Finite groups with smooth one fixed point actions on spheres
Finite groups with smooth one fixed point actions on spheres
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球体上具有平滑单定点作用的有限群
DOI：
10.1515/form.10.4.479
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发表时间：
1998
影响因子：
--
通讯作者：
M. Morimoto
中科院分区：
文献类型：
--
作者：
E. Laitinen;M. Morimoto
自1946年以来，紧李群能否光滑作用于具有一个不动点的球面上一直是一个悬而未决的问题。本文完全解决了有限群的问题：这些群正是那些能在无不动点的圆盘上平滑作用的群，这类群由R. Oliver确定。我们的主要工具是Burnside环和Grothendieck-Witt环（在某种程度上是经典的），以及最近发展的一种允许中维奇异集的等变手术理论。
Abstract Since 1946 it has been an open question which compact Lie groups can act smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are exactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular sets developed recently.