One fixed point actions on spheres

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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.
*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.
*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.
*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 for any finite non-solvable group $G$, there exists a smooth action of $G$ on some sphere with exactly one fixed point.
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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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*Laitinen and Morimoto {{cite|Laitinen&Morimoto1998}} constructed for any finite [[Group_actions_on_disks#Oliver_group|Oliver group]] $G$, a smooth action of $G$ on some sphere with exactly one fixed point.
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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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Revision as of 18:30, 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 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.

2 Problem

Which compact Lie groups G can act smoothly on some sphere with exactly one fixed point?

3 Results so far

  • Stein [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.
  • Petrie [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.
  • Laitinen, Morimoto, and Pawałowski [Laitinen&Morimoto&Pawalowski1995] proved that any finite non-solvable group G can act smoothly on some sphere with exactly one fixed point.
  • Laitinen and Morimoto [Laitinen&Morimoto1998] proved that any finite Oliver group G can acts smoothly on some sphere with exactly one fixed point.

4 Further discussion

...

5 References

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