2004 Fiscal Year Final Research Report Summary
Research on families of fixed point sets of G-manifolds in transformation group theory
| Project/Area Number |
15540079
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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, Faculty of Science, Professor, 理学部, 教授 (00034744)
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| Co-Investigator(Kenkyū-buntansha) |
ANDO Yoshifumi Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (80001840)
MIYAZAWA Yasuyuki Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (60263761)
SHIMA Hirohiko Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (70028182)
NAITOH Hiroo Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10127772)
NAKAUCHI Nobumitu Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50180237)
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| Project Period (FY) |
2003 – 2004
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| Keywords | cut-and-paste / SK-equivalence / SK-group / G-manifolds / fixed points / Euler characteristics / fibring over the circle / cobordism |
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| Research Abstract |
1.The cut-and-paste operation defines an equivalence relation on the set of smooth G-manifolds. This relation is called SK-equivalence. The set of SK-equivalence classes becomes a semigroup with the addition induced from the disjoint union of G-manifolds. Its Grothendieck group is called the SK-group of G-manifolds. We obtain a necessary and sufficient condition for the divisibility in the SK-group, if G is the cyclic group of order 2, or a finite abelian group of odd order. The condition is described in terms of the Euler characteristics of fixed point sets of G-manifolds.
2.A necessary and sufficient condition for a closed smooth manifold to be cobordant to the total space of fiber bundle over the circle is well-known In our research we extend this (non-equivariant) result to the equivariant case by making use of the result stated above..
3.Two linear representations of a group G are called Smith equivalent., if those two representations can occur as the tangential representations at fixed points of a homotopy G-sphere with exactly two fixed points. There are vast literatures on the question of which groups do and which groups do not have non-isomorphic Smith equivalent representations. Some of them give an affirmative answer and some of them give a negative answer. In our research we show that the question is affirmative if we restrict our attention to the restricted actions of a subgroup of index 2..
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