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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.

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
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