Tangential representations of one-fixed-point actions on spheres and Smith equivalence (Q2339651)

Let \(G\) be a finite group and \(V\) be a real \(G\)-module. If there exists a homotopy sphere \(\Sigma\) with smooth \(G\)-action such that the \(G\)-fixed point set \(\Sigma^G\) consists of exactly one point, say \(a\), and \(T_a(\Sigma) \cong V\), then \(V\) is said to be of one fixed point type. Further, \(G\) is called an Oliver group if \(G\) does not have a normal series \(P \trianglelefteq H \trianglelefteq G\) such that \(P\) and \(G/H\) are of prime power order and \(H/P\) is cyclic. The main result of the paper under review is the following theorem. Theorem 1.1. Let \(G\) be a finite Oliver group and \(V\) be a real \(G\)-module of one fixed point type. Then there exists a standard sphere \(S\) with smooth \(G\)-action such that \(S^G=\{a\}\) and \(T_a(S) \cong V\). The author also investigates the relation between tangential \(G\)-representations of smooth one fixed point actions on spheres and Smith equivalence of real \(G\)-modules.