Linearity of dimension functions for semilinear \(G\)-spheres (Q2781356)

The author studies the dimension function of a semilinear \(G\)-sphere, where by a semilinear \(G\)-sphere he means a closed smooth \(G\)-manifold \(S\) such that the \(H\)-fixed point set \(S^H\) is a homotopy sphere or empty set for every closed subgroup \(H\) of \(G\). He proves that if \(G\) is a finite nilpotent group of order \(p^nq^m\), where \(p\) and \(q\) are primes, then the dimension function of any semilinear \(G\)-sphere is equal to that of a linear \(G\)-sphere (Theorem A). On the other hand, he proves that if \(G\) is a nonsolvable compact Lie group, then there exists a semilinear \(G\)-sphere whose dimension function is not virtually linear; i.e., it is not the difference of the dimension functions of two linear \(G\)-spheres (Theorem B).