Milnor fibration theorem for differentiable maps

• 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.

Ämnesord

NATURVETENSKAP  -- Fysik -- Acceleratorfysik och instrumentering (hsv//swe)

NATURAL SCIENCES  -- Physical Sciences -- Accelerator Physics and Instrumentation (hsv//eng)

Nyckelord

Milnor fibration

d-regularity

Differentiable singularities

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