The existence of semialgebraic slices and its applications

Abstract

Let G be a compact semialgebraic group and M a semialgebraic G-set. We prove that there exists a semialgebraic slice at every point of M. Moreover M can be covered by finitely many semialgebraic G-tubes. As an application we give a different proof that every semialgebraic G-set admits a semialgebraic G-embedding into some semialgebraic orthogonal representation space of G, which has been proved in [15].