One fixed point actions on spheres
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* $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$: constructed by Stein {{cite|Stein1977}}. | * $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$: constructed by Stein {{cite|Stein1977}}. | ||
− | * $G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups | + | * $G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups: constructed by Petrie {{cite|Petrie1982}}. |
− | * $G=SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime | + | * $G=SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime: announced by Petrie {{cite|Petrie1982}}. |
* $G=S^3$ or $SO(3)$: constructed by Petrie {{cite|Petrie1982}}. | * $G=S^3$ or $SO(3)$: constructed by Petrie {{cite|Petrie1982}}. | ||
* $G$ is a finite non-solvable group: constructed by Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto&Pawalowski1995}}. | * $G$ is a finite non-solvable group: constructed by Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto&Pawalowski1995}}. | ||
− | * $G$ is an arbitrary finite [[Group_actions_on_disks#Oliver_group|Oliver group]]: constructed | + | * $G$ is an arbitrary finite [[Group_actions_on_disks#Oliver_group|Oliver group]]: constructed by Laitinen and Morimoto {{cite|Laitinen&Morimoto1998}}. |
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Revision as of 21:39, 3 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
In connection with their work on fiberings with singularities, Montgomery and Samelson [Montgomery&Samelson1946] made a comment that when a compact Lie 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. Contrary to this speculation, suppose that a compact Lie group acts smoothly on a sphere with exactly one fixed point. By removing from an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of on the disk . It follows from the work of Oliver [Oliver1975] and [Oliver1976] that the identity connected component is non-abelian or the quotient group is an Oliver group. So, it is natural to ask whether in fact any such a group has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true.
2 Problem
Conjecture 2.1. For a compact Lie group , the following three statements are equivalent.
- There exists a smooth action of on some sphere with exactly one fixed point.
- There exists a smooth action of on some disk without fixed points.
- is a non-abelian group or the quotient group is an Oliver group.
3 Results so far
For the following groups , smooth one fixed point actions on spheres have been constructed or their existence announced:
- or with : constructed by Stein [Stein1977].
- is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups: constructed by Petrie [Petrie1982].
- and , where is a power of an odd prime: announced by Petrie [Petrie1982].
- or : constructed by Petrie [Petrie1982].
- is a finite non-solvable group: constructed by Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawalowski1995].
- is an arbitrary finite Oliver group: constructed by Laitinen and Morimoto [Laitinen&Morimoto1998].
4 Further discussion
Except for the two cases solved by Petrie [Petrie1982] where or , it is not known wheter any compact Lie group with non-abelian admits a smooth one fixed point action of some sphere.
5 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
- [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
- [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
- [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