The Invariance of Milnor's Number Implies the Invariance of the Topological Type

Introduction. We are interested in analytic families of n-dimensional hypersurfaces having an isolated singularity at the origin. In this paper we only consider the case when the Milnor's number of the singularity at the origin does not change in this family. Under this hypothesis with n = 1, H. Hironaka conjectured that the topological type of the singularity does not change. We give a proof of this conjecture in the more general case of C?? family of n-dimensional hypersurfaces of dimension n=?2. The hypothesis n7^2 comes from the fact we are using n-cobordism theorem. Actually a more general conjecture should be the following: