Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f   can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0)f:(X,0)→(C,0) and g:(X,0)→(C,0)g:(X,0)→(C,0). We give applications when g is a generic linear form and when f and g have isolated singularities.