Semi-algebraic triangulations of orbit spaces ofreal algebraic G-sets

Abstract

It has been proved by T. Matumoto and M. Shiota that there exist a subanalytic traingulation of orbit space in the subanalytic category. In this thesis we consider semi-algebraic actions of compact Lie groups on semialgebraic sets. We first show the existence of semi-algebraic slice of semi-algebraic G-set M which is semi-algebraically embedded in a representation space. Then we show that for such a compact M the set M(H) is semialgebraic. Using this we show that if there exists a G-invariant semi-algebraic map f from M to some euclidean space Rn whose induced map f from the orbit space M/G to the image f(M) is a homeomorphism, then the orbit space has a unique semi-algebraic triangulation compatible with orbit types of M. We also find the same results as for arbitrary algebraic G-set M without compactness of M and existence of the map f.