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Topological Methods in Nonlinear Analysis

A Thom isotopy theorem for nonproper semialgebraic maps
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A Thom isotopy theorem for nonproper semialgebraic maps

Authors

  • Luis Renato Gonçalves Dias https://orcid.org/0000-0003-1054-870X
  • Giovanny Barrera Ramos https://orcid.org/0009-0007-2880-9898

DOI:

https://doi.org/10.12775/TMNA.2025.050

Keywords

Bifurcation set, semialgebraic sets, semialgebraic maps, stratified $\rho$ non-regular values set, Verdier stratification

Abstract

We prove a version of the Thom isotopy theorem for nonproper semialgebraic maps $f\colon X\rightarrow \mathbb R^m$, where $X \subset\mathbb R^n$ is a semialgebraic set and $f$ is the restriction to $X$ of a smooth semialgebraic map $F\colon\mathbb R^n\rightarrow \mathbb R^m$.

References

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S.T. D̄inh and Z. Jelonek, Thom isotopy theorem for nonproper maps and computation of sets of stratified generalized critical values, Discrete and Comput. Geom. 65 (2021), 279–304.

T. Gaffney, Fibers of polynomial mappings at infinity and a generalized Malgrange condition, Compos. Math. 119 (1999), no. 2, 157–167.

C.G. Gibson and K. Wirthmüller, A.A. Du Plessis and E.J.N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, vol. 552, Springer–Verlag, Berlin, New York, 1976.

J.P. Henry, M. Merle and C. Sabbah, Sur la condition de Thom stricte pour un morphisme analytique complexe, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), no. 2, 227–268.

Z. Jelonek, On the generalized critical values of a polynomial mapping, Manuscripta Math. 110 (2003), 145–157.

K. Kurdyka, P. Orro and S. Simon, Semialgebraic Sard theorem for generalized critical values, J. Differential Geom. 56(1) (2000), 67–92.

J.M. Lee, Smooth Manifolds, Springer, New York, 2012, pp. 1–31.

T.L. Loi, Lecture 2: Stratifications in o-minimal structures, The Japanese–Australian Workshop on Real and Complex Singularities – JARCS III, Proc. Centre Math. Appl. Austral. Nat. Univ. 43 (2010), 31–39.

A. Parusiński and L. Păunescu, Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity, Adv. Math. 309 (2017), 254–305.

P.J. Rabier, Ehresmann fibrations and Palais–Smale conditions for morphisms of Finsler manifolds, Annals of Mathematics 146 (1997), no. 3, 647–691.

A. Valette, Stratified C p -semialgebraic triviality, Manuscripta Math. 166 (2021), no. 1–2, 315–322.

J.-L. Verdier, Stratifications de Whitney et théoreme de Bertini–Sard, Invent. Math. 36 (1976), no. 1, 295–312.

H. Whitney, Tangents to an analytic variety, Hassler Whitney Collected Papers, Boston, MA, Birkhäuser, Boston, 1965, pp. 537–590.

Topological Methods in Nonlinear Analysis

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Published

2026-05-18

How to Cite

1.
DIAS, Luis Renato Gonçalves and RAMOS, Giovanny Barrera. A Thom isotopy theorem for nonproper semialgebraic maps. Topological Methods in Nonlinear Analysis. Online. 18 May 2026. pp. 1 - 16. [Accessed 4 June 2026]. DOI 10.12775/TMNA.2025.050.
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