Hostname: page-component-54dcc4c588-gwv8j Total loading time: 0 Render date: 2025-10-04T12:25:28.830Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Białynicki-Birula, A. and Rosenlicht, M.. Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc. 13 (1962), 200–203.10.1090/S0002-9939-1962-0140516-5CrossRefGoogle Scholar
Braun, F., Dias, L. R. G. and Venato-Santos, J.. On global invertibility of semialgebraic local diffeomorphisms. Topol. Methods Nonlinear Anal. 58 (2021), 713–730.10.12775/TMNA.2021.004CrossRefGoogle Scholar
Braun, F., Dias, L. R. G. and Venato-Santos, J.. Surjectivity of linear operators and semialgebraic global diffeomorphisms. J. Anal. Math. 150 (2023), 789–802.10.1007/s11854-023-0286-zCrossRefGoogle Scholar
Braun, F. and Fernandes, F.. Very degenerate polynomial submersions and counterexamples to the real Jacobian conjecture. J. Pure Appl. Algebra. 227 (2023), 107345, 10.10.1016/j.jpaa.2023.107345CrossRefGoogle Scholar
Campbell, L. A.. The asymptotic variety of a Pinchuk map as a polynomial curve. Appl. Math. Lett. 24 (2011), 62–65.CrossRefGoogle Scholar
Fernandes, A. and Jelonek, Z.. On real polynomial local homeomorphisms, [math.GT] arXiv:1807.00855.Google Scholar
Fernandes, A., Maquera, C. and Venato-Santos, J.. Jacobian conjecture and semi-algebraic maps. Math. Proc. Cambridge Philos. Soc. 157 (2014), 221–229.10.1017/S0305004114000279CrossRefGoogle Scholar
Fernandes, F.. A new class of non-injective polynomial local diffeomorphisms on the plane. J. Math. Anal. Appl. 507 (2022), 125736, 5.10.1016/j.jmaa.2021.125736CrossRefGoogle Scholar
Fernandes, F. and Jelonek, Z.. The Pinchuk example revisited. C. R. Math. Acad. Sci. Paris. 362 (2024), 449–452.10.5802/crmath.570CrossRefGoogle Scholar
Godbillon, C.. Éléments de topologie algébrique. 249, (Paris: Hermann, 1971).Google Scholar
Guillemin, V. and Pollack, A.. Differential topology. xvi+222, (Prentice-Hall, Inc, Englewood Cliffs, N.J, 1974).Google Scholar
Hardt, R.. Semi-algebraic local-triviality in semi-algebraic mappings. Am. J. Math. 102 (1980), 291–302.10.2307/2374240CrossRefGoogle Scholar
Hatcher, A.. Algebraic topology. xii+544, (Cambridge University Press, Cambridge, 2002).Google Scholar
Jelonek, Z.. Geometry of real polynomial mappings. Math. Z. 239 (2002), 321–333.10.1007/s002090100298CrossRefGoogle Scholar
Jelonek, Z.. On the preimage of a sphere by a polynomial mapping. Manuscripta Math. 166 (2021), 271–276.10.1007/s00229-020-01239-6CrossRefGoogle Scholar
Lee, J. M.. Introduction to smooth manifolds. Grad. Texts in Math., Vol.218, xvi+708 (Springer, New York, 2013).Google Scholar
Lima, E. L.. The Jordan–Brouwer separation theorem for smooth hypersurfaces. Am. Math. Mon. 95 (1988), 39–42.CrossRefGoogle Scholar
Pinchuk, S.. A counterexample to the strong real Jacobian conjecture. Math. Z. 217 (1994), 1–4.10.1007/BF02571929CrossRefGoogle Scholar
Plastock, R.. Homeomorphisms between Banach spaces. Trans. Amer. Math. Soc. 200 (1974), 169–183.CrossRefGoogle Scholar
Xavier, F.. The global inversion problem: a confluence of many mathematical topics. Mat. Contemp. 35 (2008), 241–265.Google Scholar