# One fixed point actions on spheres

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}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 there exists such an action of $G$$G$ if and only if the identity connected component $G_0$$G_0$ of $G$$G$ is non-abelian or the quotient group $G/G_0$$G/G_0$ is an Oliver group.

## 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.