Group actions on spheres
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| − | == | + | == History == |
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| − | In 1946, Montgomery and Samelson \cite{Montgomery&Samelson1946} | + | In 1946, Montgomery and Samelson made a comment \cite{Montgomery&Samelson1946} that when a compact group $G$ 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 \cite{Stein1977} has obtained for the first time a counterexample to this speculation. For $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$, he constructed a smooth action of $G$ on the sphere $S^7$. Then Petrie \cite{Petrie1982} described smooth one fixed point actions on spheres in the case the acting group $G$ is a finite abelian group of odd order and with or more non-cyclic Sylow subgroups, as well as for $G=S^3$ or $SO(3)$. Moreover, he announced the existence of such actions for groups the non-solvable groups $SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$ where $q\geq 5$ is a power of an odd prime. |
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| − | == | + | == == |
| − | + | {{beginthm|Theorem|\cite{Laitinen&Morimoto&Pawalowski1995}}} | |
| + | For any finite non-solvable group $G$ there exists a smooth action of $G$ on some sphere with exactly one fixed point. | ||
| + | {{endthm}} | ||
| + | |||
| + | {{beginthm|Theorem|\cite{Laitinen&Morimoto1998}\label{thm:one_fixed_point}}} | ||
| + | For any finite Oliver group $G$ there exists a smooth action of $G$ on some sphere with exactly one fixed point. | ||
| + | {{endthm}} | ||
| + | |||
| + | {{beginthm|Corollary|\label{cor:one_fixed_point}}} | ||
| + | A finite group $G$ has a smooth one fixed point action on a sphere if and only if $G$ is an Oliver group. | ||
| + | {{endthm}} | ||
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| + | The question whether for any compact Oliver group of positive dimension, the conclusion of Theorem \ref{thm:one_fixed_point} holds remains open. | ||
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== References == | == References == | ||
Revision as of 13:52, 26 November 2010
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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 
. 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 or more non-cyclic Sylow subgroups, as well as for 
or 
. Moreover, he announced the existence of such actions for groups the non-solvable groups 
and 
where 
is a power of an odd prime.
2
Theorem 2.1 [Laitinen&Morimoto&Pawalowski1995]. For any finite non-solvable group $G$ there exists a smooth action of $G$ on some sphere with exactly one fixed point.
Theorem 2.2 [Laitinen&Morimoto1998]. For any finite Oliver group $G$ there exists a smooth action of $G$ on some sphere with exactly one fixed point.
Corollary 2.3 . A finite group $G$ has a smooth one fixed point action on a sphere if and only if $G$ is an Oliver group.
The question whether for any compact Oliver group of positive dimension, the conclusion of Theorem 2.2 holds remains 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
