Finite cyclic subgroups determine the spectrum of the equivariant 𝐾-theory

Abstract

Equivariant maps inducing an equivalence of the categories of components of the fixed point sets of topologically cyclic subgroups are considered. It is shown that they are the same as those inducing an equivalence of the categories of components of the fixed point sets of finite cyclic subgroups. It follows that equivariant maps inducing a bijection of maximal ideals of the appropriate equivariant K K -theory rings coincide with those which give bijection on the sets of all prime ideals. As a corollary we obtain that a group homomorphism inducing bijection of maximal ideals of the representation rings is an isomorphism.