Group actions on spheres
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− | Assume that a compact Lie group $G$ acts smoothly on a sphere with exactly one fixed point $x$. Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of $x$. As a result, we obtain a smooth fixed point free action of $G$ on a disk | + | Assume that a compact Lie group $G$ acts smoothly on a sphere with exactly one fixed point $x$. Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of $x$. As a result, we obtain a smooth fixed point free action of $G$ on a disk, and so, $G$ is an Oliver group. |
{{beginthm|Corollary|\label{cor:one_fixed_point}}} | {{beginthm|Corollary|\label{cor:one_fixed_point}}} |
Revision as of 15:30, 26 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
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 Solution for finite groups
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.
Following Laitinen and Morimoto [Laitinen&Morimoto1998], we say that a finite is an Oliver group if has a smooth fixed point free action on disk, i.e., satisfies the algebraic condition.
Theorem 2.2 [Laitinen&Morimoto1998]. For any finite Oliver 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.
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