Study of equivariant homology surgery theory and its applications

等变同源手术理论及其应用研究

• 批准号: 15540076
• 负责人: MORIMOTO Masaharu
• 金额: $ 2.24万
• 依托单位: OKAYAMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540076/
• 关键词: surgery theory; homology; quivariant manifold; transformation groups; sphere; fixed point set; equivariant connected sum; ホモロジー同値; 多様体; 被覆空間; 群作用; Mackey函手; Burnside環

(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.

（1）令G为有限的组，y一个具有平稳g的同源性磁盘，f：x→y a g框架地图，h h一个G. g。卡佩尔·沙内森及其小组的二次形式。我们证明了广义组是原始组的同构。使用这个新的概念，我们为上述G连接的总和提供了一个总体手术对象。（3）对于z _ {（p）}，其中p是一个质量，使得y的基本组的顺序是，我们规定了等效的同源观察组对壁群是同构的。因此，我们可以开发对等于同源手术对象组的诱导限制理论。（4）使用上述结果，我们为Gap Oliver组G.（5）我们确定了Gap Oliver Group G. G-固定点集的删除插入理论。

项目成果

Deleting and inserting fixed point sets ondisks under the strong gap condition

强间隙条件下磁盘上定点集的删除和插入

• 发表时间: 2004
• 期刊: K-theory Preprint Archives 707
• 作者: Anthony Bak;Masaharu Morimoto;Masaharu Morimoto
• 通讯作者: Masaharu Morimoto

Equivariant surgery theory to obtain homology equivalences and its applications

获得同源等价的等变手术理论及其应用

• 发表时间: 2005
• 期刊: Abstracts of 52nd Topology Symposium
• 作者: Anthony Bak;Masaharu Morimoto;Masaharu Morimoto
• 通讯作者: Masaharu Morimoto

X.M.Ju, K.Matsuzaki, M.Morimoto: "Induction theory of equivariant-surgery-obstruction groups"京都大学 数理解析研究所講究録. 1343. 129-143 (2003)

X.M.Ju、K.Matsuzaki、M.Morimoto：“等变手术阻塞群的归纳理论”京都大学数学科学研究所 Kokyuroku 1343. 129-143 (2003)

An exact sequence of Grothendieck-Witt rings

Grothendieck-Witt 环的精确序列

• 发表时间: 2004
• 期刊: RIMS Kokyuroku 1393
• 作者: Anthony Bak;Masaharu Morimoto
• 通讯作者: Masaharu Morimoto

Induction theory of equivariant-surgery-obstruction groups

等变手术阻塞群的归纳理论

• 发表时间: 2003
• 期刊: 数理解析研究所講究録 1343
• 作者: XianMeng Ju;Katsuhiko Mathuzaki;Masaharu Morimoto
• 通讯作者: Masaharu Morimoto