Milnor tube fibrations of composite map germs

Autor(es)

Araujo dos Santos, Raimundo ; Saeki, Osamu

• Tipo de documento: Artigo Científico
• Fonte: RESEARCH IN THE MATHEMATICAL SCIENCES; v. 12, n. 4, p. 12-pg., 2025-11-16.

Resumo

Let F:(RM,0)->(RN,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:(\mathbb {R}<^>M,0)\rightarrow (\mathbb {R}<^>N,0)$$\end{document} and G:(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}$$G:(\mathbb {R}<^>N,0) \rightarrow (\mathbb {R}<^>K,0)$$\end{document}, M >= 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}$$M \ge N \ge K \ge 2$$\end{document}, be non-constant real analytic map germs with isolated critical values. In this paper we study the topology of the Milnor tube fibrations of the map germs F, G and their composition H=G degrees F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H = G\circ F$$\end{document}, under the tame condition. More precisely, we show that the Milnor fiber FH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_H$$\end{document} of H is homotopy equivalent to the product FFxFG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_F \times \mathcal {F}_G$$\end{document} of the Milnor fibers of F and G, and that the boundary partial derivative FH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \mathcal {F}_H$$\end{document} of FH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_H$$\end{document} is homotopy equivalent to partial derivative(FFxFG)=(partial derivative FFxFG)boolean OR(FFx partial derivative FG)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial (\mathcal {F}_F \times \mathcal {F}_G) = (\partial \mathcal {F}_F \times \mathcal {F}_G) \cup (\mathcal {F}_F \times \partial \mathcal {F}_G)$$\end{document}. Furthermore, if each component of partial derivative(FFxFG)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial (\mathcal {F}_F \times \mathcal {F}_G)$$\end{document} is simply connected and M-K >= 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M - K \ge 6$$\end{document}, then the homotopy equivalences can be replaced by diffeomorphisms in the above statements. (AU)

• Processo FAPESP: 19/21181-0 - Novas fronteiras na Teoria de Singularidades
• Beneficiário: Regilene Delazari dos Santos Oliveira
• Modalidade de apoio: Auxílio à Pesquisa - Temático