# One fixed point actions on spheres

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

## 1 Introduction


## 2 Problem

Let $G$$G$ be a compact Lie group such that the identity connected component $G_0$$G_0$ is non-abelian or the quotient group $G/G_0$$G/G_0$ is an Oliver group. It is natural to ask whether $G$$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$$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

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

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

## 4 Further discussion

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

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