Thom property and Milnor–Lê fibration for analytic maps

Let (X, 0) be the germ of either a subanalytic set X⊂Rn$X \subset {\mathbb {R}}^n$ or a complex analytic space X⊂Cn$X \subset {\mathbb {C}}^n$, and let f:(X,0)→(Kk,0)$f: (X,0) \rightarrow ({\mathbb {K}}^k, 0)$ be a K${\mathbb {K}}$‐analytic map‐germ, with K=R${\mathbb {K}}={\mathbb {R}}$ or C${\mathbb {C}}$, respectively. When k=1$k=1$, 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 k>1$k>1$ 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 fg¯:(X,0)→(C,0)$f \bar{g}: (X,0) \rightarrow ({\mathbb {C}},0)$ on a complex analytic surface X⊂Cn$X \subset {\mathbb {C}}^n$ with arbitrary singularity.