1999 Fiscal Year Final Research Report Summary
Research on stable homotopy groups of finite complexes
| Project/Area Number |
10640082
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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 |
SHIMURA Katsumi Faculty of Science, KOCHI UNIVERSITY, Associate Professor, 理学部, 助教授 (30206247)
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| Co-Investigator(Kenkyū-buntansha) |
YAGITA Nobuaki Ibaraki University Faculty of Education, Professor, 教育学部, 教授 (20130768)
YOSIMURA Zen-ichi Nagoya Institute of Technology, Professor, 工学部, 教授 (70047330)
HEMMI Yutaka Faculty of Science, KOCHI UNIVERSITY, Professor, 理学部, 教授 (70181477)
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| Project Period (FY) |
1998 – 1999
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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 the Bousfield localization of finite complexes with respect to the Morava K-theories, and the other is to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) of the Bousfield localized Moore spectrum LィイD22ィエD2M with respect to K(2).
For the first one, we obtain some information on the localization with respect to K(1) from the view point of KOィイD2*ィエD2-quasi equivalence. We understand that the K(n)ィイD2*ィエD2 homologies reflect many properties of the space itself by observing Lie groups and H-spaces. Yagita and his coauthors show that some properties obtained from the structure of Morava K(n)ィイD2*ィエD2-homology imply the similar properties obtained from the one of BPィイD2*ィエD2.-homology, whose converse is the standard philosophy when we study this kind of homotopy theory.
For the second, we determined the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) in the first year. In the second year we concentrate to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) at the prime 3 and succeeded. Note that πィイD2*ィエD2( LィイD22ィエD2SィイD10ィエD1) has already determined at a prime > 3. The results made Hopkins set up the chromatic splitting conjecture, and the result at the prime 3 gives a counter example of it. This shows that the structure of the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) of the localized sphere spectrum at the prime 3 is different from the one at a prime > 3. At the prime 2, the structure seems very complex and we determine only the EィイD22ィエD2-term of the Adams-Novikov spectral sequence converging to πィイD2*ィエD2(LィイD22ィエD2M).
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