Construction of one-fixed-point actions on spheres of nonsolvable groups II

Abstract

Let $G$ be a finite group. If $n \leq 5$ then any $n$-dimensional homotopy sphere never admits a smooth action of $G$ with exactly one fixed point. Let $A_{n}$ and $S_{n}$ denote the alternating group and the symmetric group on some $n$ letters. If $n \geq 6$ then the $n$-dimensional sphere possesses a smooth action of $A_{5}$ with exactly one fixed point. Let $V$ be an $n$-dimensional real $G$-representation with exactly one fixed point. It is interesting to ask whether there exists a smooth $G$-action with exactly one fixed point on the $n$-dimensional sphere such that the associated tangential $G$-representation is isomorphic to $V$. In this paper, we study this problem for nonsolvable groups $G$ and real $G$-representations $V$ satisfying certain hypotheses. Applying a theory developed in this paper, we can prove that the $n$-dimensional sphere has an effective smooth action of $S_{5}$ with exactly one fixed point if and only if $n = 6$, 10, 11, 12, or $n \geq 14$ and that the $n$-dimensional sphere has an effective smooth action of $A_{5} \times Z$ with exactly one fixed point if $n$ satisfies $n \geq 6$ and $n \neq 9$, where $Z$ is a group of order 2.