The topology of the non-singular level set and the variation operator of a singularity

Abstract

Let ƒ:(ℂⁿ, 0)→(ℂ, 0) be a singularity, that is the germ of a holomorphic function, with an isolated critical point at the origin. It follows from implicit function theorem that in a neighbourhood of the origin in the space ℂⁿ the level set ƒ^-1 (ε) for ε≠0 is a non-singular analytic manifold and the level set ƒ^-1(0) is a non-singular manifold away from the origin. At the point 0∈ℂⁿ the level set has a singular point.