Relative Bruce-Roberts number and Chern obstruction

Abstract: Let $(X,0)$ be the germ of an equidimensional analytic set in $(\mathbb C^n,0)$ and $f=(f_1,f_2)$ 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 $X,\, X\cap f_2^{-1}(0),\, X\cap f_2^{-1}(0)\cap f_1^{-1}(0)$ are ICIS. In the last section, for a 2-dimensional ICIS $(X,0) \subset (\mathbb C^n,0),$ 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 $f=(f_1, f_2): (X,0) \to (\mathbb C^2,0).$ A formula for $c(f|_X)$ was first given in [21].