Group actions on spheres
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 1 Topological actions
 2 Smooth actions
 2.1 Actions with exactly one fixed point
 2.1.1 History
In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact group acts smoothly on a sphere in such a way as to have one fixed point, it is likely that there must be a second fixed point. In 1977, Stein [Stein1977] has obtained for the first time a counterexample to this speculation. For or with , he constructed a smooth action of on the sphere with exactly one fixed point. Then Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for or . Moreover, he announced the existence of such actions for the non-solvable groups and , where is a power of an odd prime.
 2.1.2 Results
Theorem 2.1 [Laitinen&Morimoto&Pawalowski1995]. For any finite non-solvable group , there exists a smooth action of on some sphere with exactly one fixed point.
Assume that a compact Lie group acts smoothly on a sphere with exactly one fixed point . Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of . As a result, we obtain a smooth fixed point free action of on a disk, and so, is an Oliver group.
Corollary 2.3. A finite group has a smooth one fixed point action on some sphere if and only if is an Oliver group.
Except for the two cases or proved by Petrie, the question whether for any compact Oliver group of positive dimension, the conclusion of Theorem 2.2 remains true is open.
 2.2 Fixed point sets
 3 References
- [Laitinen&Morimoto&Pawalowski1995] E. Laitinen, M. Morimoto and K. Pawałowski, Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres, Comment. Math. Helv. 70 (1995), no.1, 10–38. MR1314939 (96b:57043) Zbl 0843.57034
- [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
- [Montgomery&Samelson1946] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. MR0015794 (7,471a) Zbl 0060.41501
- [Petrie1982] T. Petrie, One fixed point actions on spheres. I, II, Adv. in Math. 46 (1982), no.1, 3–14, 15–70. MR676986 (84b:57027) Zbl 0502.57021
- [Stein1977] E. Stein, Surgery on products with finite fundamental group, Topology 16 (1977), no.4, 473–493. MR0474336 (57 #13982) Zbl 0383.57014