Cohomology Theory of Topological Transformation Groups
Cohomology Theory of Topological Transformation Groups
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DOI：
10.1007/978-3-642-66052-8
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发表时间：
1975
影响因子：
12.1
通讯作者：
W. Hsiang
中科院分区：
文献类型：
--
作者：
W. Hsiang
历史上，应用代数拓扑学研究拓扑变换群起源于LE 1的工作。布劳威尔对周期变换，稍后，在美丽的不动点定理的P。A. Smith关于同调球面上素周期映射的一个定理。在比较史密斯的不动点定理与它的前辈，布劳威尔和莱夫谢茨的不动点定理，人们发现，这是可能的，至少在同调球的情况下，升级的结论仅仅存在（或不存在）的实际确定的同调型的不动点集，如果映射被假定为素周期。PA史密斯的先驱结果清楚地表明了在代数拓扑学框架内研究拓扑变换群的一个富有成效的总体方向。自然地，史密斯不动点定理之后的直接问题是在用更一般的拓扑类型的空间代替同调球面的方向和用更一般的紧群代替群tl的方向上推广它。
Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of LE 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of PA Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.