Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks
Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks
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等变手术理论：球和圆盘上不动点流形的删插定理
DOI：
10.1023/a:1007710504681
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
M. Morimoto
中科院分区：
文献类型：
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作者：
M. Morimoto
该论文给出了一种删除和插入定点流形的工具，以实现有限奥利弗群在球体和圆盘上的平滑作用。对于球面上的有限不可解群，E. Laitinen 和 K. Pawałowski 的联合文章中已经给出了类似的结果。从定点数据的角度来看，它对于对球体上的平滑动作进行分类非常有用。本文采用的方法是与 Burnside 环中的元素相关的等变手术和等变连通和。消除手术障碍的思想如下：设G是非素数次幂阶的有限群，C是可收缩的有限G-CW复形，σ是作为几何对象f的障碍类而产生的K理论群中的元素。通常认为，对于大整数 m，(1−[C]) mσ 变得微不足道，其中 [C] 是 Burnside 环 (G) 中由 C 表示的元素。人们期望代数对象 (1−[C]) mσ 可实现为与 (1−[C]) m 相关的 f 的 G 连通和的阻碍类。由于这里的情况确实如此，我们可以通过取 f 的 G 连通和来消除障碍 σ。
The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawałowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finite G-CW complex, and σ an element in a K-theoretic group arising as an obstruction class of geometric object f. It often holds that (1−[C]) mσ becomes trivial for large integers m, where [C] is the element represented by C in the Burnside ring (G). One expects that the algebraic object (1−[C]) mσ is realizable as the obstruction class of G-connected sum of f’s related to (1−[C]) m. Since it is true for the case here, we can kill the obstruction σ by taking G-connected sum of f’s.