Equivariant vector fields on spheres
Equivariant vector fields on spheres
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球体上的等变向量场
DOI：
10.1090/s0002-9947-1983-0701504-9
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发表时间：
1983
影响因子：
1.3
通讯作者：
Unni Namboodiri
中科院分区：
文献类型：
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作者：
Unni Namboodiri
我们解决了以下问题：如果G是紧李群，S（M）是R[G]-模M的单位球面，那么在S（M）上可以找到多少个G-不变的正交向量场？我们将这个数称为M的G域数。在对M的合理假设下，我们将这个问题归结为确定两个G-向量丛的差在真实的射影空间的KOG-理论的某个子商中何时为零。在这个一般的设置中，我们解决了2-群，奇数阶群和阿贝尔群的问题。如果M也有“足够的”轨道类型（例如，所有的），那么我们解决了任意有限群的问题。我们还证明了在关于M的温和假设下，G-域的数目只依赖于M的不动点集的维数。致谢。我想借此机会向彼得·梅表示衷心的感谢，感谢他不断的指导和鼓励，以及他对本文的仔细阅读;向吉姆·麦克卢尔表示衷心的感谢，感谢他提出了许多有益的想法和富有启发性的谈话;向梅尔·罗森伯格表示衷心的感谢，感谢他作为一个单独的听众提出了所有的建议和改进。导论.在1940年，霍普夫[14]和后来的埃克曼[9]观察到赫维茨和拉东的一个定理可以用来构造球面上的正交向量场。证明了若n =（2a + 1）2c+4d，其中0，c为3，则可在Sn上构造p（n）-1个正交向量场，其中p（n）= 8d + 2C是第n个Hurwitz-Radon数. 20年后，亚当斯[1]证明了这是最好的可能性; Sn-上不存在p（n）正交向量场。本文的目的是解决以下问题。设G是紧李群，S（M）是真实的G-模M的单位球面，那么在S（M）上能找到多少个G-不变正交向量场（或G-场）？我们称这个数为M的G-域数。以前的工作已经完成了这个问题的贝克尔[6]。他处理的情况下，G是有限的，并自由行动的S（M）。他证明，在温和的假设下，G-域数的M只取决于G和真实的维数的M。他还证明了它与M的E-域数一致，其中2是G的任何2-Sylow子群。编辑于1982年3月22日收到。1980年数学学科分类初级57 R25、57 S25;次级55 R25、55 R50。“提交人于1981年12月23日在一场车祸中不幸丧生。编辑们感谢彼得·梅和吉姆·贝克尔的帮助，使这篇论文得以出版。重印请求应向芝加哥大学的Peter May提出。? 1983年美国数学学会0002-9947/82/00001355/$10.25
We address the following question: If G is a compact Lie group and S(M) is the unit sphere of an R[G]-module M, then how many orthonormal G-invariant vector fields can be found on S(M)? We call this number the G-field number of M. Under reasonable hypotheses on M, we reduce this question to determining when the difference of two G-vector bundles vanishes in a certain subquotient of the KOG-theory of a real projective space. In this general setting, we solve the problem for 2-groups, for odd-order groups, and for abelian groups. If M also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on M, the G-field number depends only on the dimensions of the fixed point sets of M. ACKNOWLEDGEMENTS. I wish to take this opportunity to express my heartfelt thanks to Peter May for his constant guidance and encouragement, and for his careful reading of this paper; to Jim McClure for many helpful ideas and enlightening conversations and to Mel Rothenberg for being a one-man audience and for all his suggestions and improvements. Introduction. In 1940 Hopf [14], and later Eckmann [9], observed that a theorem of Hurwitz and Radon could be used to construct orthonormal vector fields on spheres. They showed that if n = (2a + 1)2c+4d, where 0 , c s 3, then p(n)-1 orthonormal vector fields can be constructed on Sn, where p(n) = 8d + 2C is the nth Hurwitz-Radon number. Twenty years later, Adams [1] showed that this was best possible; there do not exist p(n) orthonormal vector fields on Sn-. The purpose of this paper is to address the following question. If G is a compact Lie group, and S(M) is the unit sphere of a real G-module M, then how many orthonormal G-invariant vector fields (or G-fields) can be found on S(M)? We shall call this number the G-field number of M. Previous work has been done on this problem by Becker [6]. He deals with the case in which G is finite and acts freely on S(M). He shows that, under mild hypotheses, the G-field number of M depends only on G and the real dimension of M. He also shows that it coincides with the E-field number of M where 2 is any 2-Sylow subgroup of G. Received by the editors March 22, 1982. 1980 Mathematics Subject Classification. Primary 57R25, 57S25; Secondary 55R25, 55R50. 'The author was tragically killed in an automobile accident on December 23, 1981. The editors thank Peter May and Jim Becker for their help in bringing this paper to publication. Requests for reprints should be addressed to Peter May, University of Chicago. ?1983 American Mathematical Society 0002-9947/82/00001355/$ 10.25