[Paper Review] Fibration theorems à la Milnor for differentiable maps with non-isolated singularities

José Luis Cisneros‐Molina, Aurélio Menegon+2 more|arXiv (Cornell University)|Feb 17, 2020

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper establishes fibration theorems analogous to Milnor's for $C^{\ u}$ real maps with non-isolated singularities, proving that $d$-regularity ensures the existence of a Milnor fibration on the sphere. It shows that modifying the target via homeomorphisms can linearize the discriminant, and if the resulting map is $d$-regular, a sphere fibration exists, generalizing classical results to non-isolated critical sets.

ABSTRACT

We prove fibration theorems à la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the existence of a Milnor fibration on the sphere. We also explain how one can modify the target space by homeomorphisms to linearize a general discriminant. Whenever the composed map is d-regular one has fibration on the sphere. Plenty of examples are discussed along the text.

Motivation & Objective

• To extend Milnor's fibration theorems to differentiable maps with non-isolated critical sets, where the critical value is not isolated.
• To identify $d$-regularity as the key condition ensuring the existence of a Milnor fibration on the sphere for such maps.
• To show that the target space can be modified via homeomorphisms to linearize the discriminant, enabling the application of $d$-regularity conditions.
• To provide a general framework for constructing locally trivial fibrations in the presence of non-isolated singularities, extending prior results on isolated critical points or values.
• To analyze a specific family of maps $(f,g) = (\sum a_i x_i^p, \sum b_i x_i^q)$ as illustrative examples of $d$-regularity and fibration behavior.

Proposed method

• Introduces the concept of $d$-regularity for maps with non-isolated singularities, defined via transversality of fibers $f^{-1}(\ell)$ to small spheres centered at the origin.
• Applies the Relative Ehresmann Fibration Theorem to establish local trivial fibrations on tubes around the origin when the transversality condition holds.
• Uses Ehresmann connections and horizontal lifts to prove that composition of fibrations preserves the fibration property under $d$-regularity.
• Constructs a canonical pencil of real analytic varieties $X_\ell = f^{-1}(\ell)$ for lines $\ell$ through the origin to analyze the geometry of the fibers near the singular set.
• Modifies the target space via homeomorphisms to linearize the discriminant, transforming a general discriminant into a linear one, thereby enabling $d$-regularity checks.
• Employs the theory of sub-analytic sets and $C^\ell$-maps to ensure the regularity and transversality conditions hold in the non-isolated setting.

Experimental results

Research questions

• RQ1 Under what conditions does a $C^\ell$ map $f: \mathbb{R}^n \to \mathbb{R}^k$ with non-isolated critical set admit a Milnor fibration on the sphere?
• RQ2 How can the concept of $d$-regularity be extended from isolated critical points to maps with non-isolated singularities?
• RQ3 Can the discriminant of a map be linearized via homeomorphism of the target space to simplify the analysis of fibration structures?
• RQ4 What is the role of the canonical pencil $X_\ell = f^{-1}(\ell)$ in determining the fibration properties of $f$ near the origin?
• RQ5 For the family of maps $(f,g) = (\sum a_i x_i^p, \sum b_i x_i^q)$, when is $d$-regularity satisfied and when does a sphere fibration exist?

Key findings

• A $C^\ell$ map $f: \mathbb{R}^n \to \mathbb{R}^k$ with non-isolated critical set admits a Milnor-L\'e fibration on the tube if and only if it satisfies the transversality property with respect to small spheres.
• $d$-regularity is both necessary and sufficient for the existence of a Milnor fibration on the sphere, where the projection is given by $f/\|f\|$.
• For maps with linear discriminant, $d$-regularity ensures that the fibers $f^{-1}(\ell)$ are transverse to all small spheres centered at the origin.
• The composition of two fibrations is a fibration if the corresponding Ehresmann connections are compatible, and this is guaranteed under $d$-regularity.
• The family of maps $(f,g) = (\sum a_i x_i^p, \sum b_i x_i^q)$ with $p,q \geq 2$ and generic coefficients $a_i, b_i$ is $d$-regular under mild conditions, ensuring a sphere fibration.
• By applying a suitable homeomorphism to the target space, any real map with a general discriminant can be transformed into one with a linear discriminant, preserving fibration properties if $d$-regularity holds.