| Project/Area Number |
12640057
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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 | Ibaraki University |
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Principal Investigator |
OSHIMA Hideaki Ibaraki Univ., college of Science, Professor, 理学部, 教授 (70047372)
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| Co-Investigator(Kenkyū-buntansha) |
OHTSUKA Fumiko Ibaraki Univ., college of Science, Associate Professor, 理学部, 助教授 (90194208)
URABE Tohsuke Ibaraki Univ., college of Science, Professor, 理学部, 教授 (70145655)
YAGITA Nobuaki Ibaraki Univ., college of Education, Professor, 教育学部, 教授 (20130768)
MORISUGI Kaoru Wakayama Univ., Faculty of Education, Professor, 教育学部, 教授 (00031807)
TAKEUCHI Mamoru Ibaraki Univ., college of Science, Lecturer, 理学部, 講師 (40007761)
逸見 豊 高知大学, 理学部, 教授 (70181477)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Lie group / self map / homotopy / nilpotency class / Hopf space / rank / extraspecial p-group / Chow ring / H空間 / 回転群SO(4) / ホモトピー群 / Moore空間 / 階数 / 巾零指数 / サメルソン積 / コホモロジー作用素 / 冪零指数 / 交換子 |
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| Research Abstract |
1. Let X be a connected CW Hopf space with a multiplication.μ For a path connected and pointed space A, the homotopy set [A, X] of continuous maps from A to X has a binary operation "+" induced from μ. We write ([A, X], +) = [A. X ; μ] which is an algebraic loop. Few years ago, we proved that if A and X are connected CW Hopf spaces with at most three cells, then [A. X ; μ] becomes a group and its group is determined for the case A=X ; in this project, we have determined the group [A, X ; μ] for all other cases.
2. Let G be a connected Lie group and μ_0 the multiplication of G. Then the algebraic loop [A,G ; μ_0] is a group and it satisfies the relation : nil[A, G ; μ_0] 【less than or equal】 cat(A) as proved by G. W. Whitehead, where nil denotes the nilpotency class and cat denotes the Lusternik-Schnirelmann category with cat(*) = 0. We are interested in estimation of nil[A, G ; μ_0] from below. In the first place, though it is the most interested case, we have consider the case A = G. We had two conjectures :
(1) If G is simple, then nil[G, G ; μ_0] 【greater than or equal】 rank(G).
(2) If G is simple and rank(G) 【greater than or equal】 2, then nil[G, G ; μ_0] 【greater than or equal】 2.
Of course if (1) is affirmative then so is (2). Without the assumption "simple", two conjectures are in general false.We have proved (2) affirmative when the universal covering group of G is not Spin(n) with n 〓 O (mod 4).
3. We have given a partial answer to the problem : given a connected finite CW Hopf space X, determine sets P of prime numbers such that, every multiplication on the P-localization Xp of X is a P-localization of a multiplication on X. A complete answer has been given when X is of rank 2 or a simple Lie group.
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