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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.

MSC:

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

Citations:

Zbl 0304.57020
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