Recognition of linear actions on spheres

Let $G$ be a finite group acting smoothly on a homotopy sphere $\Sigma^m$. We wish to establish necessary and sufficient conditions for the given $G$-action on $\Sigma$ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a $G$-homeomorphism $\gamma:\Sigma\to S^m(\rho)$, where ${S^m}(\rho ) \subset {\mathbf{R}^{m + 1}}(\rho )$ denotes the unit sphere in an orthogonal representation space $\mathbf{R}^{m + 1}(\rho )$ for $G$. In order for a $G$-action on $\Sigma$ to be topologically equivalent to a linear action it is clearly necessary that:

(i) For each subgroup $H$ of $G$ the fixed-point set $\Sigma^H$ is homeomorphic to a sphere, or empty.

(ii) For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq k$, of $G$ the pair $(\Sigma^{H},\,\cup_{i=1}^{k}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $(S^{n},\,\cup_{i=1}^{k}S_{i}^{n_{i}})$, where each $S_i^{{n_i}},\,1 \le i \le k$, is a standard $n_i$-subsphere of $S^n$.

In this paper we consider the case where the fixed-point set $\Sigma^G$ is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need:

(ii)$^{\ast}$ For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq p$, of $G$ such that ${\operatorname{dim}}\,{\Sigma ^{{H_i}}} = {\operatorname{dim}}\,{\Sigma ^H} - 2$, the pair $\Sigma^{H},\,\cup_{i=1}^{p}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $({S^n},\, \cup _{i = 1}^pS_i^{n - 2})$, where each $S_i^{n - 2},\,1 \le i \le p$, is a standard $(n-2)$-subsphere of $S^n$.

Our main results are then that, in the case when $G$ is abelian, conditions (i) and (ii)$^{\ast}$ are necessary and sufficient for a given $G$-action on $\Sigma$ to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.