Project/Area Number  10640077 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Wakayama University 
Principal Investigator 
MORISUGI Kaoru Faculty of Education, WAKAYAMA UNIVERSITY, Professor, 教育学部, 教授 (00031807)

CoInvestigator(Kenkyūbuntansha) 
HEMMI Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
OSHIMA Hideaki Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
KAWAKAMI Tomohiro Faculty of Education, WAKAYAMA UNIVERSITY, Associate Professor, 教育学部, 助教授 (20234023)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,300,000 (Direct Cost : ¥2,300,000)
Fiscal Year 1999 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Keywords  Hspaces / Samelson products / Whitehead products / fiber bundles / homotopy groups / self maps / H空間 / Samelson積 / Whitehead積 / ファイバー束 / ホモトピー群 / 自己写像 / ホップ空間 / ホモトピー / サメルソン積 / ホップ構成 / ホトヘッド積 
Research Abstract 
The summary of research results is as follows. (a)Let X be an Hopf space. Oshima showed that the homotopy set of self maps of X [X, X] forms a group when X has at most 3cells and he also determined the group structure of [X, X] for such spaces X. (b)We studied the group structure [ΣX, ΣX], where ΣX is the suspension space of X. We determined the group structure in case that X = SU(3) and Sp(2). They are typical example of above space X. In general, the suspension map [X, X] → [ΣX, ΣX] is not a homomorphism. (c)For general Hopf spaces, we can consider the Samelson products in [Y, X] when Y is a CWcomplexes. The we found a formula which relates the Samelson product in [Y, X] and its suspension in [Y, ΩΣX] by using the generalized Hopf construction. The above item is an application of this formula. (d)It is important to know what nilpotency the group [X, X] for a Hopf space X. However another important is to study the composition structure of the group [X, X]. We determined this composition structure for X = SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied. (e)Let MィイD1nィエD1 = SィイD1n1ィエD1 UィイD22ィエD2eィイD1nィエD1 be the mod 2 Moore space. There is a canonical projection map P : MィイD1nィエD1 → SィイD1nィエD1. Given an element α∈ πィイD2κィエD2(SィイD1nィエD1), we studied when α has a lift to MィイD1nィエD1. One of our results is that the Whitehead square [ιィイD2nィエD2, ιィイD2nィエD2] does not have a lift for n ≠1, 3, or 7. This is the joint work with J.Mukai in Shinshu University.
