Smooth compact Lie group actions on disks
Smooth compact Lie group actions on disks
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磁盘上平滑紧凑的李群动作
DOI：
10.1007/bf01301634
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发表时间：
1976
影响因子：
0.8
通讯作者：
R. Oliver
中科院分区：
文献类型：
--
作者：
R. Oliver
在两篇较早的论文（[9]和[10]）中，作者研究了圆盘上有限群的光滑作用;特别是描述了任何给定的有限群的这种作用可能出现的不动点集（直到同伦型）。文[9]的主要结果是：对任意非素幂阶的有限群G，存在一个整数no，使得对任意有限CW复形F，G在圆盘上有光滑作用当且仅当：g（F）1（mod no）.文[9]和[10]对所有这类有限群计算了no的个数.本文证明了对任意紧李群G，若G/Go不是素数幂阶或Go不是环面（Go表示G的单位分支），则存在具有相同性质的数nG.很容易看出，如果Go是一个群G，则no= no/co;对于具有非交换单位分量的群G，构造了G在圆盘上的光滑不动点自由作用，证明了n G-1.更具体地说，磁盘上的行动，使任何子群的不动点集是一个磁盘或空的研究。
In two earlier papers ([9] and [10]), the author studied smooth actions of finite groups on disks; in particular describing the fixed point sets (up to homotopy type) which can occur for such actions of any given finite group. The main result in [9] was that for any finite group G not of prime power order, there exists an integer no such that for any finite CW complex F, G has a smooth action on a disk with fixed point set having the homotopy type of F if and only if; g (F) 1 (mod no).The number no was calculated for all such finite groups in [9] and [10]. In this paper, it is shown that a number n G with the same properties exists for any compact Lie group G such that G/Go is not of prime power order or G o is not a torus (G o denotes the identity component of G). It is easily seen that no= no/co if Go is a toms; for groups G with non-abelian identity component smooth fixedpoint free actions of G on disks are constructed to show that n G--1. More specifically actions on disks such that the fixed point set of any subgroup is either a disk or empty are studied.