Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normalized Milnor Fibrations for Real Analytic Maps

José Luis Cisneros Molina, Aurélio Menegon

TL;DR

The paper addresses the obstacle that real analytic maps with isolated critical value do not always admit a normalized Milnor fibration on the sphere. It introduces conic target deformations $h$ to render such maps $d_h$-regular, proving that the normalized fibration $\frac{h^{-1}\circ f}{\|h^{-1}\circ f\|}$ is a smooth locally trivial fibration on the sphere and is equivalent to both the Milnor-Lê fibration on the tube and the Milnor fibration on the sphere. The construction hinges on conic vector fields $\vec{v}_\alpha$ and lifting arguments to produce suitable parameters $\alpha$ for $f$, with a compactness argument guaranteeing a neighborhood of parameters where $d_h$-regularity holds. Consequently, every real analytic map-germ with isolated critical value and the transversality property has a normalized Milnor fibration after a suitable target change of coordinates, aligning real and complex fibrations topologically without altering the singularity type. This result reveals a closer topological parallel between real and complex singularities than previously recognized and provides a general positive answer to the normalized Milnor fibration problem in the real setting.

Abstract

Milnor's fibration theorem and its generalizations play a central role in the study of singularities of complex and real analytic maps. In the complex analytic case, the Milnor fibration on the sphere is always given by the normalized map $f/|f|$. In contrast, for real analytic maps the existence of such a normalized Milnor fibration generally fails, even when a Milnor--Le fibration exists on a tube. For locally surjective real analytic maps $f:(R^n,0)->(R^k,0)$ with isolated critical value, the existence of a Milnor--Le fibration on a tube is guaranteed under a transversality condition. However, the associated fibration on the sphere need not be given by the normalized map $f/||f||$, unless an additional regularity condition (d-regularity) is imposed. In this paper we show that this apparent obstruction is not intrinsic. More precisely, we prove that for any such map satisfying the transversality property, there exists a homeomorphism $h:(R^k,0)->(R^k,0)$ of the target space such that the composition $h^{-1}f$ becomes d-regular. As a consequence, the normalized map $(h^{-1}f)/||h^{-1}f||$ defines a smooth locally trivial fibration on the sphere, which is equivalent to both the Milnor--Le fibration on the tube and the Milnor fibration on the sphere. Our result reveals a closer topological parallel between real and complex analytic singularities than previously recognized, without changing the topological type of the singularity.

Normalized Milnor Fibrations for Real Analytic Maps

TL;DR

The paper addresses the obstacle that real analytic maps with isolated critical value do not always admit a normalized Milnor fibration on the sphere. It introduces conic target deformations to render such maps -regular, proving that the normalized fibration is a smooth locally trivial fibration on the sphere and is equivalent to both the Milnor-Lê fibration on the tube and the Milnor fibration on the sphere. The construction hinges on conic vector fields and lifting arguments to produce suitable parameters for , with a compactness argument guaranteeing a neighborhood of parameters where -regularity holds. Consequently, every real analytic map-germ with isolated critical value and the transversality property has a normalized Milnor fibration after a suitable target change of coordinates, aligning real and complex fibrations topologically without altering the singularity type. This result reveals a closer topological parallel between real and complex singularities than previously recognized and provides a general positive answer to the normalized Milnor fibration problem in the real setting.

Abstract

Milnor's fibration theorem and its generalizations play a central role in the study of singularities of complex and real analytic maps. In the complex analytic case, the Milnor fibration on the sphere is always given by the normalized map . In contrast, for real analytic maps the existence of such a normalized Milnor fibration generally fails, even when a Milnor--Le fibration exists on a tube. For locally surjective real analytic maps with isolated critical value, the existence of a Milnor--Le fibration on a tube is guaranteed under a transversality condition. However, the associated fibration on the sphere need not be given by the normalized map , unless an additional regularity condition (d-regularity) is imposed. In this paper we show that this apparent obstruction is not intrinsic. More precisely, we prove that for any such map satisfying the transversality property, there exists a homeomorphism of the target space such that the composition becomes d-regular. As a consequence, the normalized map defines a smooth locally trivial fibration on the sphere, which is equivalent to both the Milnor--Le fibration on the tube and the Milnor fibration on the sphere. Our result reveals a closer topological parallel between real and complex analytic singularities than previously recognized, without changing the topological type of the singularity.
Paper Structure (6 sections, 13 theorems, 49 equations, 2 figures)

This paper contains 6 sections, 13 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The $d_h$-regularity for analytic maps with isolated critical value
  3. Conic vector fields
  4. Suitable vector fields
  5. Case 1:
  6. Case 2:

Key Result

Theorem 1

Let $f \colon (\mathbb{R}^n,0) \to (\mathbb{R}^k,0)$, with $2 \le k \le n$, be a locally surjective real analytic map with an isolated critical value at $0$. Assume moreover that $f$ satisfies the transversality property. Then there exists a homeomorphism such that the map is a smooth locally trivial fibration. Moreover, this fibration is equivalent to both the Milnor--Lê fibration on the tube e

Figures (2)

  • Figure 1: The zeros of the conic vector fields $\vec{v}_\alpha$ in ${\mathbb{R}}^2$ given by $\alpha=(-\frac{1}{2},-\frac{1}{2})$.
  • Figure 2: The conic vector fields $\vec{v}_\alpha$ in ${\mathbb{R}}^2$ given by $\alpha=(-\frac{1}{2},-\frac{1}{2})$ on $\mathring{{\mathbb{B}}}_{\frac{1}{2}}^k$.

Theorems & Definitions (34)

  • Theorem
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Corollary 2.4
  • Example 2.5
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 24 more