Surjectivity of certain injective semialgebraic transformations of \({\mathbb{R}}\) n
It is proved that every injective semialgebraic mapping \(F: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) such that \(i(graph(F))=E\cap i({\mathbb{R}}^ n\times {\mathbb{R}}^ n)\) for some algebraic compactification (X,i) of \({\mathbb{R}}^ n\times {\mathbb{R}}^ n\) and for an arcwise symmetric subset \(E\subset X\), is surjective. This extends the known Białynicki-Birula and Rosenlicht theorem for regular endomorphisms of \({\mathbb{R}}^ n.\) The definition and all needed properties of arcwise symmetric sets are included. The proof is based on the theorem saying that the space \({\mathbb{R}}^ n\) is not homotopically equivalent to the complement of its non-void arcwise symmetric semialgebraic subset. Some examples are discussed.
scientific article; zbMATH DE number 4043990