Relative Bruce-Roberts number and Chern obstruction

2023/11/06 by Bárbara K. Lima Pereira, Pereira, Bárbara K. Lima, María Aparecida Soares Ruas +3
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2311.03548

Abstract

Let (X,0) be the germ of an equidimensional analytic set in (\mathbb Cⁿ,0) and f=(f₁,f₂) a map-germ into the plane defined on X. In this work, we investigate topological invariants associated to the pair (f,X), among them, the Euler obstruction of f, Eu_f,X(0), and under convenient assumptions, the Chern number 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 giving by the Euler obstruction and the Chern numbers. Closed formulas are given when X, X∩ f₂^-1(0), X∩ f₂^-1(0)∩ f₁^-1(0) are ICIS. In the last section, for a 2-dimensional ICIS (X,0) ⊂ (\mathbb Cⁿ,0), we apply our results to give an alternative description for the number of cusps c(f|_X) of an stabilization of an \mathcal A-finite map-germ f=(f₁, f₂): (X,0) → (\mathbb C²,0). A formula for c(f|_X) was first given in [21].

Related

• The geometrical information encoded by the Euler obstruction of a map
• Bruce-Roberts numbers and indices of vector fields on an ICIS
• The local Euler obstruction and topology of the stabilization of\n associated determinantal varieties
• On invariants of a map germ from n-space to 2n-space
• Chern obstructions for collections of 1-forms on singular varieties