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One fixed point actions on low-dimensional spheres. (English) Zbl 0735.57021

The paper deals with smooth (and, more general, locally linear) group actions on low-dimensional disks and Euclidean spaces without fixed points and on spheres with exactly one fixed point.
First, using gauge-theoretic techniques of S. K. Donaldson [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)]and R. Fintushel and R. J. Stern [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)], the authors prove that if \(M\) is a closed oriented 3- manifold such that the fundamental group of \(M\) has no nontrivial representation in \(SU(2)\), and \(G\) is a finite group acting locally linearly on \(M\), then the number of isolated fixed points in \(M\) is either 0 or 2. This allows them to note that every locally linear action of a finite group \(G\) on a three-dimensional disk \(D^ 3\), as well as on Euclidean space \({\mathbb{R}}^ 3\), has a fixed point. Since locally linear actions on 3-manifolds are smoothable by the result of S. Kwasik and K. B. Lee [Math. Proc. Camb. Philos. Soc. 104, 253-260 (1988; Zbl 0672.57023)], the authors deal here with smooth actions.
Further results are obtained using the methods of group representations and surgery theory. The authors prove that every locally linear action of a finite group on \(D^ 4\) has a fixed point, and they note that with respect to fixed point free actions on \({\mathbb{R}}^ 4\) the situation is still unclear. Then, they prove that \({\mathbb{R}}^ 5\) admits a smooth fixed point free action of the alternating group \(A_ 5\) on five letters, and \(D^ 5\) admits a fixed point free \(PL\) action of \(A_ 5\times {\mathbb{Z}}_ k\) for \(k>1\).
For group actions on spheres, the authors prove three theorems. The first theorem asserts that a closed integral homology 4-sphere admits no locally linear finite group actions with exactly one fixed point. Moreover, a discrete fixed point set of such an action is either empty or consists of two points [cf. M. Furuta, Topology 28, No. 1, 35-38 (1989; Zbl 0682.57023); S. Demichelis, Enseign. Math., II. Ser. 35, No. 1/2, 107-116 (1989; Zbl 0682.57022); and M. Morimoto, Osaka J. Math. 25, 575-580 (1988)]. The second theorem asserts that a closed integral homology 5-sphere admits no locally linear finite group actions with exactly one fixed point. The third theorem shows that commencing with dimension six the situation is entirely different. In fact, it asserts that for each \(n\geq6\), there is a locally linear action of \(A_ 5\) on the standard sphere \(S^ n\) with exactly one fixed point. As the authors note, it is not known whether any of these locally linear actions are smoothable. However, the work of M. Morimoto [Proc. Japan Acad., Ser. A 63, 95-97 (1987; Zbl 0724.57026), and Lect. Notes Math. 1375, 240-258 (1989; Zbl 0693.57018)], as well as the work of A. Bak and M. Morimoto [Equivariant surgery on compact manifolds with half dimensional singular sets, Preprint, 1991] show that for each \(n\geq6\), \(S^ n\) admits a smooth action of \(A_ 5\) with exactly one fixed point.

MSC:

57S25 Groups acting on specific manifolds
57S17 Finite transformation groups
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References:

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