| Project/Area Number |
17540082
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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 | Yamaguchi University |
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Principal Investigator |
KOMIYA Katsuhiro Yamaguchi University, Graduate School of Sci.and Eng., Proffesor, 大学院理工学研究科, 教授 (00034744)
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| Co-Investigator(Kenkyū-buntansha) |
ANDO Yoshifumi Yamaguchi University, Graduate School of Sci.and Eng., Proffesor, 大学院理工学研究科, 教授 (80001840)
MIYAZAWA Yasuyuki Yamaguchi University, Graduate School of Sci.and Eng., Associate Proffesor, 大学院理工学研究科, 助教授 (60263761)
NAITO Hiroo Yamaguchi University, Graduate School of Sci.and Eng., Proffesor, 大学院理工学研究科, 教授 (10127772)
NAKAUCHI Nobumitsu Yamaguchi University, Graduate School of Sci.and Eng., Associate Proffesor, 大学院理工学研究科, 助教授 (50180237)
HATAYA Yashusi Yamaguchi University, Graduate School of Sci.and Eng., Assistant, 大学院理工学研究科, 助手 (20294621)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2006: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | transformation group theory / G-manifold / fixed points submanifold / characteristic number / Euler characteristic / normal representation / オイラー類 / Borsuk-Ulamの定理 / cut and paste / SK群 |
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| Research Abstract |
1. The cut-and-paste operation defines an equivalence relation on the set of smooth G-manifolds, G a compact Lie group. This relation is called SK-equivalence. The set of SK-equivalence classes becomes a semiring with the addition induced from the disjoint union and the product induced from the Cartesian product of two of G-manifolds. Its Grothendieck ring is called the SK-ring of G-manifolds. We obtain a necessary and sufficient condition for the decomposability in the SK-ring, if G is the cyclic group of order 2. Here the decomposability means for a given G-manifold to be Sk-equivalent to the product of two G-manifolds with a lower dimension than M. The condition is obtained in terms of the Euler characteristics of G-manifolds and its fixed points submanifolds.
2. For a closed subgroup H of G, M^H denotes the fixed points submanifold of M by the restricted H-action. We already know various types of arithmetic congruences for the Euler characteristics of G-manifold and its fixed points submanoifolds. Making use of these results we obtain the following facts: if G is abelian, M is of odd-dimension, and M^G is of 0-dimension, i.e., M^G consists of finite isolated fixed points, then the number of points of M^G is even. Moreover, if the index 2 subgroup H of G is unique, then the tangential representations at fixed points are pairwisely isomorphic to each other as representations of H.
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