# zbMATH — the first resource for mathematics

Equivariant semi-algebraic triangulations of real algebraic $$G$$-varieties. (English) Zbl 0902.14039
Let $$M \subset {\mathbb R}^m$$ denote a real algebraic variety and assume that a compact real Lie group $$G$$ acts algebraically on $$M$$. A semi-algebraic triangulation of the orbit space is defined to be a countable locally finite simplicial complex $$K$$ of finite dimension in some $${\mathbb R}^n$$ together with a homeomorphism $$M/G \to | K |$$ such that the map $$M \to M/G \to | K |$$ is semi-algebraic. As a first result the authors obtain that there is a semi-algebraic triangulation of $$M/G$$ that is compatible with the orbit types. On the one hand, this result is applied to introduce the structure of a certain equivariant $$CW$$-complex on $$M$$ and to obtain an equivariant simple homotopy type for compact $$M$$. On the other hand, the first result is used to prove that for finite $$G$$ there exists an equivariant semi-algebraic triangulation of $$M$$, that induces a semi-algebraic triangulation of $$M/G$$, compatible with the orbit types.

##### MSC:
 14P10 Semialgebraic sets and related spaces 55Q91 Equivariant homotopy groups 14L30 Group actions on varieties or schemes (quotients) 22E99 Lie groups 57S25 Groups acting on specific manifolds
Full Text: