Finite groups with smooth one fixed point actions on spheres. (English) Zbl 0905.57023

Which compact Lie groups can act smoothly on a sphere with exactly one fixed point? This question had been posed by D. Montgomery and H. Samelson in 1946. For finite groups, it is finally solved in this paper: A finite group admits a smooth one fixed point action on a sphere if and only if it can act smoothly on a disk without fixed points. The latter class of groups had been determined by R. Oliver [Comment. Math. Helv. 50, 155-177 (1975; Zbl 0304.57020)]. The proof uses a modification of equivariant surgery techniques as developed by T. Petrie and his school. Equivariant surgery is applied to a \(G\)-normal map into the sphere of a \(G\)-module \({\mathbb R}\oplus V\) with a \(G\)-representation \(V\) constructed using regular representations of \(G\) and of certain of its quotient groups. A careful study of the gap properties of this module based on Burnside ring properties shows that there may at worst occur middle-dimensional singular sets. This fact makes it possible and necessary to apply a new version of \(G\)-surgery developed by A. Bak and M. Morimoto. The authors show that all occuring surgery obstructions can be killed using carefully chosen and constructed equivariant connected sums of \(G\)-normal maps. New induction theorems are used as important tools.


57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
57R67 Surgery obstructions, Wall groups
57R85 Equivariant cobordism
19A22 Frobenius induction, Burnside and representation rings
57R65 Surgery and handlebodies


Zbl 0304.57020
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