Topological Triviality in Families of Functions on Subanalytic Sets and Complex Analytic Spaces

Abstract

Abstract We study families of analytic functions defined on either subanalytic sets or complex analytic spaces. We give sufficient conditions for a family depending linearly on one parameter to have constant topological type, extending classical results due to Lê and Ramanujam [9] and Parusiński [12]. In the particular case of isolated singularity families defined on an ICIS, we prove that the $\mu $-constancy implies constant topological type.