Recognition of linear actions on spheres

Abstract

Let G G be a finite group acting smoothly on a homotopy sphere Σ m \Sigma ^m . We wish to establish necessary and sufficient conditions for the given G G -action on Σ \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 G -homeomorphism γ : Σ → S m ( ρ ) \gamma :\Sigma \to S^m(\rho ) , where S m ( ρ ) ⊂ R m + 1 ( ρ ) {S^m}(\rho ) \subset {\mathbf {R}^{m + 1}}(\rho ) denotes the unit sphere in an orthogonal representation space R m + 1 ( ρ ) \mathbf {R}^{m + 1}(\rho ) for G G . In order for a G G -action on Σ \Sigma to be topologically equivalent to a linear action it is clearly necessary that: (i) For each subgroup H H of G G the fixed-point set Σ H \Sigma ^H is homeomorphic to a sphere, or empty. (ii) For any subgroups H H and H ⊊ H i , 1 ≤ i ≤ k H \subsetneq {H_i},\,1 \leq i \leq k , of G G the pair ( Σ H , ∪ i = 1 k Σ H i ) (\Sigma ^{H},\,\cup _{i=1}^{k}\Sigma ^{H_{i}}) is homeomorphic to a standard pair ( S n , ∪ i = 1 k S i n i ) (S^{n},\,\cup _{i=1}^{k}S_{i}^{n_{i}}) , where each S i n i , 1 ≤ i ≤ k S_i^{{n_i}},\,1 \le i \le k , is a standard n i n_i -subsphere of S n S^n . In this paper we consider the case where the fixed-point set Σ G \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 H and H ⊊ H i , 1 ≤ i ≤ p H \subsetneq {H_i},\,1 \leq i \leq p , of G G such that dim Σ H i = dim Σ H − 2 {\operatorname {dim}}\,{\Sigma ^{{H_i}}} = {\operatorname {dim}}\,{\Sigma ^H} - 2 , the pair Σ H , ∪ i = 1 p Σ H i ) \Sigma ^{H},\,\cup _{i=1}^{p}\Sigma ^{H_{i}}) is homeomorphic to a standard pair ( S n , ∪ i = 1 p S i n − 2 ) ({S^n},\, \cup _{i = 1}^pS_i^{n - 2}) , where each S i n − 2 , 1 ≤ i ≤ p S_i^{n - 2},\,1 \le i \le p , is a standard ( n − 2 ) (n-2) -subsphere of S n S^n . Our main results are then that, in the case when G G is abelian, conditions (i) and (ii) ∗ ^{\ast } are necessary and sufficient for a given G G -action on Σ \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.