2005 Fiscal Year Final Research Report Summary
Study on rank filetration of finite H-spaces
| Project/Area Number |
15540074
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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 | WAKAYAMA UNIVERSITY |
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Principal Investigator |
MORISUGI Kaoru WAKAYAMA UNIVERSITY, Faculty of Education, Professor, 教育学部, 教授 (00031807)
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| Co-Investigator(Kenkyū-buntansha) |
OSHIMA Hideaki Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
HEMMI Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
WATANABE Takashi Osaka Prefecture University, Faculty of Liberal Arts and Sciences, Professor, 総合教育研究機構, 教授 (20089957)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Hopf spaces / Lie groups / Whitehead products / homotopy groups of spheres / self maps / nilpotent class / Moore spaces / mod p cohomology |
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| Research Abstract |
The summary of research results is as follows.
1 It is important to know what nilpotency the group [X,X] has for a Hopf space X. However another important thing is to study the composition structure of the group [X,X]. Morisugi determined the relationship between the composition structure and the group structure of [X,X] for X=SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.
Let M^n be the mod 2 Moore space. For n【greater than or equal】3 it is known that there exists a lift <η_n>^^^^of the suspension of the Hopf map, η_n:S^<n+1>→S^n. We investigated the order of the Whitehead product [<η_n>^^^^,<η_n>^^^^] in π_<2n+1>(M^n).
2 Let G be the simple Lie group of classical type. Oshima showed that the group [G,G] is non-commutative for almost all cases of G mentioned above. And for some cases of G, he determined the nilpotency class of [G,G].
Let X be a Hopf complex. In this case [X,X] is, so called, an algebraic loop, that is, it has a binary operation with both left and right inverses. Oshima also investigated how they differs from each other.
3 Hemmi showed that the possible even dimensional generators of mod 3 cohomology rings of finite Hopf spaces occurs only in dimension 8 or 20. And he almost determined the structure of such mod 3 cohomology rings. He also showed that under some conditions, there is no Hopf space X with H(X;Z/p)≡Λ(x,p^1x,…p^<p-2>x)
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