Equivalence of Milnor and Milnor-L\^e fibrations for real analytic maps

TL;DR

This paper generalizes Milnor's fibration theorem for real analytic maps with isolated critical points, establishing conditions under which the fibration on the tube is equivalent to the one on the sphere, using a specially constructed vector field.

Contribution

It extends Milnor-Lê fibration equivalence results to maps with linear discriminant and constructs a vector field that inflates the tube to the sphere in a controlled manner.

Findings

Established equivalence between d-regularity and fibration on the sphere for maps with linear discriminant.

Constructed a vector field satisfying properties analogous to Milnor's in the complex case.

Generalized Milnor's fibration theorem to a broader class of real analytic maps.

Abstract

In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$, where $n \geq p$, with an isolated critical point at the origin has a fibration on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1}) \to S_\delta^{p-1}$. Constructing a vector field such that, (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to the sphere, to get a fibration $\varphi\colon S_\epsilon^{n-1} \setminus f^{-1}(0) \to S^{p-1}$, but the projection is not necessarily given by $f/ \|f\|$ as in the complex case. In the case $f$ has isolated critical value, in [9] it was proved that if the fibres inside a small tube are transverse to the sphere $S_\epsilon$, then it has a fibration on the tube. Also in [9], the concept of $d$-regularity was defined, it turns out that $f$ is $d$-regular if and only if the map $f/|f\|\colon S_\epsilon^{n-1} \setminus f^{-1}(0) \to S^{p-1}$ is a fibre bundle equivalent to the one on the tube. In this article, we prove the corresponding facts in a more general setting: if a locally surjective map $f$ has a linear discriminant $\Delta$ and a fibration on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1} \setminus \Delta) \to S_\delta^{p-1} \setminus \Delta$, then $f$ is $d$-regular if and only if the map $f/ \|f\|\colon S_\epsilon^{n-1} \setminus f^{-1}(\Delta) \to S^{p-1} \setminus \mathcal{A}$ (with $\mathcal{A}$ the radial projection of $\Delta$ on $S^{p-1}$) is a fibre bundle equivalent to the one on the tube. We do this by constructing a vector field $\tilde{w}$ which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that $f/ \|f\|$ is constant on the integral curves of $\tilde{w}$.