# Group actions on spheres

## 2 Smooth actions

### 2.1 Actions with exactly one fixed point

#### 2.1.1 History


#### 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.