Surjectivity of certain injective semialgebraic transformations of \({\mathbb{R}}\) n
type: Item
instance of: scholarly article
title: Surjectivity of certain injective semialgebraic transformations of \({\mathbb{R}}\) n
zbMATH Open document ID: 0641.14010
DOI: 10.1007/BF01161750
author: Krzysztof Kurdyka
author: Kamil Rusek
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173997
review text: 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.
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 4043990
zbMATH Keywords: injective semialgebraic mapping
zbMATH Keywords: regular endomorphisms
Wikidata QID: Q114234051
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4043990