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On the non-properness set of a local homeomorphism

Part of: Families, fibrations Affine geometry Theory of singularities and catastrophe theory

Published online by Cambridge University Press:  30 September 2025

Francisco Braun Open the ORCID record for Francisco Braun [Opens in a new window] ,
Luis Renato Gonçalves Dias Open the ORCID record for Luis Renato Gonçalves Dias [Opens in a new window]  and
Jean Venato-Santos Open the ORCID record for Jean Venato-Santos [Opens in a new window]
Francisco Braun*
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, São Paulo, Brazil (franciscobraun@ufscar.br)
Luis Renato Gonçalves Dias
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, Brazil (lrgdias@ufu.br and jvenatos@ufu.br)
Jean Venato-Santos
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, Brazil (lrgdias@ufu.br and jvenatos@ufu.br)
*
*Corresponding author.
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Abstract

We prove that the non-properness set of a local homeomorphism $\mathbb R^n \to \mathbb R^n$ cannot be ambient homeomorphic to an affine subspace of dimension n − 2. This particularly provides a partial positive answer to a conjecture of Jelonek, that claims the global invertibility of polynomial local diffeomorphisms having non-properness set with codimension greater than 1. Our reasons to obtain this result lead us to some properties of the non-properness set when it has codimension 2, the heart of Jelonek’s conjecture. We also provide a global injectivity theorem related to this conjecture that turns out to generalize previous results of the literature.

Keywords

global injectivitylocal homeomorphismsnon-properness setAlexander dualitycovering map

MSC classification

Primary: 14R15: Jacobian problem
Secondary: 14P10: Semialgebraic sets and related spaces 14D06: Fibrations, degenerations 58K30: Global theory

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 15
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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