Recognition of Linear Actions on Spheres

Abstract

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.