2007 Fiscal Year Final Research Report Summary
The study of Borsuk-Ulam type theorems in transformation group theory
| Project/Area Number |
17540075
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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 | Kyoto Prefectural University of Medicine (2006-2007)
Osaka University (2005) |
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Principal Investigator |
NAGASAKI Ikumitsu Kyoto Prefectural University of Medicine, Graduate School of Medical Science, Professor (50198305)
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| Co-Investigator(Kenkyū-buntansha) |
HARA Yasuhiro Osaka University, Graduate School of Science, Assistant Professor (10294141)
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| Project Period (FY) |
2005 – 2007
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| Keywords | transformation groups / eouivariant map / isovariant map / equivariant homotopy / isovariant homotopy / Borsuk-Ulam type theorem / maiming degree |
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| Research Abstract |
In this study, we have studied Borsuk-Ulam type theorems from the viewpoint of transformation group theory. The classical Borsuk-Ulam theorem says that for any continuous map f from the n-sphere to the n-Euclidean space, there is a point x such that f(x)=f(-x). This result is restated as follows: There is no C_2-equivariant map from the n-sphere to the (n-1)-sphere with the antipodal action. From this point of view, the study of Borsuk-Ulam type theorems is that of the existence problem of equivariant or isovariant maps.
We first clarified the relation of equivariant maps and the cohomological indices, and using them we obtained some Borsuk-Ulam type theorems for equivariant maps; in particular, the Borsuk-Ulam type theorem holds for Stiefel manifolds with free actions of an elementary abelian p-group. Moreover, we obtained some Borsuk-Ulam type theorems for isovariant maps; for example, if there exists an isovariant map from a homology sphere with free finite group action to a representation sphere, then a Borsuk-Ulam type inequality holds. This inequality is expected to give some applications to various fields.
We next studied the converse of Borsuk-Ulam type theorems, and showed from equivariant obstruction theory that the converse holds for some cases including the classical case.
This gives a new insight to the study of Borsuk-Ulam type theorems, and provides new problems such as the classification problem of isovariant homotopy classes and multidegrees. Thus the future development of this research is expected.
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Research Products
(60 results)
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[Presentation] 不動点定理と一致点定理2007
Author(s)
原 靖浩
Organizer
数理解析研究所研究集会「変換群の理論とその応用」
Place of Presentation
京都大学数理解析研究所
Year and Date
2007-05-30
Description
「研究成果報告書概要(和文)」より
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