Relative Bruce-Roberts number and Chern obstruction

Autor(es)

Pereira, Barbara K. Lima ; Ruas, Maria Aparecida Soares ; Santana, Hellen

• Tipo de documento: Artigo Científico
• Fonte: REVISTA MATEMATICA COMPLUTENSE; v. N/A, p. 24-pg., 2025-05-27.

Resumo

Let (X, 0) be the germ of an equidimensional analytic set in (Cn,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {C}}<^>n,0)$$\end{document} and F=(f,g1,& mldr;,gp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F=(f,g_1,\ldots ,g_p)$$\end{document} a map-germ into Cp+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}<^>{p+1}$$\end{document} defined on X. In this work, we investigate topological invariants associated to the pair (F, X), among them, the Euler obstruction of F, EuF,X(0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Eu_{F,X}(0),$$\end{document} and under convenient assumptions, the Chern obstruction of families of differential forms associated to F. The topological information provided by these invariants is useful, although difficult to calculate. The aim of the paper is to introduce the Bruce-Roberts and the relative Bruce-Roberts numbers as useful algebraic tools to capture the topological information given by the Euler obstruction and the Chern obstruction. Closed formulas are given when X,X boolean AND F-1(0),X boolean AND G-1(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X,\, X\cap F<^>{-1}(0),\, X\cap G<^>{-1}(0)$$\end{document} are ICIS, for G=(g1,& mldr;,gp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(g_1,\ldots ,g_p)$$\end{document}. In the last section, for a 2-dimensional ICIS (X,0)subset of(Cn,0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,0) \subset ({\mathbb {C}}<^>n,0),$$\end{document} we apply our results to give an alternative description for the number of cusps c(f|X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(f|_X)$$\end{document} of a stabilization of an A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}-finite map-germ f=(f1,f2):(X,0)->(C2,0). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=(f_1, f_2): (X,0) \rightarrow ({\mathbb {C}}<^>2,0).$$\end{document} A formula for c(f|X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(f|_X)$$\end{document} was first given in Massey (Topology 35(4):969-1003, 1996). (AU)

• Processo FAPESP: 22/06968-6 - Equisingularidade e invariantes associados à topologia de funções com singularidade não-isolada
• Beneficiário: Hellen Monção de Carvalho Santana
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado
• Processo FAPESP: 22/08662-1 - Os números de Bruce-Roberts e as variedades logarítmicas características
• Beneficiário: Bárbara Karolline de Lima Pereira
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado
• Processo FAPESP: 23/04460-8 - O número de Bruce-Roberts e a variedade logarítmica característica
• Beneficiário: Bárbara Karolline de Lima Pereira
• Modalidade de apoio: Bolsas no Exterior - Estágio de Pesquisa - Pós-Doutorado
• Processo FAPESP: 19/21181-0 - Novas fronteiras na Teoria de Singularidades
• Beneficiário: Regilene Delazari dos Santos Oliveira
• Modalidade de apoio: Auxílio à Pesquisa - Temático