One-fixed-point actions on spheres and Smith sets (Q342971)

Let \(G\) be a finite group and Sm\((G)\) the subset of the real representation ring RO\((G)\) consisting of elements \(x= [V]- [W]\), where \(V\) and \(W\) are Smith equivalent real \(G\)-modules. If \(G\) has a normal subgroup \(N\) such that \(G/N\) is cyclic of order 8, then Sm\((G)\) is not additively closed. For a prime \(p\), let \(P_p(G)\) be the set of all \(p\)-subgroups of \(G\) and \(G^{\{p\}}\) the smallest normal subgroup of \(G\) with \(p\)-power index. Let \(P(G)\) be the union of \(P_p(G)\) over \(p\) and Sm\((G)_{p(G)}\) the primary Smith set of \(G\), that is all \(x= [V]- [W]\in \text{Sm}(G)\) such that the restrictions of \(V\) and \(W\) to any group in \(P(G)\) are equal.NEWLINENEWLINE This paper introduces an additively closed subset of Sm\((G)_{p(G)}\). Let VO\((G)\) be the family of all real \(G\)-modules obtainable as a tangential representation on a homotopy sphere with exactly one fixed point satisfying the \(P(G)\)-weak gap condition. If DO\((G)=\{ [V]- [W]\mid V, W\in \text{VO}(G)\}\cup\{0\}\), then DO\((G)_{p(G)}\subset \text{Sm}(G)_{p(G)}\). The first major theorem in this paper asserts that DO\((G)\) is an additive subgroup of RO\((G)\) and so DO\((G)_{p(G)}\) is an additive subgroup of Sm\((G)_{p(G)}\). The second and third major theorems offer conditions under which DO\((G)_{p(G)}= \text{Sm}(G)_{p(G)}\).