The Number of Fixed Points of Cyclic Group Actions
The Number of Fixed Points of Cyclic Group Actions
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DOI：
10.1112/blms/16.3.295
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发表时间：
1984-05
影响因子：
0.9
通讯作者：
J. Ewing;C. Kosniowski
中科院分区：
文献类型：
--
作者：
J. Ewing;C. Kosniowski
本文应用光滑流形上群作用的一个基本“积分性定理”证明了两个密切相关的结果。第一个问题涉及Z/p的光滑作用，p是奇素数，有两个孤立的不动点。变换群中的一个基本结果是Atiyah和Bott [1，定理7.15]的结果：如果Z/p作用在具有两个孤立不动点的球面S2”上，则Z/p在切空间上的每个不动点处的表示是同构的。这也许是令人惊讶的是，同样的结果是真实的任何光滑流形，提供的维数不是太小。定理。设p是奇素数，Z/p光滑地作用在具有两个孤立不动点的光滑流形M2 n上.假设n^ p-2。则Z/p在切空间上的每个不动点的表示是同构的。
In this note we apply a basic" integrality theorem" for group actions on smooth manifolds to prove two closely related results. The first concerns smooth actions of Z/p, p an odd prime, with two isolated fixed points. A basic result in transformation groups is the result of Atiyah and Bott [1, Theorem 7.15] that if Z/p acts on a sphere S2" with two isolated fixed points then the representations of Z/p on the tangent space at each fixed point are isomorphic. It is perhaps surprising that the same result is true for any smooth manifold, provided the dimension is not too small.THEOREM. Let p be an odd prime and suppose Z/p acts smoothly on a closed, oriented, smooth manifold M2n with exactly two isolated fixed points. Suppose n^ p—2. Then the representations of Z/p on the tangent space at each fixed point are isomorphic.