ON THE ACTION OF A FINITE GROUP ON Sn X S'n
ON THE ACTION OF A FINITE GROUP ON Sn X S'n
复制标题
论Sn X Sn上有限群的作用
DOI：
10.2307/1969910
复制
发表时间：
1957
影响因子：
4.9
通讯作者：
P. E. Conner
中科院分区：
文献类型：
--
作者：
P. E. Conner
众所周知，如果有限群在球面上自由作用，则每个阿贝尔子群都是循环的（[1，p.352]，[4]）。我们希望在这个简短的说明中，将这一点推广到有限群自由作用于球面与自身的乘积的情况。自由行动这个术语的意思是，群中除了单位元之外，没有任何元素具有固定点。此外，我们将验证福克斯（[2]）的猜想有关纽结群的考虑并非无关的那些用于推导定理1。如果G是一个在Sn × Sn上自由作用的有限群，则G的交换子群的秩都不大于2。这个定理显然是一种推广，但我们目前还没有通用的方法。假设G自由作用在Sn X Sn上，则存在一个谱序列，其E2-项为
It is known that if a finite group acts freely on a sphere, then every abelian subgroup is cyclic ([1, p. 352], [4]). We wish in this brief note to give a generalization of this to the case of a finite group acting freely on the product of a sphere with itself. By the term act freely we mean that no element of the group other than the identity has a fixed point. Also we shall verify a conjecture of Fox ([2]) about knot groups by considerations not unrelated to those used in deriving THEOREM 1. If G is a finite group acting freely on Sn X Sn, then no abelian subgroup of G has rank greater than two. This theorem clearly suggests generalizations, but we have no general approach at this time. Suppose that G acts freely on Sn X Sn, then there is a spectral sequence whose E2-term is