Topological Equisingularity of Function Germs with 1-dimensional Critical Set

In this paper we prove several topological equisingularity theorems for holomorphic function germs with 1-dimensional critical set. The topological equisingularity notions that we will handle are the following. Denote by Bǫ and Sǫ the ball and the sphere of radius ǫ centered at the origin of C. Let f : (C, O) → C be a holomorphic germ. The embedded link of f is the topological type of the pair (Sǫ, f (0) ∩ Sǫ) for ǫ sufficiently small; the abstract link is the topological type of f(0)∩ Sǫ. Two germs f and g are topologically Requivalent if there is a germ of self-homeomorphism φ of (C, O) such that g = f◦φ; if we only have that φ(f(0)) = g(0) then we say that f and g have the same topological type. By the conical structure of singularities having the same embedded link implies having the same topological type. A family ft : (C , O) → C of holomorphic germs depending continuously on a parameter t varying in a manifold T is topologically R-equisingular if there is a family of self-homeomorphisms φt depending continuously in t and parametrised over T such that ft◦φt = ft0 for a fixed t0 ∈ T and any t ∈ T . Topological equisingularity has proved to be a very subtle subject, with several long standing open questions. For instance Zariski’s Multiplicity Question asks whether two germs with the same topological type must have the same multiplicity (Question A, [14]). Suppose that ft has an isolated singularity at the origin and that the Milnor number of ft is independent on t. Assume n 6= 3. Lê and Ramanujam [7] proved that the diffeomorphism type of the Milnor fibration of f and of the embedded link are independent on t. Later King [6] and Timourian [13] proved that for any t ∈ T there is a neighborhood U of t in T such that the restriction of ft over U is topologically R-equisingular. As the Milnor number is a topological invariant, if n 6= 3, it is an invariant characterising topological equisingularity for germs with isolated singularities. Answering whether families with constant Milnor number are topologically equisingular for n = 3 is a major open problem. Topological equisingularity for general germs (having not necessarily isolated singularities) turns out to be much more difficult. As an illustration of this, and also to motivate the results of this paper, let us mention some of the latest developments. In [9], [10] Massey introduced the Lê numbers, a set of polar invariants attached to a germ and to a coordinate system. Suppose that the family ft depends holomorphically on a complex parameter t, that the critical set of ft has dimension