| Full text | |
| Author(s): |
Pereira, Barbara K. Lima
;
Ruas, Maria Aparecida Soares
;
Santana, Hellen
Total Authors: 3
|
| Document type: | Journal article |
| Source: | REVISTA MATEMATICA COMPLUTENSE; v. N/A, p. 24-pg., 2025-05-27. |
| Abstract | |
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) | |
| FAPESP's process: | 22/06968-6 - Equisingularity and invariants associated to the topology of functions with non-isolated singularity |
| Grantee: | Hellen Monção de Carvalho Santana |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 22/08662-1 - The Bruce-Roberts numbers and the logarithmic characteristic variety |
| Grantee: | Bárbara Karolline de Lima Pereira |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 23/04460-8 - The Bruce-Roberts number and the logarithmic characteristic variety |
| Grantee: | Bárbara Karolline de Lima Pereira |
| Support Opportunities: | Scholarships abroad - Research Internship - Post-doctor |
| FAPESP's process: | 19/21181-0 - New frontiers in Singularity Theory |
| Grantee: | Regilene Delazari dos Santos Oliveira |
| Support Opportunities: | Research Projects - Thematic Grants |