Thom property and Milnor–Lê fibration for analytic maps

Abstract

Let (X, 0) be the germ of either a subanalytic set or a complex analytic space , and let be a ‐analytic map‐germ, with or , respectively. When , there is a well‐known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when that is not always the case. In this paper, we give conditions which guarantee that the image of f is well‐defined as a set‐germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function‐germs of type on a complex analytic surface with arbitrary singularity.