[Paper Review] Topological equisingularity of function germs with 1-dimensional critical set

Javier Fernández de Bobadilla|ArXiv.org|Mar 21, 2006

Meromorphic and Entire Functions15 references3 citations

TL;DR

This paper introduces the concept of equisingularity at the critical set for holomorphic function germs with 1-dimensional critical sets, proving that such equisingularity implies topological R-equisingularity. It shows that constant generic Lê numbers imply equisingularity at the critical set, resolving a question of D. Massey, and constructs new examples of topologically equisingular families with non-flat, singular critical set deformations.

ABSTRACT

We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Le numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey). We use this to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnold's series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.

Motivation & Objective

• To address open questions in topological equisingularity for holomorphic function germs with non-isolated singularities, particularly when the critical set is 1-dimensional.
• To resolve a question posed by D. Massey regarding the topological equisingularity of families with constant generic Lê numbers.
• To redefine and characterize topological stems, linking them to Arnold’s series of singularities, and to clarify the role of Lê numbers in equisingularity.
• To analyze the deformation behavior of critical sets in topologically equisingular families, especially when the critical set is not flatly deformed.
• To construct new examples of topologically equisingular families where the critical set is singular in the special fiber but smooth in the generic fiber, challenging classical expectations.

Proposed method

• Introduces the notion of equisingularity at the critical set, defined via a family of self-homeomorphisms that trivialize the critical set and preserve the stratification by transverse Milnor number.
• Uses Lê numbers—polar invariants associated with a coordinate system—as a key tool to detect equisingularity at the critical set, particularly the generic Lê number.
• Applies results from the theory of Milnor fibrations and embedded links, leveraging the fact that constant Milnor number implies constant homotopy type of abstract links for $ n \geq 4 $.
• Employs weighted homogeneous structures and parametrizations (e.g., $ (x,y,z) \mapsto (t^{-1}x, t^{-1}y, t^{-1}z) $) to analyze critical sets and verify transversal Milnor number constancy.
• Constructs explicit families of functions (e.g., $ f_t = (ty - xz)^9 + (tx^3 - y^2z)^4 + (y^3 - x^4)^3 + (t^3x - z^3)^{12} $) to demonstrate equisingularity at the critical set and topological R-equisingularity.
• Uses stabilization via $ g_t = f_t + u^a + v^b $ to ensure the dimension conditions for applying the main equisingularity theorem, particularly for $ n \geq 5 $.

Experimental results

Research questions

• RQ1 Does a family of holomorphic function germs with 1-dimensional critical set and constant generic Lê numbers imply topological R-equisingularity?
• RQ2 Can the concept of topological stem be redefined and characterized in terms of equisingularity at the critical set, especially in relation to Arnold’s series of singularities?
• RQ3 Is it possible to have topologically equisingular families where the critical set undergoes a non-flat deformation with a singular special fiber and smooth generic fiber?
• RQ4 What conditions ensure that equisingularity at the critical set implies topological equisingularity, especially when the Milnor number or multiplicity is not constant?
• RQ5 Can one construct non-equimultiple, topologically equisingular families with prescribed critical set structure and transversal Milnor number?

Key findings

• A family of holomorphic function germs with 1-dimensional critical set is topologically R-equisingular if it is equisingular at the critical set, as defined via trivialization of the critical set and preservation of the transverse Milnor number stratification.
• Constant generic Lê numbers imply equisingularity at the critical set, and hence topological R-equisingularity, providing a positive answer to a question of D. Massey.
• The family $ f_t = (ty - xz)^9 + (tx^3 - y^2z)^4 + (y^3 - x^4)^3 + (t^3x - z^3)^{12} $ has a critical set parametrized by $ (t^{-1}x, t^{-1}y, t^{-1}z) $, is singular at $ t = 0 $, and smooth for $ t \neq 0 $, yet remains equisingular at the critical set.
• The transversal Milnor number is 6 at all points of the critical set except the origin, where it is 18, yet the family remains equisingular at the critical set due to control by lower-order terms.
• Stabilization via $ g_t = f_t + u^a + v^b $ preserves equisingularity at the critical set and ensures topological R-equisingularity for $ n \geq 5 $, confirming the robustness of the framework.
• The paper constructs the first known example of two topologically R-equivalent functions with 1-dimensional critical sets where one has a smooth critical set and the other has a singular critical set at the origin, demonstrating a new deformation behavior in topological equisingularity.