2005 Fiscal Year Final Research Report Summary
Study of algebraic structures on homotopy sets
| Project/Area Number |
15540058
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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 University, College of Science, Professor, 理学部, 教授 (70047372)
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| Co-Investigator(Kenkyū-buntansha) |
YAGITA Nobuaki Ibaraki University, College of Education, Professor, 教育学部, 教授 (20130768)
URABE Tohsuke Ibaraki University, College of Science, Professor, 理学部, 教授 (70145655)
OHTSUKA Fumiko Ibaraki University, College of Science, Assistant Professor, 理学部, 助教授 (90194208)
AIBA Akira Ibaraki University, College of Science, Assistant Professor, 理学部, 助教授 (90202457)
TAKEUCHI Mamoru Ibaraki University, College of Science, Lecturer, 理学部, 講師 (40007761)
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| Project Period (FY) |
2003 – 2005
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| Keywords | homotopy / Lie grop / H-space / H-map / Samelson product / localization / nilpotency class of groups / 群のべき零指数 |
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| Research Abstract |
Given a compact connected Lie group G,[G,G] denotes the set of homotopy classes of self maps of G, and it inherits a group structure from G.
1. Our first target is the following Conjecture : If G is simple, then the nilpotency class of [G,G] is greater than or equal to the rank of G. We have proved the conjecture for G with small rank. But we can still give neither a proof nor a counter example of the conjecture.
2. One of our results is the following theorem which proves a weak form of the conjecture.
Theorem. The group [G,G] is commutative if and only if G is isomorphic to one of T^n (n【greater than or equal】0), T^m×S^3 (0【less than or equal】m【less than or equal】2) and SO(3), where T^n is the n-dimensional torus.
3. Let ε_#(G) be the group of homotopy classes of self homotopy equivalences of G which induce the identity on homotopy groups. We have proved that the following three statements are equivalent : (1) The group ε_#(G) is trivial ; (2) the left distributive law aο(b+c)=aοb+aοc holds in [G,G], where the group multiplication in [G,G] is denoted by + ; and (3) G is isomorphic to one of T^n, S^3 and SO(3).
4. Let G be semi-simple and of rank 2. We have determined the group structure of [G,G] except for the case G〓SO(3)×SO(3).
5. Let G_2 be the exceptional Lie group of rank 2 and φ:S^3×S^<11>→G_2 the canonical map. We have determined odd prime numbers p such that the localization φ_<(p)>:S^3_<(p)>×S^<11>_<(p)>→G_<2(p)> is an H-map.
6. The head investigator, Arkowitz and Strom have studied relations between typical subgroups of ε(X), where ε(X) is the group of homotopy classes of self homotopy equivalences of the space X.
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