Journal of the
Korean Mathematical Society JKMS

Let $G$ be a compact Lie group and $M$ a semialgebraic $G$ space in some orthogonal representation space of $G$. We prove that if $G$ is finite then $M$ has an equivariant semialgebraic triangulation. Moreover this triangulation is unique. When $G$ is not finite we show that $M$ has a semialgebraic $G$ $CW$ complex structure, and this structure is unique. As a consequence compact semialgebraic $G$ space has an equivariant simple homotopy type.

Keywords: action, semialgebraic set, triangulation, $CW$ complex structure, algebraic variety

MSC numbers: 14P10, 22E99, 57Q91