ONE FIXED POINT ACTIONS AND HOMOLOGY 3-SPHERES
ONE FIXED POINT ACTIONS AND HOMOLOGY 3-SPHERES
复制标题
1 个定点作用和同调 3 球体
DOI：
10.2307/2375090
复制
发表时间：
1995
影响因子：
1.7
通讯作者：
R. Schultz
中科院分区：
文献类型：
--
作者：
S. Kwasik;R. Schultz
最著名的三维流形之一是庞加莱同调三维球面1（2，3，5）;这个闭三维流形与标准球面S3具有相同的同调群，但不是单连通的（参见[Pnc，p. 106]或[Br 2，Section 1.8]）。在过去的世纪中，大量的研究表明，这种流形具有许多显著的性质，[KiS]中总结了1（2，3，5）的几个等价描述。从变换群的观点来看，一个值得注意的性质是1（2，3，5）是唯一允许紧李群传递作用的非单连通同调球面[Brl]。流形1（2，3，5）在类球面流形上的群作用的正则性问题中也有重要的意义。球面上有限群的线性（或正交）作用通常被视为此类作用的最简单和最规则的例子，而变换群的主题之一是确定球面流形上的任意作用与线性作用相似的程度。线性作用的一个显著特征是它们的不动点集（如果非空）也是球面。P.A.史密斯（cf。[Br 2]）表明，在这方面，同调球面上的连续p-群作用都类似于线性作用;即，不动点集总是Zp-同调（上同调）球面。另一方面，在p-群族之外，有许多奇异的群作用。也许这类最基本的例子涉及1（2，3，5）。若A是旋转群SO 3的子群，由正二十面体的对称性给出（使得A同构于交错群A5），则商空间SO 3/A是Poincare同调3-球面，A5的诱导作用是？A只有一个不动点。这个例子表明，有限群在同调球面上的一般作用可能与线性作用有很大的不同，在过去的三十年里，关于奇异有限群作用的大部分工作至少部分是由这种群作用的存在所激发的。
One of the best known three-dimensional manifolds is the Poincare homology three-sphere 1(2, 3,5); this closed 3-manifold has the same homology groups as the standard sphere S3 but is not simply connected (see [Pnc, p. 106] or [Br2, Section 1.8]). Numerous investigations during the past century have shown that this manifold has many remarkable properties; several equivalent descriptions of 1(2,3,5) are summarized in [KiS]. From the viewpoint of transformation groups, one noteworthy property is that 1(2, 3, 5) is the only nonsimply connected homology sphere admitting a transitive action of a compact Lie group [Brl]. The manifold 1(2, 3,5) also figures importantly in regularity questions for group actions on spherelike manifolds. Linear (or orthogonal) actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions, and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions. One distinguishing feature of linear actions is that their fixed point sets (if nonempty) are also spheres. The well-known results of P. A. Smith (cf. [Br2]) show that continuous p-group actions on homology spheres all resemble linear actions in this respect; i.e., the fixed point sets are always Zp-homology (cohomology) spheres. On the other hand, outside the family of p-groups, there are many exotic group actions. Perhaps the most elementary example of this type involves 1(2, 3,5). If A is the subgroup of the rotation group SO3 given by the symmetries of a regular icosahedron (so that A is isomorphic to the alternating group A5), then the quotient space S03/A is the Poincare homology 3-sphere and the induced action of A5 ? A has exactly one fixed point. This example suggests that general actions of finite groups on homology spheres can be quite different from linear actions, and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action.