2007 Fiscal Year Final Research Report Summary
| Project/Area Number |
16540076
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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 | Osaka Prefecture University (2005-2007)
Osaka Women's University (2004) |
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Principal Investigator |
IRIYE Kouyemon Osaka Prefecture University, Gradate School of Science, Professor (40151691)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Takashi Osaka Prefecture University, Faculty of Liberal arts and Sciences, Professor (20089957)
YAMAGUCHI Atsushi Osaka Prefecture University, Faculty of Liberal arts and Sciences, Professor (80182426)
KATO Kiriko Osaka Prefecture University, Graduate School of Science, Associate Professor (00347478)
YOSHITOMI Kentaro Osaka Prefecture University, Faculty of Liberal arts and Sciences, Lecturer (10305609)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Phantom map / Loop space / rational homotopy eauivalence / exceptional Lie groups / symplectic group / stable homotopy |
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| Research Abstract |
Through our research we obtained the following three major results.
1. We proved that for every simply connected finite complex X there is a map which induces rational homotopy equivalence from the loop space on X to a product of some odd dimensional spheres and loop spaces on odd dimensional spheres. This result implies that the phantom map out of the loop space on a simply connected finite complex into a finite type space is trivial.
2. If a space X satisfies one of the following three conditions, then the stable suspension order of the universal phantom map out of the space X is infinite.
(1) The fundamental group of X is finite and homotopy groups are vanish in all sufficiently large dimensions.
(2) X is the classifying space of non-trivial Lie group.
(3) X is the loop space on the Lie group whose integral homology group has torsion element.
3. The loop space on symplectic group Sp (n) is stably indecomposable if n is greater than or equal to 2.
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