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- Cisneros-Molina, José Luis, et al.
(author)
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Fibration theorems à la Milnor for analytic maps with non-isolated singularities
- 2023
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In: São Paulo Journal of Mathematical Sciences. - : Springer Nature. - 1982-6907 .- 2316-9028.
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Journal article (peer-reviewed)abstract
- We study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions (f, g) : Rn→ R2 of the type (f,g)=(∑i=1naixip,∑i=1nbixiq), where ai, bi∈ R are constants in generic position and p, q≥ 2 are integers.
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| 2. |
- Cisneros-Molina, José Luis, et al.
(author)
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Errata to equivalence of Milnor and Milnor–Lê fibrations for real analytic maps
- 2021
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In: International Journal of Mathematics. - : World Scientific Pub Co Pte Ltd. - 0129-167X .- 1793-6519.
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Other publication (peer-reviewed)abstract
- In the proof of Theorem 3.7 of the original paper Internat. J. Math. 30(14) (2019) 1950078, 1–25, two inequalities are used that do not hold in general. In this note, we prove an extra propositions which allows us to give a proof of Theorem 3.7 without using the aforementioned inequalities. Hence, all the results in the original paper are valid. We have also posted a corrected version in the arXiv.
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| 3. |
- Cisneros-Molina, Jose Luis, et al.
(author)
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Milnor fibration theorem for differentiable maps
- 2024
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In: Research in the Mathematical Sciences. - : Springer Nature. - 2522-0144. ; 11:2
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Journal article (peer-reviewed)abstract
- In Cisneros-Molina et al. (Sao Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations a la Milnor (in the tube and in the sphere) for real analytic maps f:(Rn,0)->(Rk,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:({\mathbb {R}}<^>n,0) \rightarrow ({\mathbb {R}}<^>k,0)$$\end{document}, where n >= k >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge k\ge 2$$\end{document}, with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (Sao Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class Cl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{\ell }$$\end{document}, l >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 2$$\end{document}, with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class Cl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{\ell }$$\end{document} between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65-70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.
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