2006 Fiscal Year Final Research Report Summary
Construction of exotic homology manifolds and generalization of Quinn index
| Project/Area Number |
16540064
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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 | Shizuoka University |
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Principal Investigator |
KOYAMA Akira Shizuoka University, Graduate School of Science and Technology, Professor, 創造科学技術大学院, 教授 (40116158)
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| Co-Investigator(Kenkyū-buntansha) |
SUGAHARA Kunio Osaka Kyoiku University, Faculty y of Education, Professor, 教育学部, 教授 (20093255)
UNO Katsuhiro Osaka Kyoiku University, Faculty of Education, Professor, 教育学部, 教授 (70176717)
YAGASAKI Tatsuhiko Kyoto Institute of Technology, Faculty of Engineering and Design, Associate Professor, 工芸学部, 助教授 (40191077)
HATTORI Yasunao Shimane University, The Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (20144553)
YOKOI Yatsuya Shimane University, The Interdisciplinary Faculty of Science and Engineering, Associate Professor, 総合理工学部, 助教授 (90240184)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Geometric Topology / Cohomological dimension / Embeddings / Symmetric Products |
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| Research Abstract |
By SP^n(X) we denote the n-fold symmetric product of a topological space X. We have investigated the problem what kind of n-dimensional compact metric spaces can be embedded into the n-fold symmetric product SP^n(X) of a one-dimensional continuum X. Our result is the following :
Theorem 1. The n-dimensional sphere S^n cannot be embedded into the n-fold symmetric product of any one-dimensional continuum X.
In order to prove Theorem we have calculated the n-dimensional cohomology group of the bouquet of the 1-sphere S^1. In fact, we showed that the n-fold symmetric product of the bouquet of the 1-sphere S^1 can be embedded into the Cartesian product of the n-fold symmetric products of 1-sphere S^1. Moreover the embedding image is the retract of the product space. Therefore we can calculate cohomology group of the symmetric product as follows :
Theorem 2. H^n(SP^n(vS^1)) is isomorphic to the direct sum of the n-dimensional cohomology groups of the n-dimensional tori.
Theorem 1 follows from Theorem 1 by Dydak-Koyama(Bull. Polish Academy of Sciences, 2000, 48(1), 51-56).
We have another notion of symmetric products. Namely, for a topological space X let F_n(X) be the set of all nonempty subsets of X whose cardinalities are at most n. We often call F_n(X) endowed the Hausdorff metric the n-fold symmetric product of X. In general, F_2n(X) is equal to SP^2(X), but if n > 2, F_n(X) is different from SP^n(X). However, as those product have similarities, we have the same problem what kind of n-dimensional compact metric spaces can be embedded into the n-fold symmetric product F_n(X) of a one-dimensional continuum X. As a folhlore we know Borsuk-Bott Theorem: F_3(S^1) is isomorphic to the 3-dimensional sphere S^3. We also investigate this theorem and give a modern proof and a generalizations.
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