Let (X, 0) be the germ of an equidimensional analytic set in \(({\mathbb {C}}^n,0)\) and \(F=(f,g_1,\ldots ,g_p)\) a map-germ into \({\mathbb {C}}^{p+1}\) 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 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 \(X,\, X\cap F^{-1}(0),\, X\cap G^{-1}(0)\) are ICIS, for \(G=(g_1,\ldots ,g_p)\) . 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 a stabilization of an \({\mathcal {A}}\) -finite map-germ \(f=(f_1, f_2): (X,0) \rightarrow ({\mathbb {C}}^2,0).\) A formula for \(c(f|_X)\) was first given in Massey (Topology 35(4):969-1003, 1996).
