One fixed point actions on spheres

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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 $G$${{Stub}} == Introduction == <wikitex>; In connection with their work on fiberings with singularities, Montgomery and Samelson {{cite|Montgomery&Samelson1946}} made a comment that when a compact Lie 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. Contrary to this speculation, suppose that a compact Lie group $G$ acts smoothly on a sphere $S^n$ with exactly one fixed point. By removing from $S^n$ an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of $G$ on the disk $D^n$. It follows from the work of Oliver {{cite|Oliver1975}} and {{cite|Oliver1976}} that the identity connected component $G_0$ is non-abelian or the quotient group $G/G_0$ is an [[Group_actions_on_disks#Oliver_group|Oliver group]]. So, it is natural to ask whether in fact any such a group $G$ has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true. </wikitex> == Problem == <wikitex>; {{beginthm|Conjecture}} For a compact Lie group $G$, the following three statements are equivalent. * There exists a smooth action of $G$ on some sphere with exactly one fixed point. * There exists a smooth action of $G$ on some disk without fixed points. * $G_0$ is a non-abelian group or the quotient group $G/G_0$ is an Oliver group. </wikitex> == Results so far == <wikitex>; For the following groups $G$, smooth one fixed point actions on spheres have been constructed. * $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}_q)$ and $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime: constructed by Petrie {{cite|Petrie1982}}. * $G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups: constructed by Petrie {{cite|Petrie1982}}. * $G=S^3$ or $SO(3)$: constructed by Petrie {{cite|Petrie1982}}. * $G$ is an arbitrary 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 by Laitinen and Morimoto {{cite|Laitinen&Morimoto1998}}. </wikitex> == Further discussion == <wikitex>; The results obtaine so far show that a finite group $G$ has a smooth one fixed point action on a sphere if and only if $G$ is an Oliver group. However, except for the two cases solved by Petrie {{cite|Petrie1982}} where $G = S^3$ or $SO(3)$, it is not known wheter a compact Lie group $G$ with non-abelian $G_0$ admits a smooth one fixed point action of some sphere. </wikitex> == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]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. Contrary to this speculation, suppose that a compact Lie group $G$$G$ acts smoothly on a sphere $S^n$$S^n$ with exactly one fixed point. By removing from $S^n$$S^n$ an invariant open disk neighborhood around the single fixed point, one obtains a smooth fixed point free action of $G$$G$ on the disk $D^n$$D^n$. It follows from the work of Oliver [Oliver1975] and [Oliver1976] that the identity connected component $G_0$$G_0$ is non-abelian or the quotient group $G/G_0$$G/G_0$ is an Oliver group. So, it is natural to ask whether in fact any such a group $G$$G$ 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 $G$$G$, the following three statements are equivalent.

• There exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.
• There exists a smooth action of $G$$G$ on some disk without fixed points.
• $G_0$$G_0$ is a non-abelian group or the quotient group $G/G_0$$G/G_0$ is an Oliver group.

3 Results so far

For the following groups $G$$G$, smooth one fixed point actions on spheres have been constructed.

• $G=SL_2(\mathbb{F}_5)$$G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$$SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$$(120, r)=1$: constructed by Stein [Stein1977].
• $G=SL_2(\mathbb{F}_q)$$G=SL_2(\mathbb{F}_q)$ and $PSL_2(\mathbb{F}_q)$$PSL_2(\mathbb{F}_q)$, where $q\geq 5$$q\geq 5$ is a power of an odd prime: constructed by Petrie [Petrie1982].
• $G$$G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups: constructed by Petrie [Petrie1982].
• $G=S^3$$G=S^3$ or $SO(3)$$SO(3)$: constructed by Petrie [Petrie1982].
• $G$$G$ is an arbitrary finite non-solvable group: constructed by Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawalowski1995].
• $G$$G$ is an arbitrary finite Oliver group: constructed by Laitinen and Morimoto [Laitinen&Morimoto1998].

4 Further discussion

The results obtaine so far show that a finite group $G$$G$ has a smooth one fixed point action on a sphere if and only if $G$$G$ is an Oliver group. However, except for the two cases solved by Petrie [Petrie1982] where $G = S^3$$G = S^3$ or $SO(3)$$SO(3)$, it is not known wheter a compact Lie group $G$$G$ with non-abelian $G_0$$G_0$ 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