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)设G是有限群，Y是具有光滑G-作用的同调圆盘，f：X→Y是G-框架映射，H是G的子群，假设Y有一个G-不动点和一个迷向子群H的点，则我们构造了G-连通和f#_G X_H D(F)：X#_G G x_H D(X)→Y。证明了广义群与原广义群同构。利用这一新概念，我们证明了上述G-连通和的手术障碍的求和公式。(3)对于Z_{(P)}，其中p是素数，使得Y的基本群阶，我们证明了等变同调障碍群与Wall群同构。(4)利用上述结果，我们证明了Gap Oliver群G的G-不动点集的删除-插入定理。(5)当G是幂零Oliver群或非平凡完全群时，我们判定流形表现为球面上光滑G-作用的G-不动点集。

项目成果

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

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

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

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