groups of the Ravenel spectra T(m), and considered the Picard groups consisting of the invertible spectra in a stable homotopy, which is closely related to the homotopy groups. We obtained the homotopy groups π_*(L_nT(m)ΛV(n-1)) for m > n^2-n, and showed that there is no invertible spectrum in the stable homotopy category of E-local spectra as long as E is connective.
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