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\(S^ 4\) does not have one fixed point actions. (English) Zbl 0736.57018
It is shown that no compact Lie group can act smoothly on \(S^ 4\) with exactly one fixed point. This extends results of M. Furuta [Topology 28, No. 1, 35-38 (1989; Zbl 0682.57023)] and W.-Y. Hsiang and E. Straume [J. Reine Angew. Math. 369, 21-39 (1986; Zbl 0583.57025)]to the non-connected case. Looking at the slice representation at a fixed point, the groups in question are reduced to subgroups of \(O(4)\). The theorem is then derived using P. A. Smith theory for subgroups of \(O(4)\), a result of R. Oliver [Comment. Math. Helv. 50, 155-177 (1975; Zbl 0304.57020)] and Furuta’s result.

57S25 Groups acting on specific manifolds
57S15 Compact Lie groups of differentiable transformations