The local topology of singular points in analytic spaces is known, by work of Milnor and others, to be determined by its link \(L_0\) . In this work, we look at holomorphic map-germs \((X,{\underline{0}}) \buildrel {f}\over {\rightarrow } ({{\mathbb {C}}},0)\) where X is an arbitrary reduced and equidimensional complex analytic space in some \({\mathbb {C}}^m\) . We compare the topology of the link \(L_0\) with that of the boundary of a local non-critical level, i.e., with the boundary of its Milnor fiber \(F_t\) . We study how the topology of the boundary \(\partial F_t\) changes as the non-critical level degenerates to the critical one.
