One fixed point actions on spheres

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
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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.
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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 there exists such
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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 $D^n$. Therefore, by the work of Oliver {{cite|Oliver1975}} and {{cite|Oliver1976}}, 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 the following conjecture.
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an action of $G$ if and only if the identity connected component $G_0$ of $G$ is non-abelian or the quotient group $G/G_0$ is an [[Group_actions_on_disks#Oliver_group|Oliver group]].
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== Problem ==
== Problem ==
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{{beginthm|Conjecture}} Let $G$ be a compact Lie group such that the identity connected component is non-abelin or the quotient group $G/G_0$ is an Oliver group. Then there exists a smooth action of $G$ on some sphere with exactly one fixed point.
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Let $G$ be a compact Lie group such that the identity connected component $G_0$ is non-abelian or the quotient group $G/G_0$ is an Oliver group. It is natural to ask whether $G$ has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true.
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{{beginthm|Conjecture}} For a compact Lie group $G$, the following three statements are equivalent.
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* There exists a smooth action of $G$ on some sphere with exactly one fixed point.
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* There exists a smooth action of $G$ on some disk without fixed points.
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* $G_0$ is a non-abelian group or the quotient group $G/G_0$ is an Oliver group.
</wikitex>
</wikitex>
== Results so far ==
== Results so far ==
<wikitex>;
<wikitex>;
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*Stein {{cite|Stein1977}} has obtained for the first time smooth one fixed point actions on spheres. 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 $S^7$ with exactly one fixed point.
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Smooth one fixed point actions of $G$ on spheres have been constructed by
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*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 three 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 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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*Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto&Pawalowski1995}} proved that any finite non-solvable group $G$ can act smoothly on some sphere with exactly one fixed point.
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* Stein {{cite|Stein1977}} for $G=SL_2(\mathbb{F}_5)$ or $SL_2(\mathbb{F}_5)\times \mathbb{Z}_r$ with $(120, r)=1$.
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*Laitinen and Morimoto {{cite|Laitinen&Morimoto1998}} proved that any finite [[Group_actions_on_disks#Oliver_group|Oliver group]] $G$ can acts smoothly on some sphere with exactly one fixed point.
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* Petrie {{cite|Petrie1982}} for $G=SL_2(\mathbb{F}_q)$ or $PSL_2(\mathbb{F}_q)$, where $q\geq 5$ is a power of an odd prime.
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* Petrie {{cite|Petrie1982}} for any finite abelian group $G$ of odd order and with three or more non-cyclic Sylow subgroups.
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* Petrie {{cite|Petrie1982}} for $G=S^3$ or $SO(3)$.
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* Laitinen, Morimoto, and Pawałowski {{cite|Laitinen&Morimoto&Pawałowski1995}} for any finite non-solvable group $G$.
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* Laitinen and Morimoto {{cite|Laitinen&Morimoto1998}} for any finite [[Group_actions_on_disks#Oliver_group|Oliver group]] $G$.
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== Further discussion ==
== Further discussion ==
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Except for the two cases solved by Petrie {{cite|Petrie1982}} where $G = SO(3)$ or $S^3$, it is not known wheter any compact Lie group $G$ with non-abelian $G_0$ admits a smooth one fixed point action of some sphere.
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The results obtained so far show that Conjecture 2.1 is true for finite groups $G$.
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{{beginthm|Theorem}} For a finite group $G$, the following three statements are equivalent.
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* There exists a smooth action of $G$ on some sphere with exactly one fixed point.
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* There exists a smooth action of $G$ on some disk without fixed points.
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* $G$ is an Oliver group.
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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 identity connected component $G_0$ admits a smooth one fixed point action of some sphere.
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[[Category:Problems]]
[[Category:Problems]]
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[[Category:Group actions on manifolds]]

Latest revision as of 17:32, 13 December 2010

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 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 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 [Oliver1975] and [Oliver1976] that there exists such an action of G if and only if the identity connected component G_0 of G is non-abelian or the quotient group G/G_0 is an Oliver group.

[edit] 2 Problem

Let G be a compact Lie group such that the identity connected component G_0 is non-abelian or the quotient group G/G_0 is an Oliver group. It is natural to ask whether G has a smooth one fixed point action on some sphere. The affirmative answer would confirm that the following conjecture is true.

Conjecture 2.1. 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.


[edit] 3 Results so far

Smooth one fixed point actions of G on spheres have been constructed by

  • Stein [Stein1977] for G=SL_2(\mathbb{F}_5) or SL_2(\mathbb{F}_5)\times \mathbb{Z}_r with (120, r)=1.
  • Petrie [Petrie1982] for G=SL_2(\mathbb{F}_q) or PSL_2(\mathbb{F}_q), where q\geq 5 is a power of an odd prime.
  • Petrie [Petrie1982] for any finite abelian group G of odd order and with three or more non-cyclic Sylow subgroups.
  • Petrie [Petrie1982] for G=S^3 or SO(3).
  • Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawałowski1995] for any finite non-solvable group G.
  • Laitinen and Morimoto [Laitinen&Morimoto1998] for any finite Oliver group G.

[edit] 4 Further discussion

The results obtained so far show that Conjecture 2.1 is true for finite groups G.

Theorem 4.1. For a finite 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 is an Oliver group.
Except for the two cases solved by Petrie [Petrie1982] where G = S^3 or SO(3), it is not known wheter a compact Lie group G with non-abelian identity connected component G_0 admits a smooth one fixed point action of some sphere.


[edit] 5 References

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