# Group actions on spheres

## 2 Smooth actions

### 2.1 Actions with exactly one fixed point

#### 2.1.1 History

In 1946, Montgomery and Samelson made a comment [Montgomery&Samelson1946] that when a compact 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. In 1977, Stein [Stein1977] has obtained for the first time a counterexample to this speculation. 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$, he constructed a smooth action of $G$$G$ on the sphere $S^7$$S^7$ with exactly one fixed point. Then Petrie [Petrie1982] described smooth one fixed point actions on spheres in the case the acting group $G$$G$ is a finite abelian group of odd order and with three or more non-cyclic Sylow subgroups, as well as for $G=S^3$$G=S^3$ or $SO(3)$$SO(3)$. Moreover, he announced the existence of such actions for the non-solvable groups $SL_2(\mathbb{F}_q)$$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.

#### 2.1.2 Results

Theorem 2.1 [Laitinen&Morimoto&Pawalowski1995]. For any finite non-solvable group $G$$G$, there exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.

Theorem 2.2 [Laitinen&Morimoto1998]. For any finite Oliver group $G$$G$, there exists a smooth action of $G$$G$ on some sphere with exactly one fixed point.

Assume that a compact Lie group $G$$G$ acts smoothly on a sphere with exactly one fixed point $x$$x$. Then, the Slice Theorem allows us to remove from the sphere an invariant open ball neighbourhood of $x$$x$. As a result, we obtain a smooth fixed point free action of $G$$G$ on a disk, and so, $G$$G$ is an Oliver group.

Corollary 2.3. A finite group $G$$G$ has a smooth one fixed point action on some sphere if and only if $G$$G$ is an Oliver group.

Except for the two cases $G=S^3$$G=S^3$ or $SO(3)$$SO(3)$ proved by Petrie, the question whether for any compact Oliver group $G$$G$ of positive dimension, the conclusion of Theorem 2.2 remains true is open.