2005 Fiscal Year Final Research Report Summary
Study of equivariant homology surgery theory and its applications
| Project/Area Number |
15540076
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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 | OKAYAMA UNIVERSITY |
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Principal Investigator |
MORIMOTO Masaharu OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (30166441)
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| Co-Investigator(Kenkyū-buntansha) |
SHIMAKAWA Kazuhisa OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (70109081)
NAKAJIMA Atsushi OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (30032824)
IKEHATA Shuichi OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (20116429)
SASAKI Toru OKAYAMA UNIVERSITY, Graduate School of Environmental Science and Technology, Lecturer, 大学院・環境学研究科, 講師 (20260664)
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| Project Period (FY) |
2003 – 2005
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| Keywords | surgery theory / homology / quivariant manifold / transformation groups / sphere / fixed point set / equivariant connected sum |
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| Research Abstract |
(1)Let G be a finite group, Y a homology disk with a smooth G-action, f : X → Y a G-framed map, H a subgroup of G. Suppose that Y has a G-fixed point and a point of isotropy subgroup H. Then we constructed G-connected sum f#_G G x_H D(f) : X #_G G x_H D(X) → Y.
(2)We generalized the notion of quadratic forms of Cappell-Shaneson and their group. We proved the generalized group is isomorphic to the original one. Using this new notion, we proved a sum formula of surgery obstructions for the G-connected sum above.
(3)For Z_{(p)} where p is a prime such that the order of fundamental group of Y, we proved that the equivariant homology obstruction group is isomorphic to the Wall group. So, we could develop induction-restriction theory of equivariant homology surgery obstruction groups.
(4)Using above results, we proved a deleting-inserting theorem of G-fixed point sets for gap Oliver group G.
(5)We decided manifolds appearing as the G-fixed point sets of smooth G-actions on spheres where G was a nilpotent Oliver group or a nontrivial perfect group.
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